Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra

Home , Antisymmetric tensor, Design matrix, Fuzzy associative matrix, Hodge star operator, Multilinear algebra, Tensor algebra

Appendix A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra

The key word “algebra” is derived from the name and the work of the nineth-century scientist Mohammed ibn Mûsâ al-Khowârizmi who was born in what is now Uzbekistan and worked in Bagdad at the court of Harun al-Rashid’s son.“al-jabr” appears in the title of his book Kitab al-jabr wail muqabala where he discusses symbolic methods for the solution of equations, K. Vogel: Mohammed Ibn Musa Alchwarizmi’s Algorismus; Das forested Lehrbuch zum Rechnen mit indischen Ziffern, 1461, Otto Zeller Verlagsbuchhandlung, Aalen 1963). Accordingly what is an algebra and how is it tied to the notion of a vector space,atensor space, respectively? By an algebra we mean asetS of elements and a finite set M of operations. Each operation .opera/k is a single-valued function assigning to every finite ordered sequence .x1;:::;xn/ of n D n.k/ elements of S a value S .opera/k .x1;:::;xk / D xl in . In particular for .opera/k.x1;x2/ the operation is called binary,for.opera/k.x1;x2;x3/ ternary, in general for .opera/k.x1;:::;xn/n-ary. For a given set of operation symbols .opera/1, .opera/2;:::, .opera/k we define a word.Inlinear algebra the set M has basically two elements, namely two internal relations .opera/1 worded “addition” (including inverse addition: subtraction) and .opera/2 worded “multiplication” (including inverse multiplication: division). Here the elements of the set S are vectors over the field R of real numbers as long as we refer to linear algebra.Incontrast,in multilinear algebra the elements of the set S are tensors over the field of real numbers R. Only later modules as generalizations of vectors of linear algebra are introduced in which the “scalars” are allowed to be from an arbitrary ring rather than the field R of real numbers. Let us assume that you as a potential reader are in some way familiar with the elementary notion of a threedimensional vector space X with elements called vectors x 2 R3, namely the intuitive space “ we locally live in”. Such an elementary vector space X is equipped with a metric to be referred to as threedimensional Euclidean.

E. Grafarend and J. Awange, Applications of Linear and Nonlinear Models, 571 Springer Geophysics, DOI 10.1007/978-3-642-22241-2, © Springer-Verlag Berlin Heidelberg 2012 572 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra

As a real threedimensional vector space we are going to give it a linear and multilinear algebraic structure. In the context of structure mathematics based upon (a) Order structure (b) Topological structure (c) Algebraic structure an algebra is constituted if at least two relations are established, namely one internal and one external. We start with multilinear algebra, in particular with the multilinearity of the tensor product before we go back to linear algebra, in particular to Clifford algebra.

p A-1 Multilinear Functions and the Tensor Space Tq

Let X be a ﬁnite dimensional linear space, e.g. a vector space over the ﬁeld R of real numbers, in addition denote by X its dual space such that n D dim X D dim X. Complex, quaternion and octonian numbers C, H and O as well as rings will only be introduced later in the context. For p; q 2 ZC being an element of positive integer numbers we introduce

Tp X X q . ; / as the p-contravariant, q-covariant tensor space or space of multilinear functions

X X f W X ::: X X ::: X ! Rp dim q dim :

1 p If we assume x ;:::;x 2 X and x1;:::;xq 2 X,then § ¤ x1 ˝ :::˝ xp ˝ x ˝ :::˝ x 2 Tp.X; X/ ¦ 1 q q ¥ holds. Multilinearity is understood as linearity in each variable.“˝” identiﬁes 1 p the tensor product, the Cartesian product of elements .x ;:::;x ; x1;:::;xq/. 9 6

Example A.1: Bilinearity of the tensor product x1 ˝ x2 For every x1; x2 2 X, x; y 2 X and r 2 R bilinearity implies

.x C y/ ˝ x2 D x ˝ x2 C y ˝ x2 .internal left-linearity/ x1 ˝ .x C y/ D x1 ˝ x C x1 ˝ y .internal right-linearity/ rx ˝ x2 D r.x ˝ x2/.external left-linearity/ x1 ˝ ry D r.x1 ˝ y/.external right-linearity/: | 8 7 p A-1 Multilinear Functions and the Tensor Space Tq 573

x x T0 The generalization of bilinearity of 1 ˝ 2 2 2 to multilinearity of x1 xp x x Tp ˝ :::˝ ˝ 1 ˝ :::˝ q 2 q is obvious.

Tp Deﬁnition A.1. (multilinearity of tensor space q ):

1 p For every x ;:::;x 2 X and x1;:::;xq 2 X as well as u; v 2 X , x; y 2 X and r 2 R multilinearity implies

2 p .u C v/ ˝ x ˝ :::˝ x ˝ x1 ˝ :::˝ xq 2 p 2 p D u ˝ x ˝ :::˝ x ˝ x1 ˝ :::˝ xq C v ˝ x ˝ :::˝ x ˝ x1

˝ :::˝ xq .internal left-linearity/ 1 p x ˝ :::˝ x ˝ .x C y/ ˝ x2 ˝ :::xq 1 p 1 p D x ˝ :::˝ x ˝ x ˝ x2 ˝ :::˝ xq C x ˝ :::˝ x ˝ y ˝ x2

˝ :::˝ xq .internal right-linearity/ 2 p ru ˝ x ˝ :::˝ x ˝ x1 ˝ :::˝ xq 2 p D r.u ˝ x ˝ :::˝ x ˝ x1 ˝ :::˝ xq/.external left-linearity/ 1 p x ˝ :::˝ x ˝ rx ˝ x2 ˝ :::˝ xq 1 p D r.x ˝ :::˝ x ˝ x ˝ x2 ˝ :::˝ xq /.external right-linearity/:

A possible way to visualize the different multilinear functions which span T0 T1 T0 T2 T1 T0 Tp f 0; 0; 1; 0; 1; 2;:::; q g is to construct a hierarchical diagram or a special tree as follows.

∈ 0 {0} 0S

x1 ∈ 1 x ∈ 0 0 1 1

x1⊗ x2∈ 2 x1 ⊗ x ∈ 1 x ⊗ x ∈ 0 0 1 1 1 2 2

x1 ⊗ x2 ⊗ x3∈ 3 x1 ⊗ x2 ⊗ x ∈ 2 x1 ⊗ x2 ⊗ x ∈ 2 x ⊗ x ⊗ x ∈ 0 0 ; 1 1 ; 1 1 ; 1 2 3 3 574 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra

When we learnt ﬁrst about the tensor product symbolised by “˝”aswellasits multilinearity we were left with the problem of developing an intuitive under- 1 2 1 standing of x ˝ x , x ˝ x1 and higher order tensor products. Perhaps it is helpful to represent ”the involved vectors” in a contravariant or in a covariant 1 1 2 3 1 2 3 basis. For instances, x D e x1 C e x2 C e x3 or x1 D e1x C e2x C e3x is 1 2 3 a left representation of a three-dimensional vector in a 3-left basis fe ; e ; e gl of contravariant type or in a 3-left basis fe1; e2; e3gl of covariant type. Think in terms of 1 x or x1 as a three-dimensional position vector with right coordinates fx1;x2;x3g or fx1;x2;x3g , respectively. Since the intuitive algebras of vectors is commutative we 1 2 3 may also represent the three-dimensional vector in a in a 3-right basis fe ; e ; e gr of contravariant type or in a 3-right basis fe1; e2; e3gr of covariant type such that 1 1 2 3 1 2 3 x D x1e C x2e C x3e or x1 D x e1 C x e2 C x e3 is a right representation of a three-dimensional position vector which coincides with its left representation thanks to commutativity. Further on, the tensor product x ˝ y enjoys the left and right representations X3 X3 1 2 3 1 2 3 i j e x1 C e x2 C e x3 ˝ e y1 C e y2 C e y3 D e ˝ e xi yj iD1 j D1 and X3 X3 1 2 3 1 2 3 i j x1e C x2e C x3e ˝ y1e C y2e C y3e D xi yj e ˝ e iD1 j D1 which coincides again since we assumed a commutative algebra of vectors. The product of coordinates .xi yj /; i; j 2f1; 2; 3g is often called the dyadic product. Please do not miss the alternative covariant representation of the tensor product x ˝ y which we introduced so far in the contravariant basis, namely

X3 X3 1 2 3 1 2 3 i j e1x C e2x C e3x ˝ e1y C e2y C e3y D ei ˝ ej x y iD1 j D1 of left type and

X3 X3 1 2 3 1 2 3 i j x e1 C x e2 C x e3 ˝ y e1 C y e2 C y e3 D x y ei ˝ ej iD1 j D1 of right type. In a similar way we produce

3;3;3X 3;3;3X i j k i j k x ˝ y ˝ z D e ˝ e ˝ e xi yj zk D xi yj zk e ˝ e ˝ e i;j;kD1 i;j;kD1 of contravariant type and p A-1 Multilinear Functions and the Tensor Space Tq 575

3;3;3X 3;3;3X i j k i j k x ˝ y ˝ z D ei ˝ ej ˝ ekx y z D x y z ei ˝ ej ˝ ek i;j;kD1 i;j;kD1 of covariant type. “Mixed covariant-contravariant” representations of the tensor 1 product x1 ˝ y are

X3 X3 X3 X3 1 j i i j x1 ˝ y D ei ˝ e x yj D x yj ei ˝ e iD1 j D1 iD1 j D1 or X3 X3 X3 X3 1 i j j i x ˝ y1 D e ˝ ej xi y D xi y e ˝ ej : iD1 j D1 iD1 j D1

1 In addition, we have to explain the notion x 2 X ; x1 2 X: While the vector x1 is an element of the vector space X , x1 is an element of its dual space. What is a dual space? Indeed the dual space X is the space of linear functions over the elements of X. For instance, if the vector space X is equipped with inner product, namely hei je j iDgij ;i;j 2 f1; 2; 3g, with respect to the base vectors fe1; e2; e3g which span the vector space X,then

X3 j ij e D ei g iD1 transforms the covariant base vectors fe1; e2; e3g into the contravariant base vectors 1 2 3 ij 1 fe ; e ; e gDspan X by means of Œg D G , the inverse of the matrix Œgij D 33 G 2 R . Similarly the coordinates gij of the metric tensor g are used for “raising” i or “lowering” the indices of the coordinates x ;xj , respectively, for instance

X3 X3 i ij j x D g xj ;xi D gij x : j D1 j D1

In a ﬁnite dimensional vector space, the power of a linear space X and its dual X? does not show up. In contrast, in an inﬁnite dimensional vector space X the dual space X? is the space of linear functionals which play an important role in functional analysis. While through the tensor product “˝” which operated on vectors, e.g. x ˝ y, we constructed the p-contravariant, q-covariant tensor space or Tp X X T2 T0 T1 space of multilinear functions q . ; / ,e.g. 0; 2; 1 , we shall generalise the p representation of the elements of Tq by means of

nDXdim X i1 ip Tp f D e ˝ :::˝ e fi1:::ip 2 0 i1;:::;ipD1 576 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra

nDXdim X i1:::iq T0 f D ei1 ˝ :::˝ eiq f 2 q i1;:::;iq D1

nDXdim X nDXdim X i ;:::;i f D ei1 ˝ :::˝ eip ˝ e ˝ :::˝ e f 1 p 2 Tp j1 jq j1;:::;jq q i1;:::;ipD1 j1;:::;jqD1 for instance

X3 X3 i j i j T2 f D e ˝ e fij D fij e ˝ e 2 0 i;jD1 i;jD1 X3 X3 ij ij T0 f D ei ˝ ej f D f ei ˝ ej 2 2 i;jD1 i;jD1 X3 X3 i i i i T1 f D e ˝ ej fj D fj e ˝ ej 2 1 i;jD1 i;jD1

i ;:::;i We have to emphasize that the tensor coordinates f ;fi1;:::;iq , f 1 p are no i1;:::;ip j1;:::;jp longer of dyadic or product type. For instance, for § ¤ ¦(2,0)-tensor : bilinear functions: ¥

Xn Xn i j i j T2 f D e ˝ e fij D fij e ˝ e 2 0 i;jD1 i;jD1

fij ¤ fi fj § ¤ ¦(2,1)-tensor : trilinear functions: ¥

Xn Xn i j k k i j T2 f D e ˝ e ˝ ekfij D fij e ˝ e ˝ ek 2 1 i;j;kD1 i;jD1

k k fij ¤ fi fj f § ¤ ¦(3,1)-tensor : ternary functions: ¥

Xn Xn i j k l l i j k T3 f D e ˝ e ˝ e ˝ el fij k D fij k e ˝ e ˝ e ˝ el 2 1 i;j;k;lD1 i;jD1

l l fij k ¤ fi fj fkf p A-1 Multilinear Functions and the Tensor Space Tq 577 holds. Table A.1 is a list of .p; q/-tensors as they appear in various sciences. Of special importance is the decomposition of multilinear functions as elements p of the space Tq into their symmetric, antisymetric and residual constituents we are going to outline.

p Table A.1 Various examples of tensor spaces Tq (p & q W rank of T2 tensor) (2,0) tensor, tensor space 0 Metric tensor Differential Gauss curvature tensor geometry Ricci curvature tensor Gravity gradient tensor Gravitation Faraday tensor, Maxwell tensor Electromagnetic Tensor of dielectric constant Tensor of permeability Strain tensor, stress tensor Continuum mechanics Energy momentum tensor Mechanics Electromagnetism General relativity Second order multipole Gravitostatics tensor Magnetostatics Electrostatics Variance-covariance Mathematical matrix statistics

p Table A.2 Various examples of tensor spaces Tq (p C q W rank of T2 tensor) (2,1) tensor, tensor space 1 Cartaan torsion tensor Differential Deometry Third order multipole Gravitostatics tensor Magnetostatics Electrostatics skewness tensor Mathematical Third momentum tensor Statistics of probability distribution Tensor of piezoelectric Coupling of stress constant and electrostatic ﬁeld 578 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra

p Table A.3 Various examples of tensor spaces Tq (p C q W rank of T3 T2 tensor) (3,1) tensor, (2,2) tensor, tensor space 1, 2 Riemann curvature tensor Differential Geometry Fourth order multipole Gravitostatics tensor Magnetostatics Electrostatics Hooke tensor Stress-strain relation Constitutive equation Continuum mechanics Elasticity, viscosticity Curtosis tensor Mathematical Fourth moment tensor of Statistics a probability distribution

A-2 Decomposition of Multilinear Functions into Symmetric Multilinear Functions, Antisymmetric Multi-linear Functions and Residual Multilinear p p p p Functions TTq D Sq ˚ Aq ˚ Rq

Tp Tp Sp q as the space of multilinear functions follows the decomposition q D q ˚ p p p p Aq ˚ Rq into the subspace Sq of symmetric multilinear functions, the subspace Aq p of antisymmetric multilinear functions and the subspace Rq of residual multilinear functions:

Box A.2i. (Antisymmetry of the symbols fi1:::ip ):

fij Dfji

fij k Dfikj;fjki Dfjik;fkij Dfkj i

fij k l Dfij l k ;fjkli Dfjkil;fklij Dfklji;fkijk Dflikj

Box A.2ii. (Symmetry of the symbols fi1:::ip ):

fij D fji

fij k D fikj;fjki D fjik;fkij D fkj i

fij k l D fij l k ;fjkli D fjkil;fklij D fklji;fkijk D flikj A-2 Decomposition of Multilinear Functions into Symmetric Multilinear Functions 579

p Box A.2iii. (The interior product of bases, span S , n D dim X D dim X D 3):

1 S1 W ei 1Š 1 S2 W .ei ˝ ej C ej ˝ ei / 2Š DW ei _ ej ; ei _ ej DCej _ ei 1 S3 W .ei ˝ ej ˝ ek C ei ˝ ek ˝ ej C ej ˝ ek ˝ ei 3Š C ej ˝ ei ˝ ek C ek ˝ ei ˝ ej C ek ˝ ej ˝ ei / DW ei _ ej _ ek

ei _ ej _ ek D ei _ ek _ ej D ek _ ei _ ej D ek _ ej _ ei D ej _ ek _ ei D ej _ ei _ ek

p Box A.2iv. (The exterior product of bases, span A , n D dim X D dim X D 3):

1 A1 W ei 1Š 1 A2 W .ei ˝ ej ej ˝ ei / 2Š DW ei ^ ej ; ei ^ ej Dej ^ ei 1 A3 W .ei ˝ ej ˝ ek ei ˝ ek ˝ ej C ej ˝ ek ˝ ei 3Š ej ˝ ei ˝ ek C ek ˝ ei ˝ ej ek ˝ ej ˝ ei / DW ei ^ ej ^ ek

ei ^ ej ^ ek Dei ^ ek ^ ej DCek ^ ei ^ ej Dek ^ ej ^ ei D ej ^ ek ^ ei Dej ^ ei ^ ek 580 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra

Box A.2v. (Sp, symmetric multilinear functions):

8 9

n C p 1 2p 1 Corollary:dimSp D , in particular if nDp,thendimSp D . p p

Box A.2vi. (Ap, antisymmetric multilinear functions):

8 9

n Corollary:dimAp D nŠ=.pŠ.n p/Š/D in particular if nDp,thendimApD1. p

Box A.2vii. (Aq, antisymmetric multilinear functions, exterior product Proposition):

(a) For every x1;:::;xi1; xiC1;:::;xq 2 X as well as x; y 2 X and r; s 2 R multilinearity implies

x1 ^ :::^ xi1 ^ .rx C sy/ ^ xiC1 ^ :::^ xq

D r.x1 ^ :::^ xi1 ^ x ^ xiC1 ^ :::^ xq

C s.x1 ^ :::^ xi1 ^ y ^ xiC1 ^ :::^ xq: 582 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra

(b) For every permutation of f1;2;:::;qg we have x x x x x x 1 ^ 2 ^ :::^ q D sign ./ 1 ^ 2 ^ :::^ q:

B ^ A D .1/qsA ^ B:

(d) For every q, 0 q n, the tensor space Aq of antisymmetric multilinear functions has dimension n dim A D D nŠ=.qŠ.n q/Š/: q q

T1 T0 T2 T0 R2 R3 As detailed examples we like to decompose f 0; 1; 0; 2g in and , respectively, into symmetric and antisymmetric constituents.

Tp Sp p Rp Example A.2. ( q D q ˚ Aq ˚ q , decomposition of multilinear functions into symmetric and antisymmetric constituents):

As a ﬁrst example of the decomposition of multilinear functions (tensor space) into symmetric and antisymmetric constituents we consider a linear space X (vector space) of dimension dim X D n D 2. Its dual space X,dimX D dim X D n D 2, is spanned by orthonormal contravariant base vectors fe1; e2g. Choose q D 0, p D 1 and p D 2.

1 2 X D span fe1; e2g versus X D span fe ; e g ( ) X2 T1 1 S1 ei e1 e2 X 0 D A D 3 f D fi D f1 C f2 2 iD1 T2 2 S2 0 D A ˚ 8 9 < X2 = T2 ei ej 0 3 f D : ˝ fij ; i;jD1 1 1 1 2 2 1 2 2 D e ˝ e f11 C e ˝ e f12 C e ˝ e f21 C e ˝ e f22 1 D e1 ˝ e1f C e2 ˝ e2f C .e1 ˝ e2 e2 ˝ e1/f 11 22 2 12 1 1 C .e1 ˝ e2 C e2 ˝ e1/f .e1 ˝ e2 e2 ˝ e1/f 2 12 2 21 1 C .e1 ˝ e2 C e2 ˝ e1/f 2 21 1 1 2 2 1 2 1 2 D e _ e f11 C e _ e f22 C e ^ e .f12 f21/=2 C e _ e .f12 C f21/=2 dim S2 D 3; dim A2 D 1: A-2 Decomposition of Multilinear Functions into Symmetric Multilinear Functions 583

As a second example of the decomposition of multilinear functions (tensor space) into symmetric and antisymmetric constituents we consider a linear space X (vector space) of dimension dim X D n D 3, spanned by orthonormal contravariant base vectors fe1; e2; e3g. Choose p D 0, q D 1 and 2. span X Dfe1; e2; e3g ( ) X3 T0 S e i e 1 e 2 e 3 X 1 D A1 D 1 3 f D i f D 1f C 2f C 3f 2 iD1 T0 D A ˚ S 2 82 2 9 < X3 = T0 e e ij 2 3 f D : i ˝ j f ; i;jD1 ( ) X3 X3 X3 i1 i2 i3 D ei ˝ e1f C ei ˝ e2f C ei ˝ e3f iD1 iD1 iD1 11 21 31 12 22 D e1 ˝ e1f C e2 ˝ e1f C e3 ˝ e1f C e1 ˝ e2f C e2 ˝ e2f 32 13 23 33 C e3 ˝ e2f C e1 ˝ e3f C e2 ˝ e3f C e3 ˝ e3f 1 1 C .e ˝ e e ˝ e /f 12 C .e ˝ e C e ˝ e /f 12 2 1 2 2 1 2 1 2 2 1 1 1 .e ˝ e e ˝ e /f 21 C .e ˝ e C e ˝ e /f 12 2 1 2 2 1 2 1 2 2 1 1 1 C .e ˝ e e ˝ e /f 23 C .e ˝ e C e ˝ e /f 23 2 2 3 3 2 2 2 3 3 2 1 1 .e ˝ e e ˝ e /f 32 C .e ˝ e C e ˝ e /f 32 2 2 3 3 2 2 2 3 3 2 1 1 C .e ˝ e e ˝ e /f 31 C .e ˝ e C e ˝ e /f 31 2 3 1 1 3 2 3 1 1 3 1 1 .e ˝ e e ˝ e /f 13 C .e ˝ e C e ˝ e /f 13 | 2 3 1 1 3 2 3 1 1 3

p p p p p p Since the subspaces Sq , Aq and Rq are independent, Sq ˚ Aq ˚ Rq denotes p p p p the direct sum of subspace Sq , Aq and Rq . Unfortunately Tq as the space of multilinear functions cannot be completely decomposed in the space of symmetric and antisymmetric multilinear functions: For instance, the dimension identities p p p nCp1 p n apply dim T D n ,dimS D . p /,dimA D .p/ with respect of a vector space X of dimension dim X D n, such that dim Rp D dim Tp dim Sp dim Ap D p nCpC1 n p n . p / .p/

A-3 Matrix Algebra, Array Algebra, Matrix Norm and Inner Product

Symmetry and antisymmetry of the symbols fi1:::ip can be visualized by the trees of Box A.2i and A.2ii. With respect to the symbols of the interior product “_”andtheexterior product “^”(“wedge product”) we are able to redeﬁne symmetric and antisymmetric functions according to Box A.2iii–vi. Tp Note the isomorphism of tensor algebra q and array algebra, namely of n (a) Œfi 2 R (onedimensional array, “column vector”, dim Œfi D n 1) nn (b) Œfij 2 R (twodimensional array, column-row array, “matrix”, dim Œfij D n n) nnn (c) Œfij k 2 R (threedimensional array, “indexed matrix”, dim Œfij k D n n n) etc. For the base space x 2 ˝ R3 to be threedimensional Euclidean we had answered the question how to measure the length of a vector (“norm”) and the angle between two vectors (“inner product”). The same question will ﬁnally been raised p Tp for tensors tq 2 q . The answer is constructively basedonthevectorization of the arrays Œfij , Œfij k , ::: Œfi1:::ip by taking advantage of the symmetry-antisymmetry structure of the arrays and later on applying the Euclidean norm and the Euclidean inner product to the vectorized array. For a 2-contravariant, 0-covariant tensor we shall outline the procedure.

(a) Firstly let F D Œfij be the quadratic matrix of dimension dim F D n n, T2 F an element of 0. Accordingly vec is the vector 2 3 fi1 6 7 6 fi2 7 6 7 F 6 : 7 F 2 vec D 6 : 7 ; dim vec D n 1 4 5 fin1

fin

which is generated by stacking the elements of the matrix F columnwise in a vector. The Euclidean norm and the Euclidean inner product of vec F ,vecG, respectively is kvec F k2 WD .vec F /T .vec F / D tr F T F ; G F T G F T G hvec F jvec iWD.vec / vec D tr :

(b) Secondly let F D Œfij D Œfji be the symmetric matrix of dimension dim F D n n, an element of S 2. Accordingly vech F (read “vector half”) is A-3 Matrix Algebra, Array Algebra, Matrix Norm and Inner Product 585

the n.n C 1/=2 1 vector which is generated by stacking the elements on and under the main diagonal of the matrix F columnwise in a vector: 2 3 f11 6 7 6 : 7 6 7 6 f 7 6 n1 7 6 7 F F T F 6 f22 7 F D Œfij D Œfji D H) vech WD 6 7 ; dim vech D n.n C 1/=2 6 : 7 6 7 6 f2n 7 4 : 5

fnn

vech F D H vec F ; dim H D n.n C 1/=2 n2:

The Euclidean norm and the Euclidean inner product of vech F , vech G, respectively is # T F D Œfij D Œfji D F H) T G D Œgij D Œgji D G kvech F k2 WD .vech F /T .vech F / F G T G hvech jvech iWD.vech F/ .vech /:

(c) Thirdly let F D Œfij DŒfji be the antisymmetric matrix of dimension dim F D n n, an element of A2. Accordingly veck F (read “vector skew”) is the n.n 1/=2 1 vector which is generated by stacking the elements under the main diagonal of the matrix F columnwise in a vector: 2 3 f21 6 7 6: 7 6 7 6f 7 6 n1 7 6 7 F F T F 6f32 7 F D Œfji DŒfji D H) veck WD 6 7 ; dim veck D n.n 1/=2 6: 7 6 7 6fn2 7 4: 5

fn1n

veck F D K vec F ; dim K D n.n 1/=2 n2:

The Euclidean norm and the Euclidean inner product of veck F , veck G, respectively 586 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra # T F D Œfij DŒfji DF H) T G D Œgij DŒgji DG kveck F k2 WD .veck F /T .veck F / : F G F T G hveck jveck iWD.veck / .veck / Youwillﬁndtheforms(a)vec F and (b) VechF in the standard book of (Harville, 1997, Chap. 16, 2001, Chap. 16) including many examples. Here we will present Example A.3 in computing the norm as well as the inner product of a 2-contravariant, 0-covariant tensor for the operator vec, vech and veck. The veck operator has been introduced by (Grafarend and Schaffrin, 1993, pp. 418–419).

Example A.3. (Norm and inner product of a 2-contravariant, 0-covariant tensor): 2 3 adg .a/ A WD4 beh5 ; dim A D 3 3 H) vec A D Œa;b;c;d;e;f;g;h;kT ; cf k dim vec A D 9 1 k vec Ak2 D .vec A/T .vec A/ D tr AT A D a2 C :::C k2 2 3 ab c .b/ A WD4 bd e5D AT ; dim A D 3 3 H) vech A D Œa;b;c;d;e;fT ; cef dim vech A D 6 1 H vech A D2 vec A 3 10 0 000 000 6 7 6 01=20 1=2 0 0 0007 6 7 6 7 H 6 001=2 000 0007 H 8 WD 6 7 ; dim D 6 9 6 00 0 010 0007 4 00 0 001=201=205 00 0 000 001 kvech Ak2 D .vech A/T .vech A/ D a2 C b2 C c2 C d 2 C e2 C f 2 (Henderson2 and Searle (1978,3 p. 68–69)) 0 a b c 6 7 6 a0d e 7 .c/ A WD4 5DAT ; dim A D 3 3 H) bd 0f cef 0 veck A D Œa;b;c;d;e;fT ; dim veck A D 6 1 veck A D K vec A A-4 The Hodge Star Operator, Self Duality 587 2 3 0100 1000 0 000 0000 6 7 6 0010 0000 1 000 00007 6 7 1 6 7 K 6 0001 0000 0 000 10007 8 WD 6 7 ; 2 6 0000 0010 0 100 00007 4 0000 0001 0 000 0 1005 0000 0000 0 001 0010 dim K D 6 16 kveck Ak2 D a2 C b2 C c2 C d 2 C e2 C f 2 |

A-4 The Hodge Star Operator, Self Duality

In Chap. 3, we took advantage of the Hodge star operator, for instance for transforming an overdetermined system of linear equations (inconsistent system) into a system of condition equations. There we referred to multilinear operations “join” and“meet”. The most important operator of the algebra of antisymmetric multilinear functions is the “Hodge star operator” which we shall present ﬁnally. In addition, we shall bring to you the surprising special feature of skew algebra called “selfduality”. p The algebra Aq of antisymmetric multilinear functions has been based on the exterior product “^” (“wedge product”). There has been created a duality operator called the Hodge star operator which is a linear map of Ap ! Anp where n D dim X D dim X denotes the dimension of the base space X R3.The basic idea of such a map of antisymmetric multilinear functions f 2 Ap into antisymmetric linear functions f 2 Anp originates according to Box A.2vii from the following situation: The multilinear base of Ap is spanned by

f1; ei1 ; ei1 ^ ei2 ;:::ei1 ^ ei2 ^ :::^ eip g once we focus on p D 0;1;2;:::n, respectively. Obviously for any dimension p number n and p-contravariant, q-covariant index of the skew tensor space Aq there is an associated cobasis, namely 8 <ˆ basis Wf1; ei1 g n D 1; p D 0; 1 W :ˆ associated cobasis Wfei1 ;1g 8 <ˆ basis Wf1; ei1 ; ei1 ^ ei2 g n D 2; p D 0; 1; 2 W :ˆ associated cobasis Wfei1 ^ ei2 ; ei2 ;1g 588 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra 8 <ˆ basis Wf1; ei1 ; ei2 ^ ei3 ; ei1 ^ ei2 ^ ei3 g n D 3; p D 0; 1; 2; 3 W :ˆ associated cobasis Wfei1 ^ ei2 ^ ei3 ; ei2 ^ ei3 ; ei3 ;1g: in general, for arbitrary n 2 N, p D 0;1;:::;n 1; n 1 ( basis W f1; ei1 ;:::;ei1 ^ :::^ ein g associated cobasis W ei1 ei2 ein1 ein ei2 ein1 ein ein1 ein ein 0f ^ ^ :::^ ^ ; ^ ::: ^ ;:::; ^ ; ;1; ) as long as we concentrate on p-contravariant Ap only. A similar set-up of basis- p associated cobasis for q-covariant Aq and mixed Aq can be made. The linear map Ap ! Anp,theHodge star operator ¨ 1 i :::i .ei1 ^ :::^ eip / WD 1 p eipC1 ^ :::^ ein ipC1:::in .n p/Š © maps by means of the permutation symbol 2 C 1 for an even permutation 6 6 of f1;2;:::;n 1; ng 6 i1 ::: ip 6 WD 6 1 for an odd permutation ipC1 ::: in 6 4 of f1;2;:::;n 1; ng 0 otherwise

– sometimes called Eddington’s epsilons – on orthonormal (“unimodular”) base of Ap onto an orthonormal (“unimodular”) base of Anp. For antisymmetric multilinear functions also called antisymmetric tensor-valued functions represented in an orthonormal (“unimodular”) base the Hodge star operator is the following linear map 8 9 < D X = 1 n Xdim Tp Ap 3 f D ei1 ^ :::^ eip f ; 0 :pŠ i1:::ip ; i1;:::;ip D1 8 9 < D X D X = 1 n Xdim n Xdim 1 p np i1:::ip ipC1 in T A 3f WD e ^ :::^ e fi :::i : 0 :.n p/Š pŠ ipC1:::in 1 p; ipC1;:::;in i1;:::;ip

As soon as the base space x 2 ˝ R3 is not covered by Cartesian coordinates, rather by curvilinear coordinates, its coordinates base A-4 The Hodge Star Operator, Self Duality 589 @ @ @ fb1; b2; b3gDfdy1;dy2;dy3g versus fb ; b ; b gD ; ; 1 2 3 @y 1 @y 2 @y 3 of contravariant versus covariant type is neither orthogonal nor normalized. It is for this reason that ﬁnally we present f ,theHodge star operator of an antisymmetric multilinear function f , also called the dual of f , in a general coordinate base.

Deﬁnition A.2. (Hodgestaroperator, the dual of an antisymmetric multilinear function):

If an antisymmetric .p; 0/ multilinear function is an element of the skew algebra Ap with respect, to a general base fbi1 ^ :::^ bip g is given 8 9 < X = 1 nDXdim f D bi1 ^ :::^ bip f :pŠ i1:::ip ; i1;:::;ip D1

then the8Hodge star operator,thedual of f , can be uniquely represented by < X X X 1 nDXdim nDXdim nDXdim 1 (a) f D bipC1 ^ :::^ bip :.n p/Š pŠ ipC1;:::;in i1;:::ip j1;:::;jp )

p i1j1 ip jp gi1:::ip ipC1:::in g :::g fj1:::jp 8 9 < nDdim X nDdim X = 1 X X 1 p (b) f D bipC1 ^ :::^ bin g f i1:::ip :.n p/Š pŠ i1:::ipipC1:::in ; (ipC1;:::;in i1;:::;ip ) nDdimPX p i1:::ip (c) .f/k1:::knp D gi1:::ipk1:::knp f i1;:::ip as an element of the skew algebra Anp in the general associated cobase fbpC1^ :::^bin g with respect to the base space x 2 X Rpn of dimension n D dim X D kl 1 p dim X and Œg D G D adj G=det G , g D jgklj. If we extend the algebra Ap of antisymmetric multilinear functions by 1 D e1 ^ :::en 2 An and e1 ^ :::^ en D 1 2 A0 D R, respectively, let us collect some properties of f ,thedual of f . Proposition A.1. (Hodgestaroperator, the dual of an antisymmetric multilinear function):

Let the linearly ordered base fe1;:::;eng be orthonormal (“unimodular”). Then the Hodge star operator of an antisymmetric multilinear function f ,the dual of f , with respect to fe1;:::eng satisﬁes the following: 590 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra

(a) maps antisymmetric p-contravariant tensor-valued functions to antisymmetric .n p/-contravariant tensor-valued functions: (WAp ! Anp 1 D e1 ^ :::^ en DW e for every 1 2 A0; e 2 A (b) e D 1 for every e 2 An;12 Ap (c) f D .1/p.np/f for every f 2 Ap (d) f ^f Dkf k2e1 ^ :::^ en with respect to the norm

X 1 nDXdim kf k2 WD f f i1:::ip : pŠ i1:::ip i1;:::;ipD1

Example A.4. (Hodge star operator n D dim X D dim X D 3, span X D fe1; e2; e3g, Ap ! Anp): n D 3; p D 0 W1 D e1 ^ e2 ^ e3 1 n D 3; p D 1 Wei1 D i1 ei2 ^ ei3 2 i2i3 2 1 1 2 3 3 2 2 3 6 e D .e ^ e e ^ e / D e ^ e 6 2 6 6 1 6 e2 e3 e1 e1 e3 e3 e1 6 D . ^ ^ / D ^ 6 2 4 1 e3 D .e1 ^ e2 e2 ^ e1/ D e1 ^ e2 2 n D 3; p D 2 Wei1 ^ ei2 D i1i2 ei3 i3 2 e1 ^ e2 D e3 6 4 e2 ^ e3 D e1 e3 ^ e1 D e2 n D 3; p D 3 Wei1 ^ ei2 ^ ei3 D 1 |

Example A.5. (Hodge star operator of an antisymmetric tensor-valued function, n D dim X D dim X D 3, Ap ! Anp):

Throughout we apply the summation convention over repeated indices. ( f “0-differential form” n D 3; p D 0 W f D fdx1 ^ dx2 ^ dx3 “3-differential form’’ A-4 The Hodge Star Operator, Self Duality 591 8 i1 ˆ f D dx fi1 “1-differential form” ˆ ˆ 1 < i1 i2 i3 f D dx ^ dx fi 2 i2i3 1 n D 3; p D 1 W ˆ ˆ 2 3 3 1 ˆ D f1dx ^ dx C f2dx ^ dx :ˆ 1 2 C f3dx ^ dx “2-differential form” 8 1 ˆ i1 i2 ˆ f D dx ^ dx fi i “2-differential form” <ˆ 2 1 2 1 n D 3; p D 2 W i1i2 i3 ˆ f D dx fi i ˆ i3 1 2 :ˆ 2 1 2 3 D f23dx C f31dx C f12dx “1-differential form” 8 ˆ 1 <ˆ f D dxi1 ^ dxi2 ^ dxi3 f “3-differential form” 6 i1i2i3 n D 3; p D 3 W ˆ 1 : f D i1i2i3 f D f “0-differential form” | 6 i1i2i3 123

Example A.6. (Hodge star operator, n D dim X D dim X D 3,“” product (cross product)):

By means of the Hodge star operator we are able to interpret the “” product (“cross product”) in threedimensional vector space. If the vectors x; y 2 X,dimX D 3, presented in the orthonormal (“unimodular”) base fe1; e2; e3g the following equivalence between x ^ y and x y holds: # i j x D ei x ; y D ej y .summation convention H) x 2 X ; y 2 X i;j 2f1; 2; 3; g/

i j x ^ y D ei ^ ej x y 1 2 2 1 2 3 3 2 D e1 ^ e2.x y x y / C e2 ^ e3.x y x y / 3 1 1 3 C e3 ^ e1.x y x y / 1 2 2 1 H) .x ^ y/ D.e1 ^ e2/.x y x y / 2 3 3 2 C.e2 ^ e3/.x y x y / e e x3y1 x1y3 k e xi yj C. 3 ^ 1/. / D ij k x y e x1y2 x2y1 e x2y3 x3y e x3y1 x1y3 . ^ / D 3. / C 1. 2/ C 2. / 2 3 3 2 3 1 1 3 1 2 2 1 D e1.x y x y / C e2.x y x y / C e3.x y x y / # ¨ x y e e xi xj D i j x y x y e e ke H) D. ^ / | i j WD ij k © 592 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra

Example A.7. (Hodge star operator, n D dim X D dim X D 4;X 2 R4, Minkwinski space, self-duality (Atiyah et al., 1978)):

By means of the Examples A-5–A-7 we like to make you familiar with (a) the Hodge star operator of an antisymmetric tensor-valued function over R3, (b) its equivalence to the “x” product (“cross product”) and (c) selfduality in a fourdimensional space. Such a selfduality plays a key role in differential geometry and physics as being emphasized by Atiyah et al. (1978). 9 6 Historical Aside Thus we have constructed an anticommutative algebra by implement- ing the “exterior product”“^”, also called “wedge product”, initiated by H. Grassmann in “Ausdehnungslehre” (second version published in 1882). See also his collected works, H. Grassmann (1911). In addition the work by G. Peano (Calcolo geometrico secondo, l’Ausdehnungslehre di Grassmann, Fratelli Bocca Editori, Torino 1888) should be mentioned here. The historical development may be documented by the work of Forder (1960). A modern version of the “wedge product” is given by Berman (1961). In particular we mention the contribution by Barnabei et al. (1985) where by avoiding the notion of the dual X of a linear space X and based upon operations like union, intersection,andcomplement – i.e. known in Boolean algebra – have developed a double algebra with exterior products of type one (“wedge product”, “the join”) and of type two (“the meet”), namely “to restore H. Grassmanns original ideas to full geometrical power”. The star operator “” has been introduced by W. V. D. Hodge, being implemented into algebra in the work Hodge (1941)and Hodge and Pedoe (1968), pp. 232–309). Here the star operation has been called “dual Grassmann coordinates”; in addition “intersections and joins”havebeen introduced. 8 7

A-5 Linear Algebra

Multilinear algebra is built on linear algebra we are going into now. At first we give a careful definition of linear algebra which secondly we deepen by the diagrams “Ass”, “Uni”and“Comm”. The subalgebra “ring with identity” which is of central importance for solving polynomial equations by means of Groebner bases,the Buchberger algorithm and the multipolynomial resultant method is our third subject. Section 4 introduces the motion of division algebra and the non-associative algebra. Fifthly, we confront you with Lie algebra (“God is a lie Group”); in particular with Witt algebra. Section 6 compares Lie algebra and Killing analysis.Hereweadd some notes on the difficulties of a composition algebra in Sect. 7. Finally in Sect. 8 A-5 Linear Algebra 593 matrix algebra is presented again, but this time as a division algebra. As examples of a division algebra as well as composition algebra we introduce complex algebra (Clifford algebra Cl.0;1/) in Sect. 9 and quaternion algebra in Sect. 10 (Clifford algebra Cl.0;2/ ) which is followed by an interesting letter of W. R. Hamilton (16 October 1943) to his son reproduced in Sect. 11. Octonian algebra (Clifford algebra with respect to H H) in Sect. 12 is an example for a “non associative” algebra as well as a composition algebra. Of course, we have reserved “Sect. 13” for the fundamental Hurwitz theorem of composition algebra and the fundamental Frobenius theorem of division algebra.

A-51 Deﬁnition of a Linear Algebra

Up to now we have succeeded to introduce the base space X of vectors x 2 X D R3 equipped with a metric and specialized to be threedimensional Euclidean.Wehave extended the base space to a tensor space, namely from vector-valued functions to n tensor-valuedfunctions on fR ;gij g. Now we proceed to give linear and multilinear functions an algebraic structure, namely by the deﬁnition of two binary operations, (two internal relations) .opera/1 D ˛, .opera/2 D and one binary operation (external relation) .opera/3 D ˇ.

Deﬁnition A.51. (linear algebra over the ﬁeld of real numbers, linearity of vector space X):

Let R be the ﬁeld of real numbers. A linear algebra over R or R-algebra consists of a set X of objects, two internal relations (either “additive”or “multiplicative”) and one external relation

.opera/1 DW ˛ W X X ! X

.opera/2 DW ˇ W R X ! X or X R ! X

.opera/3 DW W X X ! X:

1 With respect to the internal relation ˛ (“join”) X as a linear space is a vector space over R,anAbelian group written “additively”or“multiplicatively”:

x; y; z 2 X

additively written multiplicatively written Abelian group Abelian group ˛.x; y/ DW x C y ˛.x; y/ DW x ı y .G1C/.x C y/ C z D x C .y C z/.G1ı/.x ı y/ ı z D x ı .y ı z/ .additive associativity/.multiplicative associativity/ 594 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra

.G2C/ x C 0 D x .G2ı/ x ı 1 D x .additive identity;.multiplicative identity; neutral element/ neutral element/ .G3C/ x C .x/ D 0 .G3ı/ x ı x1 D 1 .additive inverse/.multiplicative inverse/ .G4C/ x C y D y C x .G4ı/ x ı y D y ı x .additive commutativity;.multiplicative commutativity; Abelian axiom/ Abelian axiom/:

The triplet of axioms f.G1C/; .G2C/; .G3C/g or f.G1ı/; .G2ı/; .G3ı/g constitutes the set of group axioms. 2 With respect to the external relation ˇ the following compatibility conditions are satisﬁed:

x; y 2 X;r; s 2 R ˇ.r; x/ DWr x

.D1C/r .x C y/ D .x C y/ r.D1ı/r .x ı y/ D .x ı y/ r D r x C r y D .r x/ ı y D x r C y r D x ı .y ı r/ .1st additive distributivity/.1st multiplicative distributivity/ .D2C/.rC s/ x D x .r C s/ .D2ı/.rı s/ x D x .r ı s/ D r x C s x D r .s x/ D x r C x s D .x r/ s .2nd additive distributivity/.2nd multiplicative distributivity/

.D3/1 x D x 1 D x .left and right identity/

3 With respect to the internal relation (“meet”) the following compatibility conditions are satisﬁed:

x; y; z 2 X;r2 R .x; y/ DW x y .G1/.x y/ z D x .y z/ .associativity w.r.t internal multiplication/ A-5 Linear Algebra 595

.D1 C/ x .y C z/ D x y C x z .x C y/ z D x z C y z .left and right additive distributivity w.r.t internal multiplication/ .D1 ı/ x .y ı z/ D .x y/ ı z .x ı y/ z D x ı .y z/ .left and right multiplicative distributivity w.r.t internal multiplication/ .D2 /r .x y/ D .r x/ y .x y/ r D x .yr/ .left and right distributivity of internal and external multiplication/

Please, in the following pay attention to the 3 axiom sets of the linear algebra. We specify now the three axiomatic sets called “associativity” (ass), “unity”(Uni) and “commutativity” (comm).

A-52 The Diagrams “Ass”, “Uni” and “Comm”

Conventionally a linear algebra is minimally constituted by the triplet (X;˛;ˇ/ where X as a linear space is a vector space equipped with the linear maps ˛ W X X ! X and ˇ W R X ! X satisfying the axioms (Ass)and(Uni) according to the following diagrams:

(Ass): (Uni): The diagram The square b × id id × b ×× × × × × a × id α id × a

α × Commutes. Commutes. (Comm): The triangle

α α 596 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra

Axiom (Ass) expresses the requirement that the multiplication ˛ is associative whereas Axiom (Uni) means that the element ˇ.1/ of X is a left as well as a right unit for ˛. The algebra .X;˛;ˇ/is commutative if in addition it satisfies the axiom x y (Comm): commutes where X;X is the flip switching the factors: X;X. ı / D y ı x. | Indeed we have expressed a set of axioms both explicitly as well as in a diagram- matic approach which minimally constitute a linear algebra .X;˛;ˇ/. In addition, beside the first internal relation ˛ called “join” we have experienced a second internal relation called “meet” which had to be made compatible with the other relations ˛ and ˇ, respectively. Actually the diagram for the axiom (Dis) is left as an exercise. Obviously we have experienced the words “addition”and“multiplication”for various binary operations. Note that in the linear algebra isomorphic to the vector space as its geometric counterpart we have not specified the inner multiplication .x; y/ 2 X. In a threedimensional vector space of Euclidean type

.x; y/ DW .x ^ y/ D x y namely the star of the exterior product x ^ y or the “cross product” x y,for x 2 R3, y 2 R3 is an example. Sometimes

.x; y/ DW Œx; y is written by rectangular brackets. 9 6 Historical Aside

Following a proposal of L. Kronecker (“Uber¨ die algebraisch auﬂosbaren¨ Gleichungen (Abhandlung, 1929)theaxiom of commutativity .G4C/ or .G4ı/ is called after N. H. Abel (Memoire sur un classe particuliere` d’equations resolable´ algebraique,´ Crelle’s J. reine angewandte Mathematik 4 (1828) 131–156 Oeuvres vol. 1, pp. 478–514, vol. 2, pp. 217–243, 329–331 edited by S. Lie and L. Sylow, Christiana 1881) who dealt with a particular class of equations of all degrees which are solvable by radicals, e.g. the cyclotomic equation xn1 D 0. N. H. Abel has proved the following general theorem: If the roots of an equation are such that all roots can be expressed as rational functions of one of them, say x, and if any two of the roots, say r1x and r2x where r1 and r2 are rational functions are connected in such a way that r2r1x D r1r2x,then the equation can be solved by radicals. Refer r2r1x D r1r2x to .G1/. 8 7 A-5 Linear Algebra 597

A-53 Ringed Spaces: The Subalgebra “Ring with Identity”

In .G2ı/ the neutral element 1 as well as in .G3ı/ the inverse element has been multiplied from the right. Similarly left multiplication .G2ı/ by the neutral element 1 as well as .G3ı/ by the inverse element are deﬁned. Indeed it can be shown that there exist exactly one neutral element which is both left-neutral and right-neutral as well as exactly one inverse element which is both left-inverse and right-inverse. A subalgebra is called a “ring with identity” if the following seven conditions hold:

A ring with identity .G3/ is a division ring if every nonzero element of the ring has a multiplicative inverse.Acommutative ring is a ring with commutative multiplication .G4/. Modules are generalizations of the vector spaces of linear algebra in which the “scalars” are allowed to be from an arbitrary ring, rather than a ﬁeld of real numbers. They will be discussed as soon as we introduce superalgebras. Now we take reference to Lemma A.53. (anticommutativity): x ı x D 0 for all x 2 X ” x ı y D y ı x for all x; y 2 X.“ı” is used in the notation “^” accordingly.

Proof.

“ H) ”x ı y C y ı x D x ı x C x ı y C y ı x C y ı y D x ı .x C y/ C y ı .x C y/ D .x C y/ ı .x C y/ D 0 “ (H ”x D y H) x ı x Dx ı x H) x ı x D 0:

Later on we refer to the following algebras. | 598 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra

A-54 Deﬁnition of a Division Algebra and Non-Associative Algebra

Deﬁnition A.23. (division algebra):

A R-algebra is called division algebra over R, if all non-null elements of X0 WD X nf0g form additionally a group with respect to inner multiplication DW x ı y, namely

x; y; z 2 Xnf0g .G1ı/.x ı y/ ı z D x ı .y ı z/ (associativity of inner multiplication) .G2ı/ x ı 1 D x (identity of inner multiplication) .G3ı/ x ı x1 D 1 (inverse of inner multiplication)

Deﬁnition A.24. (non-associative algebra):

A weaking of a R-algebra is the non-associative algebra over R,iftheaxioms 1 , 2 and 3 of linear algebra according to Deﬁnition A.21 hold with the exception of .G1ı/, that is the associativity of inner multiplication is cancelled.

A-55 Lie Algebra, Witt Algebra

“perhaps god is a lie group” Many physicists believe that all modern physics is based on the operation called “Lie algebra”. Indeed more than 1000 textbooks and papers are written on this most important subject. Indeed many physicists believe in “perhaps god is a lie group”. Here, we can give a very short notice of “Lie algebra”. We skip a note of comparison “Lie algebra” and its brother “Killing analysis”.

Deﬁnition A.25. (Lie algebra):

A non-associative algebra is called Lie algebra over R, if the following operations with respect to inner multiplication WD x ı y hold: A-5 Linear Algebra 599

.L1/ x ı x D 0 .L2/ .x ı y/ ı z C .y ı z/ ı x C .z ı x/ ı y D 0: .Jacobi identity/

The examples of the Lie algebra are numerous. As a special Lie algebra we present the Witt algebra which is applied to Laurent polynomials.

Example A.8. (Witt algebra on the ring of Laurent polynomials (Chen, 1995)):

The Witt algebra W is the complex Lie algebra of polynomial ﬁelds on the unit circle S1. An element of W is a linear combination of the elements of the form in @ e @ ,where is a real parameter, and the Lie bracket of W is given by @ @ @ eim ;ein D i.n m/ei.mCn/ : @ @ @

A-56 Deﬁnition of a Composition Algebra

Various algebras are generated by adding an additional structure to the minimal set of axioms of a linear algebra blocked by (a), (b) and (c). Later on we use such an additional structure for matrix algebra.

Deﬁnition A.26. (composition algebra):

A non-associative algebra with 1 as identity of inner multiplication is called composition algebra over R, if there exists a regular quadratic form Q W X ! R which is compatible with the corresponding operations that is the following operations hold:

.K1/QW X ! R is a regular quadratic form; x;y;z 2 R;r2 R Q.r x/ D r2 Q.x/.quadratic form/ Q.x C y C z/ D Q.x C y/C Q.x C z/ C Q.y C z/ Q.x/ Q.y/ Q.z/ Q.r x C y/ r Q.x C y/ D .r 1/ Œr Q.x/ Q.y/ Q.x/ D 0 ” x D 0.regularity/ .K2/Q.x ^ y/ D Q.x/ Q.y/ .multiplicativity/ Q.1/ D 1 600 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra

The quadratic form introduced by Deﬁnition A.55 leads to the topological notion of scalar products, norm and metric we already used:

Lemma A.56. (scalar product, norm, metric):

In a composition algebra with a positive-deﬁnite quadratic form a scalar product (“inner product”) is deﬁned by the bilinear form hji W X X ! R with

1 hxjyiWD ŒQ.x C y/ Q.x/ Q.y/I 2

a norm is deﬁned by kkWX ! R with

kxkWDCŒQ.x/1=2

and metric is deﬁned by the bilinear form

.x; y/ WD CŒQ.x y/1=2:

Thus to the algebraic structure a topological structure is added, if in addition

.K3/ Q.x/ 0

for all x 2 X holds. Proof. § ¤ ¦(i) scalar product ¥

hji W X X ! R is a scalar product since 1 .1/ hxjyiD ŒQ.x C y/ Q.x/ Q.y/ 2 1 D ŒQ.y C x/ Q.y/ Q.x/ Dhyjxi .symmetry/ 2 1 .2/ hx C yjziD ŒQ.x C y C z/ Q.x C y/ Q.z/ 2 1 D ŒQ.x C z/ Q.x/ Q.z/ 2 1 C ŒQ.y C z/ Q.y/ Q.z/ 2 DhxjziChyjzi .additivity/ A-5 Linear Algebra 601

1 .3/ hrxjyiD ŒQ.rx C y/ Q.rx/ Q.y/ 2 1 D rŒQ.x C y/ Q.x/ Q.y/ 2 D r hxjyi .homogenity/ 1 .4/ hxjxiD ŒQ.x C x/ Q.x/ Q.x/ 2 1 D ŒQ.2x/ 2 Q.x/ D Q.x/ 0.positivity/ 2 § ¤ ¦(ii) norm ¥

kkWX ! R is a norm since .N 1/ kxkDCŒQ.x/1=2 0 (positivity) and kxkD0 ” x D 0 .N 2/ krxkDCŒQ.rx/1=2 DCŒr 2 Q.x/1=2 Djrjkxk (homogenity) .N 3/ kx C ykDCŒQ.x C y/1=2 DCŒQ.x/ C Q.y/ C 2hxjyi1=2 h i 1=2 DC kxk2 Ckyk2 C 2kxkkyk kxkCkyk: .Cauchy-Schwarz’ inequality/“triangle inequality”/ £ ¢(iii) metric ¡

W X X ! R is a metric since

.M1/ .x; y/ DCŒQ.x y/1=2 Dkx yk0.positivity/ and .x; y/ D 0 ” x y D 0 ” x D y .M2/ .x; y/ Dkx ykDj1jky xkD.y; x/.symmetry/ .M3/ .x; y/ Dkx ykDk.x z/ C .z y/k kx zkCkz xkD.x; z/ C .z; y/ (triangle inequality) | 602 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra

We treat in the next section a special case of a division algebra, namely matrix algebra.

A-6 Matrix Algebra Revisited, Generalized Inverses

It is known that every nonsingular matrix A has a unique inverse, usually denoted by A1 such that AA1 D A1A D I,whereI is the unit matrix. A matrix has a inverse only if it is square, and even then only if it is nonsingular, or alternatively if its columns or rows are linearly independent. Over the many years past it was felt the need of some kind of partial inverse of a matrix that is singular or even rectangularsingular matrix. By the term generalized inverse of a given matrix A we shall mean a matrix X associated in some way with the matrix A that (a) exists for a class of matrices larger than the class of nonsingular matrices, (b) has some of the properties of the usual inverse, and (c) reduces to the usual inverse when A is non singular. Some others have used the term “pseudoinvese” rather than general inverse without specifying the chosen type of general inverse. For a given matrix A, the matrix equation AXA D A alone characterizes those generalized inverses X that are of use in analyzing the solutions of the linear system Ax D b. For other purposes, other relationships play an essential role. Thus if we are concerned with least squares properties, the equation AXA D A is not enough and must be supplemented by other relations. There results a more restricted class of generalized inverses. Here we are limited to generalized inverses finite matrices,but extension to infinite-dimensional space and to differential and integral operations are very well known. E.H. Moore is attributed to have written one of the first papers on the topic of generalized inverses called by him the “general reciprocal” of any finite matrix,square or rectangular. His first publication on the subject was an abstract talk given at a meeting of the American Mathematical Society which appeared in 1920. Details of his talk were published only in 1935 (Moore, 1920), Moore (1935); (Moore and Nashed, 1974) after Moore’s death! Little note was taken of Moore’s discovery for 30 years after his first publication, during which time generalized inverses were taken for matrices by C. L. Siegel and for operators by Tseng (1936), Tseng (1949a)–Tseng (1949c), Tseng (1956), Murray and von Neumann (1936)and many others. A surprising revival of interest in general inverses appeared in 1950s when least squares properties were analyzed. The important properties were realized by Bjerhammar (1951a), Bjerhammar (1951b), (1968), one swedish colleague teaching LESS and geodetic applications of the subject. In 1955 R. Penrose extended A. Bjerhammar’s results on linear systems and showed that E.H. Moore inverse 0 satisfied four equations of type (a) AXA D A,(b)XAX D X,(c)AX D AX,and 0 (d) XA D XA nowadays called the axioms of the Moore-Penrose inverse,often denoted by AC. There are excellent textbooks on matrix algebra and generalized inverses of matrices including many examples upto 500! We like to mention the excellent text A-6 Matrix Algebra Revisited, Generalized Inverses 603 of Ben-Israel and Greville (1974), Bjerhammar (1973), Gere and Weaver (1965), Graybill (1963), Mardia et al. (1979), Nashed (1974), Rao and Mitra (1971), Rao and Rao (1998), Scarle (1982), Styan (1983). Here we will be unable to compete with the dense information given by these special texts. We assume that the reader is familiar with the standard notion of a matrix as a rectangular array of numbers offered by any undergraduate text of applied mathematics. Familiarity of various notion, for instance the dimension of a matrix of type n m, multiplication and addition of two matrices of type (a) “Cayley” or simply the matrix product C WD A:B,(b)“Kronecker-Zehfuss” denoted by C WD B ˝ A,(c)“Katri-Rao” denoted by C WD B ˇ A, and (d)“Hadamard” attributed by K WD G H. How are the various matrix product defined? Who defined the various algebras? Matrix algebra is the special division algebra over the field of real numbers. There is a natural generalization in terms of complex numbers for instance. The three axioms

nm Let A D Œaij 2 R be a rectangular matrix

1 A; B; C 2 Rnm; first set of axiom ˛.A; B/ DW A C B .G1C/.A C B/ C C D A C .B C C / .G2C/ A C 0 D A .G3C/ A A D 0 .G4C/ A C B D B C A

2 A; B 2 Rnm ; second set of axiom r; s 2 R;ˇ.r;A/ DW r A .D1C/r .A C B/ D r A C r B .D2C/.rC s/ A D r A C sA .D3/ 1 A D A

3 “multiplication of matrices” (a) “Cayley-product” (just “the matrix product”) # nl A D Œaij 2 R ; dim A D n l H) C WD A B D Œc 2 Rnm ; lm ij B D Œbij 2 R ; dim B D l m Pl dim C D n mcij WD aikbkl: kD1 604 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra

The product was introduced by Cayley (1857); see also his Collected Works, vol. 2, 475–496. A historical perspective is given in Feldmann (1962). (b) “Kronecker-Zehfuß-product” # nm A D Œaij 2 R ; dim A D n m kl B D Œbij 2 R dim B D k l

knlm H) C WD B ˝ A D Œcij 2 R ;

dim C D kn lm; B ˝ A WD Œbij A:

The product was early referenced to L. Kronecker by Mac Duffee (1946). The other reference is Zehfuss (1858). See also Henderson et al. (1983) and Horn and Johnson (1990). Reference is also made to Steeb (1991)and Graham (1981). (c) “Khatri-Rao-product” (two rectangular matrices of identical column number) # nm A D Œa1;:::;am 2 R ; dim A D n m km B D Œb1;:::;bm 2 R ; dim B D k m

knm H) C WD B ˇ A WD Œb1 ˝ a1;:::;bm ˝ am 2 R dim C D kn m:

“two rectangular matrices of identical column numbers are multiplied.” The product was introduced by Khatri and Rao (1968). (d) “Hadamard product” (two rectangular matrices of the same dimension, elementwise product) # nm G D Œgij 2 R ; dim G D n m mm H D Œhij 2 R ; dim H D n m nm H) K WD G H D Œkij 2 R ; dim K D n m; kij WD gij hij .no summation/:

“two rectangular matrices of the same dimension deﬁne the elementwise product.” The product was introduced by Hadamard (1899). See also Moutard (1894), as well as Hadamard (1949)andSchur (1911)andHorn and Johnson (1991)andStyan (1973). A-6 Matrix Algebra Revisited, Generalized Inverses 605

Example: (a) 2 3 23 123 A D 2 Z23; B D 4 455 2 Z32 H) 456 67

(b) 28 34 H) A B D 2 Z22 .“integer numbers”/: 64 79 2 3 12 123 A D 2 Z23 ; B D 4 345 2 Z32 H) B ˝ A D Œb A 456 ij 56 2 3 123 246 6 7 6 4 5 6 810127 22 3 3 6 7 12 6 7 6 7 44 5 5 6 368 48127 Z66 D 34 ˝ A D 6 7 2 6 12 15 18 16 20 24 7 56 6 7 4 5 10 15 6 12 18 5 20 25 30 24 30 36

(c) 2 3 123 123 A D 2 Z23; B D 4 4565 2 Z33 456 789 22 3 2 3 2 3 2 3 2 3 2 33 1 1 2 2 3 3 H) B ˇ A D 44 4 5 ˝ 4 5 ; 4 5 5 ˝ 4 5 ; 4 6 5 ˝ 4 55 7 4 8 5 9 6 2 3 149 6 7 6 410187 6 7 6 7 6 410187 Z63 D 6 7 2 6 16 25 36 7 4 716275 23 40 54 606 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra

(d) 123 234 G D 2 Z23 ; H D 2 Z23 456 567

2612 H) G H D Œg h D 2 Z23: ij ij 20 30 42

A-61 Special Matrices: Helmert, Hankel, and Vandemonte

All the quoted textbooks review effectively special matrices of various types: symmetric, antisymmetric, diagonal, unity, zero, triangular, idempotent, normal, orthogonal, orthonormal, positive-definite, positive-semidefinite, permutation, and commutation matrices. Here, we review only orthonormal matrices of various representations. The Helmert representation of orthonomal matrix is given in various forms. In addition, we give examples for an orthogonal matrix, for instance the Hankel matrix of power sums and the Vandermonde matrix. Definition (orthogonal matrix) :

The matrix A is called orthogonal if AA0 and A0A are diagonal matrices. (The rows and columns of A are orthogonal.) Deﬁnition (orthonormal matrix): The matrix A is called orthonormal if AA0 D A0A D I. (The rows and columns of A are orthonormal.) Facts (representation of a 22 orthonormal matrix) X 2 SO.2/:A2 2 orthonormal matrix X 2 SO.2/ is an element of the special orthogonal group SO(2) deﬁned by

22 0 SO.2/ WD fX 2 R j X X D I2 and det X DC1g

8 ˇ 9 ˇ 2 2 < ˇ x1 C x2 D 1 = x1 x2 R22 ˇ :X D 2 ˇ x1x3 C x2x4 D 0; x1x4 x2x3 DC1; x3 x4 ˇ 2 2 x3 C x4 D 1

(a) cos sin X D 2 R22;2 Œ0; 2 sin cos is a trigonometric representation of X 2 SO.2/. A-6 Matrix Algebra Revisited, Generalized Inverses 607

(b) " p # x 1 x2 X D p 2 R22;x2 ŒC1; 1 1 x2 x is an algebraic representation of X 2 SO.2/ p p 2 2 2 2 2 2 x11 C x12 D 1; x11x21 C x12x22 Dx 1 x C x 1 x D 0; x21 C x22 D 1 :

(c) " # 1x2 2x 2 C 2 X D 1Cx 1Cx 2 R22;x2 R 2x 1x2 1Cx2 1Cx2 is called a stereographic projection of X (stereographic projection of SO(2) S1 onto L1). (d) 0x X D .I C S/.I S/1; S D ; 2 2 x0 where S DS0 is a skew matrix (antisymmetric matrix), is called a Cayley-Lipschitz representation of X 2 SO.2/.(e)X 2 SO.2/ is a commutative group (“Abel”) (Example: X1 2 SO.2/, X2 2 SO.2/,thenX1X2 D X2X1)(SO(n)forn D 2 is the only commutative group, SO(nj n ¤ 2) is not “Abel”). Facts (representation of an n n orthonormal matrix) X 2 SO.n/: An n n orthonormal matrix X 2 SO.n/ is an element of the special orthogonal group SO(n) deﬁned by

nn 0 SO.n/ WD fX 2 R jX X D In and det X DC1g:

As a differentiable manifold SO(n) inherits a Riemann structure from the ambient 2 2 space n with a Euclidean metric (vec X0 2 Rn ; dim vec X0 D n2). Any atlas of the special orthogonal group SO(n) has at least four distinct charts and there is one with exactly four charts. (“minimal atlas”: Lusternik-Schnirelmann category)(a)

1 X D .In C S/.In S/ ; where

S DS0 608 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra is a skew matrix (antisymmetric matrix), is called a Cayley-Lipschitz representation of X 2 SO.n/.(nŠ=2.n 2/Š is the number of independent parameters/coordinates of X) (b) If each of the matrices R1;:::;Rk is an n n orthonormal matrix,then their product

R1R2 Rk1Rk 2 SO.n/ is an n n orthonormal matrix. Facts (orthonormal matrix: Helmert representation): 0 Let a D Œa1;:::;an represent any row vector such that ai ¤ 0 .i 2f1;:::;ng/ is any row vector whose elements are all nonzero. Suppose that we require an n n orthonormal matrix, one row which is proportional to a0. In what follows one such 0 0 matrix R is derived.LetŒr 1;:::;r n represent the rows of R and take the ﬁrst row 0 0 0 r 1 to be the row of R that is proportional to a .Takethesecond row r 2 to be proportional to the n-dimensional row vector

2 Œa1; a1=a2; 0; 0;:::;0; .H 2/

0 the third row r 3 proportional to 2 2 a1; a2; .a1 C a2/=a3; 0; 0;:::;0 .H3/

0 0 and more generally the ﬁrst through nth rows r 1;:::;r n proportional to " # Xk1 2 a1; a2;:::;ak1; ai =ak; 0; 0;:::;0 .Hn 1/ iD1 for k 2f2;:::;ng; respectively conﬁrm to yourself that the n-1 vectors (Hn1) are orthogonal to each 0 0 0 other and to the vector a . In order to obtain explicit expressions for r 1;:::;r n it 0 remains to normalize a and the vectors (Hn1). The Euclidean norm of the kth of the vectors (Hn1)is 8 9 ! 1=2 ( ! ! )

0 0 Accordingly for the orthonormal vectors r 1;:::;r n we ﬁnally ﬁnd " # Xn 1=2 0 2 (1st row) r 1 D ai .a1;:::;an/ iD1 A-6 Matrix Algebra Revisited, Generalized Inverses 609

2 3 1=2 6 7 6 a2 7 0 6 !k !7 (kth row) r k D 6 7 4 kP1 Pk 5 2 2 ai ai iD1 iD1 ! Xk1 2 ai a1; a2;:::;ak1; ; 0; 0;:::;0 a k iD1 2 3 1=2 " # 6 2 7 Xn1 2 0 6 an 7 ai .nth row/ r D 6 7 a ; a ;:::;a ; : n 4 nP1 Pn 5 1 2 n 1 a 2 2 iD1 n ai ai : iD1 iD1

The recipe is complicated: When a0 D Œ1; 1;:::; 1; 1,theHelmert factors in the 1st row, :::; kthrow,...,nthrowsimplifyto

0 1=2 n r 1 D n Œ1;1;:::;1;12

0 1=2 n r k D Œk.k 1/ Œ1;1;:::;1;1 k; 0; 0; : : : ; 0; 0 2

0 1=2 n r n D Œn.n 1/ Œ1; 1; :::; 1; 1 n 2 R :

The orthonormal matrix 2 3 0 r 1 6 0 7 6 r 2 7 6 7 6 7 6 0 7 6 r k1 7 6 0 7 2 SO.n/ 6 r k 7 6 7 6 7 4 0 5 r n1 0 r n is known as the Helmert matrix of order n. (Alternatively the transposes of such a matrix are called the Helmert matrix.) Example (Helmert matrix of order 3):

2 p p p 3 1= 31= 31=3 6 7 6 p p 7 4 1= 2 1= 205 2 SO.3/: p p p 1= 61= 6 2= 6

Check that the rows are orthogonal and normalized. 610 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra

Example (Helmert matrix of order 4):

2 3 1=2 1=2 1=2 1=2 6 p p 7 6 7 6 1= 2 1= 20 07 6 p p p 7 2 SO.4/: 4 1= 61=6 2= 605 p p p p 1= 12 1= 12 1= 12 3= 12

Check that the rows are orthogonal and normalized. Example (Helmert matrix of order n):

2 p p p p p p 3 1= n1=n1=n1=n 1= n1=n 6 p p 7 6 7 6 1= 2 1= 20 0 007 6 p p p 7 6 7 6 1= 61=6 2= 60 007 6 7 2 SO.n/: 6 7 6 7 6 p 1 p 1 p 1 p 1 .n 1/ 0 7 4 .n1/.n2/ .n1/.n2/ .n1/.n2/ .n1/.n2/ 5 p 1 p 1 p 1 p 1 p 1n n.n1/ n.n1/ n.n1/ n.n1/ n.n1/

Check that the rows are orthogonal and normalized. An example is the nth row

1 1 .1 n/2 n 1 1 2n C n2 CC C D C n.n 1/ n.n 1/ n.n 1/ n.n 1/ n.n 1/ n2 n n.n 1/ D D D 1; n.n 1/ n.n 1/ where .n 1/ terms 1=Œn.n 1/ have to be summed. Deﬁnition (orthogonal matrix):

nm A rectangular matrix A D Œaij 2 R is called “a Hankel matrix”ifthenCm 1 distinct elements of A, 2 3 a 6 11 7 6 a 7 6 21 7 6 7 6 7 4 an11 5 an1 an2 anm only appear in the ﬁrst column and last row. A-6 Matrix Algebra Revisited, Generalized Inverses 611

Example: Hankel matrix of power sums Let A 2 Rnm be a nm rectangular matrix (n 6 m) whose entries are power sums. 2 3 Pn Pn Pn 2 m 6 ˛i xi ˛i xi ˛i xi 7 6 iD1 iD1 iD1 7 6 Pn Pn Pn 7 6 2 3 mC1 7 6 ˛i xi ˛i xi ˛i xi 7 6 iD1 iD1 iD1 7 A WD 6 7 6 : : : : 7 6 : : : : 7 4 Pn Pn Pn 5 n nC1 nCm1 ˛i xi ˛i xi ˛i xi iD1 iD1 iD1

A is a Hankel matrix. Deﬁnition (Vandermonde matrix): Vandermonde matrix: V 2 Rnn 2 3 11 1 6 7 6 x x x 7 6 1 2 n 7 V WD6 : : : : 7 ; 4 : : : : 5 n1 n1 n1 x1 x2 xn Y det V D .xi xj /: i;j n i>j

Example: Vandermonde matrix V 2 R33 2 3 111 6 7 V WD 4 x1 x2 x3 5 ; det V D .x2 x1/.x3 x2/.x3 x1/: 2 2 2 x1 x2 x3

Example: Submatrix of a Hankel matrix of power sums Consider the submatrix nm P D Œa1;a2;:::;an of the Hankel matrix A 2 R .n 6 m/ whose entries are power sums. The determinant of the power sums matrix P is ! ! Yn Yn 2 det P D ˛i xi .det V/ ; iD1 iD1 where det V is the Vandermonde determinant. Example: Submatrix P 2 R33 of a 3 4 Hankel matrix of power sums (n D 3, m D 4) 612 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra

2 3 ˛ x C ˛ x C ˛ x ˛ x2 C ˛ x2 C ˛ x2 ˛ x3 C ˛ x3 C ˛ x3 ˛ x4 C ˛ x4 C ˛ x4 6 1 1 2 2 3 3 1 1 2 2 3 3 1 1 2 2 3 3 1 1 2 2 3 3 7 6 7 A D 6 2 2 2 3 3 3 4 4 4 5 5 5 7 4 ˛1x1 C ˛2x2 C ˛3x3 ˛1x1 C ˛2x2 C ˛3x3 ˛1x1 C ˛2x2 C ˛3x3 ˛1x1 C ˛2x2 C ˛3x3 5 3 3 3 4 4 4 5 5 5 6 6 6 ˛1x1 C ˛2x2 C ˛3x3 ˛1x1 C ˛2x2 C ˛3x3 ˛1x1 C ˛2x2 C ˛3x3 ˛1x1 C ˛2x2 C ˛3x3 2 3 ˛ x C ˛ x C ˛ x ˛ x2 C ˛ x2 C ˛ x2 ˛ x3 C ˛ x3 C ˛ x3 6 1 1 2 2 3 3 1 1 2 2 3 3 1 1 2 2 3 3 7 6 7 P D Œa ;a ;a 6 2 2 2 3 3 3 4 4 4 7 : 1 2 3 4 ˛1x1 C ˛2x2 C ˛3x3 ˛1x1 C ˛2x2 C ˛3x3 ˛1x1 C ˛2x2 C ˛3x3 5 3 3 3 4 4 4 5 5 5 ˛1x1 C ˛2x2 C ˛3x3 ˛1x1 C ˛2x2 C ˛3x3 ˛1x1 C ˛2x2 C ˛3x3

A-62 Scalar Measures of Matrices

All reference textbooks review various scalar measures of matrices like • Linear independence • Column and row rank • Rank identities to which we refer. Permanently, we referred to a special technique called IPM (“Inverse Partitioned Matrix”) of a symmetric matrix we will review now. Facts: (Inverse Partitional Matrix/IPM/of a symmetric matrix): Let the symmetric matrix A be partitioned as " # A A A WD 11 12 ; A0 D A ; A0 D A : 0 11 11 22 22 A 12 A22

Then its Cayley inverse A1 is symmetric and can be partitioned as well as 2 3 1 A11 A12 A1 D 4 5 0 A 12 A22 2 3 1 0 1 1 0 1 1 0 1 1 ŒI C A11 A12.A22 A 12A11 A12/ A 12A11 A11 A12.A22 A 12A11 A12/ D 4 5 ; 0 1 1 0 1 0 1 1 .A22 A 12A11 A12/ A 12A11 .A22 A 12A11 A12/

1 if A11 exists; 2 3 1 A11 A12 A1 D 4 5 0 A 12 A22 2 3 0 1 1 0 1 1 1 .A11 A 12A22 A12/ .A11 A 12A22 A12/ A12A22 D 4 5 ; 1 0 0 1 1 1 0 1 0 1 1 A22 A 12.A11 A 12A22 A12/ ŒI C A22 A 12.A11 A12A22 A 12/ A12A22 A-6 Matrix Algebra Revisited, Generalized Inverses 613

1 if A22 exists:

0 1 0 1 S11 WD A22 A 12A11 A12 and S22 WD A11 A 12A22 A12 are the minors determined by properly chosen rows and columns of the matrix A called “Schur complements” such that " # " # 1 A A .I C A1A S1A0 /A1 A1A S1 A1 D 11 12 D 11 12 11 12 11 11 12 11 0 1 0 1 1 A 12 A22 S11 A 12A11 S11

1 if A11 exists; " # " # 1 A A S1 S1A A1 A1 D 11 12 D 22 22 12 22 0 1 0 1 1 0 1 1 A 12 A22 A22 A 12S22 ŒI C A22 A 12S22 A12A22

1 if A22 exists; are representations of the Cayley inverse partitioned matrix A1 in terms of “Schur complements”. The formulae S11 and S22 were ﬁrst used by J. Schur (1917). The term “Schur complements” was introduced by Haynsworth (1968). Albert (1969) replaced the Cayley inverse A1 by the Moore-Penrose inverse AC. For a survey we recommend Cottle (1974), Ouellette (1981), Carlson (1986)andStyan (1985). Proof. For the proof of the “inverse partitioned matrix” A1 (Cayley inverse) of the partitioned matrix A of full rank we apply Gauss elimination (without pivoting).

AA1 D A1A D I " mm ml A A A11 2 R ; A12 2 R A D 11 12 ; A0 D A ; A0 D A 0 11 11 22 22 0 lm ll A 12 A22 A 12 2 R ; A22 2 R " mm ml B B B11 2 R ; B12 2 R A1 D 11 12 ; B0 D B ; B0 D B 0 11 11 22 22 0 lm ll B 12 B22 B 12 2 R ; B22 2 R 2 0 0 A11B11 C A12B 12 D B11A11 C B12A 12 D Im .1/ 6 6 A B C A B D B A C B A D 0 .2/ AA1 D A1A D I , 6 11 12 12 22 11 12 12 22 4 0 0 0 0 A 12B11 C A22B 12 D B 12A11 C B22A 12 D 0 .3/ 0 0 A 12B12 C A22B22 D B 12A12 C B22A22 D Il :.4/ 614 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra

1 Case (a) W A11 exists

“forward step” # 0 0 1 A11B11 C A12B 12 D Im .ﬁrst left equation:multiply by A 12A11 / 0 0 A 12B11 C A22B 12 D 0 .secondrightequation) # A0 B A0 A1A B0 D A0 A1 , 12 11 12 11 12 12 12 11 0 0 A 12B11 C A22B 12 D 0 " A B C A B0 D I , 11 11 12 12 m 0 1 0 0 1 .A22 A 12A11 A12/B 12 DA 12A11 0 0 1 1 0 1 ) B 12 D.A22 A 12A11 A12/ A 12A11 0 1 0 1 B 12 DS11 A 12A11 or " #" # " # I 0 A A A A m 11 12 D 11 12 : 0 1 0 0 1 A 12A11 Il A 12 A22 0A22 A 12A11 A12

0 1 Note the “Schur complement” S11 WD A22 A 12A11 A12. “backward step” # 0 A11B11 C A12B 12 D Im 0 0 1 1 0 1 B 12 D.A22 A 12A11 A12/ A 12A11 1 0 0 1 ) B11 D A11 .Im A12B 12/ D .Im B12A 12/A11 1 0 1 1 0 1 B11 D ŒIm C A11 A12.A22 A 12A11 A12/ A 12A11 1 1 1 0 1 B11 D A11 C A11 A12S11 A 12A11

A11B12 C A12B22 D 0 .second left equation/ 1 1 0 1 1 ) B12 DA11 A12B22 DA11 A12.A22 A 12A11 A12/ 0 1 1 , B22 D .A22 A 12A11 A12/ 1 B22 D S11 :

1 Case (b) W A22 exists

“forward step” A-6 Matrix Algebra Revisited, Generalized Inverses 615

# A11B12 C A12B22 D 0 .third right equation) 0 1 A 12B12 C A22B22 D Il .fourth left equation: multiply by A12A22 / # A B C A B D 0 , 11 12 12 22 1 0 1 A12A22 A 12B12 A12B22 DA12A22 " A0 B C A B D I , 12 12 22 22 l 1 0 1 .A11 A12A22 A 12/B12 DA12A22 1 0 1 0 1 ) B12 D.A11 A12A22 A 12/ A 12A22 1 1 B12 DS22 A12A22 or " #" # " # I A A1 A A A A A1A0 0 m 12 22 11 12 D 11 12 22 12 : 0 0 0Il A 12 A22 A 12 A22

Note the “Schur complement”

1 0 S22 WD A11 A12A22 A 12:

“backward step” # 0 A 12B12 C A22B22 D Il 1 0 1 1 B12 D.A11 A12A22 A 12/ A12A22 1 0 0 0 1 ) B22 D A22 .Il A 12B 12/ D .Il B 12A12/A22 1 0 1 0 1 1 B22 D ŒIl C A22 A 12.A11 A12A22 A 12/ A12A22 1 1 0 1 1 B22 D A22 C A22 A 12S22 A12A22 0 0 A 12B11 C A22B 12 D 0 .third left equation/ 0 1 0 1 0 1 0 1 ) B 12 DA22 A 12B11 DA22 A 12.A11 A12A22 A 12/

1 0 1 B11 D .A11 A12A22 A 12/ , | 1 B11 D S22 :

0 0 The representations B11; B12; B21 D B 12; B22 in terms of A11; A12; A21 D A 12; A22 have been derived by Banachiewicz (1937). Generalizations are referred to Ando (1979), Brunaldi and Schneider (1963), Burns et al. (1974), Carlson (1980), Meyer (1973)andMitra (1982), Li and Mathias (2000). We leave the proof of the following fact as an exercise. 616 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra

Fact (Inverse Partitioned Matrix /IPM/ of a quadratic matrix): Let the quadratic matrix A be partitioned as " # A A A WD 11 12 : A21 A22

Then its Cayley inverse A1 can be partitioned as well as " # " # 1 A A A1 C A1A S1A A1 A1A S1 A1 D 11 12 D 11 11 12 11 21 11 11 12 11 ; 1 1 1 A21 A22 S11 A21A11 S11

1 if A11 exists " # " # 1 A A S1 S1A A1 A1 D 11 12 D 22 22 12 22 ; 1 1 1 1 1 1 A21 A22 A22 A21S22 A22 C A22 A21S22 A12A22

1 if A22 exists and the “Schur complements” are deﬁned by

1 1 S11 WD A22 A21A11 A12 and S22 WD A11 A12A22 A21:

Facts: Cayley Inverse, DG matrix identity, SMW matrix identity, sum of two matrices

.DG id/ BD.A C CBD/1 D .B1 C DA1C/1DA1 (Duncan-Guttman matrix identity):

.SMW id/.A C CBD/1 D A1 A1C.B1 C DA1C/1DA1 (Sherman - Morrison - Woodbury matrix identity):

Duncan (1944) calls (DG) the Sherman-Morrison-Woodbury matrix identity.Ifthe matrix A is singular consult Henderson and Searle (1981a), Henderson and Searle (1981b), Ouellette (1981), Hager (1989), Stewart (1977)andRiedel (1992). (SMW) has been noted by Duncan (1944)andGuttman (1946): The result is directly derived from the identity

.A C CBD/.A C CBD/1 D I ) A.A C CBD/1 C CBD.A C CBD/1 D I .A C CBD/1 D A1 A1CBD.A C CBD/1 A-6 Matrix Algebra Revisited, Generalized Inverses 617

A1 D .A C CBD/1 C A1CBD.A C CBD/1 DA1 D D.A C CBD/1 C DA1CBD.A C CBD/1 DA1 D .I C DA1CB/D.A C CBD/1 DA1 D .B1 C DA1C/BD.A C CBD/1 .B1 C DA1C/1DA1 D BD.A C CBD/1:

Up to now we presented IPM of a symmetric matrix. Alternatively, we review rank factorization of a symmetric and positive semidefinite/definite matrix. Facts (rank factorization): (a) If the n n matrix is symmetric and positive semidefinite, then its rank factorization is " # G1 0 0 A D G 1 G 2 ; G2

where G1 is a lower triangular matrix of the order O.G1/ D rk A rk A with rk G2 D rk A, whereas G2 has the format O.G2/ D .n rk A/ rk A: In this case we speak of a Choleski decomposition. (b) In case that the matrix A is positive deﬁnite, the matrix block G2 is not needed anymore: G1 is uniquely determined. There holds

1 1 0 1 A D .G1 / G1 :

Various notions, for instance submatrices adjoint, traces and determinants, as scalar measures of a matrix are given by D. A. Harville (2001, examples, Sects. 2,5 and 13) to which we refer. We already introduced vector-valued matrix forms by means of the operations vec, vech and veck. Please pay attention to the vec-forms (a) vecA B C0 D .A ˝ C/vecB (b) .A/0vecB D tr.A B0/ (c) vec xy0 D x ˝ y

0 0 (d) vec(A B/ D .B ˝ In/vecA D .B ˝ A/vecIm nm mq D .Iq ˝ A/vecB; for all A 2 R ; B 2 R

given by Pukelsheim (1977), pp. 324, 326 or Harville (2001), examples, Sect. 16, 22 pp. 618 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra

A-63 Three Basic Types of Generalized Inverses

In Chaps. 1, 3 and 5 we obtained the complete class of solution of equation “Ax D y00 and showed that every solution can be expressed in the form x D Gy where G is the properly chosen generalized inverse. Here we inquire to ﬁnd solutions for such generalized inverses which are of the following type

.i/rkA D rk.AA0/ D n W “minimum norm”

Consistent system of observation equations: Underdetermine. Theorem (“minimum norm”: Rao and Mitra (1971), pp. 44–47)

Let G be a generalized inverse of the matrix A such that Gy is a minimum norm solution of the equation Ax D y;rkA D rk.AA0/ D n. Then it is necessary and sufﬁcient that Min (a) kxk2 DkG k2 D or y Ax D y (b) AGA D A and .GA/0 D GA.or (c) A D A0.AA0/ (reﬂexive generalized inverse which is minimum norm). mr n o Min If we use the weighted generalized inverse of the type kxk2 D x0Gx Ax D y we get the Gx-MINOS solution of Chap. 1, case (a), case (b) and case (c).

.ii/rkA D rk.A0A/ D m W “generalized least-squares solution for Ax D y”

Inconsistent system of linear observation equations: overdetermine. Theorem (“Least-squares”: Rao and Mitra (1971), pp. 48–50)

Let G be a matrix (not necessarily a generalized inverse) such that Gy is a least- squares solution of Ax D y such that rkA D rk.A0A/ D m. Then it is necessary and sufﬁcient that (a) ky Axk2 D min.inconsistent/ or (b) AGA D A and .GA/0 D GA: or (c) A D .A0A/A (reﬂexive generalized inverse which is least-squares). lr n Min 2 If we use the weighted generalized inverse of the type ky AxkG D o x y 0 .y Ax/ Gy .y Ax/ we enjoy receiving the Gy-LESS solution of Chap. 3, case (a), case (b) and case (c). A-7 Complex Algebra, Quaternion Algebra, Octonian Algebra, Clifford Algebra 619

.iii/rkA

Inconsistent system of linear observation equations with a datum defect: overdetermine-underdetermined. ut Here we start with a deﬁnition Deﬁnition (“MINOLESS”: Rao and Mitra (1971), p. 51) G is said to be a minimum norm, least-squares generalized inverse of the matrix A, Y if G is Al and for any y 2

kGy km kxkm for all x 2fx jky Axkm ky Azkm for all z 2 X

m n where k:kn and k:km are norms in X and Y respectively. Finally we will refer to the basic results generating “MINOLESS”. Theorem (“MINOLESS”: Rao and Mitra (1971), pp. 50–54,64)

Let G be a matrix such that Gy is a minimum norm, least-squares solution of the inconsistent equation Ax D y C i. Then it is necessary and sufﬁcient that x D ACy namely, (a) AGA D A,(b).AG/0 D AG,(c)GAG/ D G,(d).GA/0 D GA (Penrose 1955). or

AC D A0AŒ.A0A/2A0 D .I C A0A/A0ŒA0A.I C A0A/AA or

Xk Xk lim lim AC D k ! 1 A0.I C AA0/k D k ! 1 .I C A0A/k A0 or kD1 kD1

lim lim AC D ! 0CA0.I C AA0/1 D ! 0C.I C A0A/1A0

If we use the weighted norms kxk2 and ky Axk2 , we receive the G -minimum Px Py x norm, Gy- least squares solution of Chap. 5, case (a), case (b) and case (c).

A-7 Complex Algebra, Quaternion Algebra, Octonian Algebra, Clifford Algebra, Hurwitz Theorem

The greatest advantage of Clifford algebra is the following fact: We are able to write down the field equations namely “rot” and “div” equation of gravitational field and electromagnetic field in one equation. The related literature is very rich and we could recommend the book by Meetz and Engl (1980). A more detailed introduction into Clifford algebra and its fascinating chess board as well as Clifford analysis is listed in Grafarend (2004) with more than 300 references. From some point of view, the variona algebra presented here are rather out of fashion. Indeed 620 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra

• Complex algebra, Clifford algebra Cl.0; 1/ • Quaternion algebra, Clifford algebra Cl.0; 2/

A-71 Complex Algebra as a Division Algebra as well as a Composition Algebra, Clifford algebra Cl.0;1/

Here we present to you “complex algebra” and refer it to Clifford algebra Cl.0;1/. The “complex algebra” C due to C. F. Gauß is a division algebra over R as well as a composition algebra.

x 2 C

0 1 0 1 2 x D e0x C e1x subject to fx ;x g2R

span C Df1; e1g

1 x; y; z 2 R2

˛.x; y/ DW x C y:

The axioms .G1C/, .G2C/, .G3C/, .G4C/ of an Abelian group apply. 2 x; y 2 R2;r;s2 R

ˇ.r; x/ DW r x:

The axioms .D1C/, .D2C/, .D3/ of additive distributivity apply. 3 One way to explicitly describe a multiplicative group with ﬁnitely many elements is to give a table listing the multiplications just representing the map W C C ! C. £ ¢multiplication diagram, Cayley diagram ¡

1 e1 1 1 e1 e1 e1 1

Here note in addition the associativity of the internal multiplication given in the table. Such a “complex algebra” C is not a Lie algebra since neither x x D 0.L1/ nor .x y/ zCC.y z/ x C .z x/ y D 0 (Jacobi identity) .L2/ hold. Just by means of the multiplication table compute

0 2 1 2 0 1 x x D 1f.x / .x / gC2e1x x 6D 0:

4 Begin with the choice

1 x1 D .1x0 e x1/ .x0/2 C .x1/2 1 in order to end up with

1x0 e x1 x x1 D .1x0 C e x1/ 1 D 1 1 .x0/2 C .x1/2 Accordingly .G1/, .G2/, .G3/ of a division algebra apply. 5 Begin with the choice

0 1 0 2 1 2 Q.x/ D Q.1x C e1x / WD .x / C .x /

in order to prove (K1), (K2) and (K3). We only focus on (K2i):

Q.x y/ D Q.x/ Q.y/.multiplicativity/

0 0 1 1 1 0 0 1 Q.x y/ D Qf1.x y x y / C e1.x y C x y /g D .x0/.y0/2 C .x1/2.y1/2 C .x1/2.y0/2 C .x0/2.y1/2 Q.x/ Q.y/ Df.x0/2 C .x1/2gf.y0/2 C .y1/2g D .x0/2.y0/2 C .x1/2.y1/2 C .x1/2.y0/2 C .x0/2.y1/2 § ¤ Q.x y/ D Q.x/ Q.y/ q:e:d: ¦ ¥ C How can be one dream about such a complex algebra ? Gauss (1887) hadp been motivated in his number theory to introduce complex numbers with i WD 1 as 0 1 1 the “imaginary unit”. Identify 1x with the “real part”ande1x D ix with the “imaginary part”ofx and we are left with the standard theory of complex numbers. 1 0 1 x is based upon the complex conjugate 1x e1x of x being divided by the norm of x. There is a remarkable isomorphism between complex numbers and complex algebra. The proper algebraic interpretation of complex numbers is in terms of Clifford algebra Cl.0;1/.Observeg.e1; e1/ D1 which interprets the binary operation of the base vector which spans the vector part of a complex number. Now translate the multiplication table into the language of the Clifford product, namely 622 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra # 1 ^ 1 D 1; 1 ^ e1 D e1 e ^ 1 D e ; e ^ e D1 "1 1 1 1 ! in order to convince yourself that the Clifford algebra Cl.0;1/ is algebraically isomorphic to the space of complex numbers. |

How can we relate complex numbers to Clifford algebra Cl1?Observeg.e1; e1/ D 1 which interprets the binary operation of the base vector which spans the vector part of a complex number. While the scalar part of a complex number is an element of A0, its vector part can be considered to be an element of A1.Thedirect sum

A0 ˚ A1 of spaces A0, A1 is algebraically isomorphic to the space of complex numbers, being an element of the Clifford algebra Cl1. |

A-72 Quaternion Algebra as a Division Algebra as well as a Composition Algebra, Clifford algebra Cl.0;2/

Next we review “quaternion algebra” quoting W. R. Hamilton and refer it to Clifford algebra Cl.0;2/.The”quaternion algebra” H due to Hamilton (1843) is a division algebra over R as well as a composition algebra:

x 2 H

0 1 2 3 0 1 2 3 4 x D 1x C e1x C e2x C e3x subject t o fx ;x ;x ;x g2R

span H Df1; e1; e2; e3g

1 x; y; z 2 R4

˛.x; y/ DW x C y:

The axioms .G1C/, .G2C/, .G3C/, .G4C/ of an Abelian additive group apply. 2 x; y 2 R4;r;s2 R

ˇ.r; y/ DW r x:

The axioms .D1C/, .D2C/, .D3/ of additive distributivity apply. A-7 Complex Algebra, Quaternion Algebra, Octonian Algebra, Clifford Algebra 623

3 One way to explicitly describe a multiplicative group with ﬁnitely many elements is to give a table listing the multiplications just representing the map W H H ! H. § ¤ multiplication table, Cayley diagram ¦ ¥

1 e1 e2 e3 1 1e1 e2 e3 e1 e1 1e3 e2 : e2 e2 e3 1e1 e3 e3 e2 e1 1

Note that in the multiplication table each antry of a group appears exactly once in each row and column. The multiplication has to be read from left to right that is, the entry at the intersection of the row headed by e1 and the column headed by e2 is the product e1 e2. Such a table is called a Cayley diagram of the multiplicative group. Here note in addition the associativity of the internal multiplication given by the table, e.g. e1 .e2 e3/ D e1 e1 D1 D e3 e3 D .e1 e2/ e3 or ! X3 X x y ı ı k k e ı k k ı k i j D 1 x y x y C k.x y C x y C ij x y / kD1 i;j;k

such that .x y/ z D x .y z/. Such a “Hamilton algebra” H is not a Lie algebra since neither x x D 0 (L1) nor .x y/ z C .y z/ x C .z x/ y D 0 (L2) (Jacobi identity) hold. Just by means of the multiplication table compute

x x D 1f.xı/2 .x1/2 .x2/2 .x3/2g

ı 1 ı 2 ı 3 C 2e1x x C 2e2x x C 2e3x x 6D 0

4 Begin with the choice

1 x1 D .1xı e x1 x x2 e x3/ .xı/2 C .x1/2 C .x2/2 C .x3/2 1 2 2 in order to to end up with

1xı e x1 e x2 e x3 x x1 D Œ1xı C e x1 C e x2 C e x3 1 2 3 D 1: 1 2 3 .xı/2 C .x1/2 C .x2/2 C .x3/2 Accordingly .G1/, .G2/, .G3/ of a division algebra apply. 5 Begin with the choice 624 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra

1 2 3 0 2 1 2 2 2 3 2 Q.x/ D Q.1 C e1x C e2x C e3x / WD .x / C .x / C .x / C .x /

in order to prove .K1/. .K2/ and .K3/. The laborious proofs are left as an exercise. How can one dream about such a “quaternion algebra” H? Hamilton (16 October 1843) invented quaternion numbers as outlined in a letter (1865) to his son A. H. Hamilton for the following reason: “If I may be allowed to speak of myself in connexion with the subject, I might do so in a way which would bring you in, by referring to an antequaternionic time, when you were a mere child, but had caught from me the conception of a Vector, as represented by a Triplet; and indeed I happen to be able to put the ﬁnger of memory upon the year and month – October, 1843 – when having recently returned from visits to Cork and Parsonstown, connected with a Meeting of the British Association, the desire to discover the laws of the multiplication referred to regained with me a certain strength and carnestness, which had for years been dormant, but was then on the point of being gratiﬁed, and was occasionally talked of with you. Every morning in the early part of the above cited month, on my coming down to breakfast, your (then) little brother William Edwin, and yourself, used to ask me, “Well, Papa, can you multiply triplets”? Whereto I was always obliged to reply, with a sad shake of the head: “No, I can only add and subtract them.”

But on the 16th day of the same month – which happened to be a Monday, and a Council day of the Royal Irish Academy – I was walking in to attend and preside, and your mother was walking with me, along the Royal Canal, to which she had perhaps driven; and although she talked with me now and then, yet an under-current of thought was going on in my mind, which gave at last a result, whereof it is not too much to say that I felt at once the importance. An electric circuit seemed to close; and a spark ﬂashed forth. The herald (as I foresaw, immediately) of many long years to come of deﬁnitely directed thought and work, by myself if spared, and at all events on the part of others, if should even be allowed to live long enough distinctly to communicate the discovery. Nor could I resist the impulse – unphilosophical as it may have been – to cut with a knife on a stone of Brougham Bridge, as we passed it, the fundamental formula with the symbols, i;j;k; namely

i 2 D j 2 D k2 D ij k D1 A-7 Complex Algebra, Quaternion Algebra, Octonian Algebra, Clifford Algebra 625 which contains the Solution of the Problem, but of course, as an inscription, has long since moldered away. A more durable notice remains, however, on the Council Books of the Academy for that day (October 16th, 1843), which records the fact, that I then asked for and obtained base to read a Paper on Quaternion,at the First General Meeting of the Session: which reading took place accordingly, on Monday the 13th of the November following.”

1 2 3 1 2 3 Obviously the vector part e1x C e2x C e3x D i x C j x C kx of a quaternion number replaces the imaginary part of a complex number, the scalar part 1x0 the real part.Thequaternian conjugate

X3 ı k 1x ekx DW x kD1 substitutes the complex conjugate of ta complex number, leading to the quaternion inverse x x1 D : Q.x/ The proper algebraic interpretation of quaternion numbers is in terms of Clifford algebra Cl.0;2/.Ifn D dim X D 2 is the dimension of the linear space X which we base the Clifford algebra on, its basis elements are

f1; e1; e2; e1 ^ e2g subject to e1 ^ e2 C e2 ^ e1 D 0; e1 ^ e1 D g.e1; e1/1 D1; e2 ^ e2 D g.e2; e2/1 D1; 2 .e2 ^ e2/ D .e1 ^ e2/ ^ .e1 ^ e2/ De2 ^ .e1 ^ e1/ ^ e1 DCe2 ^ e2 D1 .e2 ^ e2/ ^ e1 De2 ^ .e1 ^ e1/ D e2 .e2 ^ e2/ ^ e2 D e1 ^ .e2 ^ e2/ De1 and e3 WD e1 ^ e2 by classical notation. Obviously

ı 1 2 3 x D 1x C e1x C e2x C e1 ^ e2x 2 Cl.0;2/ is an element of Clifford algebra Cl.0;2/. There is an algebraic isomorphism between the quaternion algebra H of vectors and the quaternion algebra of matrices, namely either M.R44/ of 4 4 real matrices or M.C22/ of 2 2 complex matrices. 626 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra

Firstly we deﬁne the 4 4 real matrix basis E and decompose it into the four constituents ˙ 0, ˙ 1, ˙ 2, ˙ 3 of 4 4 real Pauli matrices which form a multiplicative group of the multiplication table of Hamilton type 2 3 1 e1 e2 e3 6 7 6 e1 1 e3 e2 7 44 E WD 4 5 D 1˙ 0 C e1˙ 1 C e2˙ 2 C e3˙ 3 2 R e2 e3 1 e1 e3 e2 e1 1

2 3 2 3 1000 010 0 6 7 6 7 6 01007 6 100 07 ˙ 0 D 4 5 ; ˙ 1 WD 4 5 ; 0010 0001 0001 001 0 2 3 2 3 0 010 00 01 6 7 6 7 6 0 0017 6 00107 ˙ 2 D 4 5 ; ˙ 3 WD 4 5 ; 1 000 01 00 0 100 10 00 § ¤ ¦multiplication table, Cayley diagram ¥

˙ 0 ˙ 1 ˙ 2 ˙ 3 ˙ 0 ˙ 0 ˙ 1 ˙ 2 ˙ 3

˙ 1 ˙ 1 ˙ 0 ˙ 3 ˙ 2 ˙ 2 ˙ 2 ˙ 3 ˙ 0 ˙ 1

˙ 3 ˙ 3 ˙ 2 ˙ 1 ˙ 0 or

˙ 0˙ 0 D ˙ 0; ˙ 0˙ i D ˙ i ˙ 0 D ˙ i for all i 2f1; 2; 3g

˙ i ˙ j Dıij ˙ 0 C ij k ˙ k for all i;j;k 2f1; 2; 3g

Let A, B, C 2 M (R44, Hamilton) constituted by means of 2 3 a1 a2 a3 a4 6 7 6 a2 a1 a4 a3 7 4 5 DW A a3 a4 a1 a2 a4 a3 a2 a1 2 3 b1 b2 b3 b4 6 7 6 b2 b1 b4 b3 7 4 5 DW B b3 b4 b1 b2 b4 b3 b2 b1 A-7 Complex Algebra, Quaternion Algebra, Octonian Algebra, Clifford Algebra 627 such that the Cayley-product 2 3 c1 c2 c3 c4 6 7 6 c2 c1 c4 c3 7 44 AB D 4 5 DW C 2 M .R ; Hamilton/ c3 c4 c1 c2 c4 c3 c2 c1

c1 D a1b1 a2b2 a3b3 a4b4

c2 D a1b2 C a2b1 C a3b4 a4b3

c3 D a1b3 a2b4 C a3b1 C a4b2

c4 D a1b4 C a2b3 a3b2 C a4b1 fulﬁlling the axioms .G1ı/, .G2ı/, .G3ı/ of a non-Abelian multiplicative group, namely

.G1ı/.AB/C D A.BC / .associativity/ .G2ı/ AI D A .identity/ .G3ı/ AA1 D I .inverse/; but .G4ı/ does not apply, in particular

A 2 2 2 2 B 2 2 2 2 det D a1 C a2 C a3 C a4; det D b1 C b2 C b3 C b4 det .AB/ D .det A/.det B/:

Secondly we deﬁne the 2 2 complex matrix basis E and decompose it into the four constituents ˙ 0, ˙ 1, ˙ 2, ˙ 3 of 2 2 complex Pauli matrices which form a multiplicative group of the multiplication table of Hamilton type " # e e e 1 C i 1 2 C i 3 0 1 2 3 22 E WD D 1˙ C e1˙ C e2˙ C e3˙ 2 C e2 C ie3 1 ie1 10 i0 ˙ 0 WD ; ˙ 1 WD ; 01 0 i 01 0i ˙ 2 WD ; ˙ 3 WD 10 i0 § ¤ ¦multiplication table, Cayley diagram ¥ 628 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra

˙ 0 ˙ 1 ˙ 2 ˙ 3 ˙ 0 ˙ 0 ˙ 1 ˙ 2 ˙ 3 ˙ 1 ˙ 1 ˙ 0 ˙ 3 ˙ 2 : ˙ 2 ˙ 2 ˙ 3 ˙ 0 ˙ 1 ˙ 3 ˙ 3 ˙ 2 ˙ 1 ˙ 0

Note that E 2 C22 is “Hermitean”. Let A, B, C 2 M .C22, Hamilton) constituted by means of " # a C ia a C ia 1 2 3 4 DW A a3 C ia4 a1 ia2 " # b C ib b C ib 1 2 3 4 DW B b3 C ib4 b1 ib2 such that the Cayley-product " # c C ic c C ic AB D 1 2 3 4 DW C 2 M .C2; Hamilton/ c3 C ic4 c1 ic2 A A B 2 2 2 2 2 2 2 2 det . B/ D .det /.det / D a1 C a2 C a3 C a4 b1 C b2 C b3 C b4 2 2 2 2 D c1 C c2 C c3 C c4

c1 C ic2 D .a1 C ia2/.b1 C ib2/ C .a3 C ia4/.b3 C ib4/

D a1b1 a2b2 a3b3 a4b4 C i.a1b2 C a2b1 C a3b4 a4b3/

c3 C ic4 D .a1 C ia2/.b3 C ib4/ C .a3 C ia4/.b1 ib2/

D a1b3 a2b4 C a3b1 C a4b2 C i.a1b4 C a2b3 a3b2 C a4b1/ fulﬁlling the axioms .G1ı/, G2ı/, .G3ı/ of a non-Abelian multiplicative group. The spinor " # " # 1 C ie s s WD 1 D 1 e2 C ie3 s2 as a vector of length zero relates to the 2 2 complex matrix basis by " # s s E D 1 2 : | s s 2 1 A-7 Complex Algebra, Quaternion Algebra, Octonian Algebra, Clifford Algebra 629

A-73 Octanian Algebra as a Non-Associative Algebra as well as a Composition Algebra, Clifford algebra with Respect to H H

The octanian algebra O also called “the algebra of octaves” due to Graves (1843) and Cayley (1845)isacomposition algebra over R as well as a non-associative algebra:

x 2 O

ı 1 2 3 4 5 6 7 x D 1x C e1x C e2x C e3x C e4x C e5x C e6x C e7x subject to fxı;x1;:::;x6;x7g2R8 span O Df1; e1; e2; e3; e4; e5; e6; e7g

1 x; y; z 2 R8

˛.x; y/ DW x C y

The axioms .G1C/, .G2C/, .G3C/, .G4C/ of an Abelian additive group apply. 2 x; y 2 R8;r;s2 R

ˇ.r; x/ DW r x

The axioms .D1C/, .D2C/, .D3/ of additive distributivity apply. 3 One way to explicitly describe a multiplicative group with ﬁnitely many elements is to give a table listing the multiplications just representing the map W O O ! O. § ¤ multiplication table, Cayley diagram ¦ ¥

1 e1 e2 e3 e4 e5 e6 e7 1 1 e1 e2 e3 e4 e5 e6 e7 e1 e1 1 e3 e2 e5 e4 e7 e6 e2 e2 e3 1 e1 e6 e7 e4 e5 e3 e3 e2 e1 1 e7 e6 e5 e4 e4 e4 e5 e6 e7 1 e1 e2 e3 e5 e5 e4 e7 e6 e1 1 e3 e2 e6 e6 e7 e4 e5 e2 e3 1 e1 e7 e7 e6 e5 e4 e3 e2 e1 1 630 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra

Note that in the multiplication table each entry of a group appears exactly once in each row and column. The multiplication has to be read from left to right that is, the entry at the intersection of the row headed by e5 is the product e3 e5. Such a table is called a Cayley diagram of the multiplicative group. Note the non-associativity of the internal multiplication given by the table, e.g. e2 .e3 e4/ 6D .e2 e3/e4, namely by means of e3 e4 D e7, e2 .e3 e4/ D e2 e7 De5 versus e2 e3 D e1, .e2 e3/ e4 D e1 e4 DCe5.Such an “octonian algebra” O is not a Lie algebra since neither x x D 0 (L1) nor .x y/z C .y z/ x C .z x/ y D 0 (L2) (Jacobi identity) hold. Just by means of the multiplication table compute

.e2 e3/ e4 C .e3 e4/ e2 C .e4 e2/ e3 D e5 6D 0:

4 does not apply. 5 Begin with the choice

ı 1 6 7 Q.x/ D Q.1x C e1x CCe6x C e7x / D .xı/2 C .x1/2 CC.x6/2 C .x7/2

in order to prove (K1), (K2), (K3). The laborious proofs are left as an exercise. The proper algebraic interpretation of octonian numbers is in terms of Clifford algebra, namely with respect to the eight dimensional set H H DW H2 where H is the usual skew ﬁeld of Hamilton’s quaternions, algebraically isomorphic to Cl.0;2/. Indeed it would have been temptating to base “octonian algebra” O on Cl.0;3/,dimCl.0;3/D 23 D 8, but its generic elements

f1; e1; e2; e3; e1 ^ e2; e2 ^ e3; e3 ^ e1; e1 ^ e2 ^ e3g are not representing the octonian multiplication table e.g.

2 e1 ^ e2 ^ e3/ D g.e1 ^ e2 ^ e3; e1 ^ e2 ^ e3/ D .e1 ^ e2 ^ e3/ ^ .e1 ^ e2 ^ e3/ De1 ^ e2 ^ e1 ^ e3 ^ e2 ^ e3 D e1 ^ e2 ^ e1 ^ e2 ^ .e3 ^ e3/ De1 ^ e2 ^ e1 ^ e2 De1 ^ .e2 ^ e2/ ^ e1 D.e2 ^ e1/ DC1

In contrast, let us introduce the pair

x WD .a; b/ 2fX j a 2 H; b 2 Hg A-7 Complex Algebra, Quaternion Algebra, Octonian Algebra, Clifford Algebra 631

0 00 1 x 2 H2; x 2 H2; x 2 H2

0 0 ˛.x; x DW x C x /

0 0 0 x C x D .a C a ; b C b /:

The axioms .G1C/, .G2C/, .G3C/, .G4C/ of an Abelian additive group apply. 0 0 2 x:x 2 H2;r;r2 R

ˇ.r; x/ DW r x r x D .r a;r b/:

The axioms .D1C/, .D2C/, .D3/ of additive distributivity apply.

0 3 x 2 H H D H2; x 2 H H D H2

0 0 .x; x / DW x x

0 0 0 0 0 x x DW .aa b b; b a C ba /

0 0 a, b denote the conjugate of a 2 H, b 2 H, respectively. If .a; b/, .a ; b / are represented by 0 1 0 1 X3 X3 X3 X3 @ ı i ı jA @ 0ı 0i 0ı 0jA 1˛ C ei ˛ ;1ˇ C ej ˇ ; 1˛ C ei 0 ˛ ;1ˇ C ej 0 ˇ iD1 j D1 i 0D1 j D1

respectively, where

span H Df1; e1; e2; e1 ^ e2 D e3g or 0 0 span H Df1 ; e10 ; e20 ; e10 ^ e20 D e30 gDf1 ; e5; e6; e7g

0 the “octonian product” x x results in 9 6 0 0 0 P3 0 x x D 1 ˛ı˛ ı ˇıˇ ı .˛k˛‘k ˇkˇ k/ kD1 P3 e ı k k ı k i 0j j 0i C kŒ˛ ˇ C ˛ ˇ C ij .˛ ˛ ˇ ˇ /; i;j;k D1 0 0 P3 0 1 ˛ıˇ ı ˛ ıˇı .˛k ˇ‘k ˛ kˇk / kD1 ! P3 e ı 0k k 0ı k j 0i 0j i C kŒ˛ ˇ C ˛ ˇ C ij .˛ ˇ ˛ ˇ / : 8 i;j;kD1 7 632 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra

the axioms .D1 C/, .D2 / of distributivity apply. The pair .1; 0/ is the neutral element. 4 Begin with the choice of

.1st/ the transpose x WD a; b/ of x 2 H2 .2nd/ x x D Q.x/ or x x D .aa C bb; 0/ x .3rd/ x1 D if x 6D 0 Q.x/

in order to end up with

x x1 D 1:

A historical perspective of octonian numbers is given by Van Der Waerden (1950). Reference is made to Graves (1848) and A. Caley: Collected Mathe- matical Papers, vol. 1, p. 127 and vol. 11, pp. 368–371. | The exceptional role of the examples on complex, quaternion and octonian algebra illustrating Clifford algebra Cl.0;1/, Cl.0;2/ as well as Clifford algebra with respect to H2 is established by the following theorems: 9 6 Theorem (“Hurwitz’ theorem of composition algebras”): A complete list of composition algebras over R consists of (a) the real numbers R, (b) the complex numbers C, (c) the quaternions H, (d) the octonians O . 8 7 9 6 Theorem (“Frobenius’ theorem of division algebras”): The only ﬁnite-dimensional division algebra over R are (˛) the real numbers R, (ˇ) the complex numbers C, () the quaternions H . 8 7

Historical Aside For details consult the historical texts Hurwitz (1898), Frobenius (1878) as well as Hazlett (1917). A more recent reference is Jacobson (1974), pp. 425 and 430. A-7 Complex Algebra, Quaternion Algebra, Octonian Algebra, Clifford Algebra 633

A-74 Clifford Algebra

We already took advantage of the notion of Clifford algebra when we referred to complex algebra, quaternion algebra,andoctonian algebra. Here we finally confront you with the definition of “orthogonal clifford algebra Cl.p;q/”. But on our way to Clifford algebra we have to generalize at first the notion of a basis, in particular its bilinear form. 9 6 Theorem (bilinear form): Suppose that the bracket hji or g.; / W XX ! R is a bilinear form on a finite dimensional linear space X, e. g. a vector space, over the field R of real numbers, in addition X its dual space such that n D dim X D dimX. There exists a basis fe1;:::;eng such that

(a) hei j ej iD0org.ei ; ej / D 0 for i 6D j 2 hei j ei iDC1org.ei ; ei / DC1 for 1 i1

The numbers r and p are determined exclusively by the bilinear form. r is called the rank, r p D q is called the index and the ordered pair .p; q/ the signature.The theorem assures that any two spaces of the same dimension with bilinear forms of the same signature are isometrically isomorphic. A scalar product (“inner product”) in this context is a nondegenerate bilinear form, i.e., a form with rank equal to the dimension of X. When dealing with low dimensional spaces as we do, we will often indicate the signature with a series of plus and minus signs and zeroes where R4 appropriate. For example, the signature of 1 may be written .CCC/ instead of .3; 1/. If the bilinear form is nondegenerate, a basis with the properties listed in corresponding Theorem is called an orthonormal basis (“unimodular”) for X with respect to the bilinear form.

Deﬁnition (orthogonal Clifford algebra Cl.p;q/): The orthogonal Clifford algebra Cl.p;q/is the algebra of polynomials generated by the direct sum of the space of multilinear functions

m n X ˚mD0 ^ . /

X X Rn on a linear space , respectively its dual over the ﬁeld of real numbers p of dimension 634 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra

m n X n dim ˚mD0 ^ . / D 2

and signature .p; q/, namely 1 ( nDdimPX nDdimPX i1 i1 i2 1f 0 C e fi1 C e ^ e fi1i2 i1D1 i1;i2D1 nDdimPX nDdimPX i1 i2 i3 i1 in C e ^ e ^ e fi1i2i3 CC e ^^ e fi1in 0i1;i2;i3D1 i1;:::;inD1 )

subject to the Clifford product, also called the Clifford dualizer, 5 2 .i/ei ^ ej Dej ^ ei for i 6D j 2 e ^ e D g.e ;e / DC1for1 i p; 6 i1 i1 i1 i1 1 .ii/ 6 4 ei2 ^ ei2 D g.ei2 ;ei2 / D1forpC 1 i2 p C q D r; e ^ e D g.e ;e / D 0forrC 1 i n; 4i3 i3 i3 i3 3 3

e1 ^ ej C ej ^ ei 2 g.ei ; ei / DC1 for 1 i1 p 6 1 1 e e 6 e e D 2g. i ; j /ıij 1subject t o 4g. i2 ; i2 / D1 for p C 1 i2 p C q D r e e g. i3 ; i3 / D 0 for r C 1 i3 n

1 being the neutral element. If ek ^ ek D 0 or g.ek;ek/ D 0 holds uniformly the orthogonal Clifford algebra Cl.p;q/ reduces to the polynomial algebra of antisym-metric multilinear functions

n m n m X X ˚mD0A D˚mD0 . / D . / n m n m X X n dim ˚mD0 A D dim ˚mD0 . / D dim . / D 2

represented by 5 2 X X 1 nDXdim 1 nDXdim 1f C ei1 f C ei1 ^ ei2 f 0 1Š i1 2Š i1i2 i1D1 i1;i2D1 X X 1 nDXdim 1 nDXdim C ei1 ^ ei2 ^ ei3 f CC ei1 ^^ein f : 3Š i1i2i3 nŠ i1in 4i1;i2;i3D1 i1;:::;inD1 3 A-7 Complex Algebra, Quaternion Algebra, Octonian Algebra, Clifford Algebra 635 9 6 Example: Clifford product: x y, x y 2 X;signX D .3; 0/

Let x y 2 X be a real three-dimensional vector space X of signature (3,0). then x y (read: “x Clifford y”) with respect to a set of the bases fe1; e2; e3g accounts for

X3 X3 X3 X3 i j i j x ^ y D ei ^ ej x y D x y ei ^ ej iD1 j D1 iD1 j D1

1 2 3 1 2 3 x ^ y D .e1x C e2x C e3x / ^.e1y C e2y C e3y /

1 1 2 2 3 3 1 2 2 1 x ^ y D 1.x y C x y C x y / C e1 ^ e2.x y x y / 2 3 3 2 3 1 1 3 C e2 ^ e3.x y x y / C e3 ^ e1.x y x y /

Indeed x ^ y as a Clifford number is decomposed into a scalar part and an antisymmetric tensor part with respect to the bilinear basis fe1 ^ e2; e2 ^ e3; e3 ^ e1g. 8 7

Tensor algebra or the algebra of multilinear functions, namely

nDXdim X nDXdim X ei1 ei ei2 1f 0 C fi1 C ˝ fi1i2 i1D1 i1;i2D1

nDXdim X nDXdim X ei1 ei2 ei3 ei1 ein C ˝ ˝ fi1i2i3 CC ˝˝ fi1in i1;i2;i3D1 i1;:::;inD1 n X X T C T 2˚mD0 ˝ . / D˝. / D ˚ " C 2h T D˚hD0 ˝ .X/.“even”/ subject to 2kC1 T D˚kD0 ˝ .X/.“odd”/ is the sum of two spaces, T C and T , respectively, in particular

Xn Xn 1 1 ClC 3 1f C ei1 ^ ei2 f C ei1 ^ ei2 ^ ei3 ^ ei4 f C ; 0 2Š i1i2 4Š i1i2i3i4 i1;i2D1 i1;i2;i3;i4 Xn Xn 1 1 Cl 3 ei1 f C ei1 ^ ei2 ^ ei3 f C : 1Š i1 3Š i1i2i3 i1D1 i1;i2;i3D1 636 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra

Obviously ClC as well as Cl are subalgebras of Cl.LettheClifford numbers z be divided into zC 2 ClC and z 2 Cl, then the properties § ¤ C C C C C ¦z ^ z 2 Cl ; z ^ z 2 Cl ; z ^ z 2 Cl ¥

prove that Cl.p;q/is graded over the cydric group Z2 Df0; 1g. | Appendix B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

We begin with sampling distributions by two sections on the transformation of random variables.Afirst confidence interval for Gauss-Laplace normal distribution 2 observations 1 known is constructed, including its famous Three Sigma Rule. The second confidence interval of Gauss-Laplace i.i.d. observations is derived for the sample mean b BLUE of 1 when the variance is known. We take advantage of the forward-backward Helmert transformation leading to Helmert chi-square 2 probability distribution (Helmert, 1875). In contrast, the sampling distributions from Gauss-Laplace i.i.d. normal distributions within the third confidence interval is analyzed for the mean, but an unknownp variance. Our basis is student’s t-distribution on the random variable n.b /=b where the sample mean b – is BLUE of whereas the sample confidence for the “true” mean ,variance 2 unknown which is based on student’s probability distribution. In detail, we derive the related Uncertainty Principle generated by the Magic Triangle of (a) length of the confidence interval, (b) coefficient of negative confidence called the uncertainty number and (c) the number of observations. For more details we refer to Lemma B.12 of W.S. Gosset. The sampling from the Gauss-Laplace normal distribution, namely the fourth confidence interval for the variance is reviewed: We introduce the missing confidence interval for the “true” variance 2: Table B.18 as a flow chart summarizes the various steps in computing a confidence interval for the variance. This time the related Uncertainty Principle is built on (a) the coefficient of complementary confidence ˛, also called uncertainty number (b) the number of observations, n, and (c) the length ı2.n; ˛/ of the confidence interval the “true” variance 2. The most detailed sampling theory from multidimensional Gauss-Laplace normal distribution, namely for the confidence region for the fixed parameters in the linear Gauss–Markov, is stated by an extensive Example B.19. We need the Principle Component Analysis PCA, also called Singular Value Decomposition (SVD) or Eigenvalue Analysis (EIGEN). Theorem B.17 summarizes the marginal distributions special linear Gauss–Markov model with datum defect. In our appendix we concentrate on multidimensional variance analysis, namely sampling form the multivariate Gauss-Laplace normal distribution. In short

E. Grafarend and J. Awange, Applications of Linear and Nonlinear Models, 637 Springer Geophysics, DOI 10.1007/978-3-642-22241-2, © Springer-Verlag Berlin Heidelberg 2012 638 B Sampling Distributions and Their Use: Confidence Intervals and Confidence Regions introduction we review the distribution of sample mean and variance-covariance. We define the k-dimensional noncentral WHISHART distribution by (B377) and the characteristic function of a vector-valued random variate of type Gauss-Laplace ^ distribution. Our highlight is the distribution of xN and SN of type WHISHART. Another highlight is the distribution related to the correlation coefficients originated by R.A. Fisher (1915), programmed by F.N. David (1954).

B-1 A First Vehicle: Transformation of Random Variables

0 If the probability density function (pdf ) of a random vector y D Œy1;:::;yn is known, but we want to derive the probability density function (pdf ) of a random 0 vector x D Œx1;:::;xn (pdf ) which is generated by an injective mapping x =g(y) or xi D gi Œy1;:::;yn for all i 2f1;:::;ng we need the results of Lemma B.1.

Lemma B.1. (transformation of pdf):

0 Let the random vector y WD Œy1;:::;yn be transformed into the random vector 0 x D Œx1;:::;xn by an injective mapping x D g.y/ or xi D gi Œy1;:::;yn for all i 2f1;:::;ng which is of continuity class C1 (ﬁrst derivatives are continuous). Let the Jacobi matrix Jx WD.@gi =@yi / be regular (det Jx ¤ 0), then the inverse 1 transformation y D g 1.x/ or yi D gi Œx1;:::;xn is unique.Letfx.x1;:::;xn/ be the unknown pdf,butfy .y1;:::;yn/ the given pdf,then ˇ ˇ 1 1 ˇ ˇ fx.x1;:::;xn/ D f.g1 .x1;:::;xn/;:::;g1 .x1;:::;xn// det Jy with respect to the Jacobi matrix 1 @yi @gi Jy WD D @xj @xj for all i; j 2f1;:::;ng holds. Before we sketch the proof we shall present two examples in order to make you more familiar with the notation.

Example B.1. (“counter example”):

The vector-valued random variable .y1;y2/ is transformed into the vector-valued random variable .x1;x2/ by means of

2 2 x1 D y1 C y2;x2 D y1 C y2 @x @x1=@y1 @x1=@y2 11 Jx WD 0 D D @y @x2=@y1 @x2=@y2 2y1 2y2 B-1 A First Vehicle: Transformation of Random Variables 639

2 2 2 2 2 x1 D y1 C 2y1y2 C y2 ;x2 C 2y1y2 D y1 C 2y1y2 C y2 2 2 x1 D x2 C 2y1y2;y2 D .x1 x2/=.2y1/ 2 1 x1 x2 2 1 2 x1 D y1 C y2 D y1 C ;x1y1 D y1 C .x1 x2/ 2 y1 2 1 y2 x y C .x2 x / D 0 1 1 1 2 1 2 s 1 x2 1 y˙ D x ˙ 1 .x2 x / 1 2 1 4 2 1 2 s 1 x2 1 y˙ D x 1 .x2 x /: 2 2 1 4 2 1 2

At ﬁrst we have computed the Jacobi matrix Jx, secondly we aimed at an inversion of the direct transformation .y1;y2/ 7! .x1;x2/. As the detailed inversion step proves, namely the solution of a quadratic equation, the mapping x D g.y/ is not injective.

Example B.2.

Suppose .x1;x2/ is a random variable having pdf exp .x x /; x 0; x 0 f .x ;x / D 1 2 1 2 x 1 2 0;otherwise:

We require to ﬁnd the pdf of the random variable

.x1 C x2;x2=x1/:

The transformation

x2 y1 D x1 C x2;y2 D x1 has the inverse

y1 y1y2 x1 D ;x2 D : 1 C y2 1 C y2

The transformation provides a one-to-one mapping between points in the first P2 quadrant of the .x1;x2/ –plane x and in the first quadrant of the .y1;y2/ –plane P2 y . The absolute value of the Jacobian of the transformation for all points in the first quadrant is 640 B Sampling Distributions and Their Use: Confidence Intervals and Confidence Regions ˇ ˇ ˇ ˇ ˇ @x @x ˇ ˇ ˇ ˇ @.x ;x / ˇ ˇ 1 1 ˇ ˇ 1 2 ˇ ˇ 1 2 ˇ ˇ @y1 @y2 ˇ ˇ .1 C y2/ y1.1 C y2/ ˇ ˇ ˇ D @x2 @x2 D ˇ 1 2 ˇ @.y ;y / ˇ ˇ y2.1 C y2/ y1.1 C y2/ Œ.1 C y2/ y2 1 2 @y1 @y2 y D y .1 C y /3 C y y .1 C y /3 D 1 : 1 2 1 2 2 2 .1 C y2/

Hence we have found for the pdf of .y1;y2/ " y1 exp.y1/ .1Cy /2 ;y1 >0;y2 >0 fy .y1;y2/ D 2 0;otherwise:

Incidentally it should be noted that y1 and y2 are independent random variables, namely

2 fy .y1;y2/ D f1.y1/f .y2/ D y1 exp.y1/.1 C y2/ : |

Proof. The probability that the random variables y1;:::;yn take on values in the region ˝y is given by Z Z

fy .y1;:::;yn/dy1 dyn:

˝y

If the random variables of this integral are transformed by the function xi D gi .y1;:::;yn/ for all i 2f1;:::;ng which map the region ˝y onto the regions ˝x, we receive Z Z

fy .y1;:::;yn/dy1 dyn

˝y Z Z ˇ ˇ 1 1 ˇ ˇ D fy .g1 .x1;:::;xn/;:::;gn .x1;:::;xn// det Jy dx1 dxn

˝x from the standard theory of transformation of hypervolume elements, namely

dy1 dyn Djdet Jy jdx1 dxn or

.dy1 ^^dyn/ Djdet Jy j.dx1 ^^dxn/:

Here we have taken advantage of the oriented hypervolume element dy1 ^^dyn (Grassmann product, skew product, wedge product) and the Hodge star operator * n n applied to the n – differential form dy1 ^^dyn 2 (the exterior algebra ). B-1 A First Vehicle: Transformation of Random Variables 641

The star * : p ! np in Rn maps a p – differential form onto a .n p/ – differential form, in general. Here p D n, n p D 0 applies. Finally we deﬁne

1 1 fx.x1;:::;xn/ WD f.g1 .x1;:::;xn/;:::;gn .x1;:::;xn//j det Jyj as a function which is certainly non-negative and integrated over ˝x to one. | In applying the transformation theorems of pdf we meet quite often the problem that the function xi D gi .y1;:::;yn/ for all i 2f1;:::;ng is given but not the inverse 1 function yi D gi .x1;:::;xn/ for all i 2f1;:::;ng. Then the following results are helpful.

Corollary B.1. (Jacobian):

If the inverse Jacobian j det JxjDjdet.@gi =@yj /j is given, we are able to compute

1 @gi .x1;:::;xn/ 1 @gi .y1;:::;yn/ 1 j det JyjDjdet jDjdet Jj Djdet j : @xj @yj

Example B.3. (Jacobian):

Let us continue Example B.2.Theinverse map 1 g1 .y1;y2/ x1 C x2 @y 11 y D 1 D ) 0 D 2 g .y1;y2/ x2=x1 @x x2=x 1= x1 2 ˇ ˇ 1 ˇ ˇ ˇ @y ˇ 1 x2 x1 C x2 jy DjJyjDˇ 0 ˇ D C 2 D 2 @x x1 x x 1ˇ ˇ 1 ˇ ˇ 2 2 1 1 ˇ @x ˇ x1 x1 jx DjJxjDjy DjJyj D ˇ 0 ˇ D D @y x1 C x2 y1 allows us to compute the Jacobian Jx from Jy.Thedirect map g .y ;y/ x y =.1 C y / x D 1 1 2 D 1 D 1 2 g2.y1;y2/ x2 y1y2=.1 C y2/ leads us to the ﬁnal version of the Jacobian.

y j DjJ jD 1 : x x 2 .1 C y2/

For the special case that the Jacobi matrix is given in a partitioned form, the results of Corollary B.3 are useful. 642 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

Corollary B.2. (Jacobian):

If the Jacobi matrix Jx is given in the partitioned form @gi U jJxjD D ; @yj V then q p 0 0 0 0 0 1 0 det Jx D j det JxJxjD det.UU / detŒVV .VU /.UU / UV

if det.UU0/ ¤ 0 q p 0 0 0 0 0 1 0 det Jx D j det JxJxjD det.VV / detŒUU UV .VV / VU ;

if det.VV0/ ¤ 0

1 j det JyjDjdet Jxj :

Proof. The Proof is based upon the determinantal relations of a partitioned matrix of type AU det D det A det.D VA 1U/if det A ¤ 0 VD AU det D det D det.A UD1U/if det D ¤ 0 VD AU det D D det A V.adjA/U; VD which have been introduced by Frobenius (1908)andSchur (1917).

B-2 A Second Vehicle: Transformation of Random Variables

Previously we analyzed the transformation of the pdf under an injective map of random variables y 7! g.y/ D x. Here we study the transformation of polar coordinates Œ1;2;:::;n1;r 2 Y as parameters of an Euclidian observation space to Cartesian coordinates Œy1;:::;yn 2 Y. In addition we introduce the hypervolume element of a sphere Sn1 Y; dim Y D n.First,wegivethree examples. Second, we summarize the general results in Lemma B.4. B-2 A Second Vehicle: Transformation of Random Variables 643

Example B.4. (polar coordinates: “2d”):

Table B.1 collects characteristic elements of the transformation of polar coordinates .1;r/ of type “longitude, radius” to Cartesian coordinates .y1;y2/; their domain 1 2 and range, the planar elements dy1;dy2 as well as the circle S embedded into E WD 2 fR ;ıklg, equipped with the canonical metric I2 D Œıkl and its total measure of arc !1. Table B.1 Cartesian and polar coordinates of a two-dimensional observation space, total measure of the arc of the circle

2 .1;r/2 Œ0; 2 0; 1Œ; .y1;y2/ 2 R

dy1dy2 D rdrd1 S1 R2 2 2 WD fy 2 jy1 C y2 D 1g

!1 D d1 D 2: 0

Example B.5. (polar coordinates: “3d”):

Table B.2 is a collectors’ item for characteristic elements of the transformation of polar coordinates .1;2;r/ of type “longitude, latitude, radius” to Cartesian coordinates .y1;y2;y3/; their domain and range, the volume element dy1;dy2;dy3 2 3 3 as well as of the sphere S embedded into E WD fR ;ıklg equipped with the canonical metric I3 D Œıkl and its total measure of surface !2. Table B.2 Cartesian and polar coordinates of a three-dimensional observation space, total measure of the surface of the circle

y1 D r cos 2 cos 1;y2 D r cos 2 sin 1;y3 D r sin 2 . ; ;r/2 Œ0; 2 ; Œ 0; rŒ; .y ;y / 2 R2 1 2 2 2 1 2 3 .y1;y2;y3/; 2 R

2 dy1dy2dy3 D r dr cos 2d1d2 S2 R 2 2 2 WD fy 2 jy1 C y2 C y3 D 1g

Z2 CZ=2

!2 D d1 d2 cos 2 D 4: 0 =2 644 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

Example B.6. (polar coordinates: “4d”):

Table B.3 is a collection of characteristic elements of the transformation of polar coordinates .1;2;3;r/to Cartesian coordinates .y1;y2;y3;y4/; their domain and 3 range, the hypervolume element dy1;dy2;dy3;dy4 as well as of the 3 - sphere S 4 4 embedded into E WD fR ;ıklg equipped with the canonical metric I4 D Œıkl and its total measure of hypersurface. Table B.3 Cartesian and polar coordinates of a four-dimensional observation space total measure of the hypersurface of the 3-sphere

y1 D r cos 3 cos 2 cos 1;y2 D r cos 3 cos 2 sin 1;

y3 D r cos 3 sin 2;y4 D r sin 3 . ; ; ;r/2 Œ0; 2 ; Œ ; Œ 0; 2Œ 1 2 3 2 2 2 2 3 2 dy1dy2dy3dy4 D r cos 3 cos 2drd3d2d1

@.y1;y2;y3;y4/ Jy WD @.1;2;3;r/ 2 3 r cos cos sin r cos sin cos r sin cos cos cos cos cos 6 3 2 1 3 2 1 3 2 1 3 2 1 7 6 Cr cos cos cos r cos sin sin r sin cos sin cos cos sin 7 D 6 3 2 1 3 2 1 3 2 1 3 2 1 7 4 0 Cr cos 3 cos 2 r sin 3 sin 2 cos 3 cos 2 5 00rcos 3 sin 3

3 2 j det Jy jDr cos 3 cos 2 S3 R4 2 2 2 2 WD fy 2 jy1 C y2 C y3 C y4 D 1g 2 !3 D 2 :

Lemma B.2. (polar coordinates, hypervolume element, hypersurface element):

2 3 2 3 y1 cos n1 cos n2 cos n3 cos 2 cos 1 6 7 6 7 6 y2 7 6 cos n1 cos n2 cos n3 cos 2 sin 1 7 6 7 6 7 6 y 7 6 cos cos cos sin 7 6 3 7 6 n1 n2 n3 2 7 6 y 7 6 7 6 4 7 6 cos n1 cos n2 cos 3 7 6 7 6 7 Let 6 7 D r 6 7 6 7 6 7 6 yn3 7 6 cos n1 cos n2 sin n3 7 6 7 6 7 6 yn2 7 6 cos n1 cos n2 7 4 5 4 5 yn1 cos n1 sin n2 yn sin n1 B-2 A Second Vehicle: Transformation of Random Variables 645 be a transformation of polar coordinates .1;2; :::,n2;n1;r/ to Cartesian coordinates .y1;y2;:::,yn1;yn/, their domain and range given by . ; ;:::; ; ;r/2 Œ0; 2 ; C Œ ; 1 2 n2 n1 2 2 2 C Œ ; C Œ 0; 1Œ; 2 2 2 then the local hypervolume element dy1 dyn n1 n2 n3 2 D r dr cos n1 cos n2 :::cos 3 cos 2dn1dn2 :::d3d2d1; as well as the global hypersurface element

CZ=2 CZ=2 Z2 .n1/=2 2 n2 !n1 D n1 WD cos n1 dn1 cos 2d2 d1; . 2 / =2 =2 0 where .X/ is the gamma function. Before we care for the proof, let us deﬁne Euler’s gamma function.

Deﬁnition B.1. (Euler’s gamma function):

Z1 .x/D et t x1dt .x > 0/

0 is Euler’s gamma function which enjoys the recurrence relation

.xC 1/ D x .x/ subject to 1 p .1/D 1 or D 2 3 1 1 1p .2/D 1D D 2 2 2 2 p p q p q .nC 1/ D nŠ D q q q p if n is integer, n 2 ZC if is a rational number, p=q 2 QC. q 646 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

Example B.7. (Euler’s gamma function): 1 p (a) .1/D 1.i/D 2 3 1 1 1p (b) .2/D 1.ii/ D D 2 2 2 2 5 3 3 3p (c) .3/D 1 2 D 2.iii/ D D 2 2 2 4 7 5 5 15p (d) .4/D 1 2 3 D 6.iv/ D D : 2 2 2 8 Proof. Our proof of Lemma B.4 will be based upon computing the image of the n1 n n1 n tangent space Ty S E of the hypersphere S E . Let us embed the n1 n hypersphere S parameterized by .1;2;:::;n2;n1/ in E parameterized n by .y1;:::;yn/, namely y 2 E ,

y D e1r cos n1 cos n2 cos 2 cos 1

C e2r cos n1 cos n2 cos 2 sin 1 C

C en1r cos n1 sin n2

C enr sin n1:

Note that 1 is a parameter of type longitude, 0 1 2, while 2;:::;n1 are parameters of type latitude, =2 < 2 < C=2;:::;=2 < n1 < C=2 (open intervals). The images of the tangent vectors which span the local tangent space are given in the orthonormal n-legfe1; e2;:::;en1; enj0g by

g1 WD D1 y De1r cos n1 cos n2 cos 2 sin 1

C e2r cos n1 cos n2 cos 2 cos 1

g2 WD D2 y De1r cos n1 cos n2 sin 2 cos 1

e2r cos n1 cos n2 sin 2 sin 1

C e3r cos n1 cos n2 cos 2

:::

gn1 WD Dn1 y De1r sin n1 cos n2 cos 2 cos 1

en1r sin n1 sin n2

C enr cos n1 B-2 A Second Vehicle: Transformation of Random Variables 647

gn WD Dn y D e1r cos n1 cos n2 cos 2 cos 1

C e2r cos n1 cos n2 sin 2 sin 1 C

C en1r cos n1 cos n2

C enr sin n1 D y=r:

n fg1;:::;gn1g span the image of the tangent space in E . gn is the hypersphere normal vector, kgnkD1. From the inner products hgi jgj iDgij , i;j 2f1;:::;ng, we derive the Gauss matrix of the metric G WD Œgij .

2 2 2 2 2 hg1jg1iDr cos n1 cos n2 cos 3 cos 2 2 2 2 2 hg2jg2iDr cos n1 cos n2 cos 3 2 hgn1jgn1iDr ;

hgnjgniD1:

The off-diagonal elements of the Gauss matrix of the metric are zero. Accordingly p p n1 n2 n3 2 det Gn D det Gn1 D r .cos n1/ .cos n2/ .cos 3/ cos 2: p p The square root det Gn, det Gn1 elegantly represents the Jacobian determinant p @.y1;y2;:::;yn/ Jy WD D det Gn: @.1;2;:::;n1;r/ p Accordingly we have found the local hypervolume element det Gn dr dn1 dn2 d3 d2 d1.Fortheglobal hypersurface element !n1,weintegrate

d1 D 2 0 CZ=2 C=2 cos 2d2 D Œsin 2=2 D 2 =2

CZ=2 1 cos2 d D Œcos sin C=2 D=2 3 3 2 3 3 3 =2 =2 648 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

CZ=2 1 4 cos3 d D Œcos2 sin 2 sin C=2 D 4 4 3 4 4 4 =2 3 =2 ::: CZ=2 1 .cos /n2d D Œ.cos /n3C=2 n1 n1 n 2 n1 =2 =2

CZ=2 1 C .cos /n4d n 3 n1 n1 =2 recursively. As soon as we substitute the gamma function, we arrive at !n1.

B-3 A First Conﬁdence Interval of GaussÐLaplace Normally Distributed Observations , 2 Known, the Three Sigma Rule

The ﬁrst conﬁdence interval of Gauss-Laplace normally distributed observations constrained to .; 2/ known, will be computed as an introductory example. An application is the Three Sigma Rule. In the empirical sciences, estimates of certain quantities derived from observations are often given in the form of the estimate plus or minus a certain amount.For instance, the distance between a benchmark on the Earth’s surface and a satellite orbiting the Earth may be estimated to be

.20; 101; 104:132 ˙ 0:023/m with the idea that the ﬁrst factor is very unlikely to be outside the range

20; 101; 104:155 m to 20; 101; 104:109m :

A cost accountant for a publishing company in trying to allow for all factors which enter into the cost of producing a certain book, actual production costs, proportion of plant overhead, proportion of executive salaries, may estimate the cost to be 21 ˙ 1; 1 Euro per volume with the implication that the correct cost very probably lies between 19.9 and 22.1 Euro per volume. The Bureau of Labor Statistics may estimate the number of unemployed in a certain area to be 2:4˙ 0:3 million at a given time though intuitively it should be between 2.1 and 2.7 million. What we are saying is that in practice we are quite accustomed to seeing estimates in the form of intervals. In order to give precision to these ideas we shall B-3 A First Conﬁdence Interval of Gauss–Laplace Normally Distributed Observations 649 consider a particular example. Suppose that a random sample x 2fR;pdfg is taken from a Gauss-Laplace normal distribution with known mean and known variance 2. We ask the key question. What is the probability of the random interval . c; C c/ to cover the mean as a quantile c of the standard deviation ? To put this question into a mathematical form we write the probabilistic two-sided interval identity.

P.x1 < X

x2DZCc 1 1 p exp .x /2 dx D 2 22 x1Dc with a left boundary l D x1 and a right boundary r D x2. The length of the interval is x2 x1 D r l.Thecenter of the interval is .x1 C x2/=2 or .Herewehave taken advantage of the Gauss-Laplace pdf in generating the cumulative probability

P.x1 < X

F.x2/ F.x1/ D F. C c/ F. C c/:

Typical values for the confidence coefficient are D 0:95 ( D 95%or1 D 5% negative confidence), D 0:99 ( D 1%or1 D 1% negative confidence) or D 0:999 ( D 999%O or 1 D 1%O negative confidence). Consult Fig. B.1 for a geometric interpretation. The confidence coefficient is a measure of the probability mass between x1 D c and x2 D C c.Fora given confidence coefficient

Zx2 f.xj; 2/dx D

x1 establishes an integral equation. To make this point of view to be better under-stood let us transform the integral equations to its standard form.

1 x 7! z D .x / , x D C z x2DZCc ZCc 1 1 1 1 p exp .x /2 dx D p exp z2 dz D 2 22 2 2 x1Dc c Zx2 ZCc f.xj; 2/dx D f.zj0; 1/dz D :

x1 c 650 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

f(x)

μ–3σ μ–2σ μ–σ μ μ+σ μ+2σ μ+3σ x

Fig. B.1 Probability mass in a two-sided conﬁdence interval x1 < X

The special Helmert transformation maps x to z,nowbeingstandard Gauss- Laplace normal: 1 is the dilatation factor, also called scale variation, but the translation parameter. The Gauss-Laplace pdf is symmetric, namely f.x/ D f.Cx/ or f.z/ D f.Cz/. Accordingly we can write the integral identity

Zx2 Zx2 f.xj; 2/dx D 2 f.xj; 2/dx D

x1 0 ZCc Zc , f.zj0; 1/dz D 2 f.zj0; 1/dz D :

c 0

The classiﬁcation of integral equations tells us that

Zz .z/ D 2 f.z/dz

0 is a linear Volterra integral equation the ﬁrst kind. B-3 A First Conﬁdence Interval of Gauss–Laplace Normally Distributed Observations 651

In case of a Gauss-Laplace standard normal pdf, such an integral equation is solved by a table. In a forward computation

Zz Zz 1 1 2 F.z/ WD f.zj0; 1/dz or ˚.z/ WD exp z dz 2 2 1 1 are tabulated in a regular grid. For a given value F.z1/ or F.z2/, z1 or z2 are determined by interpolation. C.F. Gauss did not use such a procedure. He took advantage of the Gauss inequality which has been reviewed in this context by Pukelsheim (1994). There also the Vysochanskii-Petunin inequality has been discussed. We follow here a two-step procedure. First, we divide the domain z 2 Œ0; 1 into two intervals z 2 Œ0; 1 and z 2 Œ1; 1. In the first interval f.z/ is isotonic, differentiable and convex, f 00.z/ D f.z/.z2 1/ < 0; while in the second interval isotonic, differentiable and concave, f 00.z/ D f.z/.z2 1/ > 0. z D 1 is the point of inflection. Second, we setup Taylor series of f.z/ in the interval z 2 Œ0; 1 at the point z D 0, while in the interval z 2 Œ1; 1 at the point z D 1 and z 2 Œ2; 1 at the point z D 2. Three examples of such a forward solution of the characteristic linear Volterra integral equation of the first kind will follow. They establish:

The One Sigma Rule

The Two Sigma Rule The Three Sigma Rule

Box B.1. (Operational calculus applied to the Gauss-Laplace probability distribution):

“generating differential equations”

f 00.z/ C 2f 0.z/ C f.z/ D 0 subject to

ZC1 f.z/dz D 1 1 652 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

“recursive differentiation” 1 1 f.z/ D p exp z2 2 2 f 0.z/ Dzf.z/ DW g.z/ f 00.z/ D g0.z/ Df.z/ zg.z/ D .z2 1/f .z/ f 000.z/ D 2zf.z/ C .z2 1/g.z/ D .z3 C 3z/f .z/ f .4/.z/ D .3z2 C 3/f .z/ C .z3 C 3z/g.z/ D .z4 6z2 C 3/f .z/ f .5/.z/ D .4z3 12z/f .z/ C .z4 6z2 C 3/g.z/ D .z5 C 10z3 15z/f .z/ f .6/.z/ D .5z4 C 30z2 15/f .z/ C .z5 C 10z3 15z/g.z/ D .z6 15z4 C 45z2 15/f .z/ f .7/.z/ D .6z5 60z3 C 90z/f .z/ C .z6 15z4 C 45z2 15/g.z/ D .z7 C 21z5 105z3 C 105z/f .z/ f .8/.z/ D .7z6 C 105z4 315z2 C 105/f .z/ C.z7 C 21z5 105z3 C 105z/f .z/ D .z8 28z6 C 210z4 420z2 C 105/f .z/ f .9/.z/ D .8z7 168z5 C 840z3 840z/f .z/ C.z8 28z6 C 210z4 420z2 C 105/g.z/ D .z9 C 36z7 378z5 C 1260z3 945z/f .z/ f .10/.z/ D .9z8 C 252z6 1890z4 C 3780z2 945/f .z/ C C.z9 C 36z7 378z5 C 1260z3 945/g.z/ D .z10 45z8 C 630z6 3150z4 C 4725z2 945/f .z/

“upper triangle representation of the matrix transforming f.z/ ! f n.z/” 2 3 2 3 2 3 f.z/ 1 0 0 0 0 0 00000 1 6 0 7 6 7 6 7 6 f .z/ 7 6 0 1 0 0 0 0 000007 6 z 7 6 7 6 7 6 7 6 f 00.z/ 7 6 1 0 1 0 0 0 000007 6 z2 7 6 7 6 7 6 7 6 000 7 6 7 6 3 7 6 f .z/ 7 6 03 01 0 0 000007 6 z 7 6 .4/ 7 6 7 6 4 7 6 f .z/ 7 6 306 0 1 0 000007 6 z 7 6 7 6 7 6 7 6 f .5/.z/ 7 D f.z/ 6 0 1501001000007 6 z5 7 : 6 7 6 7 6 7 6 f .6/.z/ 7 6 150450150 100007 6 z6 7 6 7 6 7 6 7 6 .7/ 7 6 7 6 7 7 6 f .z/ 7 6 0 105 0 105 0 21 0 10 007 6 z 7 6 .8/ 7 6 7 6 8 7 6 f .z/ 7 6 105 0 420 0 210 0 2801007 6 z 7 4 f .9/.z/ 5 4 0 945 0 1260 0 378 0 36 0 105 4 z9 5 f .10/.z/ 945 0 4725 0 3150 0 630 0 45 0 1 z10 B-3 A First Conﬁdence Interval of Gauss–Laplace Normally Distributed Observations 653

B-31 The Forward Computation of a First Conﬁdence Interval of GaussÐLaplace Normally Distributed Observations: ; 2 Known

We can avoid solving the linear Volterra integral equation of the ﬁrst kind if we push forward the integration for a ﬁxed value z.

Example B.8. (Series expansion of the Gauss-Laplace integral, ﬁrst interval):

Let us solve the integral

Z1 .z D 1/ WD 2 f.z/dz

0 in the first interval 0 z 1 by Taylor expansion with respect to the successive differentiation of f.z/ outlined in Box B.1 and the specific derivatives f n.0/ given in Table B.4. Based on those auxiliary results, Box B.2 presents us the detailed interpretation. First, we expand exp.z2=2/ up to order O.14/. The specific Taylor series are uniformly convergent. Accordingly, in order to compute the integral, second we integrate termwise up to order O.15/. For the specific value z D 1,we have computed the coefficient of confidence .1/ D 0:683.Theresult

Box B.2. A speciﬁc integral

“expansion of the exponential function”

x x2 x3 xn exp x D 71 C C C CC C O.n/ 8jxj <1 1Š 2Š 3Š nŠ 1 x WD z2 2 1 1 1 1 1 1 1 exp z2 D 1 z2 C z4 z6 C z8 z10 C z12 C O.14/ 2 2 8 48 384 3840 46080 654 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

“the speciﬁc integral”

Zz Zz 1 2 1 1 1 1 f.z/dz D p exp z dz D p z z3 C z5 z7 2 2 6 40 336 0 0 1 1 1 C z9 z11 C z13 C O.15/ 3456 42240 599040

“the speciﬁc values z D 1”

Z1 .1/ D 2 f.z/dz

0 2 1 1 1 1 1 1 D p 1 C C C C O.15/ 2 6 40 336 3456 42240 599040 2 D p .1 0:166;667 C 0:025;000/ 2 0:002;976 C 0:000;289 0; 000;024 C 0:000;002/ 2 D p 0:855;624 D 0:682;689 2 D 0:683

“coefﬁcient of conﬁdence”

1 0:683 D 1 317:311 103 D 1 : 3

Table B.4 (Special values of derivatives- Gauss-Laplace probability distribution):

z D 0 1 p D 0:398; 942 2 1 f.0/ D p ;f0.0/ D 0; 2 1 1 f 00.0/ Dp ; f 00.0/ D0:199; 471 2 2Š B-3 A First Conﬁdence Interval of Gauss–Laplace Normally Distributed Observations 655

3 1 f 000.0/ D 0; f .4/.0/ DCp ; f .4/.0/ DC0:049; 868 2 4Š 15 1 f .5/.0/ D 0; f .6/.0/ Dp ; f .6/.0/ D0:008; 311: 2 6Š

Example B.9. (Series expansion of the Gauss-Laplace integral, 2nd interval):

Let us solve the integrals

Z2 Z1 Z2 .z D 2/ WD 2 f.z/dz D 2 f.z/dzC2 f.z/dz

0 0 1

Z2 .z D 2/ D .z D 1/ C 2 f.z/dz;

1 namely in the second interval 1 z 2. First,wesetupTaylor series of f.z/ “around the point z D 1”. The derivatives of f.z/ “at the point z=1” are collected up to order 10 in Table B.5. Second, we integrate the Taylor series termwise and receive the specific integral of Box B.3. Note that termwise integrated is permitted since the Taylor series are uniformly convergent. The detailed computation up to order O(12) has led us to the coefficient of confidence .2/ D 0:954.Theresult

P. 2 < X <C 2/ D 0:954 is known as the Two Sigma Rule. 95.4% of the sample interval 2; C 2 Œ, 0.046% outside. If we make 22 experiments, one experiment is outside the 2 interval.

Box B.3. (A speciﬁc integral):

“expansion of the exponential function” 1 1 f.z/ WD p exp z2 2 2 1 1 1 f.z/ D f.1/C f 0.1/.z 1/ C f 00.1/.z 1/2 CC f 10.1/.z 1/10 1Š 2Š 10Š C O.11/ 656 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

“the speciﬁc integral”

Z2 1 1 1 1 f.z/dz D f.1/.z 1/ C f 0.1/.z 1/2 C f 00.1/.z 1/3 2 1Š 3 2Š 1 1 1 1 1 C f 000.1/.z 1/4 C f .4/.1/.z 1/5 4 3Š 5 4Š 1 1 1 1 C f .5/.1/.z 1/6 CC f .10/.1/.z 1/11 C O.12/ 6 5Š 11 10Š case z D 2

Z2 .2/ D .1/C 2 f.z/dz D.1/C 2.0:241; 971

1 0:120; 985 C 0:020; 122 0:004; 024 0:002; 016 C 0:000; 768 C 0:000; 120 0:000; 088 0:000; 002 0:000; 050 C O.12// D 0:682; 690 C 0:271; 632 D 0:954

“coefﬁcient of conﬁdence”

1 0:954 D 1 45:678 103 D 1 : 22

Table B.5 (Special values of derivatives – Gauss-Laplace probability distribution):

z D 1 1 1 p D 0:398; 942; exp D 0:606; 531 2 2 1 1 f.1/ D p exp D 0:241; 971 2 2 1 1 f 0.1/ Df.1/Dp exp D0:241; 971; f 00.1/ D 0 2 2 2 1 f 000.1/ D 2f .1/ D p exp D 0:482; 942 2 2 B-3 A First Conﬁdence Interval of Gauss–Laplace Normally Distributed Observations 657

1 f 000.1/ DC0:080; 490 3Š f .4/.1/ D2f .1/ D0:482; 942 1 f .4/.1/ D0:020; 122 4Š 6 1 f .5/.1/ D6f .1/ Dp exp D1:451; 826 2 2 1 f .5/.1/ D0:012; 098 5Š 16 1 f .6/.1/ D 16f .1/ D p exp D 3:871; 536 2 2 1 f .6/.1/ D 0:005; 377 6Š 20 1 f .7/.1/ D 20f.1/ D p exp D 4:829; 420 2 2 1 f .7/.1/ D 0:000; 958 7Š f .8/.1/ D132f .1/ D31:940; 172 1 f .8/.1/ D0:000; 792 8Š f .9/.1/ D28f.1/ D6:775; 188 1 f .9/.1/ D0:000; 019 9Š f .10/.1/ D8234f.1/ D1992:389 1 f .10/.1/ D0:000; 549: 10Š

Example B.10. (Series expansion of the Gauss-Laplace integral, third interval):

Let us solve the integrals

Z3 Z1 Z2 Z3 .z D 3/ WD 2 f.z/dz D 2 f.z/dz C 2 f.z/dz C 2 f.z/dz

0 0 1 2 Z3 .z D 3/ D .z D 1/ C .z D 2/ C 2 f.z/dz;

2 658 B Sampling Distributions and Their Use: Confidence Intervals and Confidence Regions namely in the third interval 2 z 3. First,wesetupTaylor series of f(z) “around the point z D 2”. The derivatives of f.z/ “at the point z D 2” are collected up to order 10 in Table B.6. Second, we integrate the Taylor series termwise and receive the specific integral of Box B.4. Note that termwise integration is permitted since the Taylor series are uniformly convergent. The detailed computation up to order O(12) leads us to the coefficient of confidence .3/ D 0:997.Theresult

P. 3 < X <C 3/ D 0:997 is known as the Three Sigma Rule. 99.7% of the sample are in the interval 3; C 3Œ, 0.003% outside. If we make 377 experiments, one experiment is outside the 3 interval.

Table B.6 (Special values of derivatives Gauss-Laplace probability distribution):

z D 2 1 p D 0:398; 942; exp .2/ D 0:135; 335 2 1 f.2/D p exp .2/ D 0:053; 991 2 f 0.2/ D2f .2/ D0:107; 982 1 f 00.2/ D 3f.2/; f 00.2/ DC0:080; 986 2Š 1 f 000.2/ D2f .2/; f 000.2/ D0:017; 997 3Š 1 f .4/.2/ D5f.2/; f .4/.2/ D0:011; 248 4Š 1 f .5/.2/ D 18f.2/; f .5/.2/ DC0:008; 099 5Š 1 f .6/.2/ D11f.2/; f .6/.2/ D0:000; 825 6Š 1 f .7/.2/ D86f .2/; f .7/.2/ D0:000; 921 7Š 1 f .8/.2/ DC249f.2/; f .8/.2/ DC0:000; 333 8Š 1 f .9/.2/ D 190f.2/; f .9/.2/ DC0:000; 028 9Š 1 f .10/.2/ D2621f.2/; f .10/.2/ D0:000; 039: 10Š B-3 A First Conﬁdence Interval of Gauss–Laplace Normally Distributed Observations 659

Box B.4 (A speciﬁc integral) “expansion of the exponential function” 1 1 f.z/ WD p exp z2 2 2 1 1 1 f.z/ D f.2/C f 0.2/.z 2/ C f 00.2/.z 2/2 CC f .10/.2/.z 2/10 1Š 3Š 10Š C O.11/

“the speciﬁc integral”

Zz 1 1 1 1 f.z/dz D f.2/.z 2/ C f 0.2/.z 2/2 C f 00.2/.z 2/3 2 1Š 3 2Š 2 1 1 1 1 C f 000.2/.z 2/4 C f .4/.2/.z 2/5 4 3Š 5 4Š 1 1 1 1 C f .5/.2/.z 2/6 CC f .10/.2/.z 2/11 C O.12/ 6 5Š 11 10Š case z D 3

Z3 .3/ D .1/C .2/C 2 f.z/dz

2 D 0:682; 690 C 0:271; 672 C 2.0:053; 991 0:053; 991 C 0:026; 995 0:004; 499 0:002; 250 C 0:001; 350 0:000; 118 C 0:000; 037 C 0:000; 003 0:000; 004 C O.12// D 0:682; 690 C 0:271; 632 C 0:043; 028 D 0:997

“coefﬁcient of conﬁdence” 1 0:997 D 1 2:65 103 D 1 : 377

B-32 The Backward Computation of a First Conﬁdence Interval of GaussÐLaplace Normally Distributed Observations: ; 2 Known

Finally we solve the Volterra integral equation of the first kind by the technique of series inversion, also called series reversion. Let us recognize that the interval 660 B Sampling Distributions and Their Use: Confidence Intervals and Confidence Regions integration of a Taylor series expanded Gauss-Laplace normal density distribution led us to a univariate homogeneous polynomial of arbitrary order. Such a univariate 2 n homogeneous polynomial y D a1x C a2x CCanx (“input”) can be reversed 2 n as a univariate homogeneous polynomial x D b1y C b2y CCbny (“output”) as outlined in Table B.7. Consult Abramowitz and Stegun (1965), p. 16 for a review, but Grafarend (1996) for a derivation based upon Computer Algebra. Table B.7 (Series inversion (Grafarend, 1996)): “input: univariate homogeneous polynomial”

2 3 4 5 6 7 8 y D a1x C a2x C a3x C a4x C a5x C a6x C a7x C O.x /

“output: reverse univariate homogeneous polynomial”

2 3 4 5 6 7 8 x D b1y C b2y C b3y C b4y C b5y C b6y C b7y C O.y /

“coefﬁcient relation”

(a) a1b1 D 1 3 (b) a1b2 Da2 5 2 (c) a1b3 D 2a2 a1a3 7 2 3 (d) a1b4 D 5a1a2a3 a1a4 5a2 9 2 2 2 4 3 2 (e) a1b5 D 6a1a2a4 C 3a1a3 C 14a2 a1a5 21a1a2a3 11 3 3 3 4 2 5 2 2 (f) a1 b6 D 7a1a2a5 C7a1a2a4 C84a1a2a3 a1a6 28a1a2a3 42a2 28a1a2a4 13 4 4 4 2 2 3 2 2 2 6 (g) a1 b7 D 6a1a2a6 C 8a1a2a5 C 4a1a4 C 120a1a2a4 C 180a1a2a3 C 132a2 5 3 2 3 3 3 4 a1a7 36a1a2a5 72a1a2a3a4 12a1a3 330a1a2a3.

Example B.11. (Solving the Volterra integral equation of the ﬁrst kind):

Let us define the coefficient of confidence D 0:90 or 90%. We want to know the quantile c0:90 which determines the probability identity

P. c < X <C c/ D 0:90:

If you follow the detailed computation of Table B.5, namely the input as well as the output data up to order O(5), you ﬁnd the quantile

c0:90 D 1:64; as well as the conﬁdence interval

P. 1:64 < X <C 1:64/ D 0:90: B-3 A First Conﬁdence Interval of Gauss–Laplace Normally Distributed Observations 661

90% of the sample are in the interval 1:64; C 1:64Œ, 10% outside. If we make 10 experiments one experiment is outside the 1.62 interval.

Table B.8 (Series inversion quantile c0:90):

(i) input

Z1 Zz .z/ D 2 f.z/dz C 2 f.z/dz D 0:682; 689 C 2Œf .1/.z 1/

0 0 1 1 1 1 C f 0.1/.z 1/2 CC f .n1/.1/.z 1/n C O.n C 1/ 2 1Š n .n 1/Š 1 1 1 Œ.z/ 0:682; 689 D f.1/.z 1/ C f 0.1/.z 1/2 C 2 2 1Š 1 1 C f .n1/.1/.z 1/n C O.n C 1/ n .n 1/Š 2 n y D a1x C a2x CCanx C O.n C 1/ x WD z 1 y WD .0:900; 000 0:682; 689/=2 D 0:108; 656

a1 WD f.1/D 0:241; 971 1 1 a WD f 0.1/ D0:241; 971 2 2 1Š 1 1 a WD f 00.1/ D 0 3 3 2Š 1 1 a WD f 000.1/ D 0:020; 125 4 4 3Š 1 1 a WD f .4/.1/ D0:004; 024 5 5 4Š 1 1 a WD f .n1/.1/: n n .n 1/Š (ii) output 1 b1 D D 4:132; 726 a1 1 b2 D 3 a2 D 8:539; 715 a1 662 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

1 2 b3 D 5 .2a2 a1a3/ D 35:292; 308 a1

1 2 3 b4 D 7 .5a1a2a3 a1a4 5a2/ D 158:070 a1

1 2 2 2 4 3 2 b5 D 9 .6a1a2a4 C 3a1a3 C 14a2 a1a5 21a1a2a3/ D 475:452; 152 a1 2 b1y D 0:449; 045; b2y D 0:100; 821; 3 4 b3y D 0:045; 273; b4y D 0:022; 032; 5 b5y D 0:007; 201 2 3 4 5 x D b1y C b2y C b3y C b4y C b5y C O.6/ D 0:624; 372

z D x C 1 D 1:624; 372 D c0:90:

At this end we would like to give some sample references on computing the “inverse error function”

Zx 1 1 y D F.x/ WD p exp z2 dz DW erfx versus 2 2 0 x D F 1.y/ D inverfy; namely Carlitz (1963) and Strecok (1968).

B-4 Sampling from the GaussÐLaplace Normal Distribution: A Second Conﬁdence Interval for the Mean, Variance Known

The second conﬁdence interval of Gauss-Laplace i.i.d. observations will be constructed for the mean b BLUUE of , when the variance 2 is known. “n”isthe size of the sample, namely to agree to the number of observations. Before we present the general sampling distribution we shall work through two examples. Example B.12 has been chosen for a sample size n D 2, while Example B.12 for n D 3 observations. Afterwards the general result is obvious and sufﬁciently motivated.

Example B.12. (Gauss-Laplace i.i.d. observations, observation space Y, dim Y)

In order to derive the marginal distributions of b BLUUE of and b 2 BIQUUE of 2 for a two dimensional Euclidean observation space we have to introduce B-4 Sampling from the Gauss–Laplace Normal Distribution: 663

= σ2 σ2 f2(x) f2(ˆ / )

x: = σˆ2 / σ2

2 Fig. B.2 Special Helmert pdf of p for one degree of freedom, p D 1 various images. First, we deﬁne the probability function of two Gauss-Laplace i.i.d. observations and consequently implement the basic .b;b 2/ decomposition into the pdf. Second, we separate the quadratic form .y 1/0.y 1/ into the quadratic form .y 1b/0.y 1b/, the vehicle to introduce b 2 BIQUUE of 2, and into the quadratic form .b /2 the vehicle to bring in b BLUUE of . Third,weaimat transforming the quadratic form .y 1b/0.y 1b/ D y0 My; rk M D 1, into the 2 specialˇ form 1=2.y1 y2/ DW x. Fourth, we generate the marginal distributions ˇ 2 2 2 f1.b ; =2/ of the mean b BLUUE of and f2. / of the sample variance b BIQUUE of 2. The basic results of the example are collected in Corollary B.6. The special Helmert pdf 2 with one degree of freedom is plotted in Fig. B.2,butthe special Gauss-Laplace normal pdf of variance 2=2 in Fig. B.3. Theﬁrstactionitem Let us assume an experiment of two Gauss-Laplace i.i.d. observations. Their pdf is given by

f.y1;y2/ D f.y1/f . y2/; 1 1 f.y ;y/ D exp Œ.y /2 C .y /2 ; 1 2 22 22 1 2 1 1 f.y ;y/ D exp .y 1/0.y 1/ : 1 2 22 22 664 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

0.4

0.3

f (z| 0, 1) 1 0.2

0.1

0 –5 –4 –3 –2 –1 0 1 2 3 4 5

2 z = (mˆ − m)/ s 2

Fig. B.3 Special Gauss-Laplace normal pdf of .b /= 2=2

The second action item 0 The coordinates of the observation vector have been denoted by Œy1;y2 D y 2 Y; dim Y D 2.Thequadratic form .y 1/0.y 1/ allows the fundamental decomposition

y 1 D .y 1b/ C 1.b / .y 1/0.y 1/ D .y 1b/0.y 1b/ C 101.b /2 .y 1/0.y 1/ D b 2 C 2.b /2:

Here, b is BLUUE of and b 2 BIQUUE of 2. The detailed computation proves our statement.

2 2 Œ.y1 b/ C .b / C Œ.y2 b/ C .b / 2 2 2 D .y1 b/ C .y2 b/ C 2.b / C 2.b /Œ.y1 b/ C .y2 b/ 2 2 2 D .y1 b/ C .y2 b/ C 2.b / C 2b.y1 b/ C 2b.y2 b/

2.y1 b/ 2.y2 b/ B-4 Sampling from the Gauss–Laplace Normal Distribution: 665

1 b D .y C y / 2 1 2 2 2 2 b D .y1 b/ C .y2 b/ :

As soon as we substitute b and b 2 we arrive at

2 2 2 2 .y1 / C .y2 / D Œ.y1 b/ C .b / C Œ.y2 b/ C .b / D b 2 C 2.b /2; since the residual terms vanish:

2b.y1 b/ C 2b.y2 b/ 2.y1 b/ 2.y2 b/ 2 2 D 2by1 2b C 2by2 2b 2y1 C 2b 2y2 C 2b 2 D 2b.y1 C y2/ 4b 2.y1 C y2/ C 4b D 4b2 4b2 4b C 4b D 0:

The third action item The cumulative pdf b 2 b 2 dF D f.y ;y/dy dy D f .b/f .x/dbdx D f .b/f dbd 1 2 1 2 1 2 1 2 2 2

2 has to be decomposed into the ﬁrst pdf f1.b j; =n/ representing the pdf of the sample mean b and the second pdf f2.x/ of the new variable x WD .y1 2 2 2 2 2 2 y2/ =.2 / D b = representing the sample variance b , normalized by .

How can the second decomposition f1f2 be understood? Let us replace the quadratic form .y 1/0.y 1/ D b 2 C 2.b /2 in the cumulative pdf 1 1 dF D f.y ;y/dy dy D exp .y 1/0.y 1/ dy dy 1 2 1 2 22 22 1 2 1 1 D exp Œb 2 C 2.b /2 dy dy 22 22 1 2

dF D f.y1;y2/dy1dy2 1 1 1 .b /2 1 1 1 b 2 D p exp p p exp dy1dy2: p 2 2=2 2 2 2 2 2 2 666 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

The quadratic form b 2, conventionally given in terms of the residual vector y 1b 0 will be rewritten in terms of the coordinates Œy1;y2 D y of the observation vector. 1 2 1 2 b 2 D .y b/2 C .y b/2 D y .y C y / C y .y C y / 1 2 1 2 1 2 2 2 1 2

1 1 1 b 2 D .y y /2 C .y y /2 D .y y /2: 4 1 2 4 2 1 2 1 2

The fourth action item

1 2 The new variable x WD 2 2 .y1 y2/ will be introduced in the cumulative pdf dF D f.y1;y2/dy1dy2. The new surface element dbdx will be related to the old surface element dy1dy2. ˇ " #ˇ ˇ ˇ ˇ D bD b ˇ b ˇ y1 y2 ˇ ddx D ˇ det ˇ dy1dy2 D jJj dy1dy2 Dy1 xDy2 x

@b 1 @b 1 b b Dy1 WD D ;Dy2 WD D @y1 2 @y2 2

@x y y @x y y D x WD D 1 2 ;Dx WD D 1 2 : y1 2 y2 2 @y1 @y2

The absolute value of the Jacobi determinant amounts to ˇ " # ˇ ˇ ˇ ˇ 1 1 ˇ y y 2 jJjDˇ det 2 2 ˇ D 1 2 ; jJj1 D : ˇ y1y2 y1y2 ˇ 2 y1 y2 2 2

In consequence, we have derived

2 dy1dy2 D dbdx D p p dbdx y1 y2 2 x based upon

1 p 1 x D .y y /2 ) 2x D .y y /: 22 1 2 1 2 In collecting all detailed partial results we can formulate a corollary. B-4 Sampling from the Gauss–Laplace Normal Distribution: 667

Corollary B.3. (marginal probability distributions of b; 2 given, and b 2):

The cumulative pdf of a set of two observations is represented by

2 dF D f.y1;y2/dy1dy2 D f1.b j; =2/f2.x/dbdx subject to b 2 2 1 1 1 . / f1.b j; ; =2/ WD p exp 2 p 2 2=2 2 b 2 1 1 1 f2.x/ D f2 WD p p exp x subject to 2 2 x 2 ZC1 ZC1

f1.b/ db D 1and f2.x/ dx D 1: 1 0 f1.b/ is the pdf of the sample mean b D .y1 C y2/=2 and f2.x/ the pdf of the 2 2 2 sample variance b D .y1 y2/ =2 D x. f1.b/ is a Gauss-Laplace pdf with 2 2 mean and variance =n, while f2.x/ is a Helmert with one degree of freedom.

Example B.13. (Gauss-Laplace i.i.d. observations, observation space Y, dim Y D 3):

In order to derive the marginal distribution of b BLUUE of and b 2 BIQUUE of 2 for a three-dimensional Euclidean observation space, we have to act in various scenes. First, we introduce the probability function of three Gauss-Laplace i.i.d. observations and consequently implement the .b; b 2/ decomposition in the pdf. Second, we force the quadratic form .y 1b/0.y 1b/ to be decomposed into b 2 and .b/2, actually a way to introduce the sample variance b 2 BIQUUE of b 2 and the sample mean b BLUUE of . Third, we succeed to transform the quadratic form b 0 b 2 2 .y 1/ .y 1/ D yMy; rkM D 2 into the canonical form z1 C z2 by means of 0 0 0 Œz1; z2 D HŒy1;y2;y3 . Fourth, we produce the right inverse Hk D H in order to 0 0 0 invert H, namely Œy1;y2;y3 D H Œz1; z2 . Fifth, in order to transform the original 2 b 0 b 2 2 2 quadratic form .y1/ .y1/ into the canonical form z1 Cz2 Cz3 we review the general Helmert transformation z D 1H.y/ and its inverse y D H0z subject to H 2 SO.3/ and identify its parameters translation, rotation and dilatation 2 (scale). Sixth, we summarize the marginal probability distributions f1.b j; =3/ 2 2 2 of the sample mean b BLUUE of and f2.2b = / of the sample variance b BIQUUE of 2. The special Helmert pdf 2 with two degrees of freedom is plotted in Fig. B.4 while the special Gauss-Laplace normal pdf of variance 2=3 in Fig. B.5. The basic results of the example are collected in Corollary B.7. 668 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

ˆ 2 2 f2(2s | s )

x := 2s2 | s2

2 Fig. B.4 Special Helmert pdf of p for two degrees of freedom, p D 2

s 2 f1(mˆ | m, ) 3

s2 (mˆ − m) / 3

b 2 Fig. B.5 Special Gauss-Laplace normal pdf of . /= 3 B-4 Sampling from the Gauss–Laplace Normal Distribution: 669

Theﬁrstactionitem Let us assume an experiment of three Gauss-Laplace i.i.d. observations. Their pdf is given by

f.y1;y2;y3/ D f.y1/f . y2/f . y3/; 1 f.y ;y;y/ D .2/3=2 3 exp Œ.y /2 C .y /2 C .y /2 ; 1 2 3 22 1 2 3 1 f.y ;y;y/ D .2/3=2 3 exp .y 1/0.y 1/ : 1 2 3 22

0 The coordinates of the observation vector have been denoted by Œy1;y2;y3 D y 2 Y, dim Y D 2. The second action item The coordinates of the observation vector have been denoted by D y 2 Y,dimY D 2. 0 The quadratic form .y1 1/ .y2 1/ allows the fundamental decomposition

y 1 D .y 1b/ C 1.b /; .y 1/0.y 1/ D .y 1b/0.y 1b/ C 101.b /2

.y 1/0.y 1/ D 2b 2 C 3.b /2:

Here, b is BLUUE of and b 2 BIQUUE of 2. The detailed computation proves our statement.

2 2 2 Œ.y1 b/ C .b / C Œ.y2 b/ C .b / C Œ.y3 b/ C .b / 2 2 2 2 D .y1 b/ C .y2 b/ C .y3 b/ C 3.b / C 2b.y1 b/

C 2b.y2 b/ C 2b.y3 b/ 2.y1 b/ 2.y2 b/ 2.y3 b/

1 b BLUUE of W b D .y C y C y / 3 1 2 3 1 b 2 BIQUUE of 2 W b 2 D Œ.y b/2 C .y b/2 C .y b/2: 2 1 2 3

As soon as we substitute b and b 2 we arrive at

2 2 2 .y1 / C .y2 / C .y3 / 2 2 2 2 2 2 2 2 D Œ.y1 b/ C .b / C Œ.y2 b/ C .b / C Œ.y3 b/ C .b / D 2b 2 C 3.b /2: 670 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

The third action item Let us begin with transforming the cumulative probability function

dF D f.y1;y2;y3/dy1dy2dy3 1 D .2/3=2 3 exp .y 1/0.y 1/ dy dy dy 22 1 2 3 1 D .2/3=2 3 exp Œ2b 2 C 3.b /2 dy dy dy 22 1 2 3 into its canonical form. In detail, we transform the quadratic form b 2 BIQUUE of 2 canonically.

1 1 2 1 2 1 1 y b D y y .y C y / D y .y C y / D y y y 1 1 3 1 3 2 3 3 1 3 2 3 3 1 3 2 3 3 1 1 1 2 1 y b D y y .y C y / D y C y y 2 2 3 2 3 1 3 3 1 3 2 3 3 1 1 1 1 2 y b D y y .y C y / D y y C y 3 3 3 3 3 1 2 3 1 3 2 3 3 as well as

4 1 1 4 2 4 .y b/2 D y2 C y2 C y2 y y C y y y y 1 9 1 9 2 9 3 9 1 2 9 2 3 9 3 1 1 4 1 4 4 2 .y b/2 D y2 C y2 C y2 y y y y C y y 2 9 1 9 2 9 3 9 1 2 9 2 3 9 3 1 1 1 4 2 4 4 .y b/2 D y2 C y2 C y2 C y y y y y y 3 9 1 9 2 9 3 9 1 2 9 2 3 9 3 1 and

2 .y b/2 C .y b/2 C .y b/2 D .y2 C y2 C y2 y y y y y y /: 1 2 3 3 1 2 3 1 2 2 3 3 1

We shall prove

b 0 b 0 2 2 R33 .y 1/ .y 1/ D y My D z1 C z2; rkM D 2; M 2 that the symmetric matrix M, 2 3 2 1 1 M D 4 121 5 1 12 B-4 Sampling from the Gauss–Laplace Normal Distribution: 671 has rank 2 or rank deﬁciency 1. Just compute jMj D 0 as a determinant identity as well as the subdeterminant ˇ ˇ ˇ ˇ ˇ 2 1 ˇ ˇ ˇ D 3 ¤ 0 12 2 3 2 3 2 1 1 y 1 1 .y 1b/0.y 1b/ D y0My D Œy ;y ;y 4 121 5 4 y 5 : 3 1 2 3 2 1 12 y3

How to transform a degenerate quadratic form to a canonical form? Helmert (1875) had the bright idea to implement what we call nowadays the forward Helmert transformation

1 z1 D p .y1 y2/ 1 2 1 z2 D p .y1 C y2 2y3/ or 2 3 1 z2 D .y2 2y y C y2/ 1 2 1 1 2 2 1 z2 D .y2 C y2 C 4y2 C 2y y 4y y 4y y / 2 6 1 2 3 1 2 2 3 3 1

2 z2 C z2 D .y2 C y2 C y2 y y y y y y / : 1 2 3 1 2 3 1 2 2 3 3 1

Indeed we found

b 0 b b 2 b 2 b 2 2 2 .y 1/ .y 1/ D .y1 / C .y2 / C .y3 / D z1 C z2:

In algebraic terms, a representation of the rectangular Helmert matrix is 2 3 " # p1 p1 0 y1 z1 12 12 4 5 z D D 1 1 2 y2 z2 p p p 23 23 23 y3 " # p1 p1 0 12 12 z D H23y; H23 WD : p1 p1 p2 23 23 23 672 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

The rectangular Helmert matrix is right orthogonal, 2 3 1 1 " # p p p1 p1 0 6 12 23 7 0 12 12 6 p1 p1 7 10 H23H 23 D 4 5 D D I2: p1 p1 p2 12 23 01 23 23 23 0 p2 23

The fourth action item

2 2 By means of the forward Helmert transformation we could prove that z1 C z2 represents the quadratic form .y1b/0.y1b/. Unfortunately, the forward Helmert transformation only allows indirectly by a canonical transformation. What would be needed is the inverse Helmert transformation z 7! y, also called backward Helmert transformation.Therectangular Helmert matrix has the disadvantage to have no Cayley inverse. Fortunately, its right inverse

0 0 1 0 R32 HR WD H 23.H23H 23/ D H 23 2 solves our problem. The inverse Helmert transformation

0 y D HRz D H 23z brings

2 .y 1b/0.y 1b/ D .y2 C y2 C y2 y y y y y y / 3 1 2 3 1 2 2 3 3 1 via 2 1 1 y1 D p z1 C p z2 6 12 23 0 6 1 1 y D H z or y2 Dp z1 C p z2 23 4 12 23 2 y3 Dp z2; 23

2 2 2 1 2 1 2 1 1 2 1 2 1 2 2 y C y C y D z C z C p z1z2 C z C z p z1z2 C z ; 1 2 3 2 1 6 2 3 2 2 6 2 3 3 2

1 2 1 2 1 1 2 1 1 2 y1y2 C y2y3 C y3y1 D z C z C p z1z2 z p z1z2 z ; 2 1 6 2 3 3 2 3 2 2 2 2 3 3 .y2 C y2 C y2 y y y y y y / D z2 C z2 D z2 C z2; 3 1 2 3 1 2 2 3 3 1 3 2 1 2 2 1 2 into the canonical form. B-4 Sampling from the Gauss–Laplace Normal Distribution: 673

The fifth action item Let us go back to the partitioned pdf in order to inject the canonical representation of the deficient quadratic form y0My; M 2 R33, rkM D 2. Here we meet first the problem to transform 1 dF D .2/3=2 3 exp Œ2b 2 C 3.b /2 dy dy dy 22 1 2 3

0 3=2 by an extended vector Œz1; z2; z3 DW z into the canonical form dF D .2/ 1 2 2 2 exp 2 .z1 C z2 C z3/ dz1dz2dz3, which is generated by the general forward Helmert transformation

z D 1H.y 1/ 2 3 2 3 2 3 p1 p1 0 z1 6 12 12 7 y1 4 5 1 6 p1 p1 p2 7 4 5 z2 D 4 5 y2 23 23 23 z3 p1 p1 p1 y3 3 3 3 or its backward Helmert transformation, also called the general inverse Helmert transformation 2 3 2 3 p1 p1 p1 2 3 y1 6 12 23 3 7 z1 4 5 D 6 p1 p1 p1 7 4 5 : y2 4 12 23 3 5 z2 y3 0 p2 p1 z3 23 3

y 1 D H0z

0 thanks to the orthonormality of the quadratic Helmert matrix H3, namely H3H 3 D 0 1 0 H 3H3 D I3 or H3 D H 3. Secondly, notice the transformation of the volume element

3 dy1dy2dy3 D d.y1 /d.y2 /d.y3 / D dz1dz2dz3; which is due to the Jacobi determinant ˇ ˇ J D 3 ˇH0ˇ D 3 jHj D 3:

Let us prove that

p b 1 z3 WD 3 .b / D p 3 674 B Sampling Distributions and Their Use: Confidence Intervals and Confidence Regions brings the first marginal density 2 1 1 1 2 f1 D bj; WD p exp 3.b / 3 p 22 2 3 into the canonical form 1 1 2 f1.z3 j0; 1/ D p exp z : 2 2 3

Let us compute b as well as 3.b /2 which concludes the proof. 2 3 y1 1 1 1 1 4 5 z3 D p ; p ; p y2 3 3 3 y3 3 1 y1 C y2 C y3 3 z3 D p 7 3 7 5 y C y C y 1 2 3 D b ) y C y C y D 3b 3 1 2 3 b 1 2 1 2 ) z3 D 3 p ; z D 3.b / : 3 3 2

Indeed the extended Helmert matrix H3 is ingenious to decompose

1 .y 1b/0.y 1b/ D z2 C z2 C z2 2 1 2 3

2 2 b2 2 b 2 into a canonical quadratic form relating z1 C z2 to and z3 to . / .Atthis point, we have to interpret the general Helmert transformation z D 1H.y /: Structure elements of the Helmert transformation

scale or dilatation 1 rotation H translation

1 2 RC produces a dilatation or a scale change, H 2 SO.3/ WD fH 2 R33 jH0H D 3 I3 and jHj DC1g a rotation (3 parameters) and 1 2 R a translation. Please, prove for yourself that the quadratic Helmert matrix is orthonormal, that is HH0 D 0 H H D I3 and jHj DC1. B-4 Sampling from the Gauss–Laplace Normal Distribution: 675

The sixth action item

Finally we are left with the problem to split the cumulative pdf into one part f1.b/ which is a marginal distribution of the arithmetic mean b BLUUE of and another 2 part f2.b/ which is a marginal distribution of the standard deviation b, b BIQUUE 2 2 of , Helmert’s 2 with two degrees of freedom. First let us introduce polar coordinates .1;r/which represent the Cartesian coordinates z1 D r cos 1, z2 D r sin 1. The index 1 is needed for later generalization to higher dimension. As a longitude, the domain of 1 is 1 2 Œ0; 2 or 0 1 2. 2 2 2 The new random variable z1 C z2 Dkzk DW x or radius r relates to Helmert’s

2b 2 1 2 D z2 C z2 D D .y 1b/0.y 1b/ : 1 2 2 2

Secondly, the marginal distribution of the arithmetic mean b, b BLUUE of , f1 b j; p db D f1.z3 j0; 1/dz3 N ; p 3 3 is a Gauss-Laplace normal distribution with mean and variance 2=3 generated by dF1 D f1 b j; p db D f1.z3 j0; 1/dz3 3 ZC1 ZC1 1 1 D .2/1=2 exp z2 dz .2/1 exp .z2 C z2/ dz dz or 2 3 3 2 1 2 1 2 1 1 p 1=2 3 3 2 f1 b j; p db D .2/ exp .b / db: 3 22

b2 2 2 2 Third, the marginal distribution of the sample variance 2 = D z1 C z2 DW x, Helmert’s 2 distribution for p D n 1 D 2 degrees of freedom, 2 2 1 p 1 x f .2b = / D f .x/ D x 2 2 2 p=2 p exp 2 .2 / 2 is generated by

ZC1 Z2 1 1 1 dF D .2/1=2 exp z2 dz d .2/1 exp x dx 2 2 3 3 1 2 2 1 0 676 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

Z2 1 1 dF D .2/1! exp x dx subject to ! D d D 2 2 2 2 2 2 1 0 1 1 dF D exp x dx 2 2 2 and

dx x WD z2 C z2 D r2 ) dx D 2rdr; dr D 1 2 2r 1 dz dz D rdrd D dxd 1 2 1 2 1 is the transformation of the surface element dz1dz2. In collecting all detailed results let us formulate a corollary.

Corollary B.7. (marginal probability distributions of b, 2 given, and of b 2):

The cumulative pdf of a set of three observations is represented by 2 dF D f.y ;y ;y /dy dy dy D f bj; f .x/dbdx 1 2 3 1 2 3 1 3 2 subject to 2 1 1 1 .b /2 f1 bj; WD p p exp 3 2 = 3 2 2=3 1 1 f .x/ D exp x 2 2 2 subject to

2 x WD z2 C z2 D 2b 2=2;dxD db 2 1 2 2 and

Z1 Z1

f1.b/ db D 1 versus f2.x/ dx D 1: 1 0 f1.b/ is the pdf of the sample mean b D .y1 C y2 C y3/=3 and f2.x/ the pdf 2 0 0 2 of the sample variance b D .y 1b/ .y 1b/=2 normalized by . f1.b/ is a B-4 Sampling from the Gauss–Laplace Normal Distribution: 677

2 2 Gauss-Laplace pdf with mean b and variance =3, while f2.x/ a Helmert with two degrees of freedom. In summary, an experiment with three Gauss-Laplace i.i.d. observations is characterized by two marginal probability densities, one for the mean b BLUUE of and another one for b 2 BIQUUE of 2: Marginal probability densities n D 3, Gauss-Laplace i.i.d. observations 2 2 2 2 b W f1 b j; p db b W f2.2b = /db 3 1 1 .b /2 b 2 b 2 D .2/1=2 p exp db D exp db 2 = 3 2 2=3 2 2 or or f1.z3 j0; 1/dz3 f2.x/dx 1 1 1 D .2/1=2 exp z2 dz D exp x dx 2 3 3 2 2 b b 2 z WD x WD 2 3 3 2

B-41 Sampling Distributions of the Sample Mean b, 2 Known, and of the Sample Variance b2

The two examples have prepared us for the general sampling distribution of the sample mean b, 2 known, and of the sample variance b 2 for Gauss-Laplace i.i.d. observations, namely samples of size n. By means of Lemma B.8 on the rectangular Helmert transformation andLemmaB.9onthequadratic Helmert transformation we prepare for Theorem B.10 which summaries both the pdfs for b BLUUE of , 2 known, for the standard deviation b and for b 2 BIQUUE of 2. Corollary B.11 focusses on the pdf of D qb where is an unbiased estimation of the standard deviation , namely EfgD.

Lemma B.3. (rectangular Helmert transformation):

n.n1/ The rectangular Helmert matrix Hn1;n 2 R transforms the degenerate quadratic form

.n 1/b 2 WD .y 1b/0.y 1b/ D y0 My; rkMD n1 subject to

1 b D 10y n 678 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions into the canonical form

b2 0 2 2 .n 1/ D z n1zn1 D z1 CCzn1:

The special Helmert transformation yn 7! Hn1;nyn D zn1 is represented by

2 Hn1;n WD 3 p1 p1 00 00 6 12 12 7 6 p1 p1 p2 0 007 6 23 23 23 7 6 7 : 6 7 6 7 p 1 p 1 p 1 p 1 p n1 0 4 .n1/.n2/ .n1/.n2/ .n1/.n2/ .n1/.n2/ .n1/.n2/ 5 p 1 p 1 p 1 p 1 p 1 p n n.n1/ n.n1/ n.n1/ n.n1/ n.n1/ n.n1/

0 0 The inverse Helmert transformation z 7! y D HRz D H z or yn D H n1nzn is 0 basedonitsright inverse which thanks to the right orthogonality Hn1;nH n;n1 D In1 coincides with its transpose,

0 0 1 0 Rn.n1/ HR D H .HH / D H 2 :

Lemma B.4. (quadratic Helmert transformation):

The quadratic Helmert matrix H 2 Rnn, also called extended Helmert matrix or augmented Helmert matrix, transforms the quadratic form

1 1 .y 1/0.y 1/ D Œ.y 1b/ C 1.b /0Œ.y 1b/ C 1.b / 2 2 subject to

1 b D 10y n by means of

z D 1H.y 1/ or y D H0.z 1/ into the canonical form

1 Xn1 .y 1/0.y 1/ D z2 CCz2 C z2 D z2 C z2 2 1 n1 n j n j D1 2 b 2 zn D n. / : B-4 Sampling from the Gauss–Laplace Normal Distribution: 679

Such a Helmert transformation y 7! z D 1H.y 1/ is represented by displaylines

H WD 2 3 p1 p1 00 00 6 12 12 7 6 7 6 p1 p1 p2 0 007 6 23 23 23 7 6 7 6 p1 p1 p1 p3 007 6 34 34 34 34 7 6 7 6 7 6 7 6 7 6 p 1 p 1 p 1 p 1 p n1 0 7 6 .n1/.n2/ .n1/.n2/ .n1/.n2/ .n1/.n2/ .n1/.n2/ 7 6 7 6 p 1 p 1 p 1 p 1 p 1 p n 7 4 n.n1/ n.n1/ n.n1/ n.n1/ n.n1/ n.n1/ 5 p1 p1 p1 p1 p1 p1 n n n n n n

Since the quadratic Helmert matrix is orthonormal, the inverse Helmert transformation is generated by

z 7! y 1 D H0z:

The proofs for Lemmas B.8 and B.9 are based on generalizations of the special cases for n D 2,ExampleB.11,andforn D 3,ExampleB.12. Any proof will be omitted here. The highlight of this paragraph is the following theorem.

Theorem B.10. (marginal probability distribution of .b; 2/ and b 2):

The cumulative pdf of a set of n observations is represented by

dF D f.y1;:::;yn/dy1 dyn 2 2 D f1.b/f4.b/dbdb D f1.b/f2.b /dbdb

1 as the product of the marginal pdf f1.b/ of the sample mean b D n 1y pand the marginal pdf f2.b/ of the sample standard deviation b D .y 1b/0.y 1b/=.n 1/, also called r.m.s.(root mean square error), or 2 2 the marginal pdf f2.b / of the sample variance b . Those marginal pdfs are represented by

dF1 D f1.b/db 2 2 dF4 D f4.b/db and dF2 D f2.b /db ; 680 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

(a) sample mean b 2 1 1 .b /2 f1.b/ D f1 bj; WD p exp n p 2 2 2=n n p n 1 1 2 z WD .b / W f1.z/dz D p exp z dz 2 2

(b) sample r.m.s. b

p WD n 1

dF4 D f4.b/db 2pp p b 2 f .b/ D b p1 4 p p=2 p exp 2 2 .2 / 2 p p b p x WD n 1 D b 2 1 f .x/dx D xp1 x2 dx 4 p=2 p exp 2 .2 / 2

dF4 D f4.x/dx

p WD n 1 1 1 b 2 f .b 2/ D pp=2b p2 p ; 2 p p=2 p exp 2 2 .2 / 2 b 2 p x WD .n 1/ D b 2 W 2 2 1 p 1 1 f .x/dx D x 2 x dx: 2 p=2 p exp 2 .2 / 2

b 2 f1.j; n / as the marginal pdf of the sample mean BLUUE of is a Gauss- 2 p Laplace pdf with mean and variance =n. f2.x/; x WD pb=,isthestandard pdf of the normalized root-mean-square error with p degrees of freedom. In b2 2 2 contrast, f2.x/; x WDp = is a Helmert Chi Square p pdf with p D n 1 degrees of freedom. Before we present a sketch of a proof of Theorem B.10 which will be run with ﬁve action items and a special reference to the ﬁrst and second vehicle, we give B-4 Sampling from the Gauss–Laplace Normal Distribution: 681

some historical comments. Kullback (1934) refers the marginal pdf f1.b/ of the “arithmetic mean” b to Poisson (1827), Haussdorff (1901) and Irwins (1937). He has also solved the problem to ﬁnd the marginal pdf of the “geometric mean”. 2 2 The marginal pdf f2.b / of the sample variance b has been originally derived by Helmert (1875), Helmert (1876a), Helmert (1876b). A historical discussion of Helmert’s distribution is offered by David (1957), Kruskal (1946), Lancaster (1965), Lancaster (1966), Pearson (1931)andSheynin (1995). The marginal pdf f4.b/has not found any interest in practice so far. The reason may be found in the effect that b is not an unbiased estimate of the standard deviation , namely EfbgD. According to Cz¨uber (1891), p. 162, Roseen` (1948), p. 37, Schmetterer (1956), p. 203, Stoom (1967), p. 199, 218, Fisz (1971), p. 240 and Richter and Mammitzsch (1973), p. 42 have documented that r p p D 2 b D qb 2 p C 1 2

is an unbiased estimation of the standard deviation , namely EfbgD. p D n 1 again denotes the number of degrees of freedom. Schaffrin (1979), p. 240 has proven that p p r .y 1b/0.y 1/ 2p 1 b p WD r D b p p 1 2p 1 2 p 2 is an asymptotic (“from above”) unbiased estimation b p of the standard deviation . Let us implement BLUUE of into the marginal pdf f.b/:

Corollary B.8. (marginal probability distributions of for Gauss-Laplace i.i.d. observations, EfgD):

The marginal pdf of , an unbiased estimation of the standard deviation, is represented by

dF4 D f4./d 0 1 0 12 p p C 1 p C 1 p B B C C 2p 2 B 2 C f ./ D p1 exp B B C 2C 4 p p=2 p @ p A 2 p 2 p @ A pC1 2 2 2 682 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

Fig. B.6 Marginal pdf for the sample standard deviation (r.m.s.) and

dF4 D f4.x/dx p C 1 p 2 2 b x WD p 2 2 p1 1 2 f4.x/ D p x exp x 2p=2 2 2 subject to p C 1 p 2 EfxgD 2 p : 2

Figure B.6 illustrates the marginal pdf for “degrees of freedom” p 2f1; 2; 3; 4; 5g. B-4 Sampling from the Gauss–Laplace Normal Distribution: 683

Proof. Theﬁrstactionitem The pdf of n Gauss-Laplace i.i.d. observations is given by

Yn f.y1;:::;yn/ D f.y1/ f.yn/ D f.yi / iD1 1 f.y ;:::;y / D .2/n=2 n exp .y 1/0.y 1/ : 1 n 22

0 The coordinates of the observation space Y have been denoted by Œy1;:::;yn D y. Note dim Y D n. The second action item The quadratic form .y 1/0.y 1/ > 0 allows the fundamental decomposition

y 1 D .y 1b/ C 1.b /; .y 1/0.y 1/ D .y 1b/0.y 1b/ C 101.b /2 101 D n

.y 1/0.y 1/ D .n 1/b 2 C n.b /2:

Here, b is BLUUE of and b 2 BIQUUE of 2. The decomposition of the quadratic .y1/0.y1/ into the sample variance b 2 and the square .b/2 of the shifted sample mean b has already been proved for n D 2 and n=3. The general result is obvious. The third action item Let us transform the cumulative probability into its canonical forms.

dF D f.y1;:::;yn/dy1 dyn 1 D .2/n=2 n exp .y 1/0.y 1/ dy dy 22 1 n 1 D .2/n=2 n exp .n 1/b 2 C n.b /2 dy dy 22 1 n z D 1H.y 1/ or y 1 D H0z 1 .y 1/0.y 1/ D z2 C z2 CCz2 C z2 : 2 1 2 n1 n Here, we have substituted the divert Helmert transformation (quadratic Helmert matrix H) and its inverse. Again 1 is the scale factor, also called dilatation, 684 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

H an orthonormal matrix, also called rotation matrix,and1 2 Rn the translation, also called shift. dF D f.y1;:::;yn/dy1 dyn 1 1 2 1 1 2 2 D p exp z exp .z CCz / dz1dz2 dzn1dzn 2 2 n .2/.n1/=2 2 1 n based upon

n dy1dy2 dyn1dyn D dz1dz2 dzn1dzn J D njH0jD njHjD n:

J again denotes the absolute value of the Jacobian determinant introduced by the ﬁrst vehicle. The fourth action item First, we identify the marginal distribution of the sample mean b. 2 dF D f bj; db: 1 1 n

According to the speciﬁc structure of the quadratic Helmert matrix zn is generated by 2 3 y1 1 1 1 6 : 7 1 y1 CCyn n zn D p ;:::; p 4 : 5 D p ; n n : n yn upon substituting

1 y CCy b D 10y D 1 n ) y CCy D nb ) n n 1 n

b b 2 1 n. / 2 . / 1 p zn D p ) z D n ;dzn D ndb: n n 2

Let us implement dzn in the marginal distribution. 1 1 2 dF1 D p exp z dzn 2 2 n B-4 Sampling from the Gauss–Laplace Normal Distribution: 685

ZC1 ZC1 1 .2/.n1/=2 exp .z2 CCz2 / dz dz ; 2 1 n1 1 n1 1 1 ZC1 ZC1 1 .2/.n1/=2 exp .z2 CCz2 / dz dz D 1 2 1 n1 1 n1 1 1 1 1 2 dF1 D p exp z dzn 2 2 n ! 1 1 1 .b /2 2 dF1 D p exp db D f1 bj; db: p 2 2 n 2 n n

The ﬁfth action item Second, we identify the marginal distribution of the sample variance b 2. We depart the ansatz

2 2 dF2 D f2.b/db D f2.b /db in order to determine the marginal distribution f2.b/ of the sample root-mean- 2 2 square errors b and the marginal distribution f2.b / of the sample variance b . A ﬁrst version of the marginal probability distribution dF2 is

Z1 1 1 2 dF2 D p exp z dzn 2 2 n 1 1 1 exp .z2 CCz2 / dz dz : .2/.n1/=2 2 1 n1 1 n1

n1 Transform the Cartesian coordinates .z1 C C zn1/ 2 R to spherical coordinates .˚1;˚2;:::;˚n2;rn1/. From the operational point of view, p D n1, the number of “degrees of freedom”, is an optional choice. Let us substitute the global hypersurface element !n1 or !p into dF2, namely

Z1 1 1 2 p exp z dzn D 1 2 2 n 1 1 1 dF D rn2 exp r2 dr 2 2.n1/=2 2 686 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

CZ=2 CZ=2 CZ=2 Z2 n3 n4 cos n2dn2 cos n3dn3 cos2d2 d1 =2 =2 =2 0 1 1 dF D rp1 exp r2 dr 2 .2/p=2 2

CZ=2 CZ=2 CZ=2 CZ=2 Z2 p2 p3 2 cos p1 cos p2dp2 cos 3d3 cos2d2 d1 =2 =2 =2 =2 0

2 2 ! D ! D .n1/=2 D p=2 n1 p n 1 p 2 2 Z=2 Z=2 p2 p3 D cos p1dp1 cos p2dp2 =2 =2

Z=2 Z=2 Z2 2 cos 3d3 cos2d2 d'1 =2 =2 0 ! 1 dF D p rp1 exp r2 dr 2 .2/p=2 2 2 1 dF D rp1 exp r2 dr : 2 p=2 p 2 2 2

The marginal distributionq of the r.m.s. f2.b/ is generated as soon as we substitute p 2 2 b the radius r D z1 CCzn1 D n 1 1=. Alternatively the marginal 2 distribution of the sample variance f2.b / is produced when we substitute the radius 2 2 2 b2 2 square r D z1 CCzn1 D .n 1/ = . Project A p p p b p r D n 1 b= D p ) dr D db

dF2 D f2.b/db 2pp 1 p f .b/ D b p1 exp b 2 : 2 p2p=2 2 2 B-4 Sampling from the Gauss–Laplace Normal Distribution: 687

Indeed, f2.b/ establishes the marginal distribution of the root-mean-square error b with p D n 1 degrees of freedom. Project B 2 dx D 2rdr b 2 b 2 6 x WD r2 D .n 1/ D p DW 2 ) 4 dx 1 dx 2 2 p dr D D p 2r 2 x

p1 1 p 1 r dr D x 2 dx 2

dF2 D f2.x/dx 1 p 1 1 f .x/ WD x 2 x : 2 p=2 p exp 2 .2 / 2

2 Finally, we have derived Helmert’s Chi Square p distribution f2.x/dx D dF2 by substituting r2, rp1 and dr in factor of x WD r2 and dx D 2rdr. Project C Replace the radical coordinate squared r2 D .n 1/b 2=2 D pb 2=2 by rescaling on the basis p=2

b 2 p x D r2 D z2 CCz2 D z2 CCz2 D .n 1/ D b 2 1 n1 1 p 2 2 p dx D db 2

2 within Helmert’s p with p D n 1 degrees of freedom

2 2 dF2 D f2.b /db 1 1 p f .b 2/ D pp=2b p2 b 2 : 2 p p=2 p exp 2 2 .2 / 2

2 2 Recall that f2.b / establishes the marginal distribution of the sample variance b with p D n 1 degrees of freedom. Both,themarginalpdf f2.b/ of the sample standard deviation b, also called root-mean-square error, and the marginal pdf 2 2 2 f2.b / of the sample variance b document the dependence on the variance and its power p . “Here is our journey’s end.”(W. Shakespeare: Hamlet) | 688 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

B-42 The Conﬁdence Interval for the Sample Mean, Variance Known

An application of Theorem B.10 is Lemma B.12 where we construct the conﬁdence interval for the sample mean b, BLUUE of ,variance 2 known, on the basis of its sampling distribution. Example B.14 is an example of a random sample of size n D 4.

Lemma B.12. (conﬁdence interval for the sample mean, variance known):

The sampling distribution of the sample mean b D n1 10y, BLUUE of ,isGauss- 2 Laplace normal, N .bj; =n/, if the observations yi ;i 2f1;:::;ng,are Gauss-Laplace i.i.d. The “true” mean is an element of the two-sided conﬁdence interval. 2b p c1˛=2; b C p c1˛=2Œ n n with conﬁdence P b p c1˛=2 <

Example B.14. (conﬁdence interval for the sample mean b, 2 known):

Suppose that a random sample 2 3 2 3 y1 1:2 6 7 6 7 6 y2 7 6 3:4 7 2 y WD 4 5 D 4 5 ; b D 2:7; D 9; r:m:s: W 3 y3 0:6 y4 5:6 of four observation is known from a Gauss-Lapace normal distribution with unknown mean and a known standard deviation D 3. b BLUUE of is the arithmetic mean. We intend to determine upper and lower limits which are rather certain to contain the unknowns parameter between them. Previously, for samples of size 4 we have known that the random variable

b b z D D p 3 n 2 B-4 Sampling from the Gauss–Laplace Normal Distribution: 689

b is normally distributedp with mean zero and unit variance. is the sample mean 2.7 and 3/2 is = n. The probability D 1 ˛ that z will be between any two arbitrarily chosen numbers c1 Dc and c2 D c is

Zc2

P fc1 < z < Cc2gD f.z/dz D D 1 ˛;

c1 ZCc P fc<2

c Zc ˛ f.z/dz D 1 ; 2 1 Zc ˛ f.z/dz D : 2 1

is the coefficient of confidence, ˛ the coefficient of negative confidence, also called complementary coefficient of confidence. The four representations of the probability D 1 ˛ to include z in the confidence interval c

Zc ˛ 1 C f.z/dz D 1 D : 2 2 1

Three values of the coefficient of confidence or its compliment ˛ are popular and listed in Table B.9. Table B.9 (Values of the coefficient of confidence):

0:950 0:990 0:999 ˛ 0:050 0:010 0:001 ˛ 2 0:025 0:005 0:000; 5 ˛ 1C 1 2 D 2 0:975 0:995 0:999; 5

In solving the linear Voltera integral equation of the ﬁrst kind

Zz 1 1 f.z/dz D 1 ˛.z/ D Œ1 C .z/: 2 2 1

Table B.10 collects the quantiles c1˛=2 given the coefficients of confidence on their complements which we listed in Table B.10. 690 B Sampling Distributions and Their Use: Confidence Intervals and Confidence Regions

Table B.10 (Quantiles for the conﬁdence interval of the sample mean, variance unknown): ˛ 1C 1 2 D 2 ˛c1˛=2 0:975 0:95 0:05 1:960 0:995 0:99 0:01 2:576 0:999; 5 0:999 0:001 3:291

Given the quantiles c1˛=2, we are going to construct the conﬁdence interval for b 2 the sample mean , the variance pto be known. For this purpose, we convert the forward transformations b ! z D n.b /= to .

b p z D n b1 WD b p c1˛=2 <

The interval b1 <

p bC c1˛=2 Zb2 Zn cZ1˛=2 ˛ f.b/db D f.b/db D f.b/db D 1 : 2 1 1 1 An animation of the coefficient of confidence and the probability functions of a confidence interval is offered by Figs. B.7 and B.8. b1 D b c1˛=2 p ; b2 D b C c1˛=2 p n n

Let us specify all the integrals to our example

cZ1˛=2 f.z/dz D 1 ˛=2 1 B-4 Sampling from the Gauss–Laplace Normal Distribution: 691

Fig. B.7 Two-sided conﬁdence interval 2 2b1;b2Œ; f .bj; =n/ σ2 pdf : f(mˆ | m, σ2) f(mˆ | m, ) a /2 g = 1-a a /2

μ mˆ mˆ1 2 mˆ1

σ σ = ˆ − m ˆ = ˆ + mˆ1 m c1−a/2 m1 m c1−a/2 n n

Fig. B.8 Two-sided conﬁdence interval, quantile c1˛=2

1:960Z 2:576Z 3:291Z f.z/dz D 0:975; f.z/dz D 0:995; f.z/dz D 0:999; 5: 1 1 1

Those data lead to a triplet of conﬁdence intervals.

case .a/ D 0:95; D 0:05; c1˛=2 D 1:960 3 3 P 2:7 1:96 < < 2:7 C 1:96 D 0:95 2 2 P f0:24 < < C5:64gD0:95

case .b/ D 0:99; D 0:01; c1˛=2 D 2:576 3 3 P 2:7 2:576<<2:7C 2:576 D 0:99 2 2 P f1:164 < < 6:564gD0:99

case .c/ D 0:999; D 0:001; c1˛=2 D 3:291 3 3 P 2:7 3:291 < < 2:7 C 3:291 D 0:999 2 2 P f2:236 < < 7:636gD0:999 :

With probability 95% the “true” mean is an element of the interval ]0.24, C5.64[. In contrast, with probability 99% the “true” mean is an element of the larger interval ]1.164, C6.564[. Finally, with probability 99.9% the “true” mean is an element of the largest interval [2.236, C7.636]. 692 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

B-5 Sampling from the GaussÐLaplace Normal Distribution: A Third Conﬁdence Interval for the Mean, Variance Unknown

In order to derive the sampling distributions for the sample mean, variance unknown, of Gauss-Laplace i.i.d. observation B51 introduces two examples (two and three observations, respectively, for generating Student’s tp distribution. Lemma B.13 reviews Student’s t-distribution of the random variable n0.b /=b where the sample mean b is BLUUE of , whereas the sample variance b 2 is BIQUUE of 2. B52 by means of Lemma B.13 introduces the confidence interval for the “true” mean variance 2 unknown, which is based on Student’s probability distribution. For easy computation, Table B.12 is its flow chart. B53 discusses The Uncertainly Principle generated by The Magic Triangle of (a) length of confidence interval, (b) coefficient of negative confidence, also called the uncertainty number, and (c) the number of observations. Various figures and examples pave the way for the routine analyst’s use of the confidence interval for the mean, variance unknown.

B-51 Student’s Sampling Distribution of the Random Variable .b /=b

Two examples for n D 2 or n D 3 Gauss-Laplacepi.i.d. observations keep us to derive Student’s t-distribution for the random variable n.b /=b where b is BLUUE of , whereas p D n1 is BIQUUE of 2. Lemma B.12 and its proof is the highlight of this paragraph in generating the sampling probability distribution of Student’s t. Example B.15 (Student’s t-distribution for two Gauss-Laplace i.i.d. observations):

First, assume an experiment of two Gauss-Laplace i.i.d. observations called y1 and y2: We want to prove that .y1 C y2/=2 and .y1 y2/2=2 or the sample mean b and 2 the sample variance b are stochastically independent. y1 and y2 are elements of the joint pdf.

f.y1;y2/ D f.y1/f .y2/ 1 1 f.y ;y / D exp Œ.y /2 C .y /2 1 2 22 22 1 2 1 1 f.y ;y / D exp .y 1/0.y 1/ : 1 2 22 22

The quadratic form .y 1/0.y 1/ is decomposed into the sample variance b 2, BIQUUE of 2, and the deviate of the sample mean b, BLUUE of , from by means of the fundamental separation B-5 Sampling from the Gauss–Laplace Normal Distribution: A Third Conﬁdence 693

y 1 D y 1b C 1.b / .y 1/0.y 1/ D .y 1b/0.y 1b/ C 101.b /2 .y 1/0.y 1/ D b 2 C 2.b /2 1 1 2 f.y ;y / D exp b 2 exp .b /2 : 1 2 22 22 22

The joint pdf f.y1;y2/ is transformed into a special form if we replace

b D .y1 C y2/=2; 1 b 2 D .y b/2 C .y b/2 D .y y /2; 1 2 2 1 2 namely 1 1 1 1 y C y f.y ;y / D exp .y y /2 exp . 1 2 /2 : 1 2 22 22 2 1 2 2 2

Obviously the product decomposition of the joint pdf documented that (y1+y2)/2 b b2 and (y1-y2)2/2 or and are independent random variables. p Second, we intend to derive the pdf of Student’s random variable t WD 2.b /=b, the deviate of the sample mean b form the “true” mean , normalized by the sample standard deviations b. Let us introduce the direct Helmert transformation p 1 y1 y2 b 2 y1 C y2 2 b z1 D p D ; z2 D D p ; 2 2 2 or " # p1 p1 z1 1 2 2 y1 D 1 1 ; z2 p p y2 2 2 as well as the inverse " # p1 p1 y1 2 2 z1 D 1 1 ; y2 p p z2 2 2 which brings the jointpdf dF D f.y1;y2/dy1dy2 D f.z1; z2/dz1dz2 into the canonical form. 694 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

1 1 2 1 1 2 dF D p exp z p exp z dz1dz2 2 2 1 2 2 2 ˇ ˇ ˇ p1 p1 ˇ 1 ˇ 2 2 ˇ dy1dy2 D ˇ ˇ dz1dz2 D dz1dz2: 2 ˇ p1 p1 ˇ 2 2

2 p The Helmert random variable x WD z1 or z1 D x replaces the random variable z1 such that 1 1 dz1dz2 D p dxdz2 2 x 1 1 1 1 1 2 dF D p p exp x p exp z dxdz2 2 2 x 2 2 2 2 is the joint pdf of x and z2. Finally, we introduce Student’s random variable p b t WD 2 b p 2 z2 D .b / ) b D p z2 2 p b p z D x D ) b D z D x 1 1

z2 p 2 2 t D p , z2 D xt ) z D xt : x 2

Let us transform dF D f.x;z2/dxdz2 to dF D f.t;x/dtdx, namely from the joint pdf of the Helmert random variable x and the Gauss-Laplace normal variate z2 to the joint pdf of the Student random variable t and the Helmert random variable x. ˇ ˇ ˇ ˇ ˇ D z D z ˇ ˇ t 2 x 2 ˇ dz2dx D ˇ ˇ dt dx Dt xDxx ˇ ˇ ˇ p p ˇ ˇ 2 3=2 ˇ x 2 x t dz2dx D ˇ ˇ dt dx ˇ 01ˇ p dz2dx D xdtdx dF D f.t;x/dtdx 1 1 1 f.t;x/ D exp .1 C t 2/x : 2 2 2 B-5 Sampling from the Gauss–Laplace Normal Distribution: A Third Conﬁdence 695

The marginal distribution of Student’s random variable t, namely dF3 D f3.t/dt, is generated by

Z1 1 1 1 f .t/ WD exp .1 C t 2/x dx 3 2 2 2 0 subject to the standard integral

Z1 1 1 1 exp.ˇx/dx D exp .ˇx/ D ˇ 0 ˇ 0 1 1 2 ˇ WD .1 C t 2/; D 2 ˇ 1 C t 2 such that

1 1 f .t/ D ; 3 2 1 C t 2 1 1 dF D dt 3 2 1 C t 2

2 and characterized by a pdf f3.t/ which is reciprocal to .1 C t /.

Example B.16. (Student’s t-distribution for three Gauss-Laplace i.i.d. observations):

First, assume an experiment of three Gauss-Laplace i.i.d. observations called y1;y2 and y3: We want to derive the joint pdf f.y1;y2;y3/ in terms of the sample mean b, BLUUE of , and the sample variance b 2, BIQUUE of 2.

f.y1;y2;y3/ D f.y1/f .y2/f .y3/ 1 1 f.y ;y ;y / D exp Œ.y /2 C .y /2 C .y /2 1 2 3 .2/3=2 3 22 1 2 3 1 1 f.y ;y ;y / D exp .y 1/0.y 1/ : 1 2 3 .2/3=2 3 22

The quadratic form .y 1/0.y 1/ is decomposed into the sample variance b 2 and the deviate sample mean b from the “true” mean by means of the fundamental separation 696 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

y 1 D y 1b C 1.b / .y 1/0.y 1/ D .y 1b/0.y 1b/ C 101.b /2 .y 1/0.y 1/ D 2b 2 C 3.b /2

dF D f.y1;y2;y3/dy1dy2dy3 1 1 3 D exp 2b 2 exp .b /2 dy dy dy : .2/3=2 3 22 22 1 2 3 p Second, we intend to derive the pdf of Student’s random variable t WD 3.b/=b, the deviate of the sample mean b from the “true” mean , normalized by the sample standard deviation b. Let us introduce the direct Helmert transformation

1 1 z1 D p .y1 C y2 / D p .y1 C y2 2/ 2 2 1 1 z2 D p .y1 C y2 2y3 C 2/ D p .y1 C y2 2y3/ 2 3 2 3 1 1 z3 D p .y1 C y2 C y3 / D p .y1 C y2 C y3 3/ 3 3 or 2 3 2 3 p1 p1 0 2 3 z 12 12 y 1 1 6 7 1 4 5 6 p1 p1 p2 7 4 5 z2 D 4 5 y2 23 23 23 z3 p1 p1 p1 y3 3 3 3 as well as its inverse 2 3 2 3 p1 p1 p1 2 3 y1 6 12 23 3 7 z1 4 5 D 6 p1 p1 p1 7 4 5 y2 4 12 23 3 5 z2 y3 0 p2 p1 z3 23 3 in general

z D 1H.y / versus .y 1/ D H0z; whichhelpustobringthejoint pdf dF.y1;y2;y3/dy1dy2dy3 Df.z1; z2; z3/dz1dz2 dz3 into the canonical form. B-5 Sampling from the Gauss–Laplace Normal Distribution: A Third Conﬁdence 697

1 1 1 dF D exp .z2 C z2/ exp z2 dz dz dz .2/3=2 2 1 2 2 3 1 2 3 ˇ ˇ ˇ p1 p1 p1 ˇ ˇ 12 23 3 ˇ 1 ˇ ˇ ˇ p1 p1 p1 ˇ dy1dy2dy3 D ˇ ˇ dz1dz2dz3 D dz1dz2dz3: 3 ˇ 12 23 3 ˇ ˇ 0 p2 p1 ˇ 23 3 q p The Helmert random variable x WD z2 Cz2 or x D z2 C z2 replaces the random q 1 2 1 2 2 2 variable z1 C z2 as soon as we introduce polar coordinates z1 D r cos 1, z2 D r 2 2 2 sin 1, z1 C z2 D r DW x and compute the marginal pdf

Z2 1 1 2 1 1 1 dF.z3;x/D p exp z dz3 exp x dx d1; 2 2 3 2 2 2 0 by means of

dx x WD z2 C z2 D r2 ) dx D 2rdr; dr D ; 1 2 2r 1 dz dz D rdrd D dxd ; 1 2 1 2 1 1 1 1 1 2 dF.z3;x/D exp x p exp z dxdz3; 2 2 2 2 3 the joint pdf of x and z. Finally, we inject Student’s random variable

p b t WD 3 ; b decomposed into p 3 z3 D .b / ) b D p z3 3 q q p p b p z2 C z2 D x D 2 ) b D p z2 C z2 D p x 1 2 2 1 2 2

p z3 1 p 2 1 2 t D p 2 , z3 D p xt ) z D xt : x 2 3 2 698 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

Let us transform dF D f.x;z3/dxdz3 todF D f.t;x/dtdx. Alternatively we may say that we transform the joint pdf of the Helmert random variable x and the Gauss- Laplace normal variate z3 to the joint pdf of the Student random variable t and the Helmert random variable x. ˇ ˇ ˇ ˇ ˇ D z D z ˇ ˇ t 3 x 3 ˇ dz3dx D ˇ ˇ dtdx Dt xDxx ˇ p ˇ ˇ x ˇ ˇ p p1 x3=2t ˇ dz3dx D ˇ 2 2 2 ˇ dtdx ˇ 01ˇ p x dz3dx D p dtdx 2 dF D f.t;x/dtdx 1 1 p 1 t 2 f.t;x/ D p p x exp .1 C /x : 2 2 2 2 2

The marginal distribution of Student’s random variable t, namely dF3 D f3.t/dt, is generated by

Z1 1 1 p 1 t 2 f3.t/ WD p p x exp 1 C x dx 2 2 2 2 2 0 subject to the standard integral

Z1 .˛C 1/ 1 1 t 2 x˛ exp.ˇx/dx D ;˛D ;ˇD 1 C ˇ˛C1 2 2 2 0 such that Z1 p 1 t 2 .3 / x exp 1 C x dx D 23=2 2 2 2 t 2 3=2 .1 C 2 / 0 3 1p D 2 2 3 p1 1 f3.t/ D 2 ; 2 2 .1C t /3=2 p 2 2 dF3 D dt: t 2 3=2 4.1 C 2 /

Again Student’s t-distribution is reciprocal t.1C t2=2/3=2. B-5 Sampling from the Gauss–Laplace Normal Distribution: A Third Conﬁdence 699

Lemma B.13. Student’s t-distribution for the derivate of the mean .b /=b (Gosset, 1908)):

0 Let the random vector of observations y D Œy1;:::;yn be Gauss-Laplace i.i.d. Student’s random variable

b p t WD n; b where the sample mean b is BLUUE of and the sample variance b 2 is BIQUUE of 2; associated to the pdf

pC1 . 2 / 1 1 f.t/ D p p : . / p t 2 .pC1/=2 2 .1 C p / p D n 1 is the “degree of freedom” of Student’sp distribution fp.t/. Gosset (1908) published the t-distribution of the ratio n.b /=b under the pseudonym “Student”: The probable error of a mean. Proof. p The joint probability distribution of the random variable zn WD n.b /= and 2 2 b2 2 the Helmert random variable x WD z1 CCzn1 D .n 1/ = represented by

dF D f1.zn/f2.x/dzndx

2 2 b b2 due to the effect that zn and z1 CCzn1 or and .n 1/ are stochastically independent. Let us take reference to the speciﬁc pdfs with n and n 1 D p degrees of freedom

dF1 D f1.zn/dzn and dF2 D f2.x/dx; p=2 1 1 2 1 1 .p2/=2 1 f1.zn/ D p exp zn and f2.x/ D p x exp x ; 2 2 .2 / 2 2 or

2 2 dF1 D f1.b/db and dF2 D f2.b /db ;

2 1 p=2 p2 p f2.b / D p b n n .b /2 p2p=2.p=2/ f1.b/ D p exp and 2 2 2 1 p exp b 2 2 2 we derived earlier. Here, let us introduce Student’s random variable 700 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

p p zn p b p x x t WD p n 1 D n or zn D p t D p t: x b n 1 p

By means of the Jacobi matrix J and its absolute Jacobi determinant jJj dt D tDt dz dz D zn x n D J n dx Dzn xDxx dx dx

2 p 3 p 1 3=2 p p p p x zn p 4 x 2 5 p 1 x J D ; jJjD p ; jJj D p 01 x p

1 we transform the surface element dzndx to the surface element jJj dtdx, namely p x dz dx D p dtdx: n p

Lemma B.14. (Gamma Function):

Our ﬁnal action item is to calculate the marginal probability distributions dF3 D f3.t/dt of Student’s random variable t, namely

dF3 D f3.t/dt Z1 p=2 2 1 1 1 .p1/=2 1 t f3.t/ WD p x exp 1 C x dx: 2p .p=2/ 2 2 p 0

Consult (Grobner¨ and Hofreiter, 1973, p. 55), for the standard integral

Z1 ˛Š .˛C 1/ x˛ exp.ˇx/dx D D and ˇ˛C1 ˇ˛C1 0 Z1 1 t 2 .pC1 / x.p1/=2 exp 1 C x dx D 2.pC1/=2 2 ; 2 p .1 C t 2 /.pC1/=2 0 p where p D n 1 is the rank of the quadratic Helmert matrix Hn. Notice a result pC1 p1 of the gamma function . 2 / D . 2 /Š. In summary, substituting the standard integral B-5 Sampling from the Gauss–Laplace Normal Distribution: A Third Conﬁdence 701

dF3 D f3.t/dt pC1 . 2 / 1 1 f3.t/ D p p . / p t 2 .pC1/=2 2 .1 C p / resulted in Student’s t distribution, namely the pdf of Student’s random variable .b /=b.

B-52 The Conﬁdence Interval for the Mean, Variance Unknown

Lemma B.12 is the basis for the construction of the confidence interval of the “true” mean, variance unknown, which we summarize in Lemma B.13,ExampleB.17 contains all details for computing such a confidence interval, namely Table B.11, a collection of the most popular values of the coefficient of confidence, as well as Tables B.12, B.13 and B.14, listing the quantiles for the confidence interval of the Student random variable with p D n 1 degrees of freedom. Figures B.9 and B.10 illustrate the probability of two-sided confidence interval for the mean, variance unknown, and the limits of the confidence interval. Table B.15 as a flow chart paves the way for the “fast computation” of the confidence interval for the “true” mean, variance unknown.

Lemma B.15. (conﬁdence interval for the sample mean, variance unknown): p The random variable t D n.b /=b characterized by the ratio of the deviate of the sample mean b D n110y, BLUUE of , from the “true” mean and the standard deviation b, b 2 D .y 1b/0.y 1b/=.n 1/ BIQUUE of the “true” variance 2,hastheStudent t-distribution with p D n 1 “degrees of freedom”. The “true” mean is an element of the two-sided conﬁdence interval

b b 2b p c1˛=2; b C p c1˛=2Œ n n with conﬁdence b b P b p c1˛=2 <

Example B.17. (conﬁdence interval for the example mean b, 2 unknown):

Suppose that a random sample 2 3 2 3 y1 1:2 6 7 6 7 6 y2 7 6 3:4 7 2 4 5 D 4 5 ; b D 2:7; b D 5:2; b D 2:3 y3 0:6 y4 5:6 b of four observations is characterized byp the sample mean D 2:7 and the sample variance b 2 D 5:2. 2.2:7 /=2:3 D n.b /=b D t has Student’s pdf with p D n1 D 3 degrees of freedom. The probability D 1˛ that t will be between any two arbitrarily chosen numbers c1 Dc and c2 DCc is

Zc2

Pfc1

c1 ZCc Pfc

Zc ˛ 1 f.t/dt D 1 D .1 C /: 2 2 1 Three values of the coefficient of confidence or its complement ˛ are popular and listed in Table B.10. Table B.11 (Values of the coefficient of confidence):

0:950 0:990 0:999 ˛ 0:050 0:010 0:001 ˛ 2 0:025 0005 0:000; 5 ˛ 1C 1 2 D 2 0:975 0:995 0:999; 5 B-5 Sampling from the Gauss–Laplace Normal Distribution: A Third Conﬁdence 703

In solving the linear Volterra integral equation of the ﬁrst kind

Zt 1 1 f.t/dt D 1 ˛.t/ D Œ1 C .t/; 2 2 1 which depends on thep degrees of freedom p D n 1, Tables B.12–B.14 collect the quantiles c1˛=2= n given the coefficients of confidence or their complements which we listed in Table B.11. p Table B.12 (Quantiles c1˛=2= n for the confidence interval of the Student random variable with p D n 1 degrees of freedom):

1 ˛=2 D .1 C /=2 D 0:975; D 0:95; ˛ D 0:05

pnp1 c pnp1 c n 1˛=2 n 1˛=2 1 2 8:99 14 15 0:554 2 3 2:48 19 20 0:468 3 4 1:59 24 25 0:413 4 5 1:24 29 30 0:373 5 6 1:05 39 40 0:320 6 7 0:925 49 50 0:284 7 8 0:836 99 100 0:198 8 9 0:769 199 200 0:139 9 10 0:715 499 500 0:088

p Table B.13 (Quantiles c1˛=2= n for the conﬁdence interval of the Student random variable with p D n 1 degrees of freedom):

1 ˛=2 D .1 C /=2 D 0:995; D 0:990; ˛ D 0:010

pnp1 c pnp1 c n 1˛=2 n 1˛=2 1 2 45:01 14 15 0:769 2 3 5:73 19 20 0:640 3 4 2:92 24 25 0:559 4 5 2:06 29 30 0:503 5 6 1:65 39 40 0:428 6 7 1:40 49 50 0:379 7 8 1:24 99 100 0:263 8 9 1:12 199 200 0:184 9 10 1:03 499 500 0:116 704 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

p Table B.14 (Quantiles c1˛=2= n for the conﬁdence interval of the Student random variable with p D n 1 degrees of freedom):

1 ˛=2 D .1 C /=2 D 0:999:5; D 0:999; ˛ D 0:001 p p pn c1˛=2 c1˛=2= np nc1˛=2 c1˛=2= n 1 2 636:619 450:158 14 15 4:140 1:069 2 3 31:598 18:243 19 20 3:883 0:868 3 4 12:941 6:470 24 25 3:725 0:745 4 5 8:610 3:851 29 30 3:659 0:668 5 6 6:859 2:800 30 41 3:646 0:655 6 7 5:959 2:252 40 41 3:551 0:555 7 8 5:405 1:911 60 61 3:460 0:443 8 9 5:041 1:680 120 121 3:373 0:307 9 10 4:781 1:512 113:291 0 p Since Student’s pdf depends on p D n 1, we have tabulated c1˛=2= n for thep conﬁdence interval we are going to construct. The Student’s random variable t D n.b /=b is solved for , evidently our motivation to introduce the conﬁdence interval b1 <

p b b b t D n ) b D p t ) D b p t b n n b b b1 WD b p c1˛=2 <

The interval b1 <

Zc cZ1˛=2 ˛ f.t/dt D f.t/dt D 1 : 2 1 1 B-5 Sampling from the Gauss–Laplace Normal Distribution: A Third Conﬁdence 705

Fig. B.9 Two-sided conﬁdence interval 2b1;b2Œ, Student’s pdf for p D 3 degrees of freedom f(t)

.n D 4/b1 D g = 1–a p a /2 a /2 b bc = n, 1 ˛=2 p b2 D b Cbc1˛=2= n

− + c1−a /2 t c1−a /2

ˆ ˆ ˆ ˆ m – sc1−a /2 / n m + sc1−a /2 / n m mˆ1 mˆ2

Fig. B.10 Two-sided conﬁdence interval for the “true” mean , quantile c1˛=2

Figures B.9 and B.10 illustrate the coefficient of confidence and the probability function of a confidence interval. Let us specify all the integrals to our example.

cZ1˛=2 f.t/dt D 1 ˛=2 1

cZ0:975 cZ0:995 cZ0:999;5 f.t/dt D 0:975; f.t/dt D 0:995; f.t/dt D0:999; 5: 1 1 1

These data substituted into Tables B.12–B.14 lead to the triplet of conﬁdence intervals for the “true” mean.

case .a/ W D 0:95; ˛ D 0:05; 1 ˛=2 D 0:975 p p D 3; n D 4; c1˛=2= n D 1:59

Pf2:7 2:3 1:59 < < 2:7 C 2:3 1:59gD0:95

Pf0:957 < < 6:357gD0:95:

case .b/ W D 0:99; ˛ D 0:01; 1 ˛=2 D 0:995 p p D 3; n D 4; c1˛=2= n D 2:92 706 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

Pf2:7 2:3 2:92<<2:7C 2:3 2:92gD0:99

Pf4:016 < < 9:416gD0:99:

case .c/ W D 0:999; ˛ D 0:001; 1 ˛=2 D 0:999; 5

p p D 3; n D 4;c1˛=2= n D 6:470 Pf2:7 2:3 6:470 < < 2:7 C 2:3 6:470gD0:999

Pf12:181 < < 17:581gD0:999:

The results may be summarized as follows: With probability 95% the “true” mean is an element of the interval 0:957, C6:357Œ. In contrast, with probability 99% the “true” mean is an element of the larger interval 4:016, C9:416Œ. Finally, with probability 99.9is an element of the largest interval 12:181, C17:581Œ. If we compare the conﬁdence intervals for the mean , 2 known,versus 2 unknown,we realize much larger intervals for the second model. Such a result is not too much a surprise, since the model “ 2 unknown” is much weaker than the model “ 2 known”.

Table B.15 (Flow chart: Conﬁdence interval for the mean , 2 unknown):

Choose a coefficient of confidence g according toTable B11.

Solve the linear Volterra integral equation of the first kind = − = + F(c1−a /2) 1 a /2 (1 g ) / 2 by Table B12, Table B13 or Table B14 for a Student pdf with p = n−1 degrees of freedom: read c1−α /2/ n.

Compute the sample mean mˆ and the sample standard deviation sˆ.

ˆˆˆˆˆ− + Compute sc1−a /2/ n and m sc1−a /2/ n, m sc1−a /2/ n. B-5 Sampling from the Gauss–Laplace Normal Distribution: A Third Conﬁdence 707

B-53 The Uncertainty Principle

Figure B.11 is the graph of the function p .nI ˛/ WD 2bc1˛=2= n; where is the length of the confidence interval of the “true” mean ; 2 unknown. The independent variable of the function is the number of observations n. The function .nI ˛/ is plotted for fixed values of the coefficient of complementary confidence ˛, namely ˛ D 10%;5%;1%: For reasons given later the coefficient of complementary confidence ˛ is called uncertainty number. The graph function .nI ˛/ illustrates two important facts. Fact #1: For a constant uncertainty number ˛, the length of the confidence interval is smaller the larger the number of observations n is chosen. Fact #2: For a constant number of observations n, the smaller the number of uncertainly ˛ is chosen, the larger is the confidence interval . Evidently, the diversive influences of (a) the length of the confidence interval ; (b) the uncertainly number ˛ and (c) the number of observations n, which we collected in The Magic Triangle of Figure B.12, constitute The Uncertainly Principle,informula

.n 1/ W k˛;

Fig. B.11 Length of the confidence interval for the mean against the number of observations 708 B Sampling Distributions and Their Use: Confidence Intervals and Confidence Regions

length of the confidence interval

uncertainty number a number of observations n

Fig. B.12 The Magic Triangle, constituents: (a) uncertainty number ˛ (b), number of observations n, (c) length of the conﬁdence interval

where k˛ is called the quantum number for the mean which depends on the uncertainty number ˛. Table B.15.2 is a list of those quantum numbers. Let us interpret the uncertainty relation .n 1/ k˛. The product .n 1/ defines geometrically a hyperbola which we approximated out of the graph of Fig. B.12. Given the uncertainty number ˛, the product .n 1/ has a smallest number, here denoted by k˛. For instance, choose ˛ D 1% such that k˛=.n 1/ or 16:4=.n1/ . For n taken values 2, 11, 101, we get the inequalities 8:2 , 1:64 , 0:164 . Table B.15.2 (Coefficient of complementary confidence ˛, uncertainty number ˛, versus quantum number of the mean k˛ (Grafarend, 1970)):

˛k˛ 10% 6:6 5% 9:6 1% 16:4 ! 0 !1

B-6 Sampling from the GaussÐLaplace Normal Distribution: A Fourth Conﬁdence Interval for the Variance

Theorem B.10 already supplied us with the sampling distribution of the sample variance, namely with the probability density function f.x/ of Helmert’s random variable x D .n 1/b 2=2 of Gauss-Laplace i.i.d. observations n of sample variance b 2, BIQUUE of 2. B61 introduces accordingly the so far missing confidence interval for the “true” variance 2. Lemma B.15 contains the details, followed by Example B.16. Tables B.17–B.19 contain the properly chosen coefficients of complementary confidence and their quantiles

c1.pI ˛=2/; c2.pI 1 ˛=2/; B-6 Sampling from the Gauss–Laplace Normal Distribution: A Fourth 709 dependent on the “degrees of freedom” p D n 1. Table B.20 as a flow chart summarizes various steps in computing a confidence interval for the variance 2. B62 reviews The Uncertainty Principle which is built on (a) the coefficient of complementary confidence ˛, also called uncertainty number, (b) the number of observations n and (c) the length 2.nI ˛/ of the confidence interval for the “true” variance 2.

B-61 The Conﬁdence Interval for the Variance

Lemma B.15 summaries the construction of a two-sided confidence interval for the “true” variance based upon Helmert’s Chi Square distribution of the random variable .n1/b 2=2 where b 2 is BIQUUE of 2.ExampleB.18 introduces a random sample of size n D 100 with an empirical variance b 2 D 20:6. Based upon coefficients of confidence and complementary confidence given in Table B.14, related confidence intervals are computed. The associated quantiles for Helmert’s Chi Square distribution are tabulated in Table B.15 . D 0:95/,TableB.18 . D 0:99/ and Table B.19 . D 0:998/: Finally, Table B.20 as a flow chart pares the way for the “fast computation” of the confidence interval for the “true” variance 2.

Lemma B.16. (conﬁdence interval for the variance):

The random variable x D .n 1/b 2=2, also called Helmert’s 2, characterized by the ration of the sample variance b 2 BIQUUE of 2,andthe“true” variance 2 2 ,hasthe' pdf of p D n 1 degrees of freedom, if the random observations 2 yi ;i 2f1;:::ng are Gauss-Laplace i.i.d.The“true” variance is an element of the two-sided conﬁdence interval

.n 1/b 2 .n 1/b 2 2 2 ; Œ c2.pI 1 ˛=2/ c1.pI ˛=2/ with confidence .n 1/b 2 .n 1/b 2 P <2 < D 1 ˛ D c2.pI 1 ˛=2/ c1.pI ˛=2/ of level 1 ˛. Tables D.8, D.18 and D.16 list the quantiles c1.pI 1 ˛=2/ and c2.pI ˛=2/ associated to three values of Table B.19 of the coefficient of complementary confidence 1 ˛. In order to make yourself more familiar with Helmert’s Chi Square distribution we recommend to solve the problems of Exercise B.1. 710 B Sampling Distributions and Their Use: Confidence Intervals and Confidence Regions

2 Exercise B.1. (Helmert’s Chi Square ' distribution):

Helmert’s random variable x WD .n 1/b 2=2 D pb 2=2 has the non-symmetric 2 ' pdf. Prove that the ﬁrst four central moments are,

(a) 1 D 0,EfxgD D p 2 (b) 2 D x D 2p p 3=2 1=4 2 2 (c) 3 D .2p/ .8=p/ (coefficient of skewness 3 =2 D 8=p/ 4 2 (d) 4 D 6 .1 C 2p/ (coefficient of curtosis 4=2 3 D 3 C 12=p/. Guide 3 p EfxgD Efb 2g .a/ 2 5 ) EfxgDp Efb 2gD 2.unbiasedness/ 3 2 2 b2 4 4 Df gD D 7 n 1 p 7 .b/ 5 ) DfxgD2p: p2 DfxgD Dfb 2g 4 Example B.18. (confidence interval for the sample variance b 2):

Suppose that a random sample .y1;:::yn/ 2 Y of size n D 100 has led to an empirical variance b 2 D 20:6. xD .n1/b 2=2 or 99 20:6=2 D 2039:4=2 has Helmert’s pdf with p D n 1 D 99 degrees of freedom. The probability D 1 ˛ that x will be between c1.pI ˛=2/ and c2.pI 1 ˛=2/ is

Zc2

Pfc1.pI ˛=2/ < x < c2.pI 1 ˛=2/gD f.x/dx D D 1 ˛

Zc2 Zc1

Pfc1.pI ˛=2/ < x < c2.pI 1 ˛=2/gD f.x/dx f.x/dx D D 1 ˛ 0 0

Zc2

f.x/dx D F.c2/ D 1 ˛=2 D .1 C /=2; 0

Zc1

f.x/dx D F.c1/ D ˛=2 D .1 /=2 0

Pfc1.pI ˛=2/ < x < c2.pI 1 ˛=2/gDF.c2/ F.c1/ D .1 C /=2 .1 /=2D b 2 c .pI ˛=2/

1 2 1 .n 1/b 2 .n 1/b 2 < < , <2 < 2 c2 .n 1/b c1 c2 c1 .n 1/b 2 .n 1/b 2 P <2 < D 1 ˛ D : c2.pI 1 ˛=2/ c1.pI ˛=2/

Since Helmert’s pdf f.xI p/ is now-symmetric there arises the question how to distribute the confidence or the complementary confidence ˛ D 1 on the confidence interval limits c1 and c2, respectively. If we setup F.c1/ D ˛=2,wedefine a cumulative probability half of the complementary confidence. If F.c2/ F.c1/ D Pfc1.pI ˛=2/ < x < c2.pI 1 ˛=2/gD1 ˛ D is the cumulative probability contained in the interval c1

Zc1 Zc2 f.x/dx D ˛=2 and f.x/dx D 1 ˛=2;

0 0 dependent on the degree of freedom p D n 1 of Helmert’s pdf f.x;p/. As soon as we have established the conﬁdence interval c1.pI ˛=2/ < x < c2.pI 1 ˛=2/ for Helmert’s random variable x D .n1/b 2=2 D pb 2=2, we are left with the problem of how to generate a conﬁdence interval for the “true” variance 2,the b2 1 1 1 sample variance given. If we take the reciprocal interval c2

Table B.16 (Values of the coefﬁcient of conﬁdence ,c1.pI ˛=2/ and c2.pI 1 ˛=2/):

˛ 0:950 0:990 0:998 ˛=2 0:050 0:010 0:002 ˛=2 0:025 0:005 0:001 1 ˛=2 0:975 0:995 0:999 712 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

In solving the linear Volterra integral equations of the ﬁrst kind

Zc1 Zc2

f.x/dx D F.c1/ D ˛=2 and f.x/dx DF.c2/ D 1 ˛=2; 0 0 which depend on the degrees of freedom p D n 1, Tables B.17–B.19 collect the quantiles c1.pI ˛=2/ and c2.pI 1 ˛=2/ for given values p and ˛.

Table B.17 (Quantiles c1.pI ˛=2/, c2.pI 1 ˛=2/ for the conﬁdence interval of the Helmert random variable with p D n 1 degrees of freedom):

˛=2 D 0:025; 1 ˛=2 D 0:975; ˛ D 0:05; D 0:95

pnc1.pI ˛=2/ c2.pI 1 ˛=2/ p n c1.pI ˛=2/ c2.pI 1 ˛=2/ 1 2 0:000 5:02 14 15 5:63 26:1 2 3 0:506 7:38 19 20 8:91 32:9 3 4 0:216 9:35 24 25 12:4 39:4 4 5 0:484 11:1 29 30 16:0 45:7 5 6 0:831 12:8 39 40 23:7 58:1 6 7 1:24 14:4 49 50 31:6 70:2 7 8 1:69 16:0 99 100 73:4 128 8 9 2:18 17:5 9 10 2:70 19:0

Table B.18 (Quantiles c1.pI ˛=2/, c2.pI 1 ˛=2/ for the conﬁdence interval of the Helmert random variable with p D n 1 degrees of freedom):

˛=2 D 0:005; 1 ˛=2 D 0:995; ˛ D 0:01; D 0:99

pnc1.pI ˛=2/ c2.pI 1 ˛=2/ p n c1.pI ˛=2/ c2.pI 1 ˛=2/ 1 2 0:000 7:88 8 9 1:34 22:0 2 3 0:010 9 10 1:73 23:6 3 4 0:072 4 5 0:207 14 15 4:07 31:3 19 20 6:84 38:6 5 6 0:412 24 25 9:89 45:6 6 7 0:676 29 30 13:1 52:3 7 8 0:989 39 40 20 65:5 49 50 27:2 78:2 99 100 66:5 139 B-6 Sampling from the Gauss–Laplace Normal Distribution: A Fourth 713

0.5

f(x)

− 0 c1(p;a /2) c2 (p;1 a /2)

2 2 2 Fig. B.13 Two-sided conﬁdence interval 2p =c2;p =c1Œ Helmert’s pdf, f(x)

Table B.19 (Quantiles c1.pI ˛=2/, c2.pI 1 ˛=2/ for the conﬁdence interval of the Helmert random variable with p D n 1 degrees of freedom):

˛=2 D 0:001; 1 ˛=2 D 0:999; ˛ D 0:002; D 0:998

pnc1.pI ˛=2/ c2.pI 1 ˛=2/ p n c1.pI ˛=2/ c2.pI 1 ˛=2/ 1 2 0:00 10:83 9 10 1:15 27:88 2 3 0:00 13:82 14 15 3:04 36:12 3 4 0:02 16:27 19 20 5:41 43:82 4 5 0:09 18:47 24 25 8:1 51:2 5 6 0:21 20:52 29 30 11:0 58:3 6 7 0:38 22:46 50 51 24:7 86:7 7 8 0:60 24:32 70 71 39:0 112:3 8 9 0:86 26:13 99 100 60:3 147:4 100 101 61:9 149:4

Those data collected in Tables B.17–B.19 lead to the triplet of conﬁdence intervals for the “true” variance

case .a/ W D 0:95; ˛ D 0:05; ˛=2 D 0:025; 1 ˛=2 D 0:975

p D 99; n D 100; c1.pI ˛=2/ D 73:4; c2.pI 1 ˛=2/ D 128 99 20:6 99 20:6 P <2 < D 0:95 128 73:4

Pf15:9 < 2 <27:8gD0:95: 714 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

2 Fig. B.14 Two-sided conﬁdence interval for the “true” variance , quantiles c1.pI ˛=2/ and c2.pI 1 ˛=2/

case .b/ W D 0:99; ˛ D 0:01; ˛=2 D 0:005; 1 ˛=2 D 0:995

p D 99; n D 100; c1.pI ˛=2/ D 66:5; c2.pI 1 ˛=2/ D 139 99 20:6 99 20:6 P <2 < D 0:99 139 66:5

Pf14:7 < 2 < 30:7gD0:99:

case .c/ W D 0:998; ˛ D 0:002; ˛=2 D 0:001; 1 ˛=2 D 0:999

p D 99; n D 100; c1.pI ˛=2/ D 60:3; c2.pI 1 ˛=2/ D 147:3 99 20:6 99 20:6 P <2 < D 0:996 147:3 60:3

Pf13:8 < 2 < 33:8gD0:998:

The results can be summarized as follows. With probability 95%, the “true” variance 2 is an element of the interval [15.9, 27.8]. In contrast, with probability 99%, the “true” variance is an element of the larger interval [14.7, 30.7]. Finally, with probability 99.8% the “true” variance is an element of the largest interval [13.8, 33.8]. If we compare the conﬁdence intervals for the variance 2, we realize much larger intervals for smaller complementary conﬁdence namely 5%, 1% and 0.2%. Such a result is subject of The Uncertainty Principle. B-6 Sampling from the Gauss–Laplace Normal Distribution: A Fourth 715

Table B.20 (Flow chart – conﬁdence interval for the variance 2):

Choose a coefficient of confidence g according to Table B16.

Solve thelinear Volterra integral equation of the first kind − − − F(c1(p;a /2)) – a / 2, F(c2(p;1 a /2)) 1 a /2 by Table B17, Table B18 or Table B19 for a Helmert pdf with p = n−1 degrees of freedom.

Compute the sample variance sˆ 2and the quantiles (n−1)sˆ 2 (n−1)sˆ 2 s2 ∈ ] [ . c2(p;a /2) c1(p;a /2)

B-62 The Uncertainty Principle

Figure B.15 is the graph of the function 1 1 2.nI ˛/ WD .n 1/b 2 ; c1.n 1I ˛=2/ c2.n 1I 1 ˛=2/ where 2 is the length of the confidence interval of the “true” variance 2. The independent variable of the functions is the number of observations n.The function 2.nI ˛/ is plotted for fixed values of the coefficient of complementary confidence ˛, namely ˛ D 5%, 1%, 0.2%. For reasons given later on the coefficient of complementary confidence ˛ is called uncertainty number. The graph of the function 2.nI ˛/ illustrates two important facts.

• Fact #1: For a contrast uncertainty number ˛; the length of the conﬁdence interval 2 is smaller, the larger number of observations n is chosen. • Fact #2: For a contrast number of observations n, the smaller the number of uncertainty ˛ is chosen, the larger is the conﬁdence interval 2.

Evidently, the divisive inﬂuences of (a) the length of the conﬁdence interval 2, (b) the uncertainty number ˛ and (c) the number of observations n, which we collect in The Magic Triangle of Fig. B.16, constitute The Uncertainty Principle, formulated by the inequality

2 .n 1/ k 2˛; 716 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

Fig. B.15 Length of the conﬁdence interval for the variance against the number of observations

length of the confidence interval

uncertainty number a number of observations n

Fig. B.16 The magic triangle, constituents: (a) uncertainty number ˛ (b), number of observations n, (c) length of the conﬁdence interval 2

2 where k 2˛ is called quantum number for the variance . The quantum number depends on the uncertainty number ˛. Let us interpret the uncertainty relation 2 2 .n 1/ k 2˛. The product .n 1/ defines geometrically a hyperbola which we approximated to the graph of Fig. B.15. Given the uncertainty number 2 ˛, the product .n 1/ has a smallest number denoted by k 2˛.Forinstance, 2 2 choose ˛ D 1% such that k 2˛=.n 1/ or 42=.n 1/ .Forthe number of observations n, for instance n D 2; 11; 101,wefindtheinequalities 42 2;4:2 2;0:42 2. Table B.21 (Coefficient of complementary confidence ˛, uncertainty number ˛, versus quantum number of the mean k 2˛ (Grafarend, 1970):

˛k 2˛ 10% 19:5 5% 25:9 1% 42:0 ! 0 !1

At the end, pay attention to the quantum numbers k 2˛ listed in Table B.21. B-7 Sampling from the Multidimensional Gauss–Laplace Normal Distribution 717

B-7 Sampling from the Multidimensional GaussÐLaplace Normal Distribution: The Conﬁdence Region for the Fixed Parameters in the Linear GaussÐMarkov Model

Example B.19. (special linear Gauss–Markov model, marginal probability distributions):

For a simple linear Gauss–Markov model EfygDA, A 2 Rnm,rkA D m, namely n D 3, m D 2, DfygDI 2 of Gauss-Laplace i.i.d. observations,weare going to compute • The sample pdf of b BLUUE of • The sample pdf of b 2 BIQUUE of 2. We follow the action within seven items. First, we identify the pdf of Gauss-Laplace i.i.d. observations y 2 Rn. Second, we review the estimations of b BLUUE of as well as b 2 BIQUUE of 2. Third, we decompose the Euclidean norm of the observation space ky Efygk2 into the Euclidean norms ky Abk2 and kb k2. Fourth, we present the eigenspace A0A analysis and the eigenspace synthesis of the 0 1 0 0 associated matrices M WD In A.A A/ A and N WD A A within the Eulidean 2 0 b 2 b 0 b norms ky A k D y My and k kA0A D . / N. /, respectively. The eigenspace representation leads us to canonical random variables .z1;:::;znm/ 2 0 relating to the norm ky Ak D y My and .znmC1;:::;zn/ relating to the norm b 2 k kN which are standard Gauss-Laplace normal. Fifth, we derive the cumulative probability of Helmert’s random variable x WD .n rkA/b 2=2 D .n m/b 2=2 and of the unknown parameter vector b BLUUE of or its canonical counterpartb BLUUE of , multivariate Gauss-Laplace normal. Action six generates the marginal pdf of b or b, both BLUUE of or , respectively. Finally, action seven leads us to 2 Helmert’s Chi Square distribution p with p D n rkA D n m (here: n m D 1) “degrees of freedom”. The first action item Let us assume an experiment of three Gauss-Laplace i.i.d. observations 0 Œy1;y2;y3 D y which constitute the coordinates of the observation space Y, dim The observationsyi, i 2f1; 2; 3g are related to a parameter space with coordinates Œ1;2 D in the sense to generate a straight line yfkgD1 C 2k. The fixed effects j ;j 2f1; 2g, define geometrically a straight line, statistically a special linear Gauss–Markov model of one variance component, namely 718 B Sampling Distributions and Their Use: Confidence Intervals and Confidence Regions

the ﬁrst moment 82 39 2 3 2 3 < y1 = 1k1 10 4 5 4 5 1 4 5 1 EfygDA E y2 D 1k2 D 11 ; rkA D 2: : ; 2 2 y3 1k3 12

The central second moment 82 39 2 3 < y1 = 100 2 4 5 4 5 2 2 DfygDIn DfygDD : y2 ; D 010 ;<0: y3 001 k represents the abscissa as a ﬁxed random, y the ordinate as the observation, naturally a random effect. Samples of the straight line are taken at k1 D 0, k2 D 1, k3 D 2, this calling for y.k1/ D y.0/ D y1, y.k2/ D y.1/ D y2, y.k3/ D y.2/ D y3, respectively. Efyg is a consistent equation. Alternatively, we may say Efyg2R.A/. The matrix A 2 R32 is rank deﬁcient by p D nrkA D 1, also called “degree of freedom”. The dispersion matrix Dfyg, the central moment of second order, is represented as a linear model, too, namely by one-variance component 2. The joint probability function of the three Gauss-Laplace i.i.d. observations

dF D f.y1;y2;y3/dy1dy2dy3 D f.y1/f .y2/f .y3/dy1dy2dy3 1 f.y ;y ;y / D .2/3=2 3 exp .y Efyg/0.y Efyg/ 1 2 3 22 will be transformed by means of the special linear Gauss–Markov model with one- variance component. The second action item For such a transformation, we need b BLUUE of and b 2 BIQUUE 2. 2 b D .A0A/1A0y; 6 6 6 33 1 5 3 6 A0A D ;.A0A/1 D ; b 6 BLUUE of W 6 35 2 3 6 33 6 y 6 1 521 1 4b D 4 y 5 ; 6 30 3 2 y3 2 1 b2 0 6 D .y A / .y A / 6 n rkA 6 2 2 6 1 b BIQUUE of W 6 D y0.I A.A0A/1A0/y; 6 n rkA n 4 1 b 2 D .y2 4y y C 2y y C 4y2 4y y C y2/: 6 1 1 2 1 3 2 2 3 3 B-7 Sampling from the Multidimensional Gauss–Laplace Normal Distribution 719

The third action item The quadratic form .y Efyg/0.y Efyg/ allows the fundamental decomposition

y EfygDy A D y A C A.b / .y Efyg/0.y Efyg/ D .y A/0.y A/ D .y A/0.y A/ C .b /0A0A.b /

.y Efyg/0.y Efyg/ D .n rkA/b 2 C .b /0A0A.b / 33 .y Efyg/0.y Efyg/ D b 2 C .b /0 .b /: 35

The fourth action item In order to bring the quadratic form .y Efyg/0.y Efyg/ D .n rkA/b 2C C.b /0A0A.b / into a canonical form, we introduce the generalised forward and backward Helmert transformation

0 HH D In z D 1H.y Efyg/ D 1H.y A/ and y EfygDH0z

.y Efyg/0.y Efyg/ D 2z0HH0z D 2z0z

1 .y Efyg/0.y Efyg/ D z0z D z2 C z2 C z2: 2 1 2 3

How to relate the sample variance b 2 and the sample quadratic form .b /0A0A.b / to the canonical quadratic form z0z? Previously, for the example of direct observations in the special linear Gauss– 2 2 2 Markov model EfygD1, DfygDIn we succeeded to relate z1 CCzn1 b2 2 b 2 b2 2 to and zn to . / . Here the sample variance , BIQUUE , as well as the quadratic form of the deviate of the sample parameter vector b from the “true” parameter vector have been represented by

1 1 b 2 D .y A/0.y A/ D y0ŒI A.A0A/1A0y n rkA n rkA n 0 1 0 rkŒIn A.A A/ A D n rkA D n m D 1 versus .b /0A0A.b /; rk.A0A/ D rkA D m: 720 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

The eigenspace of the matrices M and N, namely

0 1 0 M WD In A.A A/ A and N WD A A 2 3 1 21 1 33 D 4 242 5 D ; 6 35 1 21 will be analyzed.

The eigenspace analysis of the The eigenspace analysis of the

matrix M j 2f1;:::;ng matrices N; N1i2f1;:::;mg 0 0 V MV" # U NU D Diag.1;:::;m/ D N 0 V 1 0 1 D 0 M ŒV1; V2 U N U D Diag.1;:::;m/ D N 1 V 2 1 1 1 D ;:::;m D 1 m D Diag. ;:::; ;0;:::;0/D 1 n m M 1 1 1 D ;:::;m D 1 m rkM D n m rkN D rkN1 D m

Orthonormality of the eigencolumns Orthonormality of the eigencolumns

0 0 V 1V1 D Inm; V 2V2 D Im

0 .nm/m V 1V2 D 0 2 R 0 2 3 2 3 U U D Im 0 h i 4 V 1 5 4 Inm 0 5 V V D 0 1 2 V 2 0 Im

0 0 v1v1 D 1 u1u1 D 1 0 0 v1v2 D 0 u1u2 D 0 ::: ::: 0 0 vn1vn D 0 um1um D 0 0 0 vnvn D 1 umum D 1

V 2 SO.n/ W eigencolumns W U 2 SO.m/ W eigencolumns W .M j In/vj D 0 .N j In/uj D 0 eigenvalues eigenvalues jM j InjD0 jN i ImjD0 in particular B-7 Sampling from the Multidimensional Gauss–Laplace Normal Distribution 721

eigenspace analysis of the matrix M; eigenspace analysis of the matrix N; rkM D n m; M 2 R33; A 2 R32;rkM D 1 rkN D m; N 2 R22; rkN D 2 D Diag.1;0;0/ M2 3 N D"Diag.0:8377; 7:1623/# 6 0:4082 0:7024 0:5830 7 6 7 0:8112 0:5847 V D ŒV1; V2 D 4 0:8165 0:5667 0:1109 5 U D 0:5847 0:8112 0:4082 0:4307 0:8049 U 2 R22 31 32 V1 2 R V2 2 R to be completed by

eigenspace synthesis of the matrix M 0 M D V MV" # eigenspace synthesis of the matrix N; N 1 0 0 1 0 V N D U U ; N D U 1 U 1 N N M D ŒV1; V2 M 0 V 2 " # 0 0 V 1 N D U Diag.1;:::;m/U D ŒV1; V2 Diag.1;:::;n m;0;:::;0/ 0 0 V 2 N 1 D U Diag.1;:::;m/U 0 M D V1 Diag.1;:::;nm/V 1 in particular " #" # 2 3 0:8112 0:5847 1 0 0:4082 h i 6 7 " 0:5847 0:8112 # 02 M D 4 0:8165 5 1 0:4082 0:8165 0:4082 N D 0:8112 0:5847 0:4082 0:5847 0:8112

1 D 1 versus 1 D 0:8377; 2 D 7:1623

0:8112 0:5847 0 0:8112 0:5847 N1 D 1 0:5847 0:8112 02 0:5847 0:8112 1 1 1 D 1 versus 1 D 1 D 1:1937; 2 D 2 D 0:1396:

The non-vanishing eigenvalues of the matrix M have been denoted by .1;:::;nm/, m eigenvalues are zero such that eigen, 0;:::;0/. The eigenvalues of the regular matrix N span eigen .N/ D .1;:::;m/. Since the dispersion matrix DfbgD.A0A/1 2 D N1 2 is generated by the inverse of the matrix A0A D N, we have computed, in addition, the eigenvalues of the matrix N1 by means of 1 1 eigen.N / D .1;:::;m/. The eigenvalues of N and N , respectively, are related by

1 1 1 1 1 D 1 ;:::;m D m or 1 D 1 ;:::;m D m : 722 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

In the example, the matrix M had only one non-vanishing eigenvalue 1 D 1.In contrast, the regular matrix N was characterized by two eigenvalues 1 D 0:8377 1 and 2 D 7:1623, its inverse matrix N by 1 D 1:1937 and 2 D 0:1396. The two quadratic forms, namely

0 0 0 b 0 b b 0 0 b y My D y VMV y versus . / N. / D . / UMU . /; build up the original quadratic form

1 1 1 .y Efyg/0.y Efyg/ D y0My C .b /0N.b / 2 2 2 1 1 D y0V V0y C .b /0U U0.b / 2 M 2 N in terms of the canonical random variables

V0y D y , y D Vy and U0.b / D b such that 1 1 1 .y Efyg/0.y Efyg/ D .y/0 y C .b / .b / 2 2 M 2 N 1 1 nXm 1 Xm .y Efyg/0.y Efyg/ D .y/2 C .b /2 2 2 j j 2 i i i j D1 iD1 1 .y Efyg/0.y Efyg/ D z2 CCz2 C z2 CCz2 : 2 1 nm nmC1 n

The quadratic form z0z splits up into two terms, namely

2 2 2 2 z1 CCznm znmC1 CCzn nXm and Xm 1 2 1 2 D .y / j D .b / 2 j 2 i i j j D1 iD1 here

1 b 2 z2 D y2 D and 1 2 1 2 1 z2 C z2 D Œ.b /2 C .b /2 ; or 2 3 2 1 1 1 2 2 2 1 z2 D .y2 4y y C 2y y C 4y2 4y y C y2/ and 1 62 1 1 2 1 3 2 2 3 3 B-7 Sampling from the Multidimensional Gauss–Laplace Normal Distribution 723 1 1 33 z2 C z2 D Œ0:8377.b /2 C 7:1623.b /2 D .b /0 .b /: 2 3 2 1 1 2 2 2 35

The ﬁfth action item

We are left with the problem to transform the cumulative probability dF D f.y1;y2; y3/dy1dy2dy3 into the canonical form dF D f.z1; z2; z3/dz1dz2dz3. Here we take 2 advantage of Corollary B.3. First, we introduce Helmert’s random variable x WD z1 b b b and the random variables 1 and 2 of the unknown parameter vector of ﬁxed b 0 0 b 2 2 2 0 effects . / A A. / D z2 C z3 DkzkA0A if we denote z WD Œz2; z3 . p p 0 b b b b dz1dz2 D det A A d1d2 D 6d1d2; according to Corollary B.3 is special representation of the surface element by means of the matrix of the metric A0A. In summary, the volume element

p dx 0 b b dz1dz2dz3 D p det A A d1d2 x leads to the ﬁrst canonical representation of the cumulative probability 0 1=2 1 1 dx jA Aj 1 b 0 0 b b b dF D p exp x p exp . / A A. / d1d2: 2 2 x 22 22

2 b2 2 2 b2 The left pdf establishes Helmert’s pdf of x D z1 D = , dx D d .In contrast, the right pdf characterizes the bivariate Gauss-Laplace pdf of kb k2. Unfortunately, the bivariate Gauss-Laplace A0A normal pdf is not given in the b 2 canonical form. Therefore, second we do correlate k kA0A by means of eigenspace synthesis. 0 0 0 1 1 0 A A D UA0AU D U Diag.1;2/U D U Diag ; U 1 2 1 1 b 2 b 0 0 b b 0 0 b k kA0A WD . / A A. / D . / U Diag ; U . / 1 2

0 1=2 p 1 jA Aj D 12 D p 12 b WD U0.b / , b D U.b / 1 1 b 2 b 0 b b 2 k kA0A D . / Diag ; . / Dk kD 1 2 724 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

b 2 b 0 33 b k kA0A D . / . / 2 35 3 1 6 0 7 D .b /0 4 1:1937 5 .b / Dkb k2 : 1 D 0 0:1396

By means of the canonical variables b D U0b we derive the cumulative probability 1 1 1 1 dF D p p exp x dx p 2 2 x 2 2 12 1 .b /2 .b /2 1 1 C 2 2 db db or exp 2 1 2 2 1 2

dF D f.x/f.b1/f .b2/dxdb1db2:

Third, we prepare ourselves for the cumulative probability dF D f.z1; z2; z3/dz1 dz2dz3. We depart from the representation of the volume element p dx 0 b b dz1dz2dz3 D p det A Ad1d2 2 x subject to

1 1 x D z2 D b 2 D y0My 1 2 2 b D .A0A/1A0y:

The Helmert random variable is a quadratic form of the coordinates .y1;y2;y3/ of the observation vector y. In contrast, b BLUUE of is a linear form of the coordinates .y1;y2;y3/ of observation vector y. The transformation of the volume element b b dxd1d2 DjJxjdy1dy2dy3 is based upon the Jacobi matrix Jx.y1;y2;y3/ 2 3 " # D1xD2xD3x 6 7 a0 1 3 J D 4 b b b 5 D x D11 D21 D31 0 1 0 b b b .A A/ A 2 3 D12 D22 D32 B-7 Sampling from the Multidimensional Gauss–Laplace Normal Distribution 725 2 3 2 3 D x y 2y C y 1 2 1 1 2 3 a D 4 D x 5 D My D 4 2y C 4y 2y 5 2 2 32 1 2 3 D3x y1 2y2 C y3

1 1 x D y0My D .y2 4y y C 2y y C 4y2 4y y C y2/ 2 62 1 1 2 1 3 2 2 3 3 1 521 .A0A/1A0 D 6 30 3 2 3 2y 4y C 2y 4y C 8y 4y 2y 4y C 2y 1 1 2 3 1 2 3 1 2 3 J D 4 521 5 x 62 30 3 p 0 1 0 0 0 1 0 det Jx D det.A A/ detŒa a a A.A A/ A a p detŒa0a a0A.A0A/1A0a det Jx D p det.A0A/

4 4 a0a D y0M0My D y0My 4 4 4 a0A.A0A/1A0a D y0M0A.A0A/1A0My 4 4 D y0ŒI A.A0A/1A0A.A0A/1A0ŒI A.A0A/1A0y D 0 4 3 3 p 2 y0My det Jx D p 2 det.A0A/ p 2 0 1 det.A A/ j det JyjDjdet Jxj D p : 2 y0My

The various representations of the Jacobian will finally lead us to the special form of the volume element p 1 1 y0My dx db db D dy dy dy 2 1 2 2 jA0Aj1=2 1 2 3 and the cumulative probability 726 B Sampling Distributions and Their Use: Confidence Intervals and Confidence Regions

1 1 dF D exp .z2 C z2 C z2/ dz dz dz 3.2/3=2 2 1 2 3 1 2 3 0 1=2 1 1 dx jA Aj 1 b 0 0 b b b D p exp x p exp . / A A. / d1d2 2 2 x 22 22 1 1 D exp .y Efyg/0.y Efyg/ dy dy dy : 3.2/3=2 2 1 2 3

The sixth action item

The ﬁrst target is to generate the marginal pdf of the unknown parameter vector b, BLUUE of .

Z1 0 1=2 1 1 dx jA Aj 1 b 0 0 b b b dF1 D p exp x p exp . / A A. / d1d2 2 2 x 22 22 0 Z1 ! 1 1 dx 1 1 X2 .b /2 dF D p exp x p p exp i i db db 1 2 2 1 2 2 2 x 2 12 2 i 0 iD1

Let us substitute the standard integral

Z1 1 1 dx p exp x p D 1; 2 2 x 0 in order to have derived the marginal probability

b 0 1 2 b b dF1 D f1.j;.A A/ /d1d2 jA0Aj1=2 1 f .b/ WD exp .b /0A0A.b / 1 22 22 b 2 b b dF1 D f1. j ; N1 /d1d2 ! 1 1 1 X2 .b /2 f .b/ D p exp i i : 1 22 2 12 iD1 i

The seventh action item The second target is to generate the marginal pdf of Helmert’s random variable b2 2 b2 2 2 x WD = , BIQUUE , namely Helmert’s Chi Square pdf p with p D n rkA (here p D 1)“degree of freedom”. B-7 Sampling from the Multidimensional Gauss–Laplace Normal Distribution 727

ZC1 ZC1 0 1=2 1 1 dx jA Aj 1 b 2 b b dF2 D p exp x p exp k k 0 d1d2: 2 2 x 22 22 A A 1 1

Let us substitute the integral

ZC1 ZC1 0 1=2 jA Aj 1 b 2 b b exp k k 0 d d D 1 22 22 A A 1 2 1 1 in order to have derived the marginal distribution

dF2 D f2.x/dx; 0 x 1 1 p2 1 f .x/ D x 2 exp x ; subject to 2 2p=2.p=2/ 2 1 p p D n rkA D n m; here W p D 1; D 2 1 1 1 f2.x/ D p p exp x : 2 x 2

The results of the example will be generalized in Lemma B.

Theorem B.11. (marginal probability distributions, special linear Gauss- Markov model):

" EfygDA A 2 Rnm; rkA D m; Efyg2R.A/ subject to 2 2 C DfygDIn 2 R defines a special linear Gauss–Markov model of fixed effects 2 Rm and 2 2 RC 0 based upon Gauss-Laplace i.i.d. observations y WD Œy1;:::;yn . " EfbgD b D .A0A/1A0y subject to DfbgD.A0A/1 2 and 2 Efb 2gDb 2 2 1 0 4 b D .y A/ .y A/ subject to 24 n rkA Dfb 2gD n rkA 728 B Sampling Distributions and Their Use: Confidence Intervals and Confidence Regions identify b BLUUE and b 2 BIQUUE of 2.Thecumulative pdf of the multidimensional Gauss-Laplace probability distribution of the observation vector y D 0 Œy1;:::;yn 2 Y

2 f.yjEfyg;DfygDIn /dy1 dyn 1 1 D exp .y Efyg/0.y Efyg/ dy dy .2/n=2 n 22 1 n

b 2 b b 2 D f1./f2.b /d1 dmdb

b b 2 2 can be split into two marginal pdfs f1./ of , BLUUE of ,andf2.b / of b , BIQUUE of 2. (i) b BLUUE of The marginal pdf of b; BLUUE of , is represented by (1st version) b dF1 D f1./d1 dm ˇ ˇ 1 1=2 1 f .b/ D ˇA0Aˇ exp .b /0A0A.b / d d or 1 .2/m=2 m 22 1 m

(2nd version)

dF1 D f1.b/d1 dm ! 1 1 Xm .b /2 f .b/ D . /1=2 exp i ; 1 .2/m=2. 2/m=2 1 2 m1 m 22 iD1 i

by means of Principal Component Analysis PCA also called Singular Value Decomposition (SVD) or Eigenvalue Analysis (EIGEN) of .A0A/1, " 0 0 0 1 D U U .A A/ U D D Diag. ; ;:::; ; / subject to 1 2 m 1 m b 0 b 0 D U U U D Im; det U DC1

2 f1.bj; / D f.b1/f .b2/ f.bm1/f .bm/ 1 1 .b /2 f.b / D p i i 8i 2f1;:::;mg: i p exp 2 i 2 2 i

The transformed ﬁxed effects .b1;:::;bm/, BLUUE of .1;:::;m/,aremutu- ally independent and Gauss-Laplace normal

2 bi N .i j i / 8i 2f1;:::;mg: B-7 Sampling from the Multidimensional Gauss–Laplace Normal Distribution 729

(third version) b 1 1 pi 2 zi WD W f1.zi /dzi D p exp zi dzi 8i 2f1;:::;mg: 2 i 2 2 (ii) b 2 BIQUUE 2 The marginal pdf of b 2, BIQUUE 2, is represented by (ﬁrst version)

p D n rkA 2 2 dF2 D f2.b /db 1 1 b 2 f .b 2/ D pp=2b p2 exp p : 2 p 2p=2.p=2/ 2 2 (second version)

dF2 D f2.x/dx b 2 p 1 x WD .n A/ D b 2 D .y A/0.y A/ rk 2 2 2 1 p 1 1 f .x/ D x 2 exp x : 2 2p=2.p=2/ 2 f2(x) as the standard pdf of the normalized sample variance is a Helmert Chi 2 Square p pdf with p D n rkA “degree of freedom”. Proof. Theﬁrstactionitem First, let us decompose the quadratic form kyEfygk2 into estimates E1fyg of Efyg.

y EfygDy E1fygC.E1fygEfyg/ y EfygDy Ab C A.b / and

.y Efyg/0.y Efyg/ D .y E1fyg/0.y E1fyg/ C .E1fygEfyg/0.E1fygEfyg/ ky Efygk2 Dky E1fygk2 CkE1fygEfygk2 .y Efyg/0.y Efyg/ D .y A/0.y A/ C .b /0A0A.b / 2 2 b 2 ky Efygk Dky A k Ck kA0A :

Here, we took advantage of the orthogonality relation. 730 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

b 0 0 0 b 0 0 0 1 0 . / A .y A/ D . / A .In A.A A/ A /y D .b /0.A0 A0A.A0A/1A0/y D 0:

The second action item Second, we implement b 2 BIQUUE of 2 into the decomposed quadratic form.

2 0 0 0 1 0 ky Ak D .y A/ .y A/ D y .In A.A A/ A /y D y0My D .n rkA/b 2

ky Efygk2 D .n rkA/b 2 C .b /0A0A.b /

ky Efygk2 D y0My C .b /0N.b /:

The matrix of the normal equations N WD A0A; rkN D rkA0A D rkA D m,and 0 1 0 the matrix of the variance component estimation M WD In A.A A/ A ,rkM D n rkA D n m have been introduced since their rank forms the basis of the generalized forward and backward Helmert transformation.

0 HH D In z D 1H.y Efyg/ D 1H.y A/ and y EfygDH0z 1 .y Efyg/0.y Efyg/ D z0H0Hz D z0z 2 1 ky Efygk2 Dkzk2: 2

The standard canonical variable z 2 Rn has to be associated with norms ky Abk b and k kA0A. The third action item Third, we take advantage of the eigenspace representation of the matrices .M; N/ and their associated norms.

0 0 0 b 0 b b 0 0 b y My D y VMV y versus . / N. / D . / UNU . /

D Diag. ;:::; ;0;:::;0/ D Diag. ;:::; /: M 1 n m versus N 1 m 2 Rn D Rnm Rm 2 Rm m eigenvalues of the matrix M are zero, but nrkA D nm is the number of its non- vanishing eigenvalues which we denote by .1;:::;nm/. In contrast, m D rkA is B-7 Sampling from the Multidimensional Gauss–Laplace Normal Distribution 731 the n-m number of eigenvalues of the matrix N, all non-zero. The canonical random variable

V0y D y , y D Vy and U0.b / D b lead to 1 1 1 .y Efyg/0.y Efyg/ D .y/ y C .b /0 .b / 2 2 M 2 N 1 1 nXm 1 Xm .y Efyg/0.y Efyg/ D .y/2 C .b /2 2 2 j j 2 i i i j D1 iD1 1 .y Efyg/0.y Efyg/ D z2 CCz2 C z2 CCz2 2 1 nm nmC1 n subject to

2 2 2 2 z1 CCznm and znmC1 CCzn nPm Pm 1 2 1 b 2 D 2 .yj / j D 2 .i i / i j D1 iD1

2 0 2 2 2 2 kzk D z z D z1 CCznm C znmC1 CCzn 1 1 D y0My C .b /0N.b / 2 2 1 1 D ky Efygk2 D .y Efyg/0.y Efyg/: 2 2

0 Obviously, the eigenspace synthesis of the matrices N D A A and M D In A.A0A/1A0 has guided us to the proper structure synthesis of the generalized Helmert transformation. The fourth action item Fourth, the norm decomposition unable us to split the cumulative probability

dF D f.y1;:::;yn/dy1 dyn

2 2 2 into the pdf of the Helmert random variable x WD z1 CCznm D .n rkA/b 2 D 2.n m/b 2 and the pdf of the difference random parameter vector 2 2 2 b 0 0 b znmC1 CCzn D . / A A. /.

dF D f.z1;:::;znm; znmC1;:::;zn/dz1 dznmdznmC1dzn 732 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

1 1 f.z ;:::;z / D exp z0z 1 n .2/n=2 2 nm m 2 2 1 1 2 2 1 D exp .z1 CCznm/ 2 2 2 1 exp .z2 CCz2 / : 2 nmC1 n

Part A

2 2 2 x WD r D z1 CCznm ) dx D 2.z1dz1 CCznmdznm/

z1 D r cos nm1 cos nm2 cos 2 cos 1

z2 D r cos nm1 cos nm2 cos 2 sin 1

znm1 D r cos nm1 sin nm2

znm D r sin nm1 2 3 z 1 1 p p 4 5 D Diag. ;:::; /V0 y 1 nm 1 znm 0 0 0 V D ŒV1; V2; V 1V1 D Inm; V 2V2 D Im; V 1V2 D 0

0 nn n.nm/ nm VV D In; V 2 R ; V1 2 R ; V2 2 R and 2 3 z nmC1 1 p p 4 5 D Diag. ;:::; /U0.b / 1 m zn altogether 2 3 z 6 1 7 6 ::: 7 " # 6 7 p p 0 6 z 7 Diag. 1;:::; nm/V 1y 6 nm 7 D 1 : 6 7 p p 0 b 6 znmC1 7 Diag. 1;:::; m/U . / 4 ::: 5 zn

The partitioned vector of the standard random variable z is associated with the norm 2 2 kznmk and kzmk , namely B-7 Sampling from the Multidimensional Gauss–Laplace Normal Distribution 733

2 2 kznmk Ckzmk 2 2 2 2 D z1 CCznm C znmC1 CCzn 1 p p p p D y0V Diag. ;:::; /Diag. ;:::; /V0 y 2 1 1 nm 1 nm 1 1 p p p p C .b /0U Diag. ;:::; /Diag. ;:::; /U0.b / 2 1 m 1 m 1 1 D y0V Diag. ;:::; /V0 y C .b /0U Diag. ;:::; /U0.b / 2 1 1 nm 1 2 1 m

nm1 nm1 nm2 D dz1dz2 dznm1dznm D r dr.cos nm1/ .cos nm2/ 2 cos 3 cos 2dnm1dnm2 d3d2d1:

The representation of the local .n m/ dimensional hypervolume element in terms of polar coordinates .1;2;:::;nm1;r/has already been given by Lemma B.4. Here, we only transform the random variable r to Helmert’s random variable x. dx x WD r2 W dx D 2rdr; dr D p ;rnm1 D x.nm1/=2 2 x

1 rnm1dr D x.nm1/=2dx: 2

Part A concludes with the representation of the left pdf in terms of Helmert’s polar coordinates

nm 1 2 dF D exp 1 .z2 CCz2 /dz dz ` 2 2 1 nm 1 nm nm 2 1 1 nm2 nm1 nm2 D x 2 dx.cos / .cos / 2 2 nm1 nm2 2 cos 3 cos 2dnm1dnm2 d3d2d1:

Part B Part B focuses on the representation of the right pdf, ﬁrst in terms of the random variables b, second in terms of the canonical random variables b.

m 1 2 1 dF D exp .z2 CCz2 / dz dz r 2 2 nmC1 n nmC1 n 1 z2 CCz2 D .b /0A0A.b / nmC1 n 2 1 dz dz D jA0Aj1=2db db : nmC1 n m=2 1 m 734 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

The computation of the local m-dimensional hyper volume element dznmC1 dzn has followed Corollary B.3 which is based upon the matrix of the metric 2A0A. The ﬁrst representation of the right pdf is given by

m 1 2 1 dF D exp .z2 CCz2 / dz dz r 2 2 nmC1 n nmC1 n m 1 2 jA0Aj1=2 1 D exp .b /0A0A.b / db db : 2 m=2 22 1 m

Let us introduce the canonical random variables .b1;:::;bm/ which are generated b 2 by the correlating quadratic form k kA0A. 1 1 .b /0A0A.b / D .b /0U Diag ;:::; U0.b /: 1 m Here, we took advantage of the eigenspace synthesis of the matrix A0A DW N and .A0A/1 DW N1. Such an inverse normal matrix is the representing dispersion matrix DfbgD.A0A/1 2 D N1 2.

0 nm 0 UU D Im U 2 SO.m/ WD fU 2 jUU D Im; jUjDC1g 0 0 N WD A A D U Diag.1;:::;m/U versus 1 0 1 0 N WD .A A/ D U Diag.1;:::;m/U subject to 1 1 1 1 1 D 1 ;:::;m D m or 1 D 1 ;:::;m D m

0 1=2 p 1 jA Aj D 1 m D p 1 m

b WD U0.b / , b WD U0.b / 1 1 b 2 b 0 0 b b 0 b k kA0A DW . / A A. / D . / Diag ;:::; . /: 1 m b b The local m-dimensional hypervolume element d1 dm is transformed to the local m-dimensional hypervolume element db1 dbm by b b d1 dm DjUjdb1 dbm D db1 dbm:

Accordingly we have derived the second representation of the right pdf f.b/.

m 1 2 1 dFr D p 2 m=2 1 m 1 1 1 .b /0Diag ;:::;y .b / db db : exp 2 1 m 2 1 m B-7 Sampling from the Multidimensional Gauss–Laplace Normal Distribution 735

Part C Part C is an attempt to merge the left and right pdf

nm 2 1 1 .nm2/=2 dF D dF dF D x dxd! ` r 2 2 n m 1 m 0 1 2 jA Aj1=2 1 exp .b /0A0A.b / db db or 2 m 2 1 m nm 2 1 1 .nm2/=2 dF D dF dF D x dxd! ` r 2 2 n m 1 m 1 2 1 1 1 1 p exp .b /0Diag ;:::; .b / db db : m 2 1 m 2 1 m 2 1 m

The local .n m 1/-dimensional hypersurface element has been denoted by d!nm1 according to Lemma B.4. The ﬁfth action item

Fifth, we are going to compute the marginal pdf of b BLUUE of . b b b dF1 D f1./d1 dm as well as

dF1 D f1.b/db1 dbm b include the ﬁrst marginal pdf f1./ and f1.b/, respectively. The deﬁnition

Z1 Z nm 1 1 2 f .b/ WD dx d! x.nm2/=2 1 nm1 2 2 0 m 1 2 jA0Aj1=2 1 exp .b /0A0A.b / 2 . 2/m=2 2 subject to

Z1 Z nm 1 1 2 dx d! x.nm2/=2 D 1 nm1 2 2 0 leads us to

m 1 2 jA0Aj1=2 1 f .b/ D exp .b /0A0A.b / : 1 2 m 2 736 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

Unfortunately, such a general multivariate Gauss-Laplace normal distribution cannot be tabulated. An alternative is offered by introducing canonical unknown parameters b as random variables. The deﬁnition

Z1 Z nm 1 1 2 f .b/ WD dx d! x.nm2/=2 1 nm1 2 2 0 m 1 2 1 . /1=2 2 m 1 2 m1 m 1 1 1 .b /0Diag ;:::; .b / exp 2 2 1 m subject to

Z1 Z nm 1 1 2 dx d! x.nm2/=2 D 1 nm1 2 2 0 alternatively leads us to ! m Xm 1 2 1 1 1 .b /2 f .b/ D p exp i 1 2 m 22 1 m iD1 i f1.b1;:::;bm/ D f1.b1/ f1.bm/ 2 1 1 .bi / f1.bi / WD p p exp 8i 2f1;:::;mg: i 2 2 i

Obviously the transformed random variables .b1;:::;ybm/ BLUUE of .1;:::;m/ are mutually independent and Gauss-Laplace normal. The sixth action item Sixth, we shall compute the marginal pdf of Helmert’s random variable x D .n rkA/b 2=2 D .n m/b 2=2, b 2 BIQUUE 2,

dF2 D f2.x/dx includes the second marginal pdf f2.x/. The deﬁnition

Z nm ZC1 ZC1 m 1 1 2 1 2 jA0Aj1=2 f .x/ WD d! x.nm2/=2 db db 2 nm1 2 2 1 m 2 m 1 1 1 exp .b /0A0A.b / 22 B-7 Sampling from the Multidimensional Gauss–Laplace Normal Distribution 737 subject to Z 2 .nm1/=2 !nm1 D d!nm1 D nm1 ; . 2 / according to Lemma B.4

ZC1 ZC1 m 1 2 jA0Aj1=2 1 db db exp .b /0A0A.b / 1 m 2 m 22 1 1

ZC1 ZC1 m 1 2 1 D dz dz exp .z2 CCz2 / 1 m 2 2 1 m 1 1 leads us to

p WD n rkA D n m p n m 1 1 n m 1 n m p D D D 2 2 2 2 2 1 p 1 1 f .x/ D x 2 exp x ; 2 2p=2.p=2/ 2 namely the standard pdf of the normalised sample variance, known as Helmert’s Chi 2 Square pdf p with p D n rkA D n m “degree of freedom”. If you substitute x D .nrkA/b 2=2 D .nm/b 2=2, dx D .nrkA/ 2db 2 D .nm/2db 2 we arrive at the pdf of the sample variance b 2, in particular

dF D f .b 2/db 2 2 2 1 1 b 2 f .b 2/ D pp=2b p2 exp p : 2 r 2p=2.p=2/ 2 2

Here is our proof’s end.

Theorem B.12. (marginal probability distributions, special linear Gauss– Markov model with datum defect):

A 2nm;rWD rkA < minfn; mg EfygDA subject to Efyg2R.A/ DfygDV 2 V 2nn;rkV D n 738 B Sampling Distributions and Their Use: Confidence Intervals and Confidence Regions defines a special linear Gauss–Markov model with datum defect of fixed effects m 2 R and a positive definite variance-covariance matrix DfygD˙ y of 0 multivariate Gauss-Laplace distributed observations y WD Œy1;:::;yn . 2 b D Ly “linear” 6 b C 2 D A y subject to 4 kLA Imk D min “minimum bias” b 0 trDfgDtr L˙ y L D min “best” and 2 Efb 2gD 2 1 6 b2 b 0˙ 1 b 4 D .y A / y .y A / subject to 24 n rkA Dfb 2gD n rkA identify b BLIMBE of and b 2 BIQUUE of 2. Part A The cumulative pdf of the multivariate Gauss-Laplace probability distribution of 0 the observation vector y D Œy1;:::;yn 2 Y

f.yjEfyg;DfygD˙ y /dy1 dyn 1 1 1 D .y Efyg/0˙ 1.y Efyg/ dy dy n=2 1=2 exp y 1 n .2/ j˙ y j 2 is transformed by

0 0 p p p p ˙ y DW Diag.1;:::;n/W D W Diag. 1;:::; n/ Diag. 1;:::; n/W p p ˙ 1=2 y WD Diag. 1;:::; n/W ˙ ˙ 1=2 0˙ 1=2 y D . y / y versus 1 1 ˙ 1 0 y D W Diag ;:::; W 1 n 0 1 1 1 1 D W Diag p ;:::; p Diag p ;:::; p W 1 n 1 n 1 1 ˙ 1=2 WD Diag p ;:::; p W 1 n ˙ 1 ˙ 1=2 0˙ 1=2 D . y / y subject to the orthogonality condition

0 WW D In 2 0˙ 1 ky Efygk˙ 1 WD .y Efyg/ y .y Efyg/ y B-8 Multidimensional Variance Analysis, Sampling from the Multivariate 739 1 1 1 1 D .y Efyg/0W0Diag p ;:::;p Diag p ;:::;p 1 n 1 n W.y Efyg/0 .y E y /0.y E y / y E y 2 D f g f g DW k f gkIn subject to the star or canonical coordinates 1 1 ˙ 1=2 y D Diag p ;:::;p Wy D y y 1 n f.yjEfy g;Dfy gDIn/dy1 dyn 1 1 D exp .y Efyg/0.y Efyg/ dy dy .2/n=2 2 1 n 1 1 D exp ky Efygk2 dy dy .2/n=2 2 1 n into the canonical Gauss-Laplace pdf. Part B The marginal pdf of b BLUUE of , is represented by (1st version) b dF1 D f1./d1 dn:

B-8 Multidimensional Variance Analysis, Sampling from the Multivariate GaussÐLaplace Normal Distribution

Let x1;:::;xN denote a random sample from as k-variate Gauss-Laplace normal distribution with as the mean vector and ˙ as the variance-covariance matrix, such that

0 .B 370/ xi WD .Xi;1;:::;Xi;k/

1 for i 2f1;:::;Ng.LetX WD .x1;:::;xN /.Thejoint density of the random sample is generated by the product of N multivariate normal densities. Let us denote the parameter space by f; ˙ I 2 Rk; ˙ positive definitegDW˝. Here we describe estimation of the parameters from a multivariate Gauss-Laplace normal distribution and describe properties of these estimates. In addition, we care for inference for a simple, multiple, and partial correlation coefficients based on the normal random sample. Of course, we have to proof that we can assume a multivariate Gauss- Laplace normal distribution. We add some methods for assessing Gauss-Laplace normality and describe suitable transformation to normality. 740 B Sampling Distributions and Their Use: Confidence Intervals and Confidence Regions

B-81 Distribution of Sample Mean and Variance-Covariance

By the representation of the sample mean of the random sample x1;:::;xN by (B371) and the sample variance-covariance matrix by (B372)–(B375), namely the k k dimensional matrix SN =.N 1/ DfSjlg for all j; l 2f1;:::;kg subject PN to X D Xi;j =N . The sample variance-covariance matrix has k variances Sjj for iD1 j 2f1;:::;kg and k.k 1/=2 possibly distinct covariances. Sjl for j; l; j; l 2 f1;:::;kg.Thegeneralized sample variance is the scalar quality jSN =.N 1/j which determines the degree of “peakedness” of the joint density of x1;:::;xN and is a normal summary measure of variability in the sample. § ¤ ¦ Table B.1 Sample mean, sample variance-covariance matrix ¥ Sample mean PN ^ xN WD xi (B371) iD1 Sample variance-covariance matrix PN PN ^ ^ 0 ^ ^0 SN WD .xi xN /.xi xN /0D xi xi N xi xN xi xN (B372) iD1 iD1 SN =.N a/ DW fSjlg for j; l 2f1;:::;kg (B373) PN ^ Sjl WD .Xi;j Xj /.Xil Xl /=.N 1/ (B374) iD1 Subject to PN ^ Xj D Xi;j N (B375) iD1

2 Given a univariate random sample X1;:::;XN from a N.; / population, that the sample mean X^ has a normal distribution with mean and variance 2=N , the static .N 1/S2= 2 is distributed as a 2 variable and the two distributions are independent. The corresponding distributional result for the k-variable situations is given by our previous result. We ﬁrst deﬁne a multivariate distribution called the Wishart distribution which is derived from the multivariate Gauss-Laplace normal distribution as its sampling distribution of the sample statistics

XN ^ ^ 0 .Xi XN /.Xi XN / .B 376/ iD1

The Wishart distribution is a multivariate extension of the chi-square distribution.

Deﬁnition B.18. Wishart distribution

A random k-dimensional matrix W is said to follow a k-dimensional noncentral Wishart distribution. Wk.˙ ;m;x/ with m degrees of freedom, parameter ˙ , B-8 Multidimensional Variance Analysis, Sampling from the Multivariate 741 P and noncentrality parameter WD 0 1 0=2 if W can be represented as, Pm P W WD xi xj where xj for all j 2f1;:::;mg are i:i:i:Nk.; / vectors. Provide j D1 m:k the density function of W is

jWj.mk1/=2 expftr.˙ 1W=2/g f.WI ; ˙ / D . / km=2 m=2 B 377 2 j˙ j k.m=2/

k.k1/=4 k 1 where k.m=2/ D ˘j D1.2 .m C 1 j// is the multivariate Gamma functions.If D 0, we say that W follows a central Wishart distribution Wk.˙ ;m/; since ˙ is p.d., it is clear that D 0 if and only if D 0. The additivity ind ˙ property of Wishart matrices states that if Wj Wk. ;mj /; j D 1;:::;J,then PJ PJ Wj Wk ˙ ; mj . j D1 j D1

End of Deﬁnition B.18: Wishart distribution

Intermezzo

Before we proceed we must introduce results about the Fourier transform of a random vector, namely called “characteristic function” being generalized in Appendix C to the “characteristic functional” for random functions, namely stochastic processes. We will introduce some lemmas and deﬁnitions.

Deﬁnition B.19. Characteristic function of a random vector

The characteristic function of a random vector X is Efexp it0xg deﬁne for every vector t.

Deﬁnition B.20. Decomposition of a complex-valued function

Let the complex-valued function g.x/ be written as

g.x/ D g1.x/ C ig2.x/.B 378/ where g1.x/ and g2.x/ are real-valued. Then the expected value of g.X/ is

Efg.X/gDEfg1.x/gCiEfg2.x/g .B 379/ in particular,

Efexp.it 0x/gDEfcos.t 0x/gCiEfsin.t 0x/g .B 380/ 742 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

Lemma B.18. Product rule for independent variables

Let X0 WD .X.1/0X.2/0/.IfX.1/ and X.2/ are independent and g.x/ D g.1/.x.1//g.2/ .x.2//,then

Efg.X/gDEfg.1/.X.1//gEfg.2/.X.2//g .B 381/

Lemma B.19. Components of independent distribution data

If the component of X are independently distributed, then

YN 0 ^ Efexp.it XN /gD Efitj Xj g .B 382/ j D1

Theorem B.20. Characteristic function of a vector-valued random variant of type Gauss-Laplace distribution

The characteristic function of X which is Gauss-Laplace distribution according to N.; ˙ / is 1 Efexp.it 0X/gDexp t 0 C t 0˙ t .B 383/ 2 for any real vector t.

Theorem B.21. Characteristic function .t/

If the random vector X has the density f.x/ and the characteristic function f.t/, then

ZC1 ZC1 1 0 N f.x/ D ::: exp.it x/ .t/d! :::d! .B 384/ .2/ 1 1 1 and

ZC1 ZC1 N 0 0 f .!/ D EfCi! XgD ::: exp.Ci! x/f .x/dx1 :::dx .B 385/ 1 1 illustrating synthesis and analysis

Here ends our intermezzo B-8 Multidimensional Variance Analysis, Sampling from the Multivariate 743

ZC1 ZC1 N 0 0 f .!/ D EfCi! XgD ::: exp.Ci! x/f .x/dx1 :::dx .B 386/ 1 1

^ Results: Distribution of xN and SN

Let x1;:::;xN denote a random sample from the Gauss-Laplace normal distribution Nk.; /. ^ (a) The distribution of xN is Nk. =N / ˙ 0 (b) For N 2, SN follows a Wishart distribution Wk. 1N 1/ ^ (c) xN and SN are independently distributed. Proof

YN 1 Efexp.it 0x^ /gD Efexp.it 0 x /=N gDexp.t 0 C t 0.˙ =N /t .B 387/ N j j 2 j D1 for N D 2 it follows 1 1 0 1 1 0 S D x .x C x / x .x C x / C x .x C x / x .x C x / 2 1 2 1 2 1 2 1 2 2 2 1 2 2 2 1 2 1 ) S D .x x /.x x /0 .B 388/ 2 2 1 2 1 2 p Obviously .x1 x2/ 2 is distributed according to Nk.0; ˙ / and S2 according to Wk.˙ ;1/ 1 1 0 S D x .x C x C x / x .x C x C x / 3 1 3 1 2 3 1 3 1 2 3 1 1 0 C x .x C x C x / x .x C x C x / 2 3 1 2 3 2 3 1 2 3 1 1 0 C x .x C x C x / x .x C x C x / .B 389/ 3 3 1 2 3 3 3 1 2 3 2 S D S C .x x^/.x x^/0 .B 390/ 3 2 3 3 2 3 2 subject to

1 x^ D .2x^ C x /.B 391/ 2 3 2 3 744 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions in general

1 x^ D .mx^ C x / and m m C 1 m mC1 m S D S C .x x^/.x x^/0 .B 392/ mC1 m m C 1 mC1 m mC1 m

B-82 Distribution Related to Correlation Coefﬁcients

Since xi for all i 2f1;:::;m C 1g are mutually independent and therefore ^ ^ xmC1; Sm are functions only of xi for i 2f1;:::;mg, it follows that xmC1; Sm ^ are independent of xmC1. ^ By the hypothesis of induction we enjoy the independence of xm and Sm

^ xmC1; Sm and xmC1 .B 393/ ^ are mutually independent. Thus, Sm is independent of xm;xmC1 and therefore ^ mC1 mC1 ^ independent of xmC1xm which follows a Nk 0; m ˙ m xmC1 xm .xmC1 ^ 0 xm/ are distributed as Wk .˙; 1/. Note that ^ ^ cov xi xm; xn D 0 (B 394) h i 0 ^ ^ 0 Which implies that x1 xN ;:::; xN xN is uncorrelated and therefore ^ ^ independent of xN . In consequence, Sn is distributed independently of xN . P Results: Maximum likelihood estimation if and Based on a random sample x1;:::;P xN from a Nk . ;˙/distribution, the maximum likelihood estimates of and are

1 XN b D x D x^ ;.B 395/ ML N i N iD1 and

XN 1 ^ 0 N 1 ˙b D x x^ x X D S .B 396/ ML N i N i N N N iD1

Proof

The likelihood function L .;˙I x1;:::;xN / has the form shown on the right b b side. The MLE’s of and ˙ are denoted by ML and ˙ ML, and are the B-8 Multidimensional Variance Analysis, Sampling from the Multivariate 745

1 values that maximize L .;˙I x1;:::;xN /.Since˙ is p.d., the distance ^ 0 1 ^ xN ˙ xN in the exponent of the likelihood function is positive ^ unless D XN .TheMLEof is then xN , since it is the value of that maximizes the likelihood function. We next maximize the following function with respect to ˙: P 0 1 1 N ^ ^ exp 2 tr ˙ iD1 xi xN xi XN b L.ML;˙/D .B 397/ .2/Nk=2 j˙jN=2 P ^ 0 With A D ˙;B D N x x^ x X ,andb D N=2, we can show that iD1 i N i N P 0 b 1 N ^ ^ b ˙ ML D N iD1 xi xN xi XN maximizes L.ML;˙/. The maximum likelihood is exp Nk b b 2 L ; ˙ ML D 0 .B 397/ ML Nk=2 b N=2 .2/ j˙ MLj which completes the proof. Corollary: Invariance property of MLE Using the invariance property of MLE’s which states that the MLE of a function b g./of a parameter is g ML , we see that

1 0 b1 1. The MLE of the function 0˙ is b ˙ bML, p MLP ML p 2. The MLE of lj , the (l,j)th element of is given by b lj , the square root of Pb the MLE of the(l,j)th entry in ML b Although the estimator bML is an unbiased estimator of ; ˙ ML is a biased estimator of ˙. analogous to the univariate case, an unbiased estimator of ˙ is b 1 b b ˙ D N 1 SN . The unbiased estimators of abd ˙ denoted by and ˙ respectively are complete sufficient statistics. Of course, you have realized that we gave no space to derive the Wishart distribution. Instead we refer to his works in Wishart (1928), Wishart (1931), Wishart (1955)aswellasWishart and Bartlett (1932). the general theory of multivariate statistical analysis including the Gauss-Laplace multivariate normal distribution, the estimation of the mean vector and the variance-covariance matrix the distribution of sample correlation coefficients, the generalized t 2 statistics and the distribution of the sample covariance matrix can be taken from the excellent review of Anderson (1958) including 413 references. finally we advice the readers to consider the works of Koch (1988a), Koch (1988b), pp. 160–173 B.8 Distributions related to correlation coefficients Traditionally we divide simple, multiple and partial correlations. Here we restrict our study to the analysis of the simple correlation based on iid bivariate Gauss-Laplace normal samples. 746 B Sampling Distributions and Their Use: Confidence Intervals and Confidence Regions Xi;j; Xi;l for all i 2f1;:::;Ng .B 399/

th th where Xi;j and Xi;l denote the j and l components of the k - l dimensional z z vector x. Suppose E Xi;j D j ;E.Xi;l/ D l , var Xi;j D j ; var .Xi;l/ D l and corr Xi;j; Xi;l D jl. The exact sampling distributionp of wasp ﬁrst derived by Fisher (1955) who showed that for D 0 the statistic N 2^ 1 is distributed as t with (N-2) degrees of freedom. He introduced the Z-transformation,

1 Z WD 1 C ^=1 ^ .B 400/ 2 and proved that even for relative small samples it is approximately Gauss-Laplace normally distributed, and that as N increases the variance of the Z-transformation rapidly becomes nearly independent of converging to 1=.N 3/. In 1938, F. N. Da v id (1938) published tables of the exact sampling distribution of ^ for D 0.0:1/0:9, N D B.1/25; 50; 100; 200; 400; ^ D1.0:01/C1, and investigated the accuracy of the approximate distributions suggested by R.H. Fisher. She found approximate formulas for the mean/variance of Z: They were “extraordinarily accurate even for the low values of N provided jj is not too near unity”. Hotdling (1953) intensively studied the mathematical properties of the distribution function of ^ and the suggested improvements of the Z-transformation in terms of a hypergeometric function. Beside discussing methods of interpolating the density he also calculated moments of ^. From the vast reference list of the subject we like to mention the contribution of Soper et al. (1955), Kraemer (1973), Wilcox (1997) and Green (1952) as well as Khan (1994). Example B: Correlation coefﬁcient, “iid” Gauss-Laplace two dimensional normal

We start from the two dimensional Gauss-Laplace normal observations .xi ;yi / for N-independent identical distributed data .“iid00/ estimated by BLUUE and BIQUUE. By a special decomposition we shift the sample distribution into two normal distributions relating to b b b2b2b b2b2 .1; 2/ on the one side and to 1 2 12 or 1 212 on the other side. b12 indicates the sample correlation coefﬁcient via the characteristic function based on Fourier-Stieltjes integration we derive the sampling distribution of the correlation coefﬁcient b12 .12/ using tables of David (1954). “N independently, identically distributed observations : iid” “Gauss-Laplace normal, two-dimensional observations: .xi ;yi / for all i 2 f1;:::;Ng” B-8 Multidimensional Variance Analysis, Sampling from the Multivariate 747

“Probability density”

1 1 1 f.x;y/D p exp 2 2 2 12 1 2.1 / " # .x / .xi /.y / .y /2 1 C 2 1 2 C 2 . / 2 2 B31 1 12 2

YN f.x1;:::;xN ;y1;:::;yN / D f.xi ;yi / D f.x1;y1/ .xN ;yN / iD1 ! N 1 D p 2 212 1 !! 1 XN .x /2 .x /.y / .y i mu /2 exp i 1 2 i 1 i 2 C 2 2.1 2/ 2 2 iD1 1 1 2 2 .B32/

1 XN BLUUE W ^ D x Dk0x=N .B33/ 1 N i iD1

1 XN BLUUE W ^ D y Dk0y=N .B34/ 2 N i iD1

BIQUUE:

XN 1 2 0 ^2 D x ^ D x k^ x k^ =.N 1/ .B35/ 1 N 1 i 1 1 1 iD1 XN 1 2 0 ^2 D y ^ D y k^ y k^ =.N 1/ .B36/ 2 N 1 i 1 1 2 iD1

1 XN ^ D .x ^/.y ^/ D .x k^/0.y k^/=.N 1/ .B37/ 12 N 1 i 1 i 2 1 2 iD1

“Decomposition” 748 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

XN XN 2 b ^ 2 ^2 ^ 2 .xi 1/ D Œ.xi 1/.1 1 / D .N1/1 CN.1 1 / .B39/ iD1 iD1

XN XN 2 b ^ 2 ^2 ^ 2 .yi 2/ D Œ.yi 2/.2 2 / D .N 1/2 C N.2 2 / .B40/ iD1 iD1 XN ^ ^ ^ ^ ^ .xi 2/.yi 2/ D .N 1/121 2 C N.1 1 /.2 2 /.B41/ iD1

! N 1 f.x1;:::;xN ;y1;:::;yN / D p 2 212 1 N . b /2 2 . b /. b / . b /2 1 1 12 1 1 2 2 C 2 2 exp 2 2 2 2.1 / 1 12 2 b2 b b b b2 N 1 1 21212 1 2 2 exp 2 2 C 2 .B42/ 2.1 12/ 1 12 2

“Product of two normal distributions” First part, marginal distribution 0 1 ZC1 ZC1 " B 1 C N . b /2 @ q A 1 1 exp 2 2 2 2.1 12/ 1 1 1 212 1 12 # ! b b b 2 212.1 1/.2 2/ .2 2/ b b C 2 d1d2 .B43/ 12 2

“Fourier-Stieltjes analysis”

Z f .!/ D .exp i!x/f.x/dx , .B44/

ZC1 1 , f.x/ D .exp i!x/f.x//f ^.!/d! .B45/ 2 1

First part of the marginal normal distribution B-8 Multidimensional Variance Analysis, Sampling from the Multivariate 749

Z ZC1 1 N 1 it p db db 1 . b /2 1 2 exp 2 2 1 1 212 1 12 2.1 12/ 1 1 N 1 it 2 b b 2 exp C 2 212 .1 1/.2 2/ 2.1 12/ 12 N 1 it 3 . b /2 . / exp 2 2 2 2 B46 2.1 12/ 2

“abbreviations”

b 2 N .1 1/ b 1 WD 2 2 D 1.1/ .B47/ 2.1 12/ 1

2 N .1 b1/ exp Cit11 D exp it1 2 2 2.1 12/ 1 b b N .1 1/.2 2/ b b 2 WD 2 212 D 2.1; 2/ .B48/ 2.1 12/ 12

N .1 b1/.2 b2/ exp Cit22 D exp Cit2 2 212 2.1 12/ 12

“First part of the marginal normal distribution”

Z ZC1 1 N 1 i! q 1 b 2 exp 2 2 .1 1/ 2 2.1 12/ 1 212 1 12 1 2 .1 C i! / 12 2 . b /. b / 1 1 2 2 1 2 1 i! 3 b 2 b c C 2 .2 2/ d1dmu2 2 .1 2 /1=2 D 12 .B49/ 2 2 1=2 Œ.1 i!1/.1 i!3/ 12.1 C i!2/ ’ “Check yourself the integral expŒ˛x2 C ˇxy y2dxdy”

“Second part of the marginal normal distribution” 750 B Sampling Distributions and Their Use: Conﬁdence Intervals and Conﬁdence Regions

N 1 2 2 b2 b b b b2 .1 12/ N 1 1 21212 1 2 2 N 1 exp 2 2 C 2 2 2 2.1 / 12 Œ.1 it1/.1 it3/ 12.1 C it2/ 12 1 2 .B50/

f.x1;:::;nN ;y1;:::;yN /dx1 :::dxN dy1 :::dyN

) f.x1;y1/:::f.xN ;yN /dx1dy1 :::dxN dyN

f.b1; b2/f .b 1;b 2;b 12/db1db2db 1db 2db 12 .B51/ f.x1;:::;xN ;y1;:::;yN / D p1.b1; b2/p2.b 1;b 2;b 12/ “ﬁrst”

N b b f1 .t1;t2;t3/ f1.1; 2/ 2 N 1 .1 / 2 pN .t ;t ;t / D 12 .B52/ 1 1 2 3 2 2 Œ.1 it1/.1 it3/ 12.1 C it2/

“second”

ZZ ZC1 2 N 1 .1 / 2 exp.i! i!/ i! / p.g ;g ;g / D 12 1 1 2 2 3 3 1 2 3 3 2 2 .2n/ Œ.1 it1/.1 it3/ 12.1 C it2/ 1

d!1d!2d!3 0 1.B53/ N 1 N 2 2 2 N 2 b .1 12/ 2 N 4 d Barccos.12 12/C p.b / D .1 b / 2 @ q A 12 12 b (B.1) .N 3/Š d.12/ 2 b2 1 1212

End of Lemma Example As a specific example, we consider the confidence interval of the confidence coefficient 0.95 on the correlation of 0.782 observed by a sample of ten. Using Graph 4 of David (1954) we find the two limits are 0.34 and 0.98 . This result leads us to the interval 0:34 < 12 <0:98(Anderson 1958, p 72). Lemma B (probability interval for the correlation coefficient, functional determinant: David 1954, pp. xxxv–xxxviii): ˇ ˇ ˇ ˇ ˇ @.x1;:::;xN ;y1;:::;yN / ˇ ˇ ˇ D N @.b1; b2I 1;:::;N 1;1;:::;N 1/

subject to the Helmert transformation B-8 Multidimensional Variance Analysis, Sampling from the Multivariate 751

1 1 1 x1 D b1 C p 1 C p 2 CC p N 1 2 1 3 2 N.N 1/ 1 1 1 x2 D b1 p 1 C p 2 CC p N 1 2 1 3 2 N.N 1/ 2 1 x3 D b1 p 2 CC p N 1 3 2 N.N 1/ ::: 1 xN D b1 p N 1 N.N 1/ 1 1 1 y1 D b2 C p 1 C p 2 CC p N 1 2 1 3 2 N.N 1/ 1 1 1 y2 D b2 p 1 C p 2 CC p N 1 2 1 3 2 N.N 1/ 2 1 y3 D b2 p 2 CC p N 1 3 2 N.N 1/ ::: 1 yN D b2 p N 1 N.N 1/ ˇ ˇ ˇ ˇ ˇ @.xi ;yi / ˇ p.b1; b2;j ;j / D p.xi ;y;i/ˇ ˇ @.b1; b2;j ;j / for all j D 1;2;:::;N 1I i D 1;2;:::;N Appendix C Statistical Notions, Random Events and Stochastic Processes

Deﬁnitions and lemmas relating to statistics, random events as well as stochastic processes are given, neglecting their proofs. First, we review statistical moments of a probability distribution of random vectors and list the Gauss-Laplace normal distribution by introducing the notion of a quasi-normal distribution. As the end of Appendix C1 we take reference to two lemmas about the Gauss-Laplace 3 rule, namely the Gauss-Laplace inequality and the Vysochainskii-Potunin inequality. Appendix C2 reviews the spatial linear error propagation as well as the general nonlinear error propagation based on y D g.x/ introducing the moments of second, third and fourth order. The special role Jacobi matrix as well as the Hesse matrix is classiﬁed. Appendix C3 reviews useful identities like Efyy0 ˝j g and Efyy0 ˝yy0g as well as D Ef.y Efyg/.y Efyg/0 ˝ .y Efyg/g relating to the matrix of obliquity and WD Ef.y Efyg/.y Efyg/0 ˝ .y Efyg/.y Efyg/0gC::: relating to the matrix of courtosis.

Appendix C.4 newly introduces scalar-valued stochastic processes of one parameter enriched by Appendix C.5 reviewing characteristics of a one parameter stochastic processes like non-stationary, stationary and cryodic stochastic processes. Extensive simple examples are presented in Appendix C.6 including ARIMA processes namely. Appendix C.7 introduce the important WIENER process of type ORNSTEIN- UHLENBECK, WIENER process with drift integrated WIENER process. The spectral analysis of one parameter is enriched by 14 examples like the DIRAC representation, spectral densities, band limited white noise.An extensive introduction into scalar-, vector-, and tensor-valued stochastic processes in Appendix C.9 refers to the characteristic functional the moment representations of stochastic processes for scalar-valued and vector-valued qualities, for statistical homogeneous and isotropic fields of multi-point systems. The Appendix is specified by Example C.9.1 on the topic of two-dimensional Euclidean networks under the postulates of homogeneity and isotropy in the statistical sense,by Example C.9.2 on criterion matrices for absolute coordinates in a space time astro- nomic frame of reference, for instance represented in scalar-valued spherical harmonic, by Example C.9.3 on space gravity spectroscopy illustrating the benefits of

E. Grafarend and J. Awange, Applications of Linear and Nonlinear Models, 753 Springer Geophysics, DOI 10.1007/978-3-642-22241-2, © Springer-Verlag Berlin Heidelberg 2012 754 C Statistical Notions, Random Events and Stochastic Processes

TAYLOR-KARMAN structured criterion matrices, by Example C.9.4 on Nonlocal prediction and Example C.9.5 on Nonlocal Time Series Analysis.

C-1 Moments of a Probability Distribution, the GaussÐLaplace Normal Distribution and the Quasi-Normal Distribution

First,wedeﬁnethemoments of a probability distribution of a vector valued random function and explain the notion of a Gauss-Laplace normal distribution and its moments. Especially we deﬁne the terminology of a quasi - Gauss-Laplace normal distribution.

Deﬁnition C.1. (statistical moments of a probability distribution):

(1) The expectation or ﬁrst moment of a continuous random vector ŒXi for all i D 1;:::;n of a probability density f.x1;:::xn/ is deﬁned as the mean vector WD Œ1;:::;n of EfXi gD i Z Z C1 C1 i WD EfXi gD ::: xi f.x1;:::;xn/dx1 :::dxn: (C1) 1 1

The ﬁrst moment related to the random vector Œei WD ŒXi EfXi g is called ﬁrst central moment i WD EfXi i gDEfXi g i D 0: (C2)

(2) The second moment of a continuous random vector ŒXi for all i D 1;:::;nof a probability density f.x1;:::;xn/ is the mean matrix Z Z C1 C1 ij WD EfXi Xj gD ::: xi xj f.x1;:::;xn/dx1 dxn: (C3) 1 1

The second moment related to the random vector Œei WD ŒXi EfXi g is called variance - covariance matrix or dispersion matrix Œij , especially variance or dispersion 2 for i D j or covariance for i ¤ j :

2 2 ii D i D V fXi gDDfXi gWDEf.Xi i / g (C4) ij D ij D C fXi ; Xj gWDEf.Xi i /.Xj j /g (C5) 0 DfxgDŒij D ŒC fXi ; Xj g D Ef.x Efxg/.x Efxg/ g: (C6)

0 x WD ŒX1;:::;Xn is a collection of the n 1 random vector. The random variables Xi ; Xj for i ¤ j are called with respect to the central moment of second order uncorrelated if ij D 0fori ¤ j .(3)Thethird central moment with respect to the random vector Œei WD ŒXi EfXi g deﬁned by C-1 Moments of a Probability Distribution, the Gauss–Laplace Normal 755

ij k WD Efei ej ekgDEf.Xi i /.Xj j /.Xk k/g (C7) contains for i D j D k the vector of obliquity with the components

3 i WD Efei g (C8)

Uncorrelation with respect to the central moment up to third order is deﬁned by 2 Y3 81 i1 ¤ i2 ¤ i3 n n Efen1 en2 en3 gD Efe j g 4 (C9) i1 i2 i3 ij and j D1 0 n1 C n2 C n3 3:

(4) The fourth central moment relative to the random vector Œei WD ŒXi EfXi g

ij k ` WD Efei ej eke`gDEf.Xi i /.Xj j /.Xk k/.X` `/g (C10) leads for i1 D i2 D i3 D i4 to the vector of curtosis with the components

4 2 2 i WD Efei g3fi g (C11)

Uncorrelation with respect the central moments up the fourth order is deﬁned by 2 Y4 81 i1 ¤ i2 ¤ i3 ¤ i4 n n Efen1 en2 en3 en4 gD Efe j g 4 (C12) i1 i2 i3 i4 ij und j D1 0 n1 C n2 C n3 C n4 4:

(5) The central moments of the nth order relative to the random vector Œei WD ŒXi EfXi g are deﬁned by

i1:::in WD Efei1 :::ein gDEf.Xi1 i1 /:::.Xin in /g (C13)

A special distribution is the Gauss-Laplace normal distribution of random vectors in Rn: Note that alternative distributions on manifolds exist in large numbers, for instance the von Mises distribution on S2 or the Fisher distribution on S3 of Chap. 7.

Deﬁnition C.2. (Gauss-Laplace normal distribution):

0 An n 1 random vector x WD ŒX1;:::;Xn is a Gauss-Laplace normal distribution if its probability density f.x1;:::xi / has the representation

1 f.x/ D .2/n=2j˙ j1=2 expŒ .x Efxg/0˙ 1.x Efxg/ (C14) 2 756 C Statistical Notions, Random Events and Stochastic Processes

Symbolically we can write x N .; ˙ / (C15) with the moment of ﬁrst order or mean vector WD Efxg and the central moment of second order or variance – covariance matrix ˙ WD Ef.x Efxg/.x Efxg/0g: The moments of the Gauss-Laplace normal distribution are given next.

Lemma C.3. (moments of the Gauss-Laplace normal distribution):

0 Let the n 1 random vector x WD ŒX1;:::;Xn follow a Gauss-Laplace normal distribution. Then all central moments of odd order disappear and the central moments of even order are product sums of the central moments of second order exclusively.

Efei1 :::ein gD08n D 2m C 1; m D 1;:::;1 (C16) Efe :::e gDfct. 2 ; ;:::;2 ;:::;2 /8n D 2m;m D 1;:::;1 (C17) i1 in i1 i1i2 i2 in 2 ij D ij ;ii D i ; (C18)

ij k D 0; (C19)

ij k ` D ij k` C ikj` C i`jk; (C20) 2 2 2 2 2 iijj D iijj C 2ij ij D i j C 2ij ;iiii D 3.i / (C21)

ij k `m D 0; (C22)

i1i2:::i2m2i2m1i2m D i1i2:::i2m2 i2m1i2m C i1i2:::i2m3i2m1 i2m2i2m C :::

Ci2i3:::i2m1 i1i2m: (C23)

0 The vector of obliquity WD Œ 1;:::; m and the vector of curtosis WD 0 Œ1;:::;m vanish. A weaker assumption compared to the Gauss-Laplace normal distribution is the assumption that the central moments up to the order four are of the form (C18)– (C21). Thus we allow for a larger class of distributions which have a similar structure compared to (C18)–(C21).

Deﬁnition C.4. (quasi-Gauss-Laplace normal distribution):

A random vector x is quasi-Gauss-Laplace normally distributed if it has a continuous symmetric probability distribution f.x/ which allows a representation of its central moments up to the order four of type (C18)–(C21). Of special importance is the computation of error bounds for the Gauss-Laplace normal distribution, for instance. As an example, we have the case called the 3 C-2 Error Propagation 757 rule which states that the probability for the random variable for X falling away, from its mean by more than three standard deviations (SDs) is at most 5%,

4 P fjX j3g < 0:05: (C24) 81 Another example is the Gauss-Laplace inequality, that bounds the probability for the deviation from the mode .

Lemma C.5. (Gauss inequality):

The expected squared deviation from the mode is

4 2 p P fjX jrg forallr 4=3 (C25) 9r2 p p P fjX jrg1 .r= 3/forallr 4=3 (C26) subject to 2 WD Ef.X /2g. Alternatively, we take advantage of the Vysochanskii–Potunin inequality.

Lemma C.6. (Vysochanskii–Potunin inequality):

The expected squared deviation from an arbitrary point ˛ 2 is

42 p P fjX ˛jrg forallr 8=3 (C27) 9r2 42 1 p P fjX ˛jrg forallr 8=3 (C28) 3r2 3 subject to 2 WD Ef.X ˛/2g: References about the two inequalities are Gauss (1823), Pukelsheim (1994).

C-2 Error Propagation

At the beginning we note some properties of operators “expectation E” and “dispersion D”. Those derivatives can be taken from Pukelsheim (1993) and Teuniseen (1989b, 1990). Afterwards we review the special and general, in particular nonlinear error propagation.

Lemma C.7. (expectation operators EfXg ):

E is deﬁned as a linear operator in the space of random variables in Rn, also called expectation operator. For arbitrary constants ˛; ˇ; ı 2 R there holds the identity 758 C Statistical Notions, Random Events and Stochastic Processes

Ef˛Xi C ˇXi C ıg: (C29)

Let A and B be two m n and m ` matrices and ı an m 1 vector of constants 0 0 x WD ŒXi ;:::;Xn and y WD ŒYi ;:::;Yn two n 1 and ` 1 random vectors such that EfAX C BX C ıgDAEfxgCBEfygCı (C30) holds. The expectation operator E is not multiplicative that is

EfXi Xj gDEfXi gEfXj gCC fXi ; Xj g¤EfXi gEfXj g; (C31) if Xi and Xj are correlated.

Lemma C.8. (special error propagation):

Let y be an n 1 dimensional random vector which depends linear of the m 1 dimensional random vector x by means of a constant n m dimensional matrix A and of a constant dimensional vector ı of the form y WD Ax C ı. Then hold the “error propagation law”

DfygDDfAx C ıgDADfxgA0: (C32)

The dispersion function D is not addition, in consequence a nonlinear operator that is

DfXi C Xj gDDfXi gCDfXj gC2C fXi ; Xj g¤DfXi gCDfXj g; (C33) if Xi and Xj are correlated. The “special error propagation law” holds for a linear transformation x ! y D Ax C ı.The“general nonlinear error propagation” will be presented by Lemmas C.7 and C.8. The detailed proofs are taken from Chap. 8, Examples.

Corollary C.9. (“nonlinear error propagation”):

Let y D g.x/ be a scalar valued function between one random variable x and one random variable y. g(x) is assumed to allow a Taylor expansion around the ﬁxed approximation point 0 W

0 1 00 2 O g.x/ D g.0/ C g .0/.x 0/ C 2 g .0/.x 0/ C .3/ 2 (C34) D 0 C 1.x 0/ C 2g.0/ C O.3/ : C-2 Error Propagation 759

Then the expectation and dispersion identities hold

1 EfygDg. / C g00. / 2 C O.3/ D g. / C 2 C O.3/; (C35) x 2 0 x x 2 x

2 2 2 2 2 2 4 DfygD1 x C 4x Œ12.x 0/ C 2 .x 0/ 2 x 3 2 4 2 O CEf.x x/ gŒ212 C 42 .x 0/ C Ef.x x/ g2 C .3/: (C36)

For the special case of a ﬁxed approximation point 0 chosen to coincide with mean value x D 0 and x being quasi - Gauss-Laplace normal distributed we arrive at the identities 1 EfygDg. / C g00. / 2 C O.4/; (C37) x 2 x x 1 DfygDŒg0./2 2 C Œg00. /2 4 C O.3/: (C38) x 2 x x The representation (C37)and(C38) characterize the nonlinear error propagation which is in general dependent of the central moments of the order two and higher, especially of the obliquity and the curtosis.

Lemma C.10. (“nonlinear error propagation”):

Let y D f.x/ be a vector-valued function between the m 1 random vector x and the n 1 random vector y. g.x/ is assumed to allow a Taylor expansion around the m 1 ﬁxed approximation vector 0 W

0 1 00 O g.x/ D g. 0/ C g . 0/.x 0/ C 2 g . 0/Œ.x 0/ ˝ .x 0/ C .3/ 1 O D 0 C J.x 0/ C 2 HŒ.x 0/ ˝ .x 0/ C .3/: (C39)

2 With the n m Jacobi matrix J WD ŒJj gi . 0/ and the n m Hesse matrix H WD 0 ŒvecH1;:::;vecHn ;

Hi WD Œ@j @k gi . 0/.i D 1;:::;nI j; k D 1;:::;m/ (C40) there hold the following expectation and dispersion identities (“nonlinear error propagation”) 1 EfygD D g. / C Hvec˙ C O.3/ (C41) y x 2 760 C Statistical Notions, Random Events and Stochastic Processes

1 DfygD† D J †J 0 C J Œ† ˝ . /0 C . /0 ˝ †H0 y 2 x 0 x 0 1 C HŒ† ˝ . / C . / ˝ †J0 2 x 0 x 0 1 C HŒ† ˝ . /. /0 C . /. /0 ˝ † 4 x 0 x 0 x 0 x 0 † 0 0 † 0 C. x 0/ ˝ ˝ . x 0/ C . x 0/ ˝ ˝ . x 0/H 1 C J Ef.x /.x /0 ˝ .x /0gH0 2 x x x 1 C H Ef.x / ˝ .x /.x /0gJ0 2 x x x 1 C H ŒEf.x /.x /0 ˝ .x /g. /0 4 x x x x 0 0 0 C. x 0/ Ef.x x/ ˝ .x x/.x x/ g 0 0 C.I ˝ . x 0// Ef.x x/.x x/ ˝ .x x/ g 0 0 0 CEf.x x/ ˝ .x x/.x x/ g.. x 0/ / ˝ I/H 1 C H f.x /.x /0 ˝ .x /.x /0g 4 x x x x vec†.vec†/0H0 C O.3/:

(C42)

In the special case that the ﬁxed approximations vector 0 coincides to the mean vector x D 0 and the random vector x is quasi-Gauss-Laplace normally distributed the following identities hold:

1 † O EfygD y D g. x/ C 2 Hvec C .4/ .C 43/ † J 0 1 † † 0 0 DfygD y D †J 4 Hvec .vec / H 1 C H Ef.x /.x /0 ˝ .x /.x /0gH0 C O.3/ 4 x x x x 1 D J†J0 C HŒ† ˝ † C Ef.x /0 ˝ † ˝ .x /g H0 C O.3/: 4 x x (C44)

C-3 Useful Identities

Notable identities about higher order moments are the following. C-3 Useful Identities 761

Lemma C.11. (identities: higher order moments):

(a) Kronecker-Zehfuss products #1

Efyy0gDEf.y Efyg/.y Efyg/0gCEfygEfyg0 (C45) Efyy0 ˝ ygDEf.y Efyg/.y Efyg/0 ˝ .y Efyg/g C Efyy0g˝EfygCEfy ˝ Efyg0 ˝ yg CEfyg˝Efyy0g2Efyg˝Efyg0 ˝ Efyg (C46) Efyy0 ˝ yy0gDG ‰ ˝ Efyg0 Efyg0 ˝ ‰ ‰0 ˝ EfygEfyg˝‰ 0 C Efyy0g˝Efyy0g CEfy0 ˝ Efyy0g˝ygCEfy ˝ ygEfy ˝ yg0 2EfygEfyg0 ˝ EfygEfyg0: (C47)

(b) Kronecker products #2

‰ WDEf.y Efyg/.y Efyg/0 ˝ .y Efyg/g (C48) G WDEf.y Efyg/.y Efyg/0 ˝ .y Efyg/.y Efyg/0g Ef.y Efyg/.y Efyg/0 ˝ Ef.y Efyg/.y Efyg/0g Ef.y Efyg/0 ˝ Ef.y Efyg/.y Efyg/0g˝.y Efyg/g Ef.y Efyg/ ˝ .y Efyg/gEf.y Efyg/0 ˝ .y Efyg/0g: (C49)

The n2 n matrix ‰ contains the components of obliquity,then2 n2 matrix G the components of curtosis relative to the n 1 central random vector y Efyg. (c) Covariances between linear and quadratic forms

C fF1y C d1; F2y C d2gWDEf.F1y C d1 EfF1y C d1g/.F2y C d2 0 (C50) EfF2y C d2g/ gDF1†F 2

(linear error propagation)

CfFy C d;y0HygWDEf.Fy C d EfFy C dg/.y0Hy Efy0Hyg/g D FŒEfyy0 ˝ y0gEfygEfy0 ˝ y0gvecH 1 D F‰ 0vec.H C H0/ C F†.H C H0/Efyg (C51) 2 762 C Statistical Notions, Random Events and Stochastic Processes

C fy0Gy; y0HygWDEf.y0Gy Efy0Gyg/.y0Hy Efy0Hyg/g D .vecG/0ŒEfyy0 ˝ yy0gEfy ˝ ygEfy0 ˝ y0gvecH 1 D Œvec.G C G0/0Gvec.H C H0/ 4 1 Œvec.G C G0/0.‰ 0 ˝ EfygC‰ ˝ Efyg0/vec.H C H0/ 2 1 C trŒ.G C G0/†.H C H0/† C Efyg0.G C G0/†.H C H0/Efyg: 2 (C52)

(d) quasi-Gauss-Laplace-normally distributed data

C fF1y C ı;F2y C ı2gDF1†F 2 (C53) (independent from any distribution)

C fFy C ı; y0HygDF †.H C H0/Efyg (C54) 1 C fy0Gy; y0HygD trŒ.G C G0/˙ .H C H0/˙ 2 CEfyg0.G C G0/˙ .H C H0/Efyg: (C55)

C-4 Scalar Ð Valued Stochastic Processes of One Parameter

We have previously reviewed the moments of a probability distribution up to order four. The deviation was based on integration from minus infinity to plus infinity, A different concept is needed if we work with distributions on manifolds. For instance, we are confronted with the problem to find a distributions for data on a circle or a sphere or on a hypersphere. As mentioned before, directional measurements,also called angular observations or longitudinal data are not Gauss-Laplace distributed, but enjoy a von Mises-Fisher distribution in S RC1 Example C.4.1: p=1 (von Mises, 1918)

1 f.j; / D Œ2I0./ expŒ cos. / (C56)

Example C.4.2: p=2, (Fisher, 1953) C-4 Scalar – Valued Stochastic Processes of One Parameter 763

f.;˚ k ; ;/D expŒcos ˚ cos cos. / C sin ˚ sin ˚ 4 sinh ˚ ˚ D exp <jX > (C57) 4 sinh

Here, we have introduced the circular normal distribution CN.;/ parameterized by the mean direction .0 2/ and by the concentration parameter . > 0/, the reciprocal of a dispersion measure. The spherical normal distribution CN. ;˚ ;/ is parameterize by the “longitudinal direction, lateral mean direction . ;˚ /” and the concentration parameter , the reciprocal dispersion measure. We note that the p-spherical normal distribution is an element of the exponential class. Most important examples of the circular normal (R. von Mises) distributions are phase observation within the Global Positioning System (“Global Problem Solver”) according to Cai et al.(2007) and the spherical normal (Fisher) distribution for spherical harmonic analysis-synthesis (Fourier-Legendre functions) describing the gravitational field within gravities, electrostatics and magnetostaties according to Fisher et al. (1981), Kent (1983), Mardia (1972), Man (2005), Mc Fadden and Jones (1981) and Roberts and Ursell (1960). There have been made studies to consider an ellipsoidal reference to analyze the proper reference field of celestial bodies like the Earth, the Moon, or any other planetary body. The ellipsoid-of-revolution is the equilibrium figure of type MacLaurin which represents to first order of the Earth,theMoon, Mars or other planets even the Jacobi triaxial reference figure is a better approximation following, for instance Chandra Sekhar (1969). We leave the question: what is the characteristic distribution of a ellipsoid-of-revolution or a triaxial ellipsoid open! A random event X is the result of a random experiment under a given condition. If these conditions change it might have an influence on the result of random experiment, namely to the probability distribution. If we consider the varying conditions on the random quantities X D X.t/, we arrive at the random effects which depend on the deterministic parameter t. We illustrate the new detailed definition of one parameter stochastic process. Example C.4.1 We measure the temperature of a point continuously by a sensor for 3 years. Naturally we model the sensor recording by a graph shown in Fig. C.1. Actually we find a random variable which is function of the continuous parameter “time t”, namely X.t/. Every year we receive a temperature profile which is a deterministic function of time t W x D x.t/;0 t 1. In this form x.t/ is understood as realization of the random variable X.t/. Neglecting the influence of climate changes the function x D x.t/ is cyclic. We introduce the generalized random experiment “continuous measurement of temperature over one year” and refer it to fX.t/j0 t tg. Actually one could represent climate changes by a trend function. The character of a generalized random experiment is very obvious since for closely measuring points ti and tiC1 range in the second up to the minute. 764 C Statistical Notions, Random Events and Stochastic Processes

Fig. C.1 Graph of the function f(t) over a period of 1 year or 12 months. Two years recording: trajectories of the stochastic process

Example C.4.2

Alternatively we consider the length L=60 mm of Nylon wires and the design condition of a variance of 0.5 mm produced by one machine under equal conditions. If we analyzed the dependence of the diameter of a function of the distance t from the initial point, we receive for any measurement of the wire a function x D x.t/;0 t L. These functions differ from wire to wire, but these is no significant difference between the functional different wires. Indeed we assume changes to the constant condition of the wire. The random variations of the diameter are in the range of ˙0.02 mm and caused by the experimental conditions. Let us refer to the random wire diameter as a function of the one parameter t,weare able to define by a “continuous measurement of the various diameter of a wire as a dependence of the initial point” the generalized random experiment and formalize it by fX.t/; 0 t Lg. In contrast to random experiments which resulted in real numbers, such as generalized random experiment generates random functions, also called stochastic process or random process. We have called our parameter “t” derived from the term “time”, but it can be any parameter for instance “position”, “placement”,“pressure”etc. Finally we are able to define a stochastic process of one parameter.

Deﬁnition C.4.2: Stochastic process C-5 Characteristic of One Parameter Stochastic Processes 765

A stochastic process of one parameter t 2 T ,theparameter space, and the state z 2 Z,thestate space, is deﬁned as the set of random quantities fX.t/; t 2 T g .

If the set of parameter is finite or measurable infinite, we refer to a stochastic process with discrete time. Such processes are summarized as a sequence of random quantities fX1; X2;:::g. If the parameter t 2 T is interval, we refer to stochastic process with continuous time. A stochastic process fX.t/; t 2 Tg is called discrete if the state space z is a finite or measurable infinite. In contrast we refer to a continuous stochastic process if he state space is an interval.

Stochastic process ¨¨ @HH ¨¨ @HH ¨¨ @ HH ¨¨ @ HH ¨ H ¨¨ © @R HHj discrete stochastic discrete stochastic continuous stochastic continuous stochastic process with process with process with process with discrete time continuous time discrete time continuous time

If we consider various realization of X.t/ for all t 2 T , we call them trajectory of the stochastic process. For a discrete stochastic process with continuous time the trajectories are “step functions”

C-5 Characteristic of One Parameter Stochastic Processes

The most important function in connection with stochastic process is the distribution function of Xt ;t 2 T, namely F.x/ D P.Xt x/. But with the set of the distributions fFt .x/; t 2 Tg a stochastic ic process is not uniquely determined. A complete characterization of a stochastic process it is necessary to have information on all n-type ft1;t2; ;tn1;tng for the discrete set N WD f1; 2; ;n 1; ng with ti 2 T and the information of the distribution function of random vectors fX.t1/; X.t2/; ; X.tn1/; X.tn/g, namely

Ft1;t2;tn1;tn .x1;x2; ;xn1;xn/ WD P.X.t1/ x1; ; X.tn/ xn/ The set of distribution functions will be introduced by Table C.5.1 as well as the trend function, the covariance function, the variance function, the correlation function and the symmetry condition. 766 C Statistical Notions, Random Events and Stochastic Processes

Table C.5.1 Characteristics of one parameter stochastic process “Distribution functions of an n-dimensional random vector”

Ft1;t2;tn1;tn .x1;x2; ;xn1;xn/ WD P fX.t1/ x1; X.t2/ x2; ; X.tn1/ xn1; X.tn/ xng (C58)

“Trend function” EfXt gD.t/ for t 2 T (C59) C1R .t/ D xft .x/dx (C60) 1

“Covariance function and stochastic process”

Cov.s; t/ WD CovfXs; Xt g D EfŒXs EfXsgŒXt EfXt gg for all s; t 2 T (C61)

Cov.s; t/ D EfXs; Xt g.s/.t/ (C62)

“Variance function”

VarfXsgDWCov.t; t/ (C63)

“Correlation function” p p .s; t/ DW CovfXs; Xt g. VarfXsg VarfXt g/ (C64)

“Symmetry”

Cov.s; t/ DW Cov.t; s/ (C65) lim Cov.s; t/ D lim .s; t/ D 0 (C66) jtsj!1 jtsj End of Table C.5.2 Of special importance are those stochastic processes which do not depend on the absolute values of ti , but are functions of the distances between the points. They are considered as functions of the relative positions.

Deﬁnition 5.1 (Stationary in the narrow sense)

A stochastic process fXt ;t 2 Tg is stationary in the narrow sense if for all n 2 f1; 2; gfor arbitrary subject to ti 2 T and ti C h 2 T holds C-5 Characteristic of One Parameter Stochastic Processes 767

Ft1;t2;tn1;tn .x1;x2; ;xn1;xn/ D Ft1Ch;t2Ch;tn1Ch;tnCh.x1;x2; ;xn1;xn/ (C67) A special property is for

.t/ D EfX.t/gD.const/ (C68) VarfX.t/gD.const/ (C69)

End of Deﬁnition 5.1

Special case n D 2, for instance, t1 D 0; t2 D t s; h D s will lead us to

F0;ts.X1x2/ D Fs;t .x1x2/ D F./for WD t s (C70) Cov.s; t/ D Cov.s; s C / D Cov./ for WD t s (C71) “Symmetry”:Cov./ D Cov./ D Cov.jj/ (C72) lim Cov./ D 0 (C73) jj!1

Another generalization is achieving of the treat stochastic process ”of higher order” Deﬁnition 5.2: (stationary in the wider sense): A stochastic process of second order is stationary in the wider sense if the conditions 1. .t/ D const and 2. Cov.jj/ D CovfXs; Xt g End of Deﬁnition 5.2 Besides the condition of stationary another support property of a stochastic process is the set of conditions which hold for increments of stochastic process fX.t/; t 2 Tg in the interval Œt1;t2 for t1

Xt2 Xt1 In general such increments may be negative. The concept of incremental stochastic processes will lead us later to the concept of Kolmogov’s structure functions which is very popular in the applied sciences. Definition 5.3: (Stationary increments): A stochastic process fX.t/; t 2 Tg is called homogenous or stationary incremental, if the increments fX.t2 C /gfX.t1 C /g for all have identical probability distributions for any fixed t1 and t2 End of Definition 5.3 The great advantage of the incremental stochastic process in applications has the origin in the fact that a stationary stochastic process is not necessary. 768 C Statistical Notions, Random Events and Stochastic Processes

Deﬁnition 5.4: (Stochastic processes with incremental properties): A stochastic process fX.t/; t 2 Tg has independent increment for all n D 3; 4; and values ft1;t2; ;tn1;tng subject to the order ft1

Xt2 Xt1 ; Xt3 Xt2 ; ; Xtn1 Xtn2 ; Xtn Xtn1 (C74) being independent. End of Deﬁnition 5.4

Try to consider that the point tn1 is placed in the future the point tn at present time, is general for n>1all points ft1;t2; ;tn1;tng in the past.TheeventattnC1 for a Markov process depends only on the events at tn: Deﬁnition 5.5: (Simple Markov process): A stochastic process fX.t/jt 2 Tg has the property of a Markov process is for all n 2f1; 2; ; g if for all n 2f1; 2; g.n C 1/ tuple ft1;t2; ;tn1;tng ordered according to ft1

Z Z Z Z P fXtnC1 2 nC1jX.tn/ 2 n; X.tn1/ 2 n1; ; X.t1/ 2 1g

D P fX.tn1/ 2 Zn1jX.tn/ 2 Zng (C75)

End of Deﬁnition 5.5 Stochastic processes with the property of a simple Markov process is characterized by independent increments.

Lemma 5.6: Markov process A Markov process is in narrow sense stationary if and only if the one-dimensional probability is independent of the time such that

F.x/ D P fXt tgDx for all t 2 T (C76)

End of Lemma 5.6 Markov process with a ﬁnite state space are called Markov chains, otherwise they are called continuous Markov chains. Deﬁnition 5.7: (Continuity in the squared mean):

A stochastic process of second order fX.t/jt 2 Tg located in the point t0 is in the squared mean continuous if C-6 Simple Examples of One Parameter Stochastic Processes 769

2 lim E.ŒX.t0 C h/ X.t0/ / D 0 (C77) h!0

The stochastic process fX.t/jt 2 Tg is in the domain T0 D T in the quadratic mean continuous if the stochastic process has this property for all t 2 T0 End of Deﬁnition 5.7

Lemma 5.8: Squared mean and its covariance function

A stochastic process of second order fX.t/jt 2 Tg is continuous in the point t0 if its covariance function Cov.s; t/ is characterized by .s; t/ D .t0t0/ and its trend function .t/ is continuous in the point t D t0.

End of Lemma 5.8

C-6 Simple Examples of One Parameter Stochastic Processes

Simple examples of one parameter stochastic process are introduced divided in two types of stochastic processes, first with continuous time and second with discrete time: (i) Process with linear realizations (ii) Cosine oscillations with random amplitude (iii) Cosine oscillations with random amplitude and random phase (iv) Superposition of two uncorrelated random functions (v) Impulse modulation (vi) Randomly retarded pulse modulation (vii) Random sequences with discrete signals (viii) Random sequences of moving averages of order n: MA.n/ (ix) Random sequences of moving averages of order n: M A.n/ (x) Random sequences of unlimited order (xi) Autoregressive sequences of first order AR.1/ (xii) Autoregressive sequences of first order r: AR.r/ (xiii) ARMA (r,s) models. We present here simple examples of stochastic processes. Statements with respect to stationarity relate always to stationarity in the wide sense.

Ex 1: Processes with Linear Scalisations

Examples of stochastic processes with continuous time 770 C Statistical Notions, Random Events and Stochastic Processes

Example 6.1 Process with linear realizations

Many linear models can be characterized by the simple linear model X.t/ D Vt C W , V and W are random effects with the mean EfX.t/jt 0g called trend functions and the variance-covariance function EfŒX.s/ EfX.s/gŒX.t/ EfX.t/g D EfŒVs C W ŒV t C wg.s/.t/ DW ˙.s;t/. A ﬁrst result is the variance-covariance function oriented to our model.

˙.s;t/ D st Var.V / C .s C t/ Cov.V; W / C Var.W / (C78)

A second result is the special case W = constant. In such a case we ﬁnd the variance- covariance function ˙.s;t/ D st Var.V / End of Example 6.1

Ex 2: Cosine Oscillations with Random Amplitudes

Example 6.2 Cosine oscillation with random amplitude Here we assume the cosine model X.t/ D A cos !t, where A is a non-negative random effect, the amplitude subject to EfAg < 1. Here we realize the trend function of oscillations of type EfX.t/gDEfAg cos !t. The variance-covariance function for such a simple model is given by EfŒX.s/EfX.s/gŒX.t/EfX.t/g D 2 EfŒA cos !sŒA cos !tg.s/.t/ D ŒEfAg2 EfAg .cos !s/.cos !t/ DW ˙.s;t/ Let us denote the variance 2 D Var.A/ such that we receive the variance- covariance function ˙.s;t/, namely

˙.s;t/ D 2.cos !s/.cos !t/ (C79)

End of Example 6.2

Ex 3: Cosine Oscillations with Random Amplitudes and Phase

Example 6.3 Cosine oscillation with random amplitude and random phase

This time we assume the cosine model X.t/ D A cos.!t C ˚/, where A is a nonnegative random effect and ﬁnite variance. The phase is a random effect, equally distributed ˚ 2 Œ0; 2, The two random effects A and ˚ are independently from each other in the statistical sense. Such a stochastic process fX.t/ 2 .1; C1/g can be considered as the outputs of the larger number of similar oscillators, arbitrarily chosen when applied at different time instances. Because of C-6 Simple Examples of One Parameter Stochastic Processes 771

Z2 1 Efcos.!t C ˚/gD cos.!t C ˚/d˚ 2 0 1 D Œsin.!t C ˚/2 D 0 (C80) 2 0 We experience a zero trend function. According the variance-covariance function we may represent by

˙.s;t/ D EfŒA cos.!s C ˚/ŒA cos.!t C ˚/gd˚ Z2 1 D EfA2g cos.!s C ˚/cos.!t C ˚/d˚ 2 0 Z2 1 1 D EfA2g fcos.t s/cosŒ!.s C t/C 2˚gd˚ (C81) 2 2 0

The ﬁrst term characterizes a constant, while the second term approaches zero such the resulting variance-covariance function is proportional to the difference DW t s

1 ˙./ D ˙.t s/ D EfA2g cos !t (C82) 2

Such a process is stationary. End of Example 6.3

Ex 4: Superposition of Two Uncorrelated Random Function

Example 6.4 Superposition of two uncorrelated random functions

Let A and B be two uncorrelated random functions characterized by EfAgD EfBgD0 and var.A/ D var.B/ D 2 < 1, The derived stochastic process is deﬁned by fX.t/; t 2 .1; C1/g, namely by

X.t/ D A cos !t C B sin !t (C83)

Because of var.X.t// D 2 < 1 we have an example of random process of second order. Due to EfX.t/gDEfAg cos !t C EfBg sin !t D 0 (C84) 772 C Statistical Notions, Random Events and Stochastic Processes

Fig. C.2 Pulse demodulation of a binary system

We are able to compute the variance-covariance function ˙.s;t/ D EfX.t/X.s/g assuming EfA; BgDEfAgEfBgD0 due to uncorrelation

˙.s;t/ D EfA2 cos !s cos !t C B2 sin !s sin !tg C EfAB cos !s sin !t C sin !s cos !tg D 2.cos !s cos !t C sin !s sin !t/ C EfABg.cos !s sin !t C sin !s cos !t/ (C85) ˙.s;t/ D 2 cos !t for DW t s (C86)

Again we gain a stochastic process which depends only on the difference t s DW which is stationary. End of Example 6.4

Ex 5: Pulse Modulation

Example 6.5 Pulse modulation

A source generates at each time interval T time units of independent from each other a sign “1” or “0” with the probability 1-p or . The transfer of a “1” or “0” appears in this way that T time units produce a pulse with the amplitude A. In this way they generate a random signal, namely a stochastic process fX.t/; t 2 .1; C1/g, with this property: 0 with probability X.t/ D A with probability .1 /:

We start from the partial sequence, for instance 101101, illustrated by Fig. C.2. We have chosen the initial value t D 0 in such a way that we start the transmission of a message. Finally we review the trend function of the process as a constant and the covariance function as a non stationary process. C-6 Simple Examples of One Parameter Stochastic Processes 773

Fig. C.3 Randomly retarded pulse demodulation of a binary system

EfX.t/gDAP fX.t/ D AgCOP fX.t/ D 0gDA.1 p/ (C87) nT s; t < .n C 1/T; n D 0; ˙1; ˙2; EfX.s/; X.t/gDEfX.s/; X.t/jX.s/ D AgP fX.s/jX.s/ D Ag C EfX.s/; X.t/jX.s/ D 0gP fX.s/jX.s/ D 0g D A2.1 p/ (C88)

For m ¤ n, the process X.s/ and X.t/ are independent due to mT s .m C 1/T and nT t<.nC 1/T . Accordingly the variance-covariance function is not stationary:

˙.s;t/ D Cov.s; t/ A2p.1 p/ for all nT t<.nC 1/T; for all n D 0; ˙1; ˙2; D 0 otherwise: (C89)

End of Example 6.5

Ex 6: Random Retarded Pulse Modulation

Example 6.6: Randomly retarded pulse modulation

With regard to our previous example of a stochastic process fX.t/; t 2 .1; C1/g, we consider another stochastic process fY.t/ DfX.t 1/g subject to t 2 .1; C1/ assuming that D 2 Œ0; T is a random variable of a trajectory fX.t/; t 2 .1; C1/g shifted to the right producing the other stochastic process fY.t/; t 2 .1; C1/g. We illustrate the transfer by a random number 101101 under the constraint D D 0.Thetrend function of such a process is characterized by EfX.t D/gDA.1 p/ as we have seen before. X.s/ and X.t/ are independent if and only if when the inequality jt sj >T holds or if s and t at the points nT C D for n D 0; ˙1; ˙2; are disjunct. Let us call the random event by B.Thecomplementary event is denoted by B. The related 774 C Statistical Notions, Random Events and Stochastic Processes

Fig. C.4 Covariance function of a randomly retarded pulse modulation

probability values are called P.B/ D .t s/=T and P.B/ D 1 .t s/=T.For all jt sjT to variance-covariance function is represented by

˙.s;t/ D EfX.s/; X.t/jBgP fBgCEfX.s/; X.t/jBgEfX.s/gEfX.t/g D EfX.s/; X.t/jBgP fBgCEfŒX.s/2gP fBgEfX.s/gEfX.t/g D ŒA.1 p/2jt sj=T C A2.1 p/.1 jt sj=T/ ŒA.1 p/2 (C90) A2p.1 p/.1 jj=T/ forjjT ˙. WD t s/ D (C91) 0 otherwise:

The random process of second order fY.t/; t 2 .1; C1/g is therefore stationary illustrated by its variance covariance-function of Fig. C.4

Example of stochastic processes with discrete time

Stochastic processes with discrete time scale can be understood as a random sequence. Here we restrict us to stationary random sequences as a ﬁrst order model. They play a dominant role in signal transform and signal transfer. End of Example 6.6

Ex 7: Random Sequences with Discrete Signals

Example C6.7 Random sequences with discrete signals

Let f ; X2; X1; X0; X1; X2; gbe a random sequence of uncorrelated identical distributed random events characterized by 2 EfXi gD0; VarfXi gD < 1;i2f0; ˙1; ˙2; Due to the restricted range of variances we model a random process of second order, its trend function is zero. C-6 Simple Examples of One Parameter Stochastic Processes 775

EfXi gD0; for all i 2f0; ˙1; ˙2; 2 for all D 0I cov./ D (C92) 0 for ¤ 0:

Its covariance function with odd numbers s and t is characterized by the form Cov.s; t/ D EfXsXt g such that Cov.s; t/ D EfXsgEfXt gD0 for s ¤ t and Cov.s; t/ D EfX2gD 2 for s D t. Obviously beside the property of stationarity of random sequences it is important the property of independent increments. End of Example 6.7

Ex 8: Random Sequences of Moving Averages of Order n W MA.n/

Example C.6.8 Random sequences of moving averages of order n: MA(n) A random signal may be given by Xn Yt D ci Xt1for allt 2f0; ˙1; ˙2;g iD0 where fXi g is a random sequence of independent effects with parameter. The natural number n and the sequence of real restricted real numbers c0;c1; ;cn1;cn are given. The construction of the Yt are inﬂuenced by the momentary date Xt as well as n previous values f ; X2; X1; X0; X1; X2; g. Such a relation is called Principle of moving averages characterized by the process of second order Xt ;t 2f0; ˙1; ˙2;g.

Xn 2 VarfYt gD ci ci < 1 for all t 2f0; ˙1; ˙2;g (C93) iD0 “trend function” EfXt gD0forallt2f0; ˙1; ˙2;g .C93/ “covariance function” 8" # 9 < Xn Xn = cov.s; t/ D EfYsYt gDE : ci Xsi Œ cj Xtj ; iD0 j D0 82 39 < X Xn = 4 n 5 D E : i D 0 ci cj Xsi Xtj ; (C94) j D0 776 C Statistical Notions, Random Events and Stochastic Processes

“if jt sj >n, then EfXsi Xtj gD0 subject to s i ¤ t j ”, “if s i D t j , then”

Xn 2 cov.s; t/ D Ef ci cjtsjCi Xsi g 0in;0jtsjin njXtsj 2 D ci cjtsjCi (C95) iD0

As a summary, the covariance function cov.s; t/ D cov.t/ subject to WD t s is as a sequence of moving average stationary 2.c c C c c Cc c /for 0jjnI cov./ D 0 jj 1 jjC1 njj n (C96) 0forjj >n:

End of Example C 6.8

Ex 9: Random Sequences with Constant Coefﬁcients of Moving Averages of Order n W MA.n/

Example C.6.9 Random sequences of moving averages of order n: MA(n)

A special case if the coefﬁcient ci in our previous example is the setup c D 1=.nC1/ for i 2f0; 1; ;n 1; ng. The sequence of moving average of order n can be deﬁned by

1 Xn Y D X for all t 2f0; ˙1; ˙2;g (C97) t n C 1 ti iD0 " 2 1 jj for 0 jjnI cov./ D nC1 nC1 (C98) 0 otherwise:

“MA” summarizes “moving average”ofordern.

End of Example C 6.9 C-6 Simple Examples of One Parameter Stochastic Processes 777

Ex 10: Random Sequences of Unlimited Order

Example C.6.10 Random sequences of unlimited order

Let be given the random sequence of unlimited order of type

X1 Yt D ci Xti for all t 2f0; ˙1; ˙2;g (C99) iD0 for real numbers ci . In order to guarantee the convergence of the series we assume

X1 ci < 1 (C100) iD0 its related covariance function can be represented

X1 2 Cov.; t/ D ˙./ D ci cjjCi for all Df0; ˙1; ˙2;g (C101) iD0 XJ 2 Var.Yt / D ˙.0/ D lim ci for all t Df0; ˙1; ˙2;g (C102) J !1 iD0

Yt D ˙.0/ D lim ci Xti for all t Df0; ˙1; ˙2;g i!1;C1 2 Cov.s; t/ D ˙./ D lim ci cjjCi J 2Œ1;C1 The result is a two-sided sequence of moving averages

End of Example C 6.10

Ex 11: Autoregression Sequence of First Order: AR.1/

Example C.6.11 Autoregressive sequences of ﬁrst order: AR(1)

The starting point is the autoregressive sequence of ﬁrst order AR(1)oftype

Yt D aYt1 C bXt for all t Df0; ˙1; ˙2;g (C103) 778 C Statistical Notions, Random Events and Stochastic Processes limited by jaj <1and b as well as the random sequence fXg.Theinstant state Yt depends only of the previous state Yt1 as well as a random disturbance Xt subject 2 2 2 to fXgD0 and Ef.Xt EfXt g/ gDb . Applying n times of the series we gain

Xn1 n i Yt D a Ytn C b a Xti (C104) iD0 X1 n i lim a D 0.C 105/ ) Yt D b a Xt (C105) n!1 iD0

“Special Case”

XJ XJ 2 i i 2 2 2i b ci DW ba W lim ba D b lim a D forJ < 1 J !1 J !1 1 a2 iD0 iD0 (C106)

For an autoregressive sequence of ﬁrst order of a stationary random process we arrive a the characteristic covariance function

XJ XJ Cov./ D .b/2 lim ai ajjCi D ajj.b/2 lim a2i (C107) J !1 J !1 iD0 iD0

.b/2 Cov./ D ajj for D 0; ˙1; ˙2; (C108) 1 a2

End of Example C 6.11

Ex 12: Autoregression Sequence of First Order r: AR.r/

Example C 6.12 autoregressive sequences of order r: AR(r) An autoregressive sequence of order r called “AR(r)” is the random sequence Yt ;t D 0; ˙1; ˙2; over the set of real numbers a1;a2; ;ar1;ar namely

Yt C a1Yt1 C a2Yt2 CCar1Yt.r1/ C ar Ytr D bXt (C109)

The set Xi is a random sequence of parameters C-6 Simple Examples of One Parameter Stochastic Processes 779

PJ P1 Is the setup Yt D lim ct Xt1 for ci < 1 for a stationary sequence J !1 iD1 iD0 transformable in an autoregressive sequence of order r ? For solving this problem we have to solve for the relation of constants

c0 D b

c1 C a1c0 D 0

c2 C a1c1 C a2c0 D 0

cr C a1cr1 CCar c0 D 0

ci C a1ci1 CCar cir D 0 (C110)

A non-trivial solution of the equation leads us to the algebraic equation

r r1 y D a1y CCar1y C ar D 0 (C111)

2 Example: y C a1y C a2 D 0 For an autoregressive sequence of second order, namely r=1, we study the equation of quadratic order, namely by the eigen values 1 and 2. For the case 1 ¤ 2 we compute the covariance function.

.1 2/jjC1 .1 2/jjC1 Cov./ D Cov.0/ 1 2 2 1 (C112) .2 1/.1 12/

Alternatively, for the case 1 D 2 D we calculate the covariance function: 1 2 Cov./ D Cov.0/ 1 C jj jj for D 0; ˙1; ˙2; (C113) 1 C 2

subject to

Cov.0/ D VarfYt g

1 C a Cov.0/ D 2 (C114) 2 2 .1 a2/Œ.1 C a2/ a1

If 1 and 2 are complex, then there exist and ! subject to

1 D exp.i!/;2 D exp.i!/ (C115) 780 C Statistical Notions, Random Events and Stochastic Processes

Cov./ D Cov.0/˛jj sin.!jjCˇ/ for D 0; ˙1; ˙2; (C116) 1 1 C 2 ˛ WD ;ˇD arctan tan ! (C117) sin ˇ 1 2

Example: Numerical example

Let us introduce the numerical example for the autoregressive sequence

Yt 0:6 Yt1 C 0:05 Yt2 D 2Xt for t D 0; ˙1; ˙2; (C118)

subject to

VarfXt gD1 (C119) Obviously the impact of the term at the time instant t-2 on Y is small, at least compared to term on the time instant t-1. The related algebraic equation of second order can be represented by

Y2 0:6 y C 0:05 D 0 (C120)

Their solution is 1 D 0:1 and 2 D 0:5. Indeed compared to one, they are small than one. The solution leads to a stationary autoregressive sequence characterized by the covariance functions

Cov./ D 7:017 .0:5/jj 1:063 .0:1/jj (C121)

subject to

D 0; ˙1; ˙2; Cov.0/ D VarfYt gD5:954 More details can be taken from Andel (1984)

End of Example C.6.12

Ex 13: ARMA (r,s) Models

Example C.6.13 ARMA(r,s) models C-7 Wiener Processes 781

As a combination of random sequences of moving averages and of auto regressive sequences we introduce the ARMA(r,s) of the order Yt Ca1Yt1CCar1Yt.r1/Car Ytr D b0Xt Cb1Xt1CCbs1Xt.s1/Cbs Xts (C122) In practice, ARMA model we used to model time series as a sequence of real numbers determined by observing the state variables in discrete time instants. Very often, time series are not realizations of stationary random sequences of type fYt ;t D 0; ˙1; ˙2;g with constant trend functions EfYt gD.t/,butthe transformation Z.t/ D EffY.t/gEfY.t/gg (C123) reduced mean

EfZi gDEŒfYt gEfYt g D 0 (C124) reduced covariance matrix

DfZgDEŒfYt1 gEfYt1 g D ŒYt2 EfYt2 g D cov.t1;t2/ (C125) leads approximately to proper ﬁt of ARMA models. They are called trend reduced moving average interregressive sequences being “nearly stationary”. Very efﬁcient software exist nowadays.

End of Example C.6.13: ARMA(r,s)

C-7 Wiener Processes

First,wedeﬁneaWiener Process by three axioms, namely by stationary increments of a random function, together with path continuity. The relation to a random walk is outlined. Especially the probability density of an ordered n-dimensional Wiener Process is reviewed. Secondly, special Wiener A Process of type (a) Ornstein- Uhlenbeck (b) WIENER Process with drift and (c) integrated WIENER Process are introduced

C-71 Deﬁnition of the Wiener Processes

The English botanician R. Brown published in the year 1828 the random movement of microscopic small organic and inorganic particles suspended in liquids illustrated in Fig. C.3. We observe a totally irregular motion of particles, nowadays called Brownian motion. Its ﬁrst analytical treatment, we owe Bachelier (1900)and 782 C Statistical Notions, Random Events and Stochastic Processes

Fig. C.5 Trajectory of a two dimensional Brownian motion of 30 s distance (Perrin 1916)

Einstein (1905). Both found that a two-dimensional Gauss-Laplace distribution can model the typical Brownian motion. They spoke a chaotic movement of microscopic small particles. Wiener (1923) built up his theory of stochastic process with independent increments to model this Brownian motion we will deﬁne ﬁrst.

Deﬁnition C.7.1: Wiener Process A stochastic process fX.t/; t 0g with continuous time and the state space Z D .1; C1/ is called Wiener Process when it can be described as a path-continuous process by the following axioms. (i) X.0/ D 0 (C 126) (ii) fX.t/; t 0g has stationary increments (C 127) (iii) for all t>0, X.t/ is Gauss-Laplace normally distributed (C 128)

End of Deﬁnition C.7.1: Wiener Because of the stationarity of the increments, the difference X.t/ X.s/ for all s; t 0 are Gauss-Laplace normally distributed with expectation EfX.t/ X.s/gD0 and variance 2jt sj. Since the increments are independent the increments are a Markov-Wiener process. If D 1, then we call the process “standard Wiener”. Continuity and differentiability

EfjX.t/ X.s/j2gDVarfX.t/ X.t/gD 2jt sj (C129) lim EfjX.t C h/ X.t/j2gDlim 2jhjD0 (C130) h!0 h! “In the mean square sense the Wiener Process is continuous” Wiener Process and the random walk C-7 Wiener Processes 783

There is a intimate contact of Wiener Process and “a random walk” of a particle. The random walk of a particle can be described by

X.t/ D .X1 C X2 CCXŒt=t/x (C131)

“In the time interval starting by x D 0, a particle moves in the time difference t by x as a length unit to the left and the right with probability 1/2”. C1 if the jump is to the right X.t/ D (C132) 1 if the jump is to the left

“t=t” is the largest even number smaller or equal “t=t”

P.Xt D 1/ D P.Xt D1/ D 1=2 (C133)

EfXi gD0 and VarfXi gD1 )

) EfX.t/gD0 and VarfX.t/gD.x/2Œt=t

t ! 0 ) EfX.t/gD0, VarfX.t/gD 2t

Multidimensional distribution and conditional probability

Let fX.t/g for t 0 be a Wiener Process. We ask the question what is distribution of the sum of independent Gauss-Laplace normally distributed incremental qualities of type

X.ti / D X.t1/ C .X.t2/X.t1// CC.X.ti /X.ti1// subject to

i D 2;3; ;n 1; n

The result will lead us to the Special Wiener Process of type Gauss Process

Lemma C.7.2: Probability density of an n-dimensional Wiener Process

Let us assume 0

ft1;t2; ;tn1;tn .X1; X2; ; X.tn1/; X.tn//

D ft1 .X1/ft2t1 .X2 X1/ ftn1tn2 .Xn1 Xn2/ftntn1 .Xn Xn1/ (C134) and has the structure

ft1;t2; ;tn1;tn .X1; X2; ; X.tn1/; X.tn// 1 x2 .x x /2 .x x /2 .x x /2 D expf Œ 1 C 2 1 CC n1 n2 C n n1 g 2 t1 t2 t1 tn1 tn2 tn tn1 p n=2 n .2/ t1.t2 t1/ .tn1 tn2/.tn tn1/ (C135)

End of Lemma C.7.2: n-dimensional Wiener Process

Proof of Lemma 7.2

For the proof we depart from the one dimensional probability density of fXt g of type Wiener Process.

First assumption

p 2 2 ft .x/ DexpŒx =.2 t/ 2t (C136)

Second assumption

Let fs;t .x1x2/ the probability density of the random vector fX.s/X.t/g subject to the order 0

fs;t .x1x2/dx1dx2 D P.X.s/ D x1; X.t/ D x2/dx1dx2 (C137)

P.X.s/ D x1; X.t/ D x2/ D P.X.s/ D x1; X.t/ X.s/ D x2 x1/dx1dx2 (C138)

fs;t .x1x2/dx1dx2 D P.X.s/ D x1/P.X.t/ X.s/ D x2 x1/dx1dx2 (C139)

“independent increments”

fs;t .x1x2/ D fs .x1/ft .x2 x1/ (C140)

Third assumption

p 2 2 2 fs;t .x1x2/ D expfŒtx1 2sx1x2 C sx1 g2 s.t s/ (C141) C-7 Wiener Processes 785

“correlation coefﬁcient, covariance function” p s; t DC s=t (C142) CovfX.s/; X.t/gD 2s for 0

CovfX.s/; X.t/ X.s/gD0 (C143) CovfX.s/; X.t/gDCovfX.s/; X.s/ C X.t/ X.s/g D CovfX.s/; X.s/gCCovfX.s/; X.t/ X.s/g D CovfX.s/; X.s/gDVarfX.s/gD 2 (C144)

“conditional probability density” Condition: X.t/ D b

fX.s/.xjX.t/ D b/ D fs;t .x; t/ ft .b/ (C145) r s 2 s f .xjX.t/ D b/ D expf.x b/ 2 .t s/g (C146) X.s/ t t

s s EfX.s/jX.t/ D bgD b; VarfX.s/jX.t/ D bgD 2 .t s/ (C147) t t max VarfX.s/jX.t/ D bg,s D t=2 (C148)

Finally, a generalization from ft .t/; fs;t .s; t/ to ft1t2tn1tn leadsustoLemma C 7.2 End of proof for Lemma C.7.2 At this end we will deﬁne the Gauss process.

Deﬁnition C.7.3: Gauss process A stochastic process X.t/; t 2 T is a Gauss Process of type (C135), if for arbitrary vectors .t1;t2; ;tn1;tn/ ordered as t1

End of Deﬁnition C.7.3: Gauss process

C-72 Special Wiener Processes: OrnsteinÐUhlenbeck, Wiener Processes with Drift, Integral Wiener Processes

Here we only introduce special Wiener Process of type 786 C Statistical Notions, Random Events and Stochastic Processes

(i) Ornstein-Uhlenbeck (ii) Wiener Process with drift and (iii) integrated Wiener Process It has to be remembered that the trajectories of a Wiener Process are nowhere differentiable. Particles in motion have an infinite velocity when modeled a Wiener Process. In order to overcome this difficulty, Ornstein and Uhlenbeck developed a stochastic model for modeling the velocity of particles, an important issue in Continuum mechanics. Definition 7.3: Ornstein-Uhlenbeck Process Let fX.t/; t 0g be a Wiener Process with the parameter . Then we call

V.t/ D Œexp.˛t/X.exp.˛t// (C149) subject to ˛>0, a stochastic process fV.t/;1

p 2 2 fV.t/.x/ D Œexp.x =2 / . 2/ (C150) Property 1: Expectations

EfV.t/gD0; VarfV.t/gD 2 (C151) Cov.s; t/ D 2Œexp.2˛.t s// for s t (C152)

Proof for the covariance function

Cov.s; t/ D Cov.V .s/; V .t// D EfV.s/V.t/g D expŒ˛.s C t/EfX.exp2˛s/X.exp2˛t/g D expŒ˛.s C t/Cov.X.exp2˛s/X.exp2˛t// D expŒ˛.s C t/2Œexp.2˛.t s// (C153)

It has to be mentioned that the Ornstein-Uhlenbeck Process is stationary in the wider sense.AsaGauss Process it is also stationary in the narrow sense.Insum- mary, the Ornstein-Uhlenbeck Process is an instationary Wiener Process which is stationary process due to the action of time transformation and normalization. In comparison to the Wiener Process,theOrnstein-Uhlenbeck has the following differences C-7 Wiener Processes 787

1. The Ornstein-Uhlenbeck Process has no independent increments 2. The trajectories of an Ornstein-Uhlenbeck Process are in the quadratic mean differentiable

Wiener Process with drift Another example for the transformation of an instationary process to a stationary process is the Wiener Process with drift.

Deﬁnition 7.4: Wiener Process with drift A stochastic process fW.t/; t 0g with independent increments is called a Wiener Process with drift, if the following properties hold: (i) W.0/ D 0 (C154) (ii) Each increment has the property W.t/ W.s/ is Gauss-Laplace normally distributed with second order statistics (iii) EfW.t/gD.t s/ (C155) (iv) VarfW.t/gD 2jt sj (C156)

End of Deﬁnition 7.4: drifted Wiener Process An equivalent formulation is next:

“fW.t/; t 0g is a Wiener Process with drift, if it has the structure

W.t/ D t C X.t/ (C157) where fX.t/; t 0g is a Wiener Process subject to 2 D VarfX.1/g and an arbitrary constant, called as drift parameter”.

Obviously, a Wiener Process with drift is generated by the super position of type “additive” of a Wiener Process with a deterministically growing decreasing part. The deterministic part is a trend function of type .t/ D t. The one dimension probability density distribution of a “Wiener Process with drift” can be characterized by p 2 fW.t/.x/ D expŒ.x t/ =.2/ Œ 2t; 1;x;C1 (C158) 788 C Statistical Notions, Random Events and Stochastic Processes

In practice, drift parameters are daily experience. We restrict ourselves with only one example. Example: Geometric Wiener Process with drift Let us introduce a stochastic process of type fZ.t/; t 0g characterized by

Z.t/ D expW.t/ (C159) called “geometric Wiener Process with drift”. Since the model W.t/ has by deﬁnition a Gauss-Laplace normal distribution the expectation of .Z.t// equals the moment generating function of W.t/. For instance,

EfexpW.t/gDexpfst. C 1=22s/g (C160) s D 1 W) EfZ.t/gDexpŒt. C 2=2/ (C161) s D 2 W

) EfZ2.t/gDexpŒ2t. C 2/ (C162) ) VarfZ.t/gDexpŒt.2 C 2/Œexp.t 2/ 1 (C163)

Integrated Wiener Process By transforming of a Wiener Process we receive a stochastic process which is very important in the analysis. We will summarize these transformations in three cases in our

Lemma 7.5: Elementary transformations os a Wiener Process Let fX.t/; t 0g be a standard Wiener Process such that the following stochastic processes are as well standard Wiener Processes: (i) fU.t/; t 0g subject to U.t/ D cX.t=c2/; c > 0 (C164) (ii) fV.t/; t 0g subject to V.t/ D X.t C h/ X.h/; h > 0 (C165) (iii) fV.t/; t 0g subject to X.t/ for all t>0 W.t/ D (C166) 0 for all t>0

End of Lemma 7.5

We skip the proof. Instead let fX.t/; t 0g be a Wiener Process: Accordingly “near all” trajectories x D x.t/ are continuous in the squared mean. In consequence, there C-7 Wiener Processes 789

Rt exist the integrals u.t/ D x.y/dy for “nearly all” trajectories x D x.t/.These 0 integrals are realizations of the random integral.

Zt u.t/ D x.y/dy (C167)

0 u.t/ is the symbol for the integral acting on “nearly all” trajectories of a Wiener Process subject to the probability distribution fU.t/; t 0g deﬁnes a stochastic process called integrated Wiener Process.

Stochastic integration

Assume the Riemenn integration for 0 D t0

Xn u.t/ D lim f ŒX.ti / X.ti1/ti (C168) n!1;ti !0 iD1 u.t/ is the lines of a sum of independent Gauss-Laplace normally distributed random variables as well Gauss-Laplace normally distributed:Anintegrated Wiener Process is being analyzed by .a/ Trend function:

Zt Zt Ef X.y/dygD EfX.y/gdy D E.t/ D 0 (C169)

0 0 .b/ Covariance function:

Zs Zt Zs Zt Covfu.s/; u.t/gDEf X.z/dz X.y/dygD EfX.z/X.y/gdyd z

0 0 0 0 Zs Zt Zs Zy Zs Zt D 2 min.y; z/dydz D 2 zdzdy C 2 ydzdy

0 0 0 0 0 y Zs Zs 1 D 2 y2dy C 2 y.t y/dy (C170) z 0 0 790 C Statistical Notions, Random Events and Stochastic Processes

2 Covfu.s/; u.t/gD .3t s/s2 for s t (C171) 6 2 Varfu.t/gD t 3 for s D t (C172) 3

An integrated wiener Process is not stationary, in general. But it can be proven that the stochastic process V.t/ WD u.t C/u.t/ for V.t/ 2fV.t/; t 0g is stationary for all >0.

White noise A stochastic process, Z.t/ 2fZ.t/; t 0g subject to

Z.t/ WD dX.t/ dt D X.t/ (C173)

dX.t/ WD Z.t/dt (C174)

cannot be computed by standard differentiation: We have to introduce stochastic integration, for instance

Zb Xn f.t/dX.t/ D lim f f.ti1/ŒX.ti / X.ti1/g (C175) n!1;ti !0 a iD1 Zb Zb f.t/dX.t/ D f.b/X.b/ f.a/X.a/ X.t/df.t/ (C176) a a

The representation reminds us to “partial integration”. As the limit of the sum of independent Gauss-Laplace normally distributed random variables we have to compute the expectation of type

Zb Ef f.t/dX.t/gD0 (C177) a C-7 Wiener Processes 791

Xn Varf f.ti1/ŒX.ti / X.ti1/ iD1 Xn 2 D f .ti1/VarfX.ti / X.ti1/g iD1 Xn Xn 2 2 2 2 D f .ti1/.ti ti1/ D f .ti1/ti (C178) iD1 iD1

The limit approaches the variance of the stochastic integral of type

Zb Zb Varf f.t/dX.t/gD 2 f 2.t/dt (C179) a a

This result leads to the deﬁnition of White Noise

Deﬁnition C.7.6: White Noise A stochastic process fZ.t/; t 0g is called “White Noise” if for any arbitrary interval Œa; b of continuously differentiable functions f.t/, namely

Zb Zb f.t/dZ.t/ D f.b/X.b/ f.a/X.a/ X.t/df.t/ (C180) a a where fX.t/; t 0g is a Wiener Process.

End of Definition C.7.6: White Noise ?Why we call stochastic process “White Noise”? The origin of the question is the definition of the “generalized derivative”. To this end we define ”the Dirac Delta function”asa“generalized function” 1=h for h=2 t Ch=2 ı.t/ WD lim (C181) h!0 0 otherwise

1 for t D 0 ı.t/ WD (C182) 0 otherwise 792 C Statistical Notions, Random Events and Stochastic Processes

Properties of Dirac’s Delta function (i)

ZC1

f.t/ı.t t0/dt D f.t0/ (C183) 1

(ii) Heaviside functions 1 for t 0 H.t/ WD (C184) 0 for t<0

“The Dirac’s Delta function is the formal derivative of the Heavide function”

ı.t/ D @H.t/=@t (C185)

(iii) Covariance function

CovfZ.s/; Z.t/gDCovŒ@X.s/=@s; @X.t/=@t @ @ @ @ D CovfX.s/; X.t/gD min.s; t/ @s @t @s @t @ D H.s t/ (C186) @s

CovfZ.s/; Z.t/gDı.s t/ (C187) the covariance function is a measure of the statistically independence between Z.s/ and Z.t/, have a function of the small distances between the point s and the point t, namely js tj. “How can we illustrate the Dirac function?” Let fN.t/;t 0g be an arbitrary numbering process following our Deﬁnition C.7.6. The instants of the process at the points T1;T2; ; admit the representation in terms of the Heavide function N 0.t/ in terms of the Dirac function:

X1 X1 N.t/ D H.t Ti /; @N.t/=@t D ı.t Ti / (C188) iD1 iD1

Figure C.6 an illustration of the numbering process. The theory of stochastic integration has led us to the concept of generalized function, namely the Dirac function and Heaviside function, and to the notion of White Noise. More general notions like the concept of Colored Noise and ﬁnally to the notion of its stochastic processes. C-8 Special Analysis of One Parameter Stationary Stochastic Process 793

Fig. C.6 Dirac numbering system

C-8 Special Analysis of One Parameter Stationary Stochastic Process

C.8: Spectral analysis of one parameter stationary stochastic processes

Statistic or stochastic processes are conventionally divided into three classes we have already touched: Class one Class two Class three general instationary stationary ergodic stationary process process process

In real life, of course, we have live the property that all natural process are instationary. It is the task of modeling real world process, for instance by a ﬁrst order trend model or by taking increments of higher order to come more closely to the case of stationary processes for higher order increments. We will come back to this problem. For formal reasons we introduce ﬁrst real-valued and complex-valued stochastic process. Second we study the structure of process with a discrete spectrum.Atthe end, we introduce stochastic process with a continuous spectrum enriched by examples.

C-81 Foundations: Ergodic and Stationary Processes

C.8.1: Foundations: Ergodic and stationary processes

We move on towards complex-valued stochastic processes deﬁned by the following axioms: 794 C Statistical Notions, Random Events and Stochastic Processes

p .a/ X.t/ D Y.t/C iZ.t/ subject to i D 1 (C189) .b/ “Real” WfY.t/; t 2 Œ1; C1g .c/ “Imaginary” WfZ.t/; t 2 Œ1; C1g p p ZN D a b; jzjD zzN D a2 C b2 “absolute value” (C190) .d/ “Trend function” W EfX.t/gEfY.t/gCiEfZ.t/g .e/ “Covariance function” W CovfX.s/; X.t/gDEfŒX.s/ EfX.s/gŒX.t/ EfX.t/gg(C191) .f / Stationary in the wider sense W .vi1/ EfX.t/gD.t/ D (constant) (C192) .vi2/ CovfX.s/; X.t/gDCov.oI s t/; WD t s (C193) .g/ Complex-valued stochastic process of second order: EfjX.t/j2g < 1 forall t 2 Œ1; C1 (C194) (C195)

Ergodicity

A stochastic process EfX.t/; t 2 Œ1; C1 is stationary in the narrow sense if for all trajectories x.t/ D y.t/ C iz.t/ there hold

ZCT 1 EfX.t/gDEfX.t0/gDW lim x.t/dt for t0 2 R (C196) T !1 2 T

In this integral relation, in order to compute the expectation value Efx.t0/g we use only all the information which is located in one realization of the stochastic process. In contrast to the general case, we have used up to zero the information of all realizations of the stochastic process. In stationary ergodic process we used only one instant.Anexcellent approximation is the presentation.

1 XN EfX.t0/gD D lim xk.t0/ (C197) N !1 N RD1

Such a definition has a simple physical interpretation: The mean of a fixed instant equals the mean of a long time interval: The mean location is identical to the time C-8 Special Analysis of One Parameter Stationary Stochastic Process 795 average. In essence, This is the property of ergodic stationary processes. Analogue we can define the covariance function for a such processes namely

ZCT 1 p Cov./ D lim Œx.t/ x.t C / dt (C198) T !1 2 T

The treatment of ergodic stationary processes is reasonable for cases in which we have only one time like realization of the stochastic process. Of course increasing the length of the interval ŒT;CTimproves the estimation of EfX.t0/gD.

Here we also take reference the Euler’s representations:

exp ˙ix D cos x ˙ i sin x (C199) 1 sin x D Œexp.ix/ exp.ix/ (C200) 2i 1 cos.x/ D Œexp.ix/ C exp.ix/ (C201) 2

C-82 Processes with Discrete Spectrum

C.8.2: Processes with discrete spectrum

We start with general structure of stationary processes with discrete spectrum: we assume that all stationary process are ergodic.

Ex 1: Special Stationary Ergodic Processes

Example 8.1: Special stationary ergodic processes

Let fX.t/; t 0g be a stochastic process structure by

lrX.t/ D X˝.t/; X a complex random variable (C202) EfXgD0; E.jXj2/<1;EfX.t/X.t C /gD˝.t/˝.t C /EfX 2g (C203) D 0 W ˝.t/˝.t/ Dj˝.t/j2 D constant (C204) “ansatz” W ˝.t/ Dj˝.t/expŒi!.t/ for !.t/ real function (C205) “ansatz” W !.t C / !.t/ ¤ f.t/ W dŒ!.t C / !.t/=dt D 0 (C206) 796 C Statistical Notions, Random Events and Stochastic Processes

or d!.t/=dt D constant (C207) “ansatz” W !.t/ is a function of two constants; I ! and W (C208)

˝.t/j˝.t/j expŒi.! C / (C209)

End of Example 8.1: Special stationary ergodic processes

Lemma 8.1: decomposition of a complex stationary ergodic process

A stochastic process fX.t/; t 0g of the structure X.t/ D X˝.t/ is stationary in the wider sense if and only if

X.t/ D X exp.i!t/ (C210) subject to EfXgD0; Efjxj2g < 1;sDW EfjXj2g; Cov./ D s exp.i!/ End Lemma 8.1 The real part of a complex stochastic process describes a cosine oscillation with a random amplitude and phase

y.t/ D a cos.!t C / (C211)

The parameter ! is called frequency.

We generalize to two stationary processes as a linear combination: we refer to X1 and X2 as two complex random variables with zero expectation and two constant ı ı parameter f!1;!1 D 0; !2;!2 D 0g. We assume that the two complex random functions X1 and X2 are uncorrelated such that EfX1X2gD0 or EfX1 X2gD0. Our intention is to get the form of the covariance function:

X.t/ D X1 exp.i!1t/ C X2 exp.i!2t/ (C212) covfX.t/X.t C /g

D EfŒX1 exp.i!1t/ C X2 exp.i!2t/ŒXN1 exp.i!1t/ C XN2 exp.i!2t/g

D EfŒX1XN2 exp.i!1/ C X1XN2 exp.i.!1 !2/t i!2/g

C EfŒX2XN1 exp.i!2/ C X2XN2 exp.i.!2 !1/t i!1/g (C213)

2 2 s1 DW EfjX1j g;s2 DW EfjX2j g (C214)

Cov./ D s1 exp.i!1/C s2 exp.i!2/ (C215) C-8 Special Analysis of One Parameter Stationary Stochastic Process 797

Ex 2: Special Correlation Function of a Stationary Stochastic Processes

Example 8.2: Special correlation function of a stationary stochastic process

A special complex stationary stochastic process may be real-valued. For instance we deﬁne 1 1 X D .A C iB/ and X DW XN D .A ıB/ (C216) 1 2 2 1 2 AandBaretwo real-valued functions. In consequence of our representation X.t/ D X1 exp.i!1t/C X2 exp.i!2t/ we specialize here to

X.t/ D A cos !t B sin !t (C217)

A, B are uncorrelated. We received the covariance function Cov./ D 2s cos ! (C218)

2 2 subject to s WD EfjX1j gDEfjX2j g End of Example 8.2: Special correlation function

For pairwiseP uncorrelated stationary stochastic processes the general ansatz of type n X.t/ D kD1 exp.i!kt/subject to !j ¤ !k for j ¤ k and i;j D 1;2;:::;n1; n and EfXkgD0 will lead us to the covariance function

Xn 2 Cov./ D sk .i!k/ and sk D EfjXkj g for k 2f1;2;:::;n 1; ng kD1 (C219) in particular

Xn 2 Cov. D 0/ D EfjX.t/j gD sk (C220) kD1 In summary, the oscillation X.t/ is an additive superposition of a harmonic oscillation. This average energy per time unit is the same of circle average energies per time unit, an appropriate interpretation of our last representation. In order to achieve a real-valued function,wehavespecializeto (a) an even number n, (b) and a pair of values Xk being conjugate complex.

Let the set f!1;!2;:::g deﬁne the spectrum. We have to divide the various spectrum into two classes: small band processes and broad band processes, an entry into ﬁlter theory with many application in all sciences. 798 C Statistical Notions, Random Events and Stochastic Processes

Ex 3: Dirac Representation of a Covariance Function

Example 8.3: Dirac representation of covariance function

We use the inﬁnitely generalized covariance function for application of a Dirac function representation.

XK ZC1 Cov./ D lim sk exp.i!/s.! !k/d! (C221) K!1 kD1 1 ZC1 Cov./ D exp.i!/s.!/d! (C222) 1 subject to XK s.!/ D lim sk ı.! !k / (C223) K!1 kD1

The generalized function s.! is denoted by “spectral density” of a stationary stochastic process, formally equivalent to the Fourier transform. End of Example 8.3: spectral density

C-83 Processes with Continuous Spectrum

C.8.3 Processes with continuous spectrum

Here we present ﬁrst the spectral decomposition of the covariance function of a stochastic process with continuous spectrum. Second we present Kolmogorov’s spectral decomposition theorem.

C8-31 Spectral decomposition of a covariance function

Let fX.t/; t 2 Rg be a complex-valued stationary stochastic process described by its covariance function. Then there exists a real-valued function S.!/ such that cov./ enjoys the representation.

ZC1 Cov./ D exp.i!/dS.!/ (C224) 1 called the spectral function of the process. Note the domain t 2 Œ1; C1

Cov.0/ D S.1/ S.1/ D EfjXj2g < 1 (C225) C-8 Special Analysis of One Parameter Stationary Stochastic Process 799

The spectral function is determined only up to an additive constant c. Most commonly c is chosen to S.1/ D 0. If there exists the first derivative S.!/ D dS.!/=d!,wefindthespectral density of the stochastic process: The covariance function of this stationary process is the Fourier transform of the spectral density. Since S.!/ has to be a decreasing function we find also its property.

ZC1 Cov./ D exp.i!/s.!/d! (C226) 1 s.!/ 0; (C227) ZC1 s.!/d! < 1 (C228) 1

The set f!;s.!/ > 0g deﬁnes the continuous spectrum of the stationary stochastic process. The term

ZC1 Cov.0/ D S.1/ S.1/ D s.!/d! (C229) 1 is called average power of the spectrum. If we describe the stationary stochastic process Xt by harmonic oscillations with random amplitude and random phase, we are able to express

exp.i!t/ D expŒit.! C 2k/ for all k 2f1;2;:::g or k 2 Œ; C (C230)

ZC Cov./ D exp.i!/s.!/d! for D 0; ˙1; ˙2;::: (C231) C1R If Cov./d < 1 for stationary stochastic processes, holds, we are able to 1 introduce the inverse Fourier transform by

ZC1 1 s.!/ D exp.i!/Cov./d (C232) 2 1 800 C Statistical Notions, Random Events and Stochastic Processes

ZC1 i exp.! / exp.i! / s.! / s.! / D 2 1 Cov./d (C233) 2 1 2 1

Ex 4: Fourier Spectral Density

Example 8.4: Fourier spectral density

Assume real-valued constants a and cjaj <1. Here we will analyze the covariance function for an autoregressive stationary sequence of ﬁrst order and compute its spectral density.

1 XC1 s.!/ D Cov./ exp.i!/ 2 D1 " # e X1 X1 D a exp.i!/C a exp.i!/ 2 D1 D0 " # c X1 X1 D a exp.i!/ C a exp.i!/ (C234) 2 D1 D0

c a exp.i!/ 1 s.!/ D C (C235) 2 1 a exp.i!/ 1 a exp.i!/

End of Example: Fourier spectral density

Assume a real-valued stochastic process whose covariance function Cov./ D Cov./, namely Cov./ D ŒCov./ C Cov./=2, can be represented by

ZC1 Cov./ D cos.!/s.!/d! (C236) 1 ZC1 Cov./ D 2 cos.!/s.!/d! .cos.!/ D cos.!// (C237) 1 C-8 Special Analysis of One Parameter Stationary Stochastic Process 801

ZC1 1 s.!/ D cos.!/Cov./d (C238) 2 1 ZC1 1 s.!/ D cos.!/Cov./d (C239) 0

Correlation time In practice the notion of “correlation time”or“correlation length” is very appropriate:

ZC1 1 WD Cov./d (C240) 0 Cov.0/ 0 or s.0/ WD (C241) 0 C1R 2 s.!/d! 0

For jj0 there is a unstable correlation or statistical dependence between X.t/ and X.t/ C .Thiscorrelation decreases for growing jj; jj >0.

Ex 5: Stationary Inﬁnite Random Sequence

Example 8.5: Stationary inﬁnite random sequence

The inﬁnite random sequence f ; X1;X0; CX1; g may be structured according to Xt D at X subject to t D 0; ˙1; ˙2;:Thecomplex random variable is structured according to EfXgD0 as well as VarfXgD 2. The random sequence f ; X1;X0; CX1; g is the wider sense stationary if and only if the constants ! within the set-up.

at D exp.it!/ for t D 0; ˙1; ˙2; (C242) are represented. In the stationary case, the harmonic oscillations Xt will be given by the random amplitude and the random phase as well as a constant frequency !. For such a case a set-up of the covariance-function will be calculated.

exp.it!/ D exp it.! C 2k/ for t 2; ˙1; ˙2; (C243) 802 C Statistical Notions, Random Events and Stochastic Processes

ZC Cov./ D exp.it!/s.!/d! for 2; ˙1; ˙2; , (C244) , 1 XC1 s.!/ D exp.it!/Cov./ (C245) 2 D1 If we specialize to X real-valued and uncorrelated we achieve a pure random sequence subject to

Cov./ D 2 , s.!/ D 2=.2/ (C246)

End of Example 8.5: random sequence

Ex 6: Spectral Density of a Random Telegraph Signal

Example 8.6: spectral density of a random telegraph signal Given the covariance function, we intend to ﬁnd a representation of the spectral density for a random telegraph signal.

Cov./ D a exp.bjj/ for a>0;b>0 (C247) ZC1 1 s.!/ D exp.i!/aexp.bjj/d! 2 1 8 9 < Z0 ZC1 = a D exp.i!/d C expŒ.b i!/d 2 : ; 1 0 a 1 1 D C (C248) 2 b i! b C i!

ab s.!/ D (C249) .!2 C b2/

The correlation time is calculated to 0 D 1=b! this result is illustrated by Figs. C.7 and C.8, namely by the spectral structure of the covariance function and its spectral density End of Example: spectral density telegraph signal C-8 Special Analysis of One Parameter Stationary Stochastic Process 803

Fig. C.7 Covariance of Example 8.6

Fig. C.8 Spectral density of Example 8.6

Ex 7: Spectral Representation of a Pulse Modulation

Example 8.7: Spectral density of a pulse modulation

We start with the covariance function of a pulse modulation and study its spectral density. a.T jj/ for jjT Cov./ D (C250) 0 for jj >T

ZCT a s.!/ D Œexp.i!/.T jj/d 2 T 8 9 < ZT ZCT ZCT = a D T exp.i!/d exp.i!/d exp.i!/d 2 : ; T 0 0 8 9 < ZCT = a 2T D sin.!/ 2 cos.!/d (C251) 2 : ! ; 0

a 1 cos !T s.!/ D (C252) !2

Figures C.7 and C.8 illustrate the covariance function and the spectral density of a pulse modulation. A slight modiﬁcation of our example proves that it is not automatic for stationarity. 804 C Statistical Notions, Random Events and Stochastic Processes

Fig. C.9 Covariance of Example 8.7

Fig. C.10 Spectral density of Example 8.7

2 a.T 2/ for jj0 Cov./ D 4 for a>0;T>0 (C253) 0 for jj <0 s.!/ cannot be the spectral density of a stationary stochastic process.

Alternatively, we like to resent the covariance function Cov./ be a Dirac delta- function ı./. In this case we ﬁnd

ZC1 1 ı./ D exp.i!/d (C254) 2 1

ZC1 1 1 s.!/ D exp.i!/ı./d D (C255) 2 2 1 by formal inversion. End of Example 8.7: pulse modulation C-8 Special Analysis of One Parameter Stationary Stochastic Process 805

Fig. C.11 Spectral density of a harmonic function

Ex 8: Spectral Density and Spectral Function of a Harmonic Oscillation

Example 8.8: Spectral density and spectral function of a harmonic oscillation

Given a harmonic oscillation with a given covariance function, we will derive the structure of the spectral density function as well as the spectral function.

Cov./ D a cos !0 (C256) ZC1 a s.!/ D exp.i!/cos.! /d 2 0 1 Z1 a D exp.i!/Œexp.i! / exp.i! /d 4 0 0 1 8 9 < Z1 Z1 = a D expŒi.! C !/ C expŒi.! !/ d (C257) 4 : 0 0 ; 1 1

a s.!/ D fı.! !/ C ı.! C !/g (C258) 2 0 0

2 0 for !<!0 4 S.!/ D a=2 for !0 !0

We illustrate the spectral density as well as the spectral function of a harmonic oscillation by Fig. C.11 as well as Fig. C.12. It has to be emphasized that all three functions Cov./; s.!/ and S.!/ are stationary functions.

End of Example 8.8: Spectral function of harmonic oscillation 806 C Statistical Notions, Random Events and Stochastic Processes

Fig. C.12 Spectral function of type step function for a harmonic oscillation

Ex 9: Spectral Density Function of a Damped Harmonic Oscillation

Example 8.9: Spectral density functions of a damped harmonic oscillation

Given the covariance function of a damped harmonic oscillation,wewanttoﬁnd the spectral density of this stationary stochastic process,

Cov./ D a exp.bjj/ cos !0 (C260) Z1 a s.!/ D exp.b/cos.!/ cos.! /d 0 0 Z1 a D exp.b/Œcos.! ! / C cos.! C !/d (C261) 2 0 0 0

ab 1 1 s.!/ D C (C262) 2 2 2 2 2 b C .! !0/ b C .! C !0/

!0 is called the eigen function of the stochastic process of type stationary. A typical example is the fading of a radio signal.

End of Example 8.9: Damped harmonic oscillation

Ex 10: Spectral Density of White Noise

Example 8.10: Spectral density of white noise

Perviously we introduced the concept of “white noise”asareal-valued stationary process with continuous time of type Wiener process. Given its covariance function, we are interested in the spectral density function.

Cov./ D 2s0ı./ (C263) C-8 Special Analysis of One Parameter Stationary Stochastic Process 807

Z1 1 s.!/ D exp.i!/2s ı./d D S (C264) 0 0 1

A comparative formulation of a white noise process is the following:

White noise is a real-valued process whose spectral density is a constant

Comment the spectral density of a white noise process has only the property s.!/ 0, but not R1 the property s.!/d! < 1. Accordingly, a white noise process with an average 1 power spectrum fulﬁls s.!/d! D1 (C265) does exist only in theory not in practice. A comparison is “the mass point”or “the point mass”, which also exist only “in theory”. Nevertheless it is reasonable approximation of any reality. A stationary process can always be considered as a excellent approximation of white noise if the covariance between Z.t/ and Z.t C/ for growing values jj extremely fast goes to zero.

Example:

Let us denote the variations of the absolute value of the force by Z.t/ acting on a particle of a liquid medium at the time t and leading to a Brownian motion. Per second, the resulting force leads to approximately 1021 collisions with other particles. Therefore the stochastic variables Z.t/ and Z.t C / are stochastically independent (“uncorrelated”) if jj is of the order of 1018 seconds. If we assume a covariance function of type Cov./ D exp.bjj/ for b>0, the number b has to be larger than 1017 seconds. End of Example 8.10: White noise

Ex 11: Band Limit of White Noise

Example 8.11: Band limited white noise

A stationary stochastic process with a constant spectral density function S.!/ D s0 for ! 2f1; C1g does not in practice. But a stationary stochastic process with the spectral density 808 C Statistical Notions, Random Events and Stochastic Processes

Fig. C.13 Spectral density of band limited white noise process

Fig. C.14 Covariance function of a band limited white noise process

s for ˝=1 ! C˝=2 s.!/ D 0 (C266) 0 otherwise exists CZ˝=2 sin.˝/=2 Cov./ D exp.i!/s D 2s (C267) o 0 ˝=2

The average power of the process is proportional to Cov .0/ D s0˝. The parameter ˝ is called the band width of the process.Awhite noise process is generated by the limit of a band limited white noise illustrated by Figs. C.13 and C.14.

End of Example 8.11: band limited white noise

C-84 Spectral Decomposition of the Mean and Variance-Covariance Function

C.8.4: Spectral decomposition of the expectation and the variance-covariance function The decomposition of the spectrum of a stationary process fY.x/;x 2 Rg subject to R 2f1; C1g applied to orthogonal increments of various order has many representations, if they do not overlap in the intervals

Œx1;x2; Œx3;x4 or Œx1;x2;x3; Œx4;x5;x6 etc C-8 Special Analysis of One Parameter Stationary Stochastic Process 809 in the formulations of double, triple etc. differences. An example is

EfŒY.x2/ Y.x1/ŒY.x4/ Y.x4/g (C268)

EfŒY.x3/ 2Y.x2/ C Y.x1/ŒY.x6/ 2Y.x5/ C Y.x4/g (C269)

Accordingly, a real-valued stochastic process with independent increments whose trend function EfY.x/gD0 is zero, has orthogonal increments. Example 8.12: Spatial best linear uniformly unbiased prediction: spatial BLUUP Spatial BLUUP depends on the exact knowledge of proper covariance function. In general, the variance-covariance function depends on three parameters fı1;ı2;ı3g called f rugged, sill, rangeg

Cov./ D .kx1 x2k/ (C270)

We assume stationarity and isotropy for the mean function as well as for the variance-covariance function. (In the next section we deﬁne isotropy)

Model W Z.x/ f.x/0ˇ C e.x/ (C271) Subject to Efe.x/gD0

Postulate one: “ Mean square continuity ”

lim EfZ.y/ Z.x/gD0 (C272) y!x Postulate two: Mean square differentiability

Z.kkCh / Z.kk/ lim n ! 0 (C273) h!0 hn n!1 If we assume “mean square differentiability”, then Z0 has a variance-covariance function. If “mean square differentiability” holds, it is also differentiated in any dimension. Here we introduce various classes of differential calculus applied to variance-covariance functions. (i) DIGGLE chaos: Cov.h/ D c exp.ajhj / (C274) (ii) MATERN class: Cov.h/ D c.˛jhjH.ajhj// (C275) “ H” denotes the “modiﬁed Bessel function of order ” (iii) Whittle class: spectral density of type

˛=2 c.˛2 C !2/ (C276) 810 C Statistical Notions, Random Events and Stochastic Processes

(iv) Stochastic process on curved spaces like the sphere or the ellipsoid: no Matern classes, the characic functions have a discrete spectrum, for instance in terms of harmonic functions.

End of Example 8.12: spatial BLUUP

Let us give some references: Brown et al. (1994), Brus (2007), Bueso et al. (1999), Christenson (1991), Diggle and Lophaven (2006), Fedorov and Muller (2007), Groeningen et al. (1999), Kleijnen (2004), Muller et al. (2004), Muller and Pazman (2003), M¨uller (2005), Omre (1987), Omre and Halvorsen (1989), Pazman and Muller (2001), Pilz (1991), Pilz et al. (1997), Pilz and Spock (2006), PilzSpock(2008), Spock¨ (1997), Spock(2008), Stein (1999), Yaglom (1987), Zhu and Stein (2006).

Up to now we assumed a stochastic process of type fX.t/; t 2 Rg, R Df1; C1g as a complex stationary process of second order. Then there exist a stochastic process of second order fU.!/;! 2 Rg with orthogonal increments which can be represented by ZC1 X.t/ D exp.i!t/dU.!/ (C277) 1 The process fU.!/;! 2 Rg of the process fX.t/; t 2 Rg is called the related spectral process. Its unknown additive constant we ﬁx by the postulate

P fU.1/ D 0gD1 (C278)

subject to

EfU.!/gDW0 (C279) EfŒjU.!/j2gS.!/ (C280) EfŒjdU.!/j2gdS.!/ (C281)

Our generalization was introduced by A. N. Kolmogorov: Compare this representation with our previous model

XK X.t/ D lim exp.i!t/Xk (C282) k!1 kD1

and C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 811

ZC1

Xt D exp.i!t/dU.!/ (C283) 1 The orthogonality of the Kolmogorov stochastic process fU.t/;t 0g corresponds in case of a discrete spectrum to the concept of uncorrelation. Convergence in the squared mean of type

C1R Pn exp.i!t/dU.!/ D lim lim exp.i!k1t/ŒU.!k / U.!k1/ a!1 1 n!C1 D b!C1 ! !0 k 1 k (C284)

emphasizes the various weighting functions of frequencies: The frequency !k1 in the range Œ!k1;!k receives the random weight U.!k/ U.!k1/. A analogue version of the inversion process leads to

ZC1 i exp.i! t/ exp.i! t/ U.! / U.! / D 2 1 X.t/dt (C285) 2 1 2 t 1

In practice, Kolmogorov stochastic processes are applied in linear ﬁlters for prog- nosis or prediction.

C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes of Multi-Parameter Systems

Appendix C.9: Scalar-, vector- and tensor-valued stochastic processes of multi- parameter systems We have experienced the generalization of a scalar-valued stochastic process due to A.N. Kolmogorov. Another generalization due to him is the concept of the structure functions for more general vector-valued and tensor-valued for increments of various order as well as their power spectrum or multi-parameter systems.

First we introduce the notion of the characteristic functional replacing the notion of the characteristic function for random function. Second,themoment representation for stochastic processes is outlined. Third we generalize the notion from scalar-valued and vector-valued stochastic processes to tensor-valued stochastic processes of multi-point parameter systems. In particular, we introduce the notion 812 C Statistical Notions, Random Events and Stochastic Processes of homogeneous and isotropic stochastic ﬁelds and extent the Taylor-Karman structured criterion matrices to distributions of the sphere.

C-91 Characteristic Functional

Let us begin with the deﬁnition of the characteristic functional.

Zb YxŒ D Efexp.iY.//gDEfig .x/Y.x/dx (C286) a Y./ as the characteristic functional elates uniquely to the random function Y.x/ of a ﬁxed complex number. If the characteristic functional of a speciﬁc random function is known we are able to derive the probability density distribution fX1;:::;XN .YX ;:::;YXN /.

Example

If a scalar-valued function in a three dimensional ﬁeld x WD .x1; x2; x3/ exists deﬁne its characteristic functional by

Z ZC1Z YY Œ WD Efexp.i Y.x1; x2; x3/.x1; x2; x3//dx1dx2 (C287) 1 The domain of integration is very often chosen to be spherical coordinates of ellipsoidal coordinates .u; v; w/ for u 2f0; 2g; v 2f=2 < v < C=2g and w 2f0

Probability distribution for random functions and the characteristic functional

As a generalization of the characteristic function for random events, the concept of the characteristic functionals for random functions or stochastic processes has been developed. Here we aim at reviewing the theory. We apply the concept of random functions for multi-functions of type scalar-valued and vector-valued functions which have been ﬁrst applied in the statistical theory turbulence.

what is the characteristic functional? We summarize in Table C.9.1 the detailed notion of a scalar-valued as well as a vector-valued characteristic functional illustrated by an example by introducing the “Fourier-Stieljes functional”. C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 813

Table C.9.1 Probability distribution of a random function and the characteristic functional

“Scalar-valued characteristic functionals”

“Random function over an interval [a,b] with N-variables”

fY.x/; x 2 RN g is called a random function if the density function

fx1;x2;:::;xN 1;xN ŒY.x1/; Y.x2/; : : : ; Y.xN 1/; Y.xN / (C288)

exist at .x1;x2;:::;xN 1;xN / points.

“Fourier representation of the distribution function”

P Y . ;:::; / WD Ef Œi. N Y /g x1;:::;xR N 1R NP exp kD1 k k C1 N (C289) D 1::: expfi kD1 kYkgfx1;:::;xN .Y1;:::;YN /dY1 :::dYN

subject

Y1 WD Y.x1/;:::;Yk WD Y.xk/ (C290)

“Existence condition”

Rb YŒ1;:::;N D .x/dx (C291) a

“Vector-valued characteristic functionals”

˛ “at any point P of the domain D place a set of base vectors e .Pi / at ﬁxed points Pi for ˛ D 1;2;:::;n 1; n; i D 1;2;:::;N 1; N .

˛ ˛ Yi D;Y.P i/je .Pi /> (C292)

“Vector-valued characteristic functionals” P n YYŒ W .x;y;z/ D ˛D1 ˛.Pi /Y.Pi / (C293)

End of Table C.9.1: characteristic functional 814 C Statistical Notions, Random Events and Stochastic Processes

C-92 The Moment Representation of Stochastic Processes for Scalar Valued and Vector Valued Quantities

The moment representation of stochastic processes is the standard tool of statistical analysis. If a random function is given by its probability function, then we find a unique representation of its moments. But the inverse process of a given moment representation converted in its probability function is not unique. Therefore we present only the direct representation of moments for a given probability function. At first we derive for a given scalar-valued probability function its moment representation in Table C.9.2. In Table C.9.3 we generalize this result for stochastic processes which are vector-valued.Anexample is presented for second order statistics, namely for the Gauss-Laplace normal distribution and its first and second order moments.

Table C.9.2 Moment representation of stochastic processes for scalar quantities

“scalar-valued stochastic processes”

“moment representation”

Y.x/;x 2 RN is a random function, for instance applied in turbulence theory. Deﬁne the set of random function fx1;x2;:::;xN 1;xN g3R with the probability distribution f.x1;x2;:::;xN 1;xN /. Then the moment representation is the polynomial

p1;p2;:::;N 1;N M WD Efx1;x2;:::;xN 1;xN g RR C1RR 1 2 N 1 N D ::: x1 x2 :::xN 1 xN f fx1;x2;:::;xN 1;xN g (C294) 1

dx1dx2 :::dxN 1dxN

(nonnegative even numbers) p1;:::;pN W 1;2;:::;pN 1;pN 2 NN

Œp WD 1 C p2 CCpN 1 C N (C295)

is called the order of the moment representation C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 815

Table C.9.2 Continued

“central moment representation”

fx1 Efx1g;x2 Efx2g;:::;xN 1 EfxN 1g;xN EfxN gg (C296) 12:::pN 1;pN 1 B WD Ef.x1 Efx1g/ ; 2 N 1 .x2 Efx2g/ ;:::;.xN 1 EfxN 1g/ ; N .xN EfxN g/ g (C297) “Characteristic function” RR ˚.1;:::;N / WD C1 ::: 1 (C298) RR P expfi k kxk gf fx1;x2;:::;xN 1;xN gdx1dx2 :::dxN 1dxN

“differentiate at point 1 D 0;:::;N .0/” h i Œ Œ 1p2:::pN 1;pN @ .i/ M D p p ˚. ;:::; / (C299) @ 1 :::@ N 1 N 1 N 1D0;:::;N D0

“semi-invariants”

‰ WD ln ˚ (C300)

h S 1;:::;N WD i Œ Œ @Œ .i/ .1/ p ‰. ;:::; / (C301) @ 1 @ 2 :::@ N 1 @ N 1 N 1 2 N 1 N 1D0;:::;N D0

“transformation of semi invariants to moments of order”

2 Si D M1;S2 D M2 M1 D B2; 3 S3 D M3 3M2M1 C 2M1 D B3 2 S4 D B1 3B2 ;S5 D B5 10B3B2 (C302)

End of Table C.9.2: Moment representation of stochastic processes 816 C Statistical Notions, Random Events and Stochastic Processes

Table C.9.3 Moment representation of stochastic processes for vector- valued quantities

“Vector-valued stochastic processes”

“moment representation”

“multi-linear functions of a vector ﬁeld Gauss-Laplace distributed” P n ˛ YYŒ W .x/ D ˛D1 ˛.Px/Y .Pi / (C303)

˛ for a set of base vectors e .Pi / at ﬁxed points P˛;˛D 1;2;:::;n 1; nI i D 1;2;:::;N 1; N

“one-point n-dimensional Gauss-Laplace distribution”

Y.P˛/ f.P˛/ ! PN 1 i i j j f.PN / D C exp 2 gij .P˛/.Y˛ EfY g/.Y˛ EfY˛g/ (C304) i;j

“gauge”

RR C1RR ::: f.P˛/dY1dY2 :::dYN 1dYN WD 1 (C305) 1

“two-point n-dimensional Gauss-Laplace distribution: covariance function”

˛;ˇ ˛ ˛ ˇ ˇ B .P˛;Pˇ/ D EfŒY .P˛/ EfY gŒY .Pˇ/ EfY gg (C306)

Y.P˛/; Y.Pˇ/ f.P˛ ;Pˇ/ (C307) P 1 N i i i i f.P˛;Pˇ/ D D exp 2 i;jD1 gij .P˛;Pˇ/.Y˛ EfY˛ g/.Yˇ EfYˇ g/ (C308)

Example : Gauss-Laplace normal distribution, character’s functional

“ansatz” C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 817

Table C.9.3 Continued

Rb Rb A./ D EfX./gWDEf .Y/Y.x/gdx D EfY.x/gdx (C309) a a

Rb Rb B./ D B.x1;x2/.x1/.x2/dx1dx2 (C310) a a

“multi-point functions”

Rb Rb Rb 1 Y.Y /Œ.x/ WD expfi .x/EfY.x/gdx 2 B.x1;x2/.x1/.x2/dx1dx2 a a a (C311)

in general: multi-point functional series Fourier-Stieljes functional series Gauss-Laplace normal distribution functional: ﬁrst and second order moments

An example is the central moment of fourth order of type two-points subject to a Gauss-Laplace normal distribution:

˛ˇ; ı ˛;ˇ ı B .P1;P2/ D B .P1/B .P2/C ˛; ˇı ˛ı ˇı CB .P1;P2/B .P1;P2/ C B .P1;P2/B .P1;P2/ (C312)

general rule by JSSERLIS: on a formula for the product moment coefﬁcient in any number of variables Biometrika 12(1918) No. 138

First and Second order moments of a probability density and the uncertainty relation

Rb ŒY.x/ EfY.x/0.x/dx D 0 (C313) a

Rb Rb B.x1;x2/.x1/.x2/dx1dx2 D 0 (C314) a a 818 C Statistical Notions, Random Events and Stochastic Processes

C-93 Tensor-Valued Statistical Homogeneous and Isotropic Field of Multi-Point Systems

First, we introduce the ideal structure of a variance-covariance matrix of multi-point systems in an Euclidean space called homogenous and isotropic in the statistical sense.

Such a notion generalized the concept of stationary functions so far introduced. Throughout we use the terminology “signal”and“signal analysis”. Let us begin with the notion of an ideal structure of the variance-covariance matrix of scalar value called signal, a multi-point function. An example is the gravity force as a one-point function, namely the distance between two-points as a two-point function or the angle between to a reference point and two targets as a three-point function. Here, we are interested in a criterion matrix of vector-vector signals which are multi-point functions. An example is the placement vector between network points determined by an adjustment process, namely a vector-valued one-point function proportional to the difference vector between network points, a vector- valued two-point function. Let S.X1; ; X˛/ be a scalar-valued function between placement vectors X1; ; X˛ with in a three dimensional Euclidean space. The variance-covariance of the scalar-valued signal at placement vectors X1; ; X˛ and X˛C1; ; Xˇ is deﬁned by P .X1; ; X˛; X˛C1; ; Xˇ/ WD EfŒS.X1; ; X˛/ EfS.X1; ; X˛/g

ŒS.X˛C1; ; Xˇ/ EfS.X˛C1; ; Xˇ/gg (C315)

ForP the special case X1; ; X˛; X˛C1; ; Xˇ the multi-point function agrees to .X1; ; X˛; X˛C1; ; Xˇ/ to the variance in the other case to the covariance.

Deﬁnition C.9.1: Homogeneity and isotropy of a scalar-valued multi-point function P The scalar-valued multi-point function .X1; ; X˛; X˛C1; ; Xˇ/ is called homogenous in the statistical sense.ifitistranslational invariant, namely X X .X1 C t; ; X˛ C t;X˛C1 C t; ; Xˇ C t/ D .X1; ; X˛; X˛C1; ; Xˇ/ P (C316) t is called the translational vector. The function is called the isotropic,ifitis rotational invariant in the sense of X X .RX1; ; RX˛; RX˛C1; ; RXˇ/ D .X1; ; X˛; X˛C1; ; Xˇ/ (C317) R is called the rotational matrix. C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 819

End of Deﬁnition C.9.1: homogeneity and isotropy

The following representation theorems apply: Lemma C.9.2: homogeneity and isotropy of a scalar-valued multi-point function

The scalar-valued multi-point function is homogenous in the statistical sense if and only if it is a function of the difference vectors X2 X1; ; Xˇ Xˇ1: X X .X1; ; X˛; X˛C1; ; Xˇ/ D .X2 X1; ; X˛C1 X˛; ; Xˇ Xˇ1/ (C318) This function is isotropic in the statistical sense, P .X1; ; X˛; X˛C1; ; Xˇ/ X D .jX1j;; < X1; X3 >; ;; jXˇj/ (C319) is a function of all scalar products including the length of the vectors X1; ; X˛; X˛C1; ; Xˇ. P The function .X1; ; X˛; X˛C1; ; Xˇ/ is homogenous and isotropic in the statistical sense if P .X1; ; X˛; X˛C1; ; Xˇ/ X D .jX2 X1j;; ;

; ;; jXˇ Xˇ1j/ (C320) is a function of all scalar products including the length of the difference vectors X2 X1; ; X˛C1 X˛; Xˇ Xˇ1. End of Lemma C.9.2: homogeneity and isotropy P As a basic result we take advantage of the property that a scalar-valued function of type multi-points is homogenous and isotropic in the statistical sense if

P .X1; ; X˛; X˛C1; ; Xˇ/ (C321) X D .jX2 X1j; jX3 X1j; ; jXˇ Xˇ2j; jXˇ Xˇ1j/

is a function of length of all difference vectors X Xı for all 1 ı< 1.For a proof we need the decomposition 820 C Statistical Notions, Random Events and Stochastic Processes

1 < X; Y > D .jXj2 CjYj2 jX Yj2/ 2 1 D .jX C Yj2 jX Yj2/ (C322) 2

Example: four-point function, var-cov function

Choose a scalar-values two-point function as the distance S.X1; X2/ between two points X1 and X2. The criterion matrix of a variance-covarianceP function under the postulate of homogeneity and isotropy in the statistical sense .X1; X2; X3; X4/ as a four-point function described by ˇ.ˇ 1/=2 D 6 quantities, for instance P .X1; X2; X3; X4/ X D .jX2 X1j; jX3 X1j; jX4 X1j; jX3 X2j; jX4 X2j; jX4 X3j/ (C323)

End of example: var-cov function, 4 points

Another example is a scalar-valued three-point function of the type angular observation A.X1; X2; X3/ between a ﬁxed point X1 and the two moving points X2 and X3. The situation of vector-valued functions between placement vectors X1; ; X˛ in a three-dimensional Euclidian space based on the representation

X3 i S.X1; ; X˛/ D Si .X1; ; X˛/e .X1; ; X˛/ (C324) iD1 is slightly more complicated. the Euclidean space is spanned by the base vectors fe1;e2;e3g ﬁxed to a reference point. The variance-covariance of a vector signals at placements X1; ; X˛ and X˛C1; ; Xˇ is deﬁned as the second order tensor X X i j WD .X1; ; X˛; X˛C1; ; Xˇ/e .X1; ; X˛/ ˝ e .X˛C1; ; Xˇ/ i;j (C325) when we have applied the sum notion convention over repeated indices. P i;j .X1; ; X˛; X˛C1; ; Xˇ/ WD EfŒSi .X1; ; X˛/EfSi .X1; ; X˛/g

ŒSj .X˛C1; ; Xˇ/ EfSj .X˛C1; ; Xˇ/gg (C326) We remark that all Latin indices run 1; 2; 3 but all Greek indices label the number of multi-points.P For the case X1 D X˛C1; ; X˛ D Xˇ we denote the multi-point function i;j .X1; ; X˛; X˛C1; ; Xˇ/ as the local variance covariance matrix for the general case nonlocal covariance C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 821

Deﬁnition C.9.3: Homogeneity and isotropy in the statistical sense of a tensor valued multi-point function P The multi point tensor valued function .X1; ; X˛; X˛C1; ; Xˇ/ is called homogenous in the statistical sense,ifitistranslational invariant in the sense of X X .X1 C t; ; X˛ C t; X˛C1 C t; ; Xˇ C t/ D .X1; ; X˛; X˛C1; ; Xˇ/ (C327) Where t denotes the translational vector P In contrast the multi-point tensor-valued function .X1; ; X˛; X˛C1; ; Xˇ/ is called isotropic in the statistical sense if it is rotational invariant in the sense of X X .RX1; ; RX˛; RX˛C1; ; RXˇ/ D R .X1; ; X˛; X˛C1; ; Xˇ/R (C328) where R denotes the rotation matrix element of the SO.3/ group End of tensor-valued multi-point functions of key importance are the following representations for tensor-valued multi-point homogenous and isotropic functions

Lemma C.9.4: homogeneity and isotropy in the statistical sense of tensor-valued symmetric multi-point functions A second order symmetric tensor-valued multi-point function is homogenous if, X X .X1; ; X˛; X˛C1; ; Xˇ/ D .X2 X1; ; X˛C1 X˛; ; Xˇ Xˇ1/ (C329) is a function of difference vector .X2 X1; ; X˛C1 X˛; ; Xˇ Xˇ1/.This function is isotropic if P .X1; ; X˛; X˛C1; ; Xˇ/ i j D 0.jX1j;; < X1; X3 >; ;; jXˇj/ıij e ˝ e Xˇ Xˇ C .jX1j;; < X1; X3 >; ;; jXˇj/ D1

Œ.X ˝ X/ C .X ˝ X / (C330) for a certain isotropic scalar-valued function 0 and for all 1 ˇ

Finally such a function is homogenous and isotropic in the statistical sense,ifand only if 822 C Statistical Notions, Random Events and Stochastic Processes P .X1; ; X˛; X˛C1; ; Xˇ/ i j D 0.jX2 X1j; jX3 X1j; ; jXˇ Xˇ2j; jXˇ Xˇ1j/ıij .e ˝ e / Xˇ Xˇ C .jX2 X1j; jX3 X1j; ; jXˇ Xˇ1j/ D2

Œ.X X1/ ˝ .X X1/ C .X X1/ ˝ .X X1/ (C331) holds for a certain homogenous and isotropic scalar-valued functions 0 and for all 2 ˇ

End of Lemma C.9.4: tensor-valued function var-cov functions

For the applications the Taylor-Karman structure for the case ˇ D 2 is most important: Corollary C.9.5: ˇ D 2, Taylor-Karman structure for two-point tensor function homogeneity and isotropy A symmetric second order two-point tensor-function is homogenous and isotropic in the statistical sense if and only if

˙.X1; X2/ i j D 0.jX2 X1j/ıij .e ˝ e /

C 22.jX2 X1j/Œ.jX1 X2j/ ˝ .jX2 X1j/ i j D 0.jX2 X1j/ıij C 22.jX2 X1j/< X2 X1; e > i j D ˙ij .X1; X2/.e ˝ e / (C332)

(summation convention) and

2 ˙ij D 0.jX2 X1j/ıij C Œ22.jX2 X1j/jX2 X1j C 0.jX2 X1j/

Xi Xi 0.jX2 X1j/ 2 (C333) jX2 X1j C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 823

˙ij .X1X2/ D ˙m.jX2 X1j/ıij (C334)

Xi Xi CfŒ˙l .jX2 X1j/ ˙m.jX2 X1j/ 2 jX2 X1j

˙m and ˙l are the homogenous and isotropic scalar-values functions

End of Corollary C.9.5: ˇ D 2, Taylor-Karman structure The representation of the symmetric second-order two-point tensor-valued function under the postulate of homogeneity and isotropy has been analyzed by G. J. Taylor (1935) and T. Karman (1937) dealing with problems of Hydrodynamics.Itwas introduced into Geodetic sciences by E. Grafarend(1970, 1972). Most often the homogenous and isotropic scalar-valued function ˙m and ˙l are called longitudinal and lateral correlation function. It has been C.C. Wang who wrote ﬁrst the symmetric second order two-point tensor-valued function being homogenous and isotropic if

i j ˙.X1X2/ D a1.X2 X1/ıij Œe X1 ˝ e X2 C a2.X2 X1/Œ.X2 X1/ ˝ .X2 X1/ (C335) holds. This representations refers to a1.X2X1/ D b1.jX2X1j/ and a2.X2X1/ D b2.jX2 X1j/ according to C. C. Wang (1970 p, 214/215) as scalar-valued isotropic functions. In case we decompose the difference vector

i j i j X2 X1 D < X2 X1je .X1/>ŒeX1 ˝ e X2 D i j D Xi Xj Œe X1 ˝ e X2 (C336)

˙ij .X1X2/ D b1.jX2 X1j/ıij C b2.jX2 X1j/Xi Xj for any choice of the basis ei1 ;ej2 .Aspecial choice is to put parallel to the basic vector X2 X1 such that

2 ˙11 D b1.jX2 X1j/ C b2.jX2 X1j/jX2 X1j (C337)

˙22 D ˙22 D b1.jX2 X1j/ (C338)

2 3 The base vector e and e are orthogonal to the “longitudinal direction” X2 X1. For such a reason b1.jX2 X1j/ WD ˙m.jX2 X1j/ (C339) and 824 C Statistical Notions, Random Events and Stochastic Processes

Fig. C15 Longitudinal and lateral correlation: Two-point function

1 b2.jX2 X1j/ WD 2 f˙l .jX2 X1j/ ˙m.jX2 X1j/g (C340) jX2 X1j of the scalar-valued isotropic functions ˙l .jX2 X1j/ and ˙m.jX2 X1j/.Theresult is illustrated in Fig. C.9.1 A two-dimensional Euclidean example is the vector-valued one-point function of a network point. Let 2 3 x x x y x x x y 6 1 1 1 1 1 2 1 2 7 6 y x y y y x y y 7 ˙ D 4 1 1 1 1 1 2 1 2 5 (C341) x2x1 x2y1 x2x2 x2y2

y2x1 y2y1 y2x2 y2y2 be the variance-covariance matrix of the two-dimensional network of two points with Cartesian coordinates. The variance-covariance matrix will be characterized by the variance D 2 of the X-coordinate of the ﬁrst point, the covariance x1x1 x1 x1y1 of the x;y coordinates of the ﬁrst point and the second point and so on. The discrete variance-covariance matrix

˙ D EfŒX EfXgŒX EfXgg (C342)

˙ij .X1; X2/ D EfŒxi .X1/ Efxi X1gŒxj Efxj X2gg (C343) or

xi .X1/ Dfx1X1;x2X1gDWfx1;y1g

xj .X2/ Dfx1X2;x2X2gDWfx2;y2g C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 825

Fig. C16 Homogenous and isotropic dispersion situation: (a) general, (b)homogenous and (c) homogenous-isotropic are the estimated coordinates of the Cartesian vectors of the ﬁrst network point X1 or the second network point X2. The two-point functions ˙ij .X1; X2/ are the coordinates labeling the tensor of second order. The local variances and covariances are usually illustrated by local error ellipses:

˙ij .X1; X2 D X1/ (C344)

and

x1x1 x1y1 x2x2 x2y2 ˙ij .X1 D X2; X2/; and (C345) y1x1 y1y1 y2x2 y2y2

The nonlocal covariances are contained in the function ˙ij .X1; X2/ for the case X1 ¤ X2, in particular x1x2 x1y2 (C346) y1x1 y1y2

Figure C.9.2 illustrates the postulates of homogeneity and isotropy for variance- covariance matrices of cartesian coordinates left, we illustrate the general dispersion situation in two-dimensional networks, of course those local dispersion ellipses. In the center, the homogenous dispersion situation: At each network points, due the translation invariance all dispersion ellipses are equal. In the right side, the homogenous and isotropic dispersion situation of type Taylor-Karman is ﬁnally illustrated. “All dispersion ellipses are equal and circular” 826 C Statistical Notions, Random Events and Stochastic Processes

Fig. C17 Correlation ﬁgures of type (a) ellipse and (b) hyperboloid

Finally we illustrate“longitudinal”and“lateral” within homogenous and isotropic networks by Fig. C16, namely the characteristic ﬁgures of type (a) ellipse and (b) hyperboloid. Those characteristic functions ˙l and ˙m are described the eigen values of the variance-covariance matrix. If the eigenvalues are positive, then the correlation ﬁgure is an ellipse. But if the two eigen values change sign (either “C”or“”or“”or“C”) the characteristic function has to be illustrated by a hyperboloid.

Ex 1: Two-Dimensional Euclidean Networks Under the Postulates of Homogeneous and Isotropy in the Statistical Sense

Example C.9.1: Two-dimensional Euclidean networks under the postulates of homogeneity and isotropy in the statistical sense Two-dimensional network in a Euclidean space are presented in terms of the postulates homogeneity and isotropy in the statistical sense. At the beginning, we assume a variance-covariance matrix of absolute cartesian net coordinates of type homogenous and isotropic, in particular with Taylor Karman structure.Given such a dispersion matrix, we derive the associated variance-covariance matrix of coordinate differences, a special case of the Kolmogorov structure function, and directions, angles and distances. In order to write more simple formulae, we assume quantities of type EfYgD0 Deﬁnition C.9.6: Variance covariance matrix of Cartesian coordinate, coordinate differences, azimuths, angular observations and distances in two-dimension Euclidean networks C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 827

Let ıxi .X1/ be absolute Cartesian coordinates ıxi .X1/ ıxi .X2/ relative cartesian coordinates or coordinate differences, ı˛.X1; X2/ the azimuths of a line observation point X1 to a target point X2, the difference ı˛.X1; X2/ to ı˛X1; X3/ as angular observations or azimuth differences and ıs.X1X2/ the distances from the point X1 to the point X2.Then

˙ij .X1; X2/ D Efıxi .X1/ıxj .X2/g (C347)

˚ij .X1; X2; X3; X4/ D EfŒıxi .X1/ ıxi .X2/Œıxj .X3/ ıxj .X4/g (C348)

.X1; X2; X3/ D EfŒı˛.X1; X2/ı˛.X3; X4/g (C349)

.X1; X2; X3; X4; X5; X6/ D EfŒı˛.X1; X2/ ı˛.X1; X3/

Œı˛.X4; X5/ ı˛.X4; X6/g

˘.X1; X2; X3; X4/ D Efıs.X1; X2/ ıs.X3; X4/g (C350) are the derived variance-covariance matrices for the increments Y.x1; ;xi / y.x1; ;xi / D ıy.x1; ;xi /fıxi :ıxi ıxj ;ı˛;ı˛i ı˛j ;ısg with respect to a model y.

End of Deﬁnition C: derived var-cov matrices

Let the variance-covariance matrix ˙ij .X1; X2/ of absolute Cartesian coordinates of the homogenous and isotropic type the characteristic isotropic functions ˙l and ˙m given. Then we derive the variance-covariance of

Lemma C.9.7: Representation of derived variance-covariance matrices of Taylor- Karman structured variance-covariance of absolute Cartesian coordinates.

Let the variance-covariance matrix of absolute Cartesian ij .X1; X2/ be isotropic and homogenous in terms of the Taylor-Karman structure be given. Then the derived variance-covariance matrices are structured according to

˚ij .X1; X2; X3; X4/ D ˙ij .X1; X3/ ˙ij .X1; X4/

˙ij .X2; X3/ ˙ij .X2; X4/ (C351)

.X1; X2; X3; X4/ D ai .X1; X2/aj .X3; X4/˚ij .X1; X2; X3; X4/ (C352)

.X1; X2; X3; X4; X5; X6/ D .X1; X2; X4; X5/ .X1; X2; X4; X6/

.X1; X3; X4; X5/ C .X1; X3; X4; X6/ (C353)

˘.X1; X2; X3; X4/ D bi .X1; X2/bj .X3; X4/˚ij .X1; X2; X3; X4/ (C354)

subject to 828 C Statistical Notions, Random Events and Stochastic Processes

˙ij .Xˇ; X˛/ D ˙m.jXˇ X˛j/ıij C Œ˙l .jXˇ X˛j/

˙m.jXˇ X˛j/bi .Xˇ; X˛/bj .Xˇ; X˛/ (C355)

a1.X˛; Xˇ/ WD Œy.P X˛/ Py.Xˇ/ Œs.P Xˇ; X˛/ (C356) 2 a2.X˛; Xˇ/ WD CŒx.P X˛/ Px.Xˇ/ ŒsP .x˛;xˇ/ (C357) 2 b1.X˛; Xˇ/ WD CŒx.P X˛/ Px.Xˇ/ ŒsP .x˛;xˇ/ D cos ˛Pˇ˛ (C358) 2 b2.X˛; Xˇ/ WD CŒy.P X˛/ Py.Xˇ/ ŒsP .x˛;xˇ/ D sin ˛Pˇ˛ (C359)

End of Lemma C.9.7: Taylor-Karman derived structures For the proof, we apply summation convention with respect to two identical indices. In addition, the derivations are based on

ı˛.X1; X2/ D a1.X1; X2/Œıx.X1/ ıx.X2/ C a2.X1; X2/Œıy.X1/ ıy.X2/ (C360)

ıs.X1; X2/ D b1.X1; X2/Œıx.X1/ ıx.X2/ C b2.X1; X2/Œıy.X1/ ıy.X2/ (C361)

In order to be informed about local variance-covariance we need special results about the situation X1 D X3; X2 D X4:

Lemma C.9.8: derived local variance-covariance matrices of a Taylor-Karman structure variance-covariance matrix of absolute Cartesian coordinates

˚ij .X1; X2/ D 2Œ˙m.0/ıij .jX2 X1j/

D ˚m.jX2 X1j/ıij C Œ˚l .jX2 X1j/

˚m.jX2 X1j/bi .X1; X2/bj .X1; X2/ (C362)

˚l .jX2 X1j/ D 2Œ˙l .0/ ˙l .jX2 X1j/ (C363)

˚m.jX2 X1j/ D 2Œ˙m.0/ ˙m.jX2 X1j/ (C364)

.X1; X2/ D 2ai .X1; X2/aj .X1; X2/Œ˙m.0/ıij ˙.X1; X2/ 2 D Œ˚m.jX2 X1j/ ŒjX2 X1j (C365) C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 829

.X1; X2; X3/ D .X1; X2/ 2.X1; X2; X1; X3/ C .X1; X3/

˚m.jX2 X1j/ ˚.jX3 X1j/ D 2 C 2 ai .X1; X2/aj .X1; X3/ jX2 X1j jX3 X1j

Œ˚ij .X1; X2/ ˚ij .X2; X3/ ˚ij .X1; X3/

< X3 X2; X3 X1 D ˚m.jX2 X1j/ 2 2 jX2 X1j jX3 X1j

< X2 X1; X3 X2 > ˚m.jX3 X2j/ 2 2 jX2 X1j jX3 X2j

< X2 X1; X3 X1 > C˚m.jX3 X2j/ 2 2 jX2 X1j jX3 X1j

CŒ˚m.jX2 X1j/ ˚.jX3 X2j/ 2 2 ŒjX2 X1j jX3 X1j < X2 X1; X3 X1 > 2 2 2 (C366) ŒjX2 X1j jX3 X1j jX3 X2j

˘.X1; X2/ 2bi .X1; X2/j˙m.0/ıij ˙ij .X1; X2/ D ˚m.jX2 X1j/ (C367)

End of Lemma C 9.8: Derived local var-cov matrices The results can be interpreted as following.

First, the scalar-valued functions .X1; X2/; .X1; X2; X3/ and ˘.X1; X2/ are homogenous and isotropic. Second,ofthelocal variance-covariance matrix of Cartesian coordinate differences has the structure of nonlocal covariance matrix of absolute Cartesian coordinates as long as we depart from a Taylor-Karman matrix: Third, not every homogenous and isotropic local variance-covariance matrix of Cartesian coordinate differences produces derived homogenous and isotropic functions as we illustrated in our next example.

Of course, not every homogenous and isotropic local variance-covariance matrix of type coordinate difference has the structure of being derived by the transformation of absolute coordinates into coordinate difference. For instance, for the special case X1 D X2 the variance-covariance matrix ˙ij .X1; X2/ due to ˙m.0/ D ˙l .0/ can be characterized by circles. The limits

˙ij .X1; X1/ D ˙ij .X1; lim X2/ D lim ˙ij .X1; X2/ (C368) X2!X1 X2!X1 are ﬁxed for any direction of approximation. However, due to ˚m.jX2 X1j/ ¤ ˚l .jX2 X1j/ lead us to symmetry breaking in a two-dimensional EUCLIDEAN 830 C Statistical Notions, Random Events and Stochastic Processes space: Though we have started from a homogenous and isotropic variance- covariance matrix of absolute Cartesian coordinates into relative coordinates or coordinate differences the variance-covariance matrix ˚ij .X1; X2; X3; X4/ is not homogenous and isotropic, a result mentioned ﬁrst by W. Baarda (1977): The error eclipses are not circular.Inathree dimensional Euclidean space.Incontrast,we receive rotational symmetric ellipsoids based on the difference vector X2 X1 with respect to the axis of rotation and circles as intersection surfaces orthogonal to the vector X2 X1.

In order to summarize our results we present

Lemma C.9.9: Circular two-point functions

The image of the two-point function ˚ij .X1; X2/ D ˚.X1; X2; X1; X2/ is a circle if and only if ˚l .jX2 X1j/ D ˚m.jX2 X1j/ or ˙l .jX2 X1j/ D m.jX2 X1j/

End of Lemma C.9.9: Circular two-point functions The result is illustrated by an example: Assume network points with a distance function jX2 X1jD1. In this distance the longitudinal and the lateral correlation functions are assumed to the values 1/2 and 1/4. Then there holds

˙l .0/ D ˙m.0/ D 1 (C369) 1 1 ˙ D ;˙ D (C370) l 2 m 4 3 3 ˚ D 1; ˚ D ;.1/D ;˘.1/D 1 (C371) l m 2 2

Lemma C.9.10: Derived local variance-covariance matrix for a variance- covariance matrix of absolute Cartesian coordinate of Special Taylor-Karman structure ˙l D ˙m Let the variance-covariance matrix of absolute Cartesian coordinates with Special Taylor-Karman structure ˙l D ˙m for homogenous and isotropic signals given. Then the derived variance-covariance matrices are structured accordingly to

˚ij .X1; X2/ D 2Œ˙m.0/ ˙m.jX2 X1j/ıij

D ˚.jX2 X1j/ıij (C372)

˚l .jX2 X1j/ D ˚m.jX2 X1j/ D 2Œ˙m.0/ ˙m.jX2 X1j/ (C373) C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 831

2 .X1; X2/ D 2 Œ˙m.0/ ˙m.jX2 X1j/ jX2 X1j 1 D ˚m.jX2 X1j/ (C374) jX2 X1j 1 .X1; X2; X3/ D 2 2 jX2 X1j jX3 X1j

Œ˚m.jX2 X1j/

˚m.jX3 X1j/

C˚m.jX3 X2j/ (C375)

˘.X1; X2/ D 2Œ˙m.0/ ˙m.jX2 X1j/ D ˚m.jX2 X1j/ (C376)

End of Lemma C 9.10: Special Taylor-Karman matrix Our example is characterized now by a homogenous and isotropic variance- covariance matrix of Cartesian absolute coordinates which are the gradient of a homogenous and isotropic scalar-valued function. In addition, we assume that the scalar valued function is an element of a two-dimensional Markov process of first order. Its variance-covariance matrix is characterized by normalized longitudinal and lateral correlation function which possess the distance function s WD jX2 X1j between network points d the characteristic distance, K0 and K1 the modified Bessel function of the second kind of zero and first order. The characteristic distance dis the only variable or degree of freedom. Within the parameter d we depend on our experience with geodetic networks. Due to the property ˙l .0/ D ˙l .0/ we call the longitudinal and lateral normalized. In order to find the scale for the variance- covariance matrix we have to multiply by the variance 2.Themodified Bessel function K0 and K1 are tabulated in Table C18 in the range 0.1(0.1)8 where we followed M. Abramowtz and I. Stegun (1970 p. 378-379). In Fig. C18 we present the characteristic functions of type absolute coordinates 2 1 2 ˙l .X/ D4x C 2K0.X/ C 4x K1.X/ C 2xK1.X/ and ˙m.X/ DC4x 1 2K0.X/2K0.X/4x K1.X/. Next is the representation of the derived functions ˚l .X/ D ˘.X/ D 2˙m.0/ 2˙l .X/; ˚m.X/ D 2˙m.0/ 2˙m.X/ and .X/ D 2 2 2x j˙m.0/ ˙m.X/jD x ˚m.X/. Figures C19 and C20 illustrate the various variance-covariance functions. An explicit example is a two-dimensional network in Fig. C21. Let point one be the center of a fixed (x,y) Cartesian coordinate system. In consequence, we measure nine points by distance observations assuming the distance between point one and point two is unity, namely p p p 1; 2; 2; 5; 8 The related characteristic functions for a characteristic distance d=1 are approximately given in Table C18. The variance-covariance matrix of absolute Cartesian 832 C Statistical Notions, Random Events and Stochastic Processes

Fig. C18 Modiﬁed Bessel function K0.X/ and K1.X/ longitudinal and lateral function of absolute Cartesian coordinates f1.X/ DW ˙l .X/ and f2.X/ DW ˙m.X/, longitudinal and lateral correlation of relative Cartesian coordinates f3 WD ˚m.X/ and f4 WD ˚l .X/, variances of directions f5 WD .X/ and distances ˘.X/ DW .X/ of a two-dimensional conservation Markov process of ﬁrst order C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 833

Fig. C19 Characteristic functions l and ˙m for Taylor-Karman structured varaince-covariance matrix of Cartesian coordinates

coordinates is relative Cartesian coordinates or coordinate differences is computed 2 with respect to D 1:09; y1y2 D 0:50; x1x2;y1y2 D 0 We added by purpose according to Fig. C19 the characteristic functions for the special homogenous and isotropic variance-covariance matrix given by ˙l D ˙m. The associated variance-covariance matrix for absolute and relative Cartesian coordinates is given in Tables C25 and C26, structured by

4d2 s 4d s s s ˙ D C 2K . / C K . / C 2. /K . / (C377) l s2 0 d s 1 d d 1 d 4d2 s 4d s ˙ D C 2K . / C K . / (C378) m s2 0 d s 1 d

Finally we present the derived variance-covariance matrices for relative Cartesian coordinates, direction s, direction differences namely angles and distances, for the special case ˙l D ˙m. 834 C Statistical Notions, Random Events and Stochastic Processes

Fig. C20 Derived variance-covariance functions for Taylor-Karman structured variance-covariance matrix of Cartesian coordinates

˚l .s/ D ˘.s/ D 2˙m.0/ 2˙m.s/; 8d 2 s 8d s s s D 2 C 4K K 4 K s2 0 d s 1 d d 1 d (C379)

˚m.s/ D ˘.s/ D 2˙m.0/ 2˙m.s/; 8d 2 s 8d s D 2 4K C K ; (C380) s2 0 d s 1 d 2 .s/ D ˘.s/ D Œ˙ .0/ ˙ .s/ ; s2 m m 2 4d2 s s s D 1 2K C 4 K (C381) s2 s2 0 d d 1 d

Example 9.1 is based on our contribution E. Grafarend and B. Schaffrin (1979) on the subject of criterion matrices of type homogenous and isotropic. Other statistical properties like anisotropic or non-isotropic correlation functions were treated by G. K. Batchelor (1946), S. Chandrasekhar (1950) and E. Grafarend (1971, 1972) on the subject of axisymmetric turbulence. C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 835

Fig. C21 Two dimensional network 836 C Statistical Notions, Random Events and Stochastic Processes

Fig. C22 Characteristic functions of a network example

Fig. C23 Cartesian coordinate data for a network example

Of course there is a rich list of references for “criterion matrices”. We list only few of them. J. E. Alberta (1974), A. B. Amiri (2004), W. Baarda (1973, 1977), K. Bill (1985), R. Bill et. al. (1986), Bischoff et. al. (2006), G. Boedecker (1977), F. Bossler et. al. (1973), F. Co mte e t. a l. (2002), P. Cross et. al. (1978), G. Gaspari et. al. (1999), E. Grafarend (1971, 1972, 1975, 1976), E. Grafarend, F. Krumm and B. Schaffrin (1985), E. Grafarend and B. Schaffrin (1979), T. Karman (1937), T. Karman et. al. (1939), R. Kelm (1976), R. M. Kerr (1985), S. Kida (1989), F. Krumm et. al (1982, 1986), S Meier (1976, 1977, 1978, 1981), S. Meier and W. Keller (1990), J. van Mierlo (1982), M. Molenaar (1982), A. B. Obuchow (1958), T. Reubelt et. al. (2003), H. P. Robertson (1940), B. Schaffrin and E. Grafarend (1977, 1981), B. Schaffrin and E. Grafarend (1982), B. Schaffrin et. al. (1977), G. Schmitt (1977, 1978 i, ii), J. F. Schoeberg (1938), J. P. Shkarofky (1968), V. I . Ta ta rsk i (1961), A. I. Taylor (1935, 1938), C. C. Wang (1970), H. Wimmer (1972, 1978), M. J. Yadrenko (1983) and A. A. Yaglom (1987).

Ex 2: Criterion Matrices for Absolute Coordinates in a Space Time Holonomic Frame of Reference, for Instance Represented in Scalar-Valued Spherical Harmonics

Example C.9.2: Criterion matrices for absolute coordinates in a space time holonomic frame of reference, for instance represented in scalar-valued spherical harmonics C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 837

Fig. C24 Derived variance-covariance functions for a special Taylor-Karman structure variance-covariance matrix of type ˙l D ˙m for Cartesian coordinates

Conventional geodetic networks have been adjusted by methods of Total Least Squares and optimized in different reference frames. The advantage of a rigourously three dimensional adjustment in a uniform reference frame is obvious, especially under the inﬂuence of subdtic geodetic networks within various Global Positioning Systems (GPS, Global Problem Solver). Numerical results in a three dimensional, Total Least Squares approach are presented for instance by K. Bergbauer et. al. (1979) K. Euler et. al. (1982), E. Grafarend (1991), G. Hein and H. Landau (1983), and by W. Jorge and H. G. Wenzel (1973).

The diagnosis of a geodetic network is rather complicated since the measure of quality, namely the variance-covariance matrix of three dimensional coordinates of Cartesian type contains 3g.3g C 1/=2 independent elements. The integer number g accounts for the number of ground stations.

Example: g = 100 838 C Statistical Notions, Random Events and Stochastic Processes

Fig. C25 Homogenous isotropic normalized variance-covariance matrix of absolute coordinates from a network example C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 839

Fig. C26 Special homogenous isotropic normalized variance-covariance matrix of absolute coordinates for a network example case ˙l D ˙m

Fig. C27 Special homogenous isotropic normalized variance-covariance matrix of relative coordinates for a network example case ˙l D ˙m

The number of independent elements with in the coordinate variance-covariance matrix amounts to 45,150. Example: g = 1,000 The number of independent elements within the coordinate variance-covariance matrix amounts to 4,501,500. Example: g = 5,000 (GPS) 840 C Statistical Notions, Random Events and Stochastic Processes

The number of independent elements within the coordinate variance-covariance matrix amounts to 125,507,500 ! In a deﬁnition, the same number of unknown variance-covariance elements of vertical deﬂections and gravity disturbances.

The open big problem In light of the great number of elements of the variance-covariance matrix it is an open question how to achieve any measure of quality of geodetic network based on GPS. We think the only way to get away from this number of millions of unknowns is to design an idealized variance-covariance matrix again called criterion matrix. Comparing the real situation of the variance-covariance with an idealized one, we receive an objective measure of quality of the geodetic network.

Here we are aiming for designing a three-dimensional criterion matrices for absolute coordinates in a holonomic frame of reference and for functionals of the “disturbing potential”. In a ﬁrst section we extensively discuss the notion of criterion matrix on a surface of a moving frame of type homogenous and isotropic on the sphere which is a generalization of the classical Taylor-Karman structure for a vectorized stochastic process. The vector values stochastic process on the sphere is found. In detail we review the representation of the criterion matrix in scalar- and vector-valued spherical harmonics.Insection two we present the criterion matrices for functionals of the disturbing potential. The property of harmonicity is taken into account for the case that a representation in terms of scalar-valued spherical harmonics exists. At the end, we present a numerical example.

Example C.9.21: Criterion matrices for absolute coordinates in a space time holonomic frame if reference The variance-covariance matrix of absolute Cartesian coordinates in the ﬁxed orthonormal frame of equatorial type, holonomic in spacetime, is transformed into a moving frame, e.g. of spherical type. In the moving frame longitudinal and lateral characteristic functions are derived. The postulates of spherical homogenity and spherical isotropy are reviewed extending to the classical Taylor-Karman structure with respect to three dimensional Cartesian coordinates into three dimensions. Finally the spherical harmonic representation of the coordinate criterion matrix is introduced.

Transformation of the coordinate criterion matrix from a ﬁxed into a moving frame of reference

F ! e W

We depart from the ﬁxed orthonormal frame fF ;F ;F g which we transform e1 e2 e3 into a moving one fe ;e ;e ; g by e1 e2 e3 C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 841

F ! e D RE .;;0/F (C382)

RE .;;0/WD R3.0/R2. 2 /R3./ is the three dimensional euler rotation matrix including geometric longitude and geometric latitude . As an example assume a representation of a placement vector x by spherical coordinates .;;r/such that e

x D F1r cos cos C F2r sin cos C F3r sin (C383)

e1 Dx; kx; kDe

e2 Dx; kx;kDe

e3 Dx;r kx;r kDer (C384) x; denotes ıx=ı etc. e ! f

Now let us rotate the orthonormal frame fe1; e2; e3; g into the orthonormal frame ff1; f2; f3; g by means of the South azimuth ˛ 2 3 f1 4 5 e ! f D R3.˛/e f2 f3 2 3 2 3 cos ˛ sin ˛0 e1 4 5 4 5 D sin ˛ cos ˛0 e2 (C385) 001e3

The transformation e ! f on the unit sphere orientates f1 into the direction of the great circle connecting the points .;;r/ and .0;0;r0/ For this reason the unit vector f1 will be called “longitudinal direction”. In contrast, f2 points in direction orthogonal to f1, but still in the tangential plane of the unit sphere. Thus we call the unit vector f2 “lateral direction”. Obviously f3 D e3 holds

F ! g ! f

There is an alternative way to connect the triads F and f. Consider the spherical position of a point in the “orbit” of the great circle passing through the points x; x0, respectively. For a given set of “Keplerian elements” f˝;i;!g - ˝ the right ascension of the ascending node, i the inclination and ! the angle measured in the “orbit” - we transform the ﬁxed orthonormal frame fF1; F2; F3g into the orthonormal “great circle moving frame” fg1; g2; g3gDff3; f1; f2; g by

F ! g D R3.!/R1.i/R3.˝/F (C386) g ! F D R3.˝/R1.i/R3.!/F (C387) 842 C Statistical Notions, Random Events and Stochastic Processes

Fig. C28 Moving and ﬁxed frame of reference on the sphere : two point correlation function

or explicitly

2 3 2 3 g1 cos ! cos ˝ sin ! cos i sin ˝ cos ! sin ˝ C sin ! cos i cos ˝ C sin ! sin i 4 5 4 5 g2 D sin ! cos ˝ cos ! cos i sin ˝ sin ! sin ˝ C cos ! cos i cos ˝ cos ! sin i g3 sin i sin ˝ sin i cos ˝ cos i (C388) 2 3 F1 4 5 : F2 (C389) F3

g3 D sin i sin ˝F1 sin i cos ˝F2 C cos iF3 (C390) represents the “great circle normal” parameterized by the longitude 2 ˝ and the latitude 2 i. g1 and g2 span the “great circle plane”, namely g1 is directed towards the spherical point x.

Figure C28 illustrates the various fames of type “ﬁxed” and “moving”. fg1; g2; g3gDff3; f1; f2; g, g2 is the vector product of g3 and g1

Transformation of variance-covariance matrices Next we compute the Cartesian coordinates variance-covariance matrix. C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 843

0 0 †.x; x / D †ij .x; x /fFi ˝ Fj g

0 0 0 †.x; x / D †ij .x; x /ffi.x/ ˝ fj .x /g

0 0 0 †.x; x / D †ij .x; x /fgi .x/ ˝ gj .x /g (C391) wherei;j"f1; 2; 3g holds 2 3 ˙x;x0 ˙x;y0 ˙x;z0 ij 0 4 5 .˙ / D ˙y;x0 ˙y;y0 ˙y;z0 (C392) ˙z;x0 ˙z;y0 ˙z;z0

0 0 0 0 .˙ ij / D EfŒxi Efxi gŒX j EfX j ggDEf"i "j g (C393) ˙.x; x0/ WD Ef".x/"T .x0/g (C394) 0 Note that the variance-covariance matrix ˙ ij .x; x0/ is a two-point tensor ﬁeld paying attention to the fact that the coordinate errors ".x/ and ".x/ are functions of the point x; x0, respectively. Now let us transform ˙ $ ˙ $ ˙.

T 0 0 0 ˙30 D RE .;;˛/˙33RE . ; ;˛ / 0 T 0 T 0 D R3.˝/R1.i/R3.!/˙33R3.! /R1 .i /R3 .˝ / T 0 0 0 ˙30 D RE .;;˛/˙ RE . ; ;˛ / (C395) T 0 0 0 T 0 0 0 D RE .;;˛/˙ R.!;i;˝/˙33R .! ; i ; ˝ / RE . ; ;˛ / T 0 0 0 ˙30 D R .! ; i ; ˝ /˙33R.!;i;˝/

T 0 0 0 T 0 0 0 0 0 0 D R .! ; i ; ˝ /RE .;;˛/˙33RE . ; ;˛ / R.! ; i ; ˝ /

A detailed write-up of the variance-covariance matrix ˙ is 2 3 ˙ll0 ˙lm0 ˙lr0 ij 0 4 5 .˙ / D ˙ml0 ˙mm0 ˙mr0 (C396) ˙rl0 ˙rm0 ˙rr0 indicating the “longitudinal” base vector f1 D fl, “lateral” base vector f2 D fm and the “radical” base vector f3 D fr D er . In terms of spherical coordinates ˙ ;˙, respectively, can be written

0 0 0 0 ˙30.x; x / D ˙ .;;r;; ;r / (C397) 0 0 0 0 0 ˙30.x; x / D ˙.!;i;˝;r;! ;i ;˝ ;r / (C398) 0 ˙ll0 D Ef"l .x/"l0 .x /g; 0 0 ;˙ll0 .x; x /ffl .x/ ˝ fl0 .x /g 844 C Statistical Notions, Random Events and Stochastic Processes

0 ˙lm0 D Ef"l .x/"m0 .x /g; 0 0 ;˙lm0 .x; x /ffl .x/ ˝ fm0 .x /g 0 ˙lr0 D Ef"l .x/"r0 .x /g; 0 0 ;˙lr0 .x; x /ffl .x/ ˝ fr0 .x /g (C399)

0 The characteristic function ˙ll0 .x; x / is called longitudinal correlation function, 0 0 ˙mm0 .x; x / lateral correlation function, ˙lm0 .x; x / cross correlation function 0 between the longitudinal error "l .x/ at the point x and the lateral error "m0 .x / at the point x0.

Spherically homogenous and spherically isotropic coordinate criterion matrices in a surface moving frame terrestrial geodetic networks constitute points on the earths surface. Once we want to develop an idealized variance-covariance matrix of the coordinates of the network points, we have to reflect the restriction given by the closed surface. A three dimensional criterion matrix of Taylor-Karman type is therefore not suited since it assumes extension of a network in three dimension up to infinity. a more suitable concept of a coordinate criterion matrix starts from the representation where the radial part is treated separately from the longitudinal-lateral part. Again the nature of the coordinate criterion matrix as a second order two-point tensor field will help us to construct homogenous and isotropic representations. The spherically homogeneity-isotropy-postulate reflects the intuitive notion os an idealized variance-covariance matrix.

Deﬁnition (spherical homogeneity and spherical isotropy of a tensor-valued two- point function): The two-point second order tensor function ˙.x; x0/ is called spherically homogenous, if it is translational invariant in the sense of

0 0 0 0 0 0 0 0 †33.! C ;i;˝;r;! C ;i ;˝ ;r / D †33.!;i;˝;r;! ;i ;˝ ;r / (C400) for all 2 Œ0;2;iD i 0;˝ D ˝0;rD r0, where !0 D ! C holds. It is called spherically isotropic, if it is rotational invariant in the sense of

0 0 0 0 ˙33.! !s ;i ;˝ ;r;! !s ;i ;˝ ;r / 0 0 0 0 D R3.ıs/˙33.! !s;i;˝;r;! !s;i ;˝ ;r /R3.ıs/ (C401)

0 0 0 for all ı"Œ0;2;rD r ; where ! D ! C , ! D ! C holds. R3.ıs/ indicates 0 a local rotation around the intersection points xs of the great circle through x; x 0 and through x ; x , respectively. xs has the common coordinates .˝;i;!s and .˝ ;i;!s with respect to the locally rotated frames.

End of Deﬁnition C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 845

Corollary (spherical homogeneity and spherical isotropy of a tensor-valued two- point function): The two-point second order tensor function ˙.x; x0/ is spherically homogenous, if it is a function of . ;i;˝;r/only:

0 0 0 ˙.!;i;˝;r;! ;i ;˝ ;r/D ˙.!;i;˝;r;! !s ; ;r/ (C402) It is spherically homogenous and isotropic, if it is a function of . ; r/ only

˙.!;i;˝;r;!0;i0;˝0;r/D ˙. ;i;˝;r/ (C403) such that

0 ij 0 ˙.x; x / Df˙0. ; r/ı C ˙22. ; r/ < x xjfi > 0 < x xjfi >g.fi ˝ fj / (C404) holds for some spherically isotropic and homogenous scalar-valued functions ˙0 and ˙22 End of Corollary

Let us interpret the deﬁnition of the corollary, especially with respect to Fig. C28. 0 0 Two points x; x are connected to the sphere of radius r D r D r0 by a great circle 0 such that D 2 arcsin.kx; x k=2r0/ is the spherical distance if these points.

Obviously a “translation” of the configuration x; x0 is generated by ! ! !C;! ! !0 C where the location of x; x0 in the great circle / “orbit” plane is measured by !;!0. The “rotation” of the configuration x; x0 is more complicated. Assume 0 therefore an alternative configuration x; x of two point connected by another 0 great circle. Note the spherical distances of x; x0 and x; x respectively, coincide. 0 The great circles intersect at xs, : The location of Xs within the plane x; x is 0 described by !s with respect to the ascending node, while within the plane x ; x by !s . the angle between the two great circles is denoted by ıs.

0 The conﬁguration x; x0 and x; x is described with respect to xs by 0 0 0 0 0 0 ! !s ;i;˝;! !s ;i ;˝ and ! !s ;i ;˝ ;! !s ;i ;˝ respectively. Now it should be obvious that the two conﬁgurations under the spherical isotropy postulate coincide once we rotate around xs by the angle ıs.

We expressed ˙ . given in the F -frame in terms of the g-frame, namely by

T T T ˙ D R3 .!/R1 .i/R3 .˝/˙ R3.˝/R1.i/R3.!/ (C405)

Introducing ! D .! !s / C !s we compute the variance-covariance matrix ˙ 846 C Statistical Notions, Random Events and Stochastic Processes

T T T T 0 0 0 ˙ D R3 .! !s /R3 .!s /R1 .i/R3 .˝/˙ R3.˝ /R1.i /R3.!s/R3.! !s / (C406)

T 0 ˙ D R3 .! !s/˙s R3.! !s/ (C407) in terms of the variance-covariance matrix ˙s in the g.xs/-frame. Now we are prepared to introduce Corollary (spherical Taylor-Karman structure):

0 Denote the variance-covariance matrix ˙ ij in the f -ande- frame by 0 ij ˙ 0 . ; r/ 0 22 f ˙ D ll f ˙ (C408) 0˙mm0 . ; r/ 0 ij ˙ 0 ˙ 0 22 e˙ D 11 12 e˙ (C409) ˙210 ˙220 and transform f ˙ ! e˙ by cos ˛ sin ˛ cos ˛0 sin ˛0 e˙ D f ˙ D (C410) sin ˛ cos ˛ sin ˛0 cos ˛0

Then

0 0 ˙110 D ˙ll0 . ; r/ cos ˛ cos ˛ C ˙mm0 . ; r/ sin ˛ sin ˛ 0 0 ˙120 D ˙ll0 . ; r/ cos ˛ sin ˛ ˙mm0 . ; r/ sin ˛ cos ˛ 0 0 ˙210 D ˙ll0 . ; r/ sin ˛ cos ˛ ˙mm0 . ; r/ cos ˛ sin ˛ 0 0 ˙220 D ˙ll0 . ; r/ sin ˛ sin ˛ C ˙mm0 . ; r/ cos ˛ cos ˛ (C411)

˙lm0 D ˙ml0 D 0 (C412) holds for spherically homogenous and isotropic scalar-valued functions ˙ll0 . ; r/, ˙mm0 . ; r/ called longitudinal and lateral correlation functions on the sphere. End of Corollary

The corollary extends results of G.J. Taylor (1935), T. Karman (1937) and C. Wang (1970 p. 214-215) and has been given for a special variance-covariance two-point tensor ﬁeld of potential type by H. Moritz (1972 p. 111). Here it has been generalized for any vector-valued ﬁeld. Note that incase of homogenity and isotropy the cross-correlation functions ˙lm0 and ˙ml0 may vanish.

The classical planar Taylor-Karman structure is constrained within. C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 847

Let us put ˛ D ˛0 such that cos ˛ cos ˛0 D cos2 ˛ D 1 sin2 ˛; sin ˛ sin ˛0 D sin2 ˛ etc. can be rewritten

2 ˙110 D ˙mm0 . / C Œ˙ll0 . / ˙mm0 . / cos ˛

˙120 D Œ˙ll0 . / ˙mm0 . / sin ˛ cos ˛ D ˙210 2 ˙220 D ˙mm0 . / C Œ˙ll0 . / ˙mm0 . / sin ˛ (C413) which is the standard Taylor-Karman form Representation of the coordinate criterion matrix in scalar spherical harmonics

Let us assume a coordinate variance-covariance matrix which is homogenous and isotropic in the sphere. With reference to (C412) we postulate a coordinate vector of potential type, in addition, namely

i " D gradR " D @i R.iD 1; 2; 3/ (C414)

@ @ @ where .@i / D . @x ; @y ; @z / nd R the scalar potential. Now the variance-covariance matrix of absolute coordinates can be represented by

ij 0 i j 0 0 0 ˙ D Ef" .x/" .x /gD@i @j 0 EfR.x/R.x /g (C415) where we used the commutator relation @i E D E@i of differentiation and integration operators. Denote the scalar-valued variance-covariance function EfR.x/R.x0/gDW K.x; x0/. Once we postulate homogeneity and isotropy of two-point function K.x; x0/ on the sphere we arrive at

K.x; x0/ D K.r; r0; / (C416)

For a harmonic coordinate error potential, @i @i R D 0, we ﬁnd for the covariance function K.r; r0; /

0 0 @i @i K.r; r ; / D Ef@i @i R.x/R.x /gD0 (C417) 0 0 @j 0 @j 0 K.r; r ; / D EfR.x/@j 0 @j 0 R.x /gD0 (C418)

Case 1 .kxkDkx0k/

0 On a sphere .kxkDkx k/ D r0 the scalar-valued variance-covariance function K. / which fulﬁls (C414 - C 417) may be represented by

X1 K. / D knPn.cos / (C419) nD0 848 C Statistical Notions, Random Events and Stochastic Processes where Pn are the Legendre polynomials. The constants kn depend, of course, on the choice of r0

Case 2 .kxk¤kx0k/

Denote .kxkDr; kx0kDr0/. the scalar-valued variance-covariance function K.r; r0; /which fulﬁls (C414 - C 417) may be represented by

K D K.r; r0; / C X1 r2 n 1 X1 rr0 n D 1kn . 0 / P .cos / C 2kn . / P .cos / (C420) rr0 n r n nD0 nD0 0 where the coefﬁcients 1kn ;2kn represent the inﬂuence of the decreasing and increasing solution, by Legendre polynomials with respect to r; r 0, respectively. We can only hope that the covariances K.r; r0; / decrease with respect to r; r 0, respectively in order to be allowed to discriminate 2kn D 0. In the following we mae this assumption, thus representing

C X1 r2 n 1 K.x; x0/ D K.r; r0; /D k . 0 / P .cos / (C421) n rr0 n nD0

Next let us assume the coordinate variance-covariance matrix (C420) is of potential type. Then longitudinal and lateral correlation functions on the sphere if radius r can be represented by

d 2 ˙ 0 . / D K. / (C422) ll d 2 1 1 ˙ 0 . / D K. / (C423) mm sin d d ˙ 0 . / D .˙ 0 sin (C424) ll d mm if .x; x0/ is homogenous and isotropic on the sphere. End of Corollary

0 d 0 0 Note that the result ˙ll D d .sin ˙mm / for the sphere corresponds to ˙ll D d 0 ds .s˙mmq/ for a harmonic ﬁeld in the plane. s D .x0 x/2 C .y0 y/2 is the Euclidean distance in the plane which corresponds to at the distance on the sphere. C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 849

Corollary (longitudinal and lateral correlation functions on the sphere of harmonic type): Let us assume that the coordinate variance-covariance matrix id homogenous, isotropic and harmonic on the sphere in the sense of the deﬁnition. Longitudinal and lateral correlation functions on the sphere can be represented by

1 X .n2 C 3n C 2/cos2 0 ˙ll . / D kn 2 C .n C 1/ : nD0 sin 2 2 cos Pn.cos / Œn C 3n C 2 PnC1.cos / sin2

2 1 C Œn C 3n C 2 PnC2.cos /g (C425) sin2 X n C 1 0 ˙mm . / D 1kn 2 Œcos Pn.cos / nD0 sin

PnC1.cos / (C426)

End of Corollary

For the proof of the above corollary we have made successive use of the recurrence 2 relation .x 1/dPn.x/=dx D.n C 1/ŒxPn.x/ PnC1.x/; x D cos ;d=d D .dx=d /.d=dx/ Dsin d=dx

Representation of the coordinate criterion matrix in vector spherical harmonics Previously we restricted the coordinate criterion matrix being homogenous and isotropic on the sphere to be of potential type. Here we generalize the coordinate criterion matrix to be of potential and “turbulent” type. Thus according to the Lame lemma – for more details see A. Ben Menahem - S. J. Singh (1981) - we represent the coordinate vector " D gradR C rotA (C427) where we gauge the vector potential A by divA D 0. In vector analysis gradR is called the longitudinal vector field, rotA the transversal vector field. In order to base the analysis only on scalar potentials, the source-free vector field rotA is decomposed SCT where S is the poloidal part while T is the toroidal part such that

" D gradR C rotrot.er S/ C rot.er T/ (C428)

.R;S;T/ are the three scalar potentials we were looking for. The variance- covariance matrix of absolute coordinates can now be structured according to 850 C Statistical Notions, Random Events and Stochastic Processes

˙.x; x0/ D Ef".x/g D grad ˝ grad0EfR.x/R.x0/g 0 0 0 Crotrot ˝ rot rot er er0 EfS.x/S.x /g 0 0 Crot ˝ rot er er0 EfT.x/T .x /g (C429) where we have assumed EfR.X/S.X0/gD0; EfR.X/T .X0/gD0; EfS.X/T .X0/g D 0 holds for a coordinate variance-covariance matrix ˙.xx0/ homogenous and isotropic on the sphere. Thus with respect to the more general representation is

˙.x; x0/ D grad ˝ grad0K 0 0 C rotrot ˝ rot rot er er0 L 0 C rot ˝ rot er er0 M (C430)

where K WD EfR.X/R.X0/g;L WD EfS.X/S.X0/g;M WD EfT.X/T .X0/g are supposed to be the scalar-valued two point covariance functions depending on .r; r0; /for .˙.x; x0// homogenous and isotropic on the sphere.

The representation of the criterion matrix .˙.x; x0// in terms of vector spherical harmonics is achieved once we introduce the spherical surface harmonics. 2 q 2.2n C 1/ .nm/Š P .sin /cos m 8m>0 6 .nCm/Š nm 6 6 p 6 enm./ D 6 2.2n C 1/Pn.sin / 8m D 0 (C431) 6 4 q .njmj/Š 2.2n C 1/ .nCjmj/Š Pnjmj.sin /sin jmj 8m<0: where Pnm.sin / are the associated Legendre functions

nCm 1 m=2 d n P .x/ D .1 D x2/ .x2 1/ (C432) nm 2nnŠ dxnCm Note that the quantum numbers run n D 0; 1; ; 1I m Dn; n C 1; ;0; ; n 1; n. The base functions enm.; / are orthonormal in the sense

Z2 CZ=2 1 d d cos e e D ı ı (C433) 4 kl nm kn lm 0 =2 C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 851

Finally a vector ﬁeld " can be represented in terms of vector spherical harmonics by

" D U a C V b C W c (C434) 2nm nm nm nm nm nm a WD e e .; / 6 nm r nm 6 6 6 6 bnm WD rgradenm.; / (C435) 4

cnm WD x grad enm.; /:

For more details on vector spherical harmonics we refer to standard textbooks, here e.g. to E. Grafarend (1982).

˙.x; x0/ D Ef".x/ ˝ ".x0/g

D Unmn0m0 anm ˝ an0m0

C Vnmn0m0 bnm ˝ bn0m0

C Wnmn0m0 cnm ˝ cn0m0 (C436) which is due to < ajb >D< ajc >D< bjc >D 0. Unmn0m0 etc. are spherical covariance functions.

Example C 9.22: Towards criterion matrices for functionals of the distributing potential in a space-time holonomic frame if reference The geodetic observational equations in their linearized form contain beside unknown Cartesian coordinate corrections in an orthonormal reference frame, being holonomic in space-time, the disturbance of the potential and its derivatives as unknowns . for a complete set-up of a criterion matrix of geodetic unknowns we have therefore present thoughts about a homogenous and isotropic variance- covariance matrix of the potential disturbance and its derivatives, The postulates of homogeneity and isotropy will lead again to the extension of an classical Talyor- Karman structure. ﬁnally the representation of the criterion matrix of potential related quantities in terms of scalar spherical harmonics is introduced. Since the potential disturbance is harmonic function, vector spherical harmonics are not needed.

We begin the transformation of the gravity criterion matrix from spherical into Cartesian coordinates. b c With reference to the linearized observational equations the unknowns .ı ; ı ; ı/b have to be equipped with a criterion matrix, reasonable to be homogenous and 852 C Statistical Notions, Random Events and Stochastic Processes isotropic on the sphere. In order to take advantage of lemma and corollaries of the second chapter, we better transform the spherical coordinates of the disturbance gravity vector ı into Cartesian ones and then apply the variance-covariance “propagation” on the transformation formula. 2 3 2 3 cos cos ˚ cos cos 4 5 4 5 ıi WD i i D sin cos ˚ sin cos sin ˚ sin 2 3 . C ı/cos. C ı / cos. C ı / 4 5 D . C ı/sin. C ı / cos. C ı / . C ı/sin. C ı / 2 3 cos cos 4 5 sin cos (C437) sin 2 3 . C ı/.cos ı sin /.cos ı sin / : 4 5 ıi D . C ı/.sin C ı cos /.cos ı sin / . C ı/.sin C ı cos / 2 3 cos cos 4 5 sin cos (C438) sin 2 3 ı cos cos C ı sin cos C ı cos sin : 4 5 ıi D ı sin cos ı cos cos ı sin sin ı sin ı cos (C439) 2 3 2 3 2 3 ı1 C sin cos C cos sin cos cos ı 4 5 : 4 5 4 5 ı2 D cos cos sin sin sin cos ı ı3 0 cos sin ı (C440) 2 3 2 3 2 3 2 2 1=2 1 ı 2 C12. C / 1 ı 1 : 6 1 2 7 4 5 4 2 2 1=2 1 5 4 5 ı2 D C1 23.1 C 2 / 2 ı (C441) ı 2 2 1=2 1 ı 3 0 .1 C 2 / 3

c b 0 The variance-covariance matrix Dfıi .x/; ıj 0 .x /g as a function of the variance- b c b b c b 0 covariance matrix Df.ı ; ı ; ı/.x/; .ı ; ı ; ı/.x /g basedonthetrans- formation ( C 438) is given. ﬁnally the Cartesian variance-covariance matrix c b 0 Dfıi .x/; ıj 0 .x /g in the ﬁxed frame F should be transformed into the moving frame f according to (C381). C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 853

P5 1 Fig. C29 Normalized covariance function K. / D knPn.cos / for kn D nC2 nD0

Thus we refer to ˙ ˙ ˙ ˙ ˙ ˙ ll0 ;lm0 ;lr0 ;ml0 ;mm0 ; ;rr0 (C442) ˙ ˙ as the longitudinal correlation function ll0 ,thelateral correlation function mm0 etc. of the gravity disturbance vector ıi . In order to construct a homogenous and isotropic criterion matrix for functionals of the disturbing potential in a surface moving frame we refer to the introduction which applies here, too. Again we make use of the deﬁnition of homogeneity and isotropy 0 of the tensor-valued two-point function ˙.X;X / on the sphere, the basic lemma and the corollary for the spherical Taylor-Karman structure. Just by adding the left hand index on ˙ within formula (C410) (C411), we arrive at the corresponding formulae for longitudinal, lateral and cross-correlation of the gravity disturbance vector ı. For the representation of the criterion matrix for functionals of the disturbing potential in scalar spherical harmonics we once more refer to (C417). All results can be transferred, especially (C 420 - C 425) once we write K.r; r0; / as the scalar-valued two-point function EfŒıcw E.ıcw/xŒıcw E.ıcw/x0g homogenous and isotropic on the sphere. Besides the ﬁrst-order functionals of the potential ıw and their variance-covariance matrices, higher order functionals, e.g. for gravity gradients, and their variance-covariance matrices can be easily obtained, a project for the future. 854 C Statistical Notions, Random Events and Stochastic Processes

P5 1 Fig. C30 Normalized covariance function K. / D knPn.cos / for kn D .nC2/.n22nC2/ nD0

A numerical example of the homogenous and isotropic two-point function K.x; x0; /on the space For demonstration purpose plots C29 - C31 show the normalized covariance functions K.X; X0/ D K.r; r0; /

P5 D K. / D knPn.cos / for case 1 nD0 0 .kXkDkmathbf X kDr0/ and three different choices of the coefficients kn. The coefficients are chosen in such a manner that their sum equals the unity (so as to normalize the covariance function) and the covariance function approaches zero as tends to . As an example for the construction of a criterion matrix, the coefficients kn could be estimated for an ideal network and its variance-covariance matrix; that is derived covariance function K.x; x0/. We have taken a part of Appendix C 9.2 from our contribution E. Grafarend, I. Krumm and B. Schaffrin (Manuscripta geodetica 10 (1988) 3-22). Spherical homogeneity and isotropy in tensor-valued two-point functions have been introduced by E. Grafarend (1976) and H. Moritz (1972, 1978).

Ex 3: Space Gravity Spectroscopy: The Beneﬁts of TaylorÐKarman Structured Criterion Matrices

Example C.9.3. Space gravity spectroscopy: the beneﬁts of Taylor-Karman structured criterion matrices (eg, P. Marinkovice et. al. (2003)) C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 855

P5 1 Fig. C31 Normalized covariance function K. / D knPn.cos / for kn D n2C3 nD0

As soon as the space spectroscopy was successfully performed, for instance by means of semi-continuous ephemeris of LEO-GPS tracked satellites, the problem of data validation appeared. It is for this purpose that a stochastic model for the homogenous isotropic analysis of measurements, obtained as “directly” measured values in LEO satellite missions (CHAMP, GRACE, GOCE), is studied. An isotropic analysis is represented by the homogenous distribution os measured values and the statistical properties of the model are calculated. In particular a correlation structure function is deﬁned by the third order tensor (Taylor-Karman tensor), for the ensemble average of a set of incremental differences in measured components. Speciﬁcally, Taylor-Karman correlation tensor is calculated with the assumption that the analyzed random function is of a “potential type”. The special class of homogenous and isotropic correlation functions is introduced. Finally, a successful application of the concept is presented in the case study CHAMP and a comparison between modeled and estimated correlations is performed.

Example C.9.31. Introduction

A signiﬁcant problem of LEO satellites, both in geometry and gravity space, is the association of quality standards to Cartesian ephemeris in terms of variance-covariance matrix valued-functions. as a pessimistic measure of the quality standards os LEO satellite ephemeris, a three dimensional Taylor-Karman structured criterion matrix has been proposed, named in honor of Taylor (1938)and Karman (1938), the founders of the statistical theory of turbulence. 856 C Statistical Notions, Random Events and Stochastic Processes

The concept of the Taylor-Karman criterion matrices was ﬁrst introduced by E. Grafarend (1979) and subsequently further developed by B. Schaffrin and E. Grafarend (1982), H. Wimmer (1982), and E. Grafarend et. al. (1985, 1986). with this contribution we extend the application of the Taylor-Karman structured matrices to the third-dimension, namely to the long-arc orbit analysis.

If we assume the vector-valued stochastic process to be the gradient of a random scalar-valued potential, in particular its longitudinal and lateral correlation function or the “correlation function along-track and across-track”, what would be the structure of a three-dimensional Taylor-Karman variance-covariance matrix.Inorderto answer this question, a three-dimensional correlation tensor and its decomposition in the case of homogeneity and isotropy is studied with the emphasis on R3. Additionally, we deal with a special class of homogenous and isotropic tensor- valued correlation functions. They are derived, analyzed and applied to the data validation process. The theoretical concept for the application of the previously discussed criterion matrix in the geometric and gravitational analysis of LEO satellite ephemeris is presented. Finally the case study CHAMP is used for the application of our theory concept followed by the results and conclusion.

Example C.9.32: Homogenous and isotropic variance-covariance tensor and correlation functions in a three-dimensional Euclidean space

Example C.9.321: Notions of homogeneity and isotropy

The notions of homogeneity and isotropy for functions on R are brieﬂy explained as the following. The general context for these two deﬁnitions involves the action of a transitive group of motions on a homogenous and space belongs to the extensive theory of Lie groups (F. W. Warner (1983), A. M. Yaglom (1987)). However, it is important to clarify the different notions of homogeneity and isotropy exist due to the variety of homogenous spaces and transitive group actions on these spaces. The terminology introduced associates the notion of homogeneity and isotropy with

Fig. C32 Graphical representation of the longitudinal and lateral correlation functions C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 857 the functions on R that are invariant under the translation group acting on R . The notion of isotropy is deﬁned for functions on R that are invariant under the orthogonal group acting on R.

Table C.9.9: Tensor-valued correlation function of second-order “two-point correlation function”

P3 P3 i j i j ˙.x ; x / D e e ˙ij .x ; x / D ˙ij .x ; x /e e (C443) i;jD1 i;jD1

r WD kx; xk and r WD x; x

xi xj ˙ij .r/ D ˙m.r/ıij C Œ˙l .r/ ˙m.r/ r2 ;i;j2f1; 2; 3g (C444)

“Cartesian versus spherical coordinates”

x1 D x WD .x1 x1 / D r cos ˇ cos ˛

x2 D y WD .x2 x2 / D r cos ˇ sin ˛ (C445)

x3 D z WD .x3 x3 / D r sin ˇ

“continuity of a potential type”

dŒr˙m.r/ d˙m.r/ ˙l .r/ D dr D ˙m.r/ C r dr (C446) End of Table C.9.9

Example C.9.322: Homogenous and isotropic variance-covariance tensor (correlation tensor) Taylor (1938) proved that he homogenous random field X.t/ correlation function ˙.r/ Ddepends only on the length r Dkrk of the vector r and not on its direction, where < ::: > denoted the ensemble average. If the correlation function ˙.r/ of the homogenous random field X.t/ in R has the property, then the field X.t/ is said to be an isotropic random field in R. The corresponding correlation function ˙.r/ is then called an isotropic correlation function in R ( or an n-dimensional isotropic correlation function). Process, which satisfy the introduced postulates of homogeneity and isotropy, are said to be (widely) stationary (M. J. Yadrenko (1983)). Note that for an isotropic random field in R, all directions in space are obviously equivalent.

The decomposition of a homogenous and isotropic variance-covariance tensor- valued function, shown in box 1, was introduced by von Karman and Howarth 858 C Statistical Notions, Random Events and Stochastic Processes

Fig. C33 Graphical representation of the correlation tensor transformation

(1938) by means of a more general and direct method than the one used by G. J. Taylor(1938). Additionally, Robertson (1940) reﬁned and reviewed the T. Karman and L. Howard equation in the light of a classical invariant tensor theory.

The decomposition of ˙ij .kx x k/, with a special emphasis on n D 3,is performed in terms of Cartesian coordinates and with respect to the orthonormal frame of reference fe1; e2; e3j0g attached to the origin 0 of a three-dimensional Euclidean space. kx xk denotes the Euclidean distance r between the two points x and x of E WD fR ;ıij . Longitudinal and lateral correlation functions ˙l and ˙m, are the structural elements of such a homogenous and isotropic tensor- valued variance-covariance function which appear in the spherical tensor ˙m.r/ıij 2 as well as in the oriented tensor Œ˙l .r/ ˙m.r/xi xj =r for all i;j 2f1; 2; 3g, see (C443) and Fig. C.32. ıij denotes the Kronecker or unit matrix, xi the Cartesian coordinate difference between the points x and x. These differences are also represented in terms of relative spherical coordinates .˛;ˇ;r/, (C444). Finally the continuity condition of a potential type is formulated by (C445) which provides the unique relation between ˙l and ˙m (G. J. Taylor (1938); H. M. Obuchow (1958); E. Grafarend (1979)). Due to its complexity, it is necessary to further elaborate on the previous equation set. The correlation tensor ˙ij .r/ was transformed to a special coordinate system 0 0 0 0 R 0 0 0 0 O x1x2x2 in instead of the initial set Ox1x2x3. The new set O x1x2x3 is selected in such a way so that its origin O0 is shifted by the vector x with respect to the origin O, as illustrated in Fig. C15.ThismeansthatO0 coincides with the terminal 0 0 point of the vector x that refers to the initial coordinates, while the axis O x1 lies along the vector x x. C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 859

Fig. C34 The behavior of Tatarski’s correlation function for different values of shape parameter .d D 900/ and scale parameter d. D 3=2/

˙ij , introduced here are explanatory functions, are the components of the correla- 0 tion tensor ˙ij .r/ in the new set of coordinates. The functions ˙ij .r/ clearly depend only on the length r Dkrk of the vector r, since the direction of r is fixed in the new set of coordinates. In the space R exist a reflection which leaves the points x 0 0 0 0 0 and x D .O / unmoved and it replaces the axis O xj by O xj ,wherej ¤ 1 is a fixed number. However, it does not change the directions of all other coordinates 0 0 axes O xl ;l 2f1; 2; 3g and l ¤ j . It follows that

0 0 ˙ij .r/ D˙ij .r/ D 0textrmfori ¤ j (C447) 0 Hence, only the diagonal elements ˙ij .r/ of ˙ij .r/ can differ from zero. Further 0 0 0 0 more, if i ¤ j and j ¤ 1, then the axis O xi by its rotation around the axis O xj , 0 0 can be transformed to the axis O x1. Hence

0 0 ˙22.r/ D ˙33.r/ (C448)

0 0 The tensor ˙ij and, consequently, ˙ij are symmetric and their components ˙ij .r/ can take at most only tow non-equal non-zero values at the already introduces 0 longitudinal correlation function ˙ii.r/ D ˙l .r/ and the lateral correlation function 0 0 ˙22.r/ D ˙33.r/ D ˙m.r/, which specify in a unique way the correlation tensor. In order to obtain the explicit form of ˙ij .r/, as the function of ˙l .r/ and ˙m.r/, the unit vectors of the old coordinate axes Ox1;Ox2;Ox3 along the axes of the 0 0 0 0 0 0 new system O x1;O x2;O x3 must be resolved and then ˙ij .r/ can be represented 0 as linear combination of the functions ˙kl.r/; .k ¤ i;l ¤ j and k; l 2f1; 2; 3g/, which leads to (C442) and (C443). 860 C Statistical Notions, Random Events and Stochastic Processes

Example C.1. homogenous and isotropic correlation functions

It was shown in the pervious section that for a homogenous and isotropic random fields defined on the Euclidean space R, the correlation between x and x depends only on the Euclidean distance kx xk. Therefore as shown in Table C32, we can distinguish a homogenous and isotropic correlation function on R with the real valued function ˙.r/ defined on Œ0; 1/ and we denote by ˚n the class of all continuous permissible functions ˙.r/ W Œ0; 1/ ! such that ˙.0/ D 1 (we are working in terms od correlation not covariance) and the symmetric function ˙.k:k/ is a positive definite on R . The characterization of classes ˚n,alsoshown in Table C.9.10, is a well known result of J. F. Schoenberg (1938). The function ˙.r/ W Œ0; 1/ ! R is an element of ˚n if and only if it abmits a representation in the form of (C450), where W is a probability measure on Œ0; 1/; J.n2/=2 is the Bessel function of the first kind of order .n 2/=2 and stands for the Gamma function. Hence in for geodesy relevant spaces R and R, (C450) reduces to the form presented in (C452).

Table C.9.10: Isotropic correlation and Schoenberg’s characterization “homogenous and isotropic correlation function” ˙.x; x/ D ˙.kx; xk/; x; x 2 R (C449) r WD kx; xk R “the characterization of the ˚n class” ˙n.r/ D ˝n.r/dW./ (C450) Œ0;1/ 2 .n2/=2 ˝n.r/ D .n=2/.r / J.n2/=2.r/ (C451) “reduction of the ˝n for n D 2 and n D 3” 1 ˝2.r/ D J0.r/ and ˝3.r/ D r sin r (C452) End of Table C.9.10

Table C.9.11: Special classes of correlation functions “Tatarski’s class”

21 r r ˙Œ.r/ D ./. d / K. d / (C 453)

“Shkarofsky’s class”

2 =2 2 1=2 . r Cı2/ K .. r Cı2/ / d2 d2 ˙Œ;ı.r/ D (C454) ı K .ı/

End of Table C.9.11 C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 861

Example C.2. Tatarski’s class of homogenous and isotropic correlation functions

Many analytical candidate models for ˙ have been suggested in the literature (for example, Buell, (1972); Haslett, (1989)), but we refer to Tatarski (1962) as being the ﬁrst who elaborated on such correlation functions which fulﬁll all the conditions presented in the previous section. The Tatarski’s correlation function class is shown in Table C.9.11 and illustrated by Fig. C.34. In addition to V. J. Tatarski’s class,a very general family of correlation function models due to J. P. Shkarofsky (1968) is introduced, that came as the generalization of Tatarski’s correlation function family. These two classes, which have been proved to be the members of ˚3 (Shkarofsky, (1968) and of ˚1 classes; Gneiting et. al. (1999)), can be applied to many geodetic problems, (e.g. Grafarend (1979); Meier (1982); Wimmer (1982); Grafarend (1985)).

In equation of Table C.9.11, K stands for a modiﬁed Bessel function of order ; d > 0 is a scale parameter, and ı>0and are shape parameters. In the case of ı D 0 J. P. Shkarofsky’s class reduces to V. J. Ta ta rsk i’s class.

The special case of Tatarski’s class appears if the shape parameter is sum of a non-negative integer k and 1=2. Then the right-hand side of the equation can be written as a product of exp.r=d/ and a polynomial of degree k in r=d (eg. T. Gneiting, (1999)). In particular, in the case of n D 3 dimensional Markov process of the p D 1 order, shown in Fig. C34, the shape parameter is expressed as D .2p C 1/=.n 1/ D 3=2 and results in the ﬂowing simpliﬁcation of (C454): r r ˙ .r/ D .1 C / exp. / (C455) Œ3=2 d d

Example C.3. Three dimensional Taylor-Karman criterion matrix in the geometric and gravitational analysis of LEO satellite ephemeris

WE have so far analyzed the theoretical background and solution for design of the homogenous and isotropic Taylor-Karman correlation tensor. The question is, how this theoretical concept applies to the geometric and gravitational analysis of LEO satellite ephemeris.

The basic idea is, that the errors in the position vectors of LEO satellites constitute a vector-valued stochastic process. Following this concept, a satellite orbit of LEO- GPS tracked satellites is an homogenous and anisotropic ﬁeld of error vectors and the error situation is described by the covariance function. As it is well known, the error situation os a newly determined position in a three-dimensional Euclidean space is the best, when the error ellipsoid is sphere (isotropy) with the 862 C Statistical Notions, Random Events and Stochastic Processes minimal radius and if the error situation is uniform over the complete satellite orbit (homogeneity). This “ideal” situation can be explained by the three-dimensional Taylor-Karman structured criterion matrix of W. Baarda - E. Grafarend (potential) type. Then the correlations between the vectors of pseudo-observations of satellite ephemeris described by the longitudinal and lateral correlation functions.

The characteristic correlation functions can be estimated by matching the correlation tensor with a three-dimensional Markov process of 1st order and with the introduction of some additional information about the underlying process. The correlation analysis is performed with the assumption that the vector valued three- dimensional random function is of “potential type” (E. Grafarend (1979)), i.e. the gradient of a random scalar function “signal s.x/”. The structure of the random function s.x/ is outlined as an n-dimensional Markov process of the pth order. Figure C35 illustrates the case n D 3 and p D 1:

One of the simples differential equation os such a process has the form given by

p .2 ˛2/ s.x/ D e.x/ (C456) where e.x/ is a white noise. If the Laplace operator can be applied to the homogenous random scalar function, then it transforms this function into a new homogenous random function, having the spectral density that corresponds to the spectral density of the correlation function of (C453) and (C454), see P. Whittle (1954), V. He in e (1955) and P. Whittle (1963). Hence the homogenous solution of C455 (if it exist) must have the spectral density that corresponds to the spectral density of the correlation function.

Table C.9.9 summarizes the representation of the homogenous and isotropic correlation functions of type (i) Signal correlation function ˙, (ii) Longitudinal correlation function ˙l and (iii) Lateral correlation function ˙m

Example C.9.34: Results and conclusions

Case study: Champ As the numerical test in this study, we processed two data sets: two three- dimensional fx.tk/; y.tk/; z.tk /g and fx.td /; y.td /; z.td /g Cartesian ephemeris time-series of CHAMP satellite orbit for the test period from day 140 to 150 of 2001 (20 May to 30 May, both inclusive). We analyzed in total 27,360 triplets of satellite positions. Both time series are indexed with a 30 s sampling rate and referenced to a kinematic (index k) and a dynamic CHAMP orbit (index d). The dynamic orbit used as a reference, provides us with ephemeris differences between the two orbits. The estimation of (“real”) auto and cross correlations between the C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 863

Fig. C35 The six point interaction in the grid; the three-dimensional Markov process of the 1st order (auto regressive process); the gray rectangle represents the same process in two-dimensions

vectors of pseudo-observations as functions of time, can be performed as Priestley (1981).

According to Tables C.9.10 and C.9.11 the Taylor-Karman structured (“ideal”) correlations are computed from the three-dimensional fx.tk/; y.tk/; z.tk /g time series. The adopted scale parameter is d D .2=3/Rchar,whereRchar is the characteristic length of the process. The characteristic length is deﬁned by the arc length of 2400 seconds (80 points for the 30 second sampling rate). The both parameters are experimentally estimated. For further details on the scale parameter and characteristic length, please see Wimmer (1982).

The numerical results of the study are graphically presented in Fig. C37. The gray area represents the estimated (“real”) correlation situation along the satellite arc as presupposed by Austen et al. (2001). The high auto and low cross-correlations between CHAMP satellite positions for approximately 20 min of an orbit arc are very evident. The Taylor-Karman structured correlation (black line), as theoreti- cally assumed, gives an upper bound of the “real” correlation situation along the satellite orbit. 864 C Statistical Notions, Random Events and Stochastic Processes

Fig. C36 The behavior of the longitudinal and lateral correlation functions, (C456) and (C 458), for different values of the scale parameter d. D 3=2/

Table C.9.12 Longitudinal and lateral correlation function for a homogenous and isotropic vector-valued random ﬁeld of potential type, 1st order Markov process “condition for a process of potential type”

Rr 2p ˙.x/ ! r2 x˙.x/ dx ! 0

d ! ˙l .r/ D dr Œr˙m.r/

“input”

21=2 r 3=2 r r r ˙.r/ D .3=2/ . d / K3=2. d / D .1 C d / exp. d / (C457)

“output”

r 2 r r r 1 r 2 ˙l .r/ D6. d / C exp . d /Œ4 C 2.d / C 6. d / C 6. d / (C458)

r 2 r r 1 r 2 ˙m.r/ D6. d / C exp . d /Œ2 C 6. d / C 6. d / (C459)

End of Table C 9.12 C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 865

Fig. C37 Graphical representation of the numerical study results (20 minute correlation length, 40 points for the 30 second sampling rate)

Ex 4: Nonlinear Prediction

Example C.9.4: Nonlinear Prediction

For direct interpolation, extrapolation or ﬁltering without trend analysis, linear prediction gives poor results, asking for a more general concept, namely nonlinear prediction.Asetoffunctional series of Volterra type for nonlinear prediction of scalar-, vector-, in general, tensor-valued signals is set-up. Optimizing the Helmert or Work Meister point error, a set of linear integral equations, generalizing the Weiner-Hopf integral equations. Multi-point-correlation functions (MPCF) are characteristic for those equations. A criterion for linear prediction, namely the vanishing of all MPCFs beside the two-point-correlation, is formulated solutions of the discrete integral equations and the prediction error tensor of rank two are reviewed. A structure analysis of PCFs of homogenous and isotropic vector ﬁelds proves the number of characteristic functions to be 2.n D 2/; 4.n D 3/; 10.n D 4/ and 26.n D 5/.

The methods of statistical prediction were developed relating to stochastic processes by two famous scientists, by the Russian A. N. Kolmorogov (1941) and by the American N. Wiener (1941), nearly at the same time during the Second World War. 866 C Statistical Notions, Random Events and Stochastic Processes

The world-wide resonance to these milestones began rather late: the work of A. N. Kolmorogov was due to the bad documentation of Eastern publications in Western countries nearly unknown, the great works of N. Wiener appeared as a classified report of section D2 of the American National Defence Research Committee. At that time the difficult formula apparatus made the acceptance rather impossible. Nowadays, the difficulties are only history, the number of publications about linear filtering, interpolation and extrapolation, reaches ten thousands. For example, the American Journal JRE, Transactions on Information Theory, New York appears regularly every year with a literature report about the State-of-the-Art. Notable are the works of R. Kalman (1960) who succeeded to solve elegant by the celebrated perdition equation, the Wiener-Hopf integral equation in transforming to a differential equation which is much easier to solve. The methods became very well known as a method of data analysis applied to Navigation of the Apollo Spacecraft.

In practice we begin with a two-step procedure. First, we analyze the observed values with respect to trend: we fit a data set to a harmonic function, to a exponential function or a polynomial with respect to “position”. Second, we analyze the data after the trend fit to a Wiener-Kolmogorov prediction assuming a linear relation between the predicted signal and the reduced unknown signal. The coefficients in this linear relation depend on the distance between the points and its direction. It is well known how to operate with the discrete approximation of the classical Wiener-Hopf integral equation.

The two step procedure accounts for the typical results that (a) the direct application of a linear Kolmogorov-Wiener prediction gives too inconsistent results, namely too large mean-square-errors of the predicted signal. The remarkable disadvantage of the two step procedure, in contrast is the immense computer work. Thus the central question is: is the two-step-procedure necessary when a concept for nonlinear prediction exists.

For all statistical prediction concepts the correlation function between various points is the key quantity. In practice, the computational problems are greatly reduced if we are able to ﬁnd out whether the predicted signal is subject to a statistical homogenous and isotropic process. It is a key information if a correlation function depends only on the distance between the involved points or is a function of the difference vector, namely isotropy and homogeneity, our special topic in a special section.

Example C.9.41: The hierarchy for the equations in nonlinear prediction of scalar- valued signals We review shortly the characteristical equations which are given in nonlinear prediction. Before hand we agree to the following notation: All Greek indices run f1; 2; ;N 1; N g counting the number of points P˛ with a signal s.P˛/.The C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 867 signal may be a scalar signal, a vector signal or a tensor signal. A natural assumption is the “simple prediction model” that the predicted signal s.P/ at the point P is a linear function of the observed signal s.P˛/ for ˛ 2f1; 2; ;N 1; N g. The coefficients which describe the linear relation are dented by a.PP˛ /.The inconsistency between the predicted signal and the observed signal is denoted by the letter “"00 leading the prediction variance Ef".P /".P /g, namely assuming that PN the difference s.P/ a.P; P˛ /s.P˛/ is random. ˛D1 Nonlinear prediction is defined by a nonlinear relation between the predicted signal and the observed signal, for instance the sum of a linear and a quadratic relation described by our previous formula. The postulate os a minimum mean square error leads to the problem to estimate the coefficients a and b. Of course, we could continue with signal functions which depends of arbitrary power series. The open question is which is signal relation of arbitrary powers.

In order to give an answer to our key question, we have to change our notion from discrete to an integral concept. We arrive in this concept if we switch over 0 to a continuous distribution of points P , namely from fP;P˛g to fr; r g.Ina one-dimensional problem we integrate over a measured line, in a two-dimensional problem we integrate over a surface. The integral set-up, in contrast, is reviewed by formula .C 461/ .C 464/ O.s3/ expresses the order of neglected terms in the functional series SŒs.r0. We refer to a type of Taylor series in a generalized Hilbert space referred to V. Volterra (1959), called Volterra functional series.The coefﬁcient functions given by .C 462/ are called as variational derivatives in a generalized Hilbert space.

A special criticism originates from the continuous version of the Volterra series. Help comes from the application of functional analysis, namely the Dirac function as a generalized function illustrated by formulae (C464) and (C465). It is easy now the switch from the continuum to the discrete, namely for a regular grid. Nowadays the integral version is “simple” to end up with a discrete formulation, introduced already by N. Wiener.

Up to now we have introduced a set-up of our prediction formulae for scalar signals. In relating our work to practical problems we have to generalize to vector-valued functions.

Table C.9.13 Discrete and continuous Volterra series for scalar signals

“discrete 2nd order Volterra series” PN PN PN s.P/ D a.P; P˛ /s.P˛/ C b.P;P˛;Pˇ/s.P˛ /s.Pˇ/ C ".P / (C460) ˛D1 ˛D1 ˇD1

“continuous 2nd order Volterra series” 868 C Statistical Notions, Random Events and Stochastic Processes R n 0 0 0 s.r/ D dR r a.rR; r /s.r / C d nr0 d nr00a.r; r0; r00/s.r0/s.r00/ CO.s3/ C ".r/.C461/ rd R “continuous 3 order Volterra series” n 0 0 0 s.r/ D dR r a.rR; r /s.r / n 0 n 00 0 00 0 00 C R d r R d r a.R r; r ; r /s.r /s.r / C d nr0 d nr00 d nr000a.r; r0; r00; r000/s.r0/s.r00/s.r000/ C O.s4/ C ".r/.C462/ “variational derivatives” ıs.r/ a.r; r0/ D .C 463/ ıs.r0/ 0 00 1 ı2s.r/ a.r; r ; r / D 0 00 2Š ıs.r /ıs.r / 1 ı3s.r/ a.r; r0; r00; r000/ D 3Š ıs.r0/ıs.r00/ıs.r000/ “from continuous to discrete via Dirac functions: example” C1R d nr0f.r0/ı.r; r0/ D f.r/ (C464) 21 C1R 0 n 0 0 0 4 !1 for r D r ; d r ı.r; r / D 1 ı.r; r / WD 1 0otherR wise: s.P/ D d nr0a.P; r0/s.r0/ (C465) PN PN n 0 0 D K˛d r ı.P; r / D a.P; P˛ /s.P˛/ ˛D1 ˛D1

End of Table C.9.12

Example C.9.42: The hierarchy of the equations in nonlinear prediction of vector- valued signals In order to take care of practice, we will present ﬁnally the concept of prediction of vector-valued signals set-up by vector-valued Volterra series of type (C641). The partial variational derivatives of the vector-valued functions, also called signal 3 Si .r/, are given by (C459) upto order O.s . We let the indices of Latin type run 1; 2; ;n1; n where n is the maximum dimension number. We apply the Einstein summation convention, namely the addition with respect to identical numbered indices.

In order to fill the deflections depend on the position r of the measurements. In order to determine the unknown coefficients of the nonlinear prediction “ansatz”, we choose risk function to be optimized. For instance, we may choose the trace of the variance function T1 D trij r D min, called Helmert point error,orthe determinant of the variance function T2 D detij r D min, called Werkmeister C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 869 point error,astwo basic invariants called Hilbert invariants. Such an optimal design leads to a linear integral equation, a system of first kind, for the unknown 0 0 00 tensors aij .r; r /, aij k .r; r ; r / etc. If we only refer to terms up to second order, namely to quadratic signal terms, we have eliminated cubic, biquadratic etc. term, the standard concept of linear prediction.

The ﬁrst nonlinear term, the quadratic terms, lead to the concept of two-point correlation functions: 0 0 ˚ij .r; r / WD Efsi .r/sj .r /g three-point correlation functions: 0 00 0 00 ˚ij k .r; r ; r / WD Efsi .r/sj .r /sk.r /g and four-point correlation functions: 0 00 000 0 00 000 ˚ij k l .r; r ; r ; r / WD Efsi .r/sj .r /sk.r /sl .r /g The term “multipoint functions” is to be understood that tensor valued functions between multipoints appear as correlation functions. With respect to linear prediction newly three-point and four-point correlation functions appear.Letus formulate the discrete version of the integral equations: here run all the bar indices .1; 2; ;Nn/up to the number Nn. Indeed, we approximated the integral (C470), (C471) etc. by (C472), (C473) etc. Indeed we switched from continuum to discrete by our summation convention by neglecting the summation signs. For instance,the P Pı two-point and three-point functions aim.r; r1/; aimn.r; r1; r2/ are written aim ;bimn in the discrete versions.

In summary, our matrices have NnŒ.Nn/2 C .N n/ elements. They correspond to the correlation functions ˚i;j ;˚i;j;k and ˚i;j;k;l . Finally we restrict us to prediction terms of to second order when we write (C473) to (C476). Here we advantage of two conventions: “Extensor” calculus developed by Craig (1930–1939) abbreviates (a) as the multiindex the series i1i2; i1. Again we sum up indices in brackets. In the new version of H. V. Craig’s multiindex we gain our characteristic equations (C476) - (C 479). The convention coefficients ai .i/ have their origin in the generalized Taylor series with the condition that higher order coefficients are getting smaller with increasing multiindex degree. If we apply the fundamental prediction equation (C475) we have to sum up the coefficients ai;i ;ai;i1; i2;ai;i1; i2; i3 etc. With the generalized variance-covariance matrix, function of the coordinates of the position vector r. The Helmert point error reduces proportional to the number of prediction coefficients as long as the inequality (C481) holds. If we fix the 0 coefficient functions aij .r; r / for the distance zero to one, the following coefficients 0 00 0 00 000 aij k .r; r ; r /; aij k l .r; r ; r ; r / are smaller than one. 870 C Statistical Notions, Random Events and Stochastic Processes

Table C.9.44 Discrete and continuous Volterra series for vectors

continuous 2nd order Volterra series

Pn R n 0 0 0 si .r/ D d r aij .r; r /sj .r / j D1 Pn Pn R R n 0 n 00 0 00 0 00 3 C d r d r aij k .r; r ; r /sj r skr C o.s / (C466) j D1 kD1 2 0 ısi .r/ 0 00 ı si .r/ aij .r; r / D 0 ;aij k .r; r ; r / D 0 00 (C467) ısj .r / ısj .r /ısk .r /

Variance function as a tensor of 2nd order ˚ R n R ij .r/ DR E Œsi .r/ d .r1/aii1 .r; r1/si1 .r1/ n n d .r1/ d .r2R/aii1i2 .r; r1; r2/si1 .r1/si2 .r2/ n R ŒsRj .r/ d .r1/ajj1 .r; r1/sj1 .r1/ n n d .r1/ d .r2/ajj1j2 .r; r1; r2/sj1 .r1/sj2 .r2/ (C468)

J1 D trij .r/ D min; J2 D detij .r/ D min (C469)

from continuous to discrete: multipoint correlation functions R 0 n 0 R˚ij .r; rR/ D d r1aim.r; r1/˚jm.r ; r1/C n n d r1 d r2aimn.r; r1; r2/˚jmn.r; r1; r2/ (C470) R 0 00 n 0 00 ˚Rij k .r; rR; r / D d r1aim.r; r1/˚jkm.r ; r ; r1/C n n 0 00 d r1 d r2aimn.r; r1; r2/˚jkmn.r ; r ; r1; r2/ (C471)

P˛ P ˛ Pı ˛ı P˛ˇ P ˛ˇ Pı ˛ˇ ı ˚jk D aim ˚jm C bimn˚jmn;˚ij k D aim ˚jkm C bimn˚jkmn (C472)

˚j k D aim˚j m C bimn˚j mn;˚ij k D aim˚j km C bimn˚j kmn (C473)

ai1 D ci1;ai2 D ci2; ;aim D cim; ;aiNn D ciNn bi11 D ciNnC1;bi12 D ciNnC2; ;bi1n D ciNn C n; ;bi1Nn D ciNn C Nn bi21 D ci2NnC1;bi22 D ci2NnC2; ;bi2n D ci2Nn C n; ;bi2Nn D ci2Nn C Nn bi31 D ci3NnC1;bi32 D ci3NnC2; ;bi3n D ci3Nn C n; ;bi3Nn D ci3Nn C Nn : :

bim1 D cimNnC1;bim2 D cimN nC2; ;bimn D cimNn C n; ; C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 871

bimN n D cimNn C Nn biNnNn D ci.Nn/2 Nn (C474)

0 0 0 0 0 0 0 0 0 0 1 ˚ D C ˚ ;C D ˚ ˚00 (C475)

˚i.j/ .ai.i/;˚.j /.i// D 0 (C476)

˚ij1 .aii1 ;˚j1i1 / .aii1i2 ;˚j1i1i2 / D0 (C477)

˚ij1j2 .aii1 ;˚j1j2i1 / .aii1i2 ;˚j1j2i1i2 / D0 (C478) R n 0 .aii1 ;˚j1i1 / D d r1aii1 .r; r1/˚j1i1 /.r ; r1/ (C479) R R n n 0 00 .aii1i2 ;˚j1j2i1i2 / D d r1 d r2aii1i2 .r; r1; r2/˚j1j2i1i2 /.r ; r ; r1; r2/ (C480)

ij .r/ D ..ıi.i/ ai.i//; ˚j.i// ..ıi.i/ ai.i//; aj.j/˚.i/.j // (C481)

2tr.ai.i/;˚j.i//>tr.ai.i/;aj.j/;˚.i/.j // (C482)

Table C.9.44: Volterra series

Example C.4. Structure analysis

Here we analyst the structure of multipoint functions of higher order with respect to statistical assumptions of homogeneity and isotropy. These assumptions made our prediction formula much simpler. If we were not assuming this statistical property, we had to compute for every point and every direction an independent higher order correlation function. For instance, if we had to calculate two-point- correlation functions in two dimensions every 1 Gon, we had to live with 1600 different correlation functions. If we would be bale to assume isotropy, the number of characteristic functions would reduce to two functions only! In the case of three- point-functions the reductions range from 3200 non-isotropic functions to four in case of isotropy. Of course, we will prove these results.

The structure of these multi-point-functions of statistically homogenous and isotropic vector-valued signals follow from the form invariance of the functions with respect to rotations and mirror functions of the coordinate system. Alternatively, we have to postulate invariance against transformations of the complete orthogonality group O.n/. Form invariance of a multi-point-correlation functions is to be 872 C Statistical Notions, Random Events and Stochastic Processes

Fig. C38 Projections: longitudinal and lateral components for two-, three- and four point correlation functions, isotropy-homogeneity assumption

understood with respect to base functions which leads to scalar quantities. Our next section offers a better understanding of these theoretical results. As a short introduction to two-point-correlation os vector-valued signals we illustrate the theory of characteristic functions for two-point-functions, for three-point-functions and four-point-functions by Fig. C38.

In case of a vector valued functions the structure of the correlation function used for linear prediction is generated the coordinates si .r/ at the point r or P1 and 0 0 sj .r / at the point r or P2 and the projected coordinate components along the connection line P1P2 (along track) and orthogonal to the line P1P2 (cross track) called “longitudinal” and “lateral” sp and sn (“parallel” p, “normal” n). These characteristic functions ˚nn resp. ˚pp given by the two-point correlation functions illustrated in Fig. C38. C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 873

Table C.15 Characteristic functions for two-point- and three-point correlation functions of vector-valued signals

“homogenous and isotropic two-point-function”

0 0 ˚nn.r/ D Efsn.r/sn.r /g;˚pp .r/ D Efsp.r/sp.r /g, ˚pn.r/ D ˚np .r/ D 0 (C483)

0 0 xi xj ˚j .r; r / D Efsi .r/sj .r /gD˚nnıij Œ˚pp .r/ ˚nn.r/ r2 (C484)

“homogenous and isotropic three-point-function” 9 0 00 ˚nnp.r1;r2/ D Efsn.r/; sn.r /; sp .r /g > 0 00 = ˚np n .r1;r2/ D Efsn.r/; sp.r /; sn.r /g 0 00 > (C485) ˚pnn.r1;r2/ D Efsp.r/; sn.r /; sn.r /g ;> 0 00 ˚ppp .r1;r2/ D Efsp.r/; sp.r /; sp.r /g

x 0 x 0 00 xi0 xi j j ˚ij k .r; r ; r / D jrr0j ıjk˚pnn.r1;r2/ C jrr0j ıik˚np n .r1;r2/C .x 0 x /.x 0 x /.x 00 x / xk00 xk i i j j k k D x 00 x ıij ˚nnp.r1;r2/ C 0 2 1 1 jrr j .x100 x1/ f˚ppp .r1;r2/ ˚pnn.r1;r2/ ˚np n .r1;r2/ ˚nnp.r1;r2/g (C486)

End of Table: characteristic functions

In Table C 16, we present only results. How is the detailed proof of our results? This is here our central interest.

We begin with the definition of the popular scalar product and the norm of a Hilbert space by equations (c 484) and (c 488), extend by the variance-covariance function (C489) by Craig calculus generalizations of a Hilbert space are given by (C490) and (C491). In terms of Craig calculus we compute Helmert’s optimal design of unknown coefficients by (C492), (C493), (C494) and (C495) and in contrast Werkmeister’s optimal design by (C496), (C497) and (C498), again of the unknown coefficients.

Table C.16 Optimal variance function

Hilbert space: scalar product, norm R .f; g/ D f!g!dP! (C487) ˝ 874 C Statistical Notions, Random Events and Stochastic Processes

p rR 2 jf2jD .f; f / D jf!j dP ! (C488) ˝

Variance function of vector-valued signals, Craig calculus

ij .r/ D ."i .r/; "j .r// D ..ıi.i/ ai.i//; ˚j.i// ..ıi.i/ ai.i//; aj.j/˚i.j// (C489)

Hilbert optimal design, Craig calculus J1 D tr ij .r/ D ..ıi.i/ ai.i//; ˚j.i// ..ıi.i/ ai.i//; aj.j/˚i.j// D min 2 ıJ1ŒaI.i/ D 0; ı J1ŒaI.i/>0 (C490) 9 dJ Œa C"b d 2J Œa C"b . 1 i.i/ i.i/ / D 0; . 1 i.i/ i.i/ / >0 > d" "D0 d"2 "D0 > > => J Œa C "b D tr ..ı .a C "b //; ˚ / 1 i.i/ i.i/ i.i/ i.i/ i.i/ j.i/ (C491) tr ..ı .a C "b //; .a C "b /˚ / > i.i/ i.i/ i.i/ j.j/ j.j/ i.j/ > > ;> dJ1Œai.i/C"bi.i/ . d" /"D0 D 0 W tr .bi.i/;.˚j.i/ aj.j/˚i.j// D 0

2 . d J1Œai.i/C"bi.i/ / >0W .b ;b ˚ />0 d"2 "D0 tr i.k/ j.l/ .k/.l/ (C492)

positive deﬁniteness of the correlation function R R n 0 n 00 0 00 00 tr d r d r bik.r; r /bji.r; r /˚kl.r; r />0 (C493)

Werkmeister optimal design, Craig calculus 9 J2 D det ij .r/ Df..ı1.i/ a1.i//; ˚1.i// ..ı1.i/ a1.i//; a1.j /˚.i/.j //g = f..ı2.i/ a2.i//; ˚2.i// ..ı2.i/ a2.i//; a2.j /˚.i/.j //g ; 2 f..ı1.i/ a1.i//; ˚2.i// ..ı1.i/ a1.i//; a2.j /˚.i/.j // g (C494) 2 ıJ2Œai.i/ D 0; ı J2Œai.i/>0 9 2 dJ1Œai.i/C"bi.i/ d J1Œai.i/C"bi.i/ > . / D 0; . / >0 > d" "D0 d"2 "D0 > => 2 3 4 J2Œai.i/ C "bi.i/ D A0 C A1" C A2" C A3" C A4" > (C495) > > 2 ;> . d J1Œai.i/C"bi.i/ / D 0 W A D 0 d"2 "D0 1 C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 875

2 . d J1Œai.i/C"bi.i/ / >0W A >0 (C496) d"2 "D0 2

End of Table: Craig optimal design

Finally, we study in more detail the concept of structure functions. We illustrate the form invariance of its base invariants which are invariant in terms of rotational and mirror symmetry. Let us call the base vector ei .r/ relating to signal si .r/ at the 0 0 0 placement vector r;ej .r / relating to signal sj .r / at placement vector r etc. The set-up of base vectors by (C449) and corresponding structure functions by (C580) are given. They lead us to the combination of base invariants ˚1;˚2 following Taylor (1935), and Karman (1937), ˚3;˚4;˚5 etc in Table (C501)–(C505). ﬁnally, we summarize up to order three the various isotropic and homogenous correlation functions in Table C.17, (C501)–(C505). They are based on the work of Karman and Howarth (1938), and Obuchow (1958). Let us present ﬁnally the number of characteristic functions for homogenous and isotropic correlation functions of higher order for vector-valued signals, collected in Table C.18.

Table C.17 Scalar products of base invariants

base invariants 9 xi xi ;xj 0 xj 0 ;xk00 xk00 ;xl000 xl000 ; etc. > > e e 0 ;ee 00 ;ee 000 ; etc. > i i i i i i > e 0 e 00 ;e0 e 000 ; etc. > i i i i => etc. (C497) x e ;xe 0 ;xe 00 ;xe 000 ; etc. > i i i i i i i i > x 0 e 0 ;x0 e 00 ;x0 e 000 ; etc. > i i i i i i > 00 00 00 000 > xi ei ;xi ei ; etc. ;> etc.

base functions 9 > ˚1.ei / D ˚i ei > 0 > ˚2.ei ;ej 0 / D ei ej 0 ˚ij 0 .r; r / = ˚ .e ;e 0 ;e 00 / D e e 0 e 00 ˚ 0 00 .r; r0; r00/ (C498) 3 i j k i j k ij k > ˚ .e ;e 0 ;e 00 ;e 000 / D e e 0 e 00 e 000 ˚ 0 00 000 .r; r0; r00; r000/ > 4 i j k l i j k l ij k l ;>

combinations of base invariants

˚1 D ˚1.ei / D ˚1ei (C499) 876 C Statistical Notions, Random Events and Stochastic Processes

0 ˚2.ei ;ej 0 / D ei ej 0 ˚ij 0 .r; r / D ei ej 0 fxi xj 0 a1.r/ C ıij 0 a2.r/g (C500) (Taylor (1935), Karman (1937))

0 00 ˚3.ei ;ej 0 ;ek00 / D ei ej 0 ek00 ˚ij 0k00 .r; r ; r / D D ei ej 0 ek00 fxi xj 0 xk00 a1.r1;r2/ C xi ıj 0k00 a2.r1;r2/C Cxj 0 ıik00 a3.r1;r2/ C xk00 ıij 0 a4.r1;r2/g (C 501) 9 0 00 000 0 00 000 0 00 000 0 00 000 > ˚4.ei ;ej ;ek ;el / D ei ej ek el ˚ij k l .r; r ; r ; r / D > > D ei ej 0 ek00 el000 fxi xj 0 xk00 xl000 a1.r1;r2;r3/ > > Cxi xj 0 ık00l000 a2.r1;r2;r3/C = Cx 0 x 000 ı 00 a .r ;r ;r / C x x 00 ı 0 000 a .r ;r ;r /C j l ik 3 1 2 3 i k j l 4 1 2 3 > Cx x 000 ı 0 00 a .r ;r ;r / C x 0 x 00 ı 000 a .r ;r ;r /C > i l j k 5 1 2 3 j k il 6 1 2 3 > x 00 x 000 ı 0 a .r ;r ;r / ı 0 ı 00 000 a .r ;r ;r / > C k l ij 7 1 2 3 C ij k l 8 1 2 3 C ;> Cıik00 ıj 0l000 a9.r1;r2;r3/ C ıj 0k00 ıil000 a10.r1;r2;r3/g (C502) 9 0 00 000 0000 ˚4.ei ;ej 0 ;ek00 ;el000 ;em0000 / D ei ej 0 ek00 el000 em0000 ˚ij 0k00l000m0000 .r; r ; r ; r ; r / D > > 0 00 000 0000 0 00 000 0000 > D ei ej ek el em fxi xj xk xl xm a1.r1;r2;r3;r4/ > > Cxi xj 0 xk00 ıl000m0000 a2C > > Cxi xj 0 xm0000 ık00l000 a3 C xi xj 0 xl0000 ık00m0000 a4 > > Cx x 000 x 0000 ı 0 00 a C > i l m j k 5 > Cx x 00 x 0000 ı 0 000 a C x x 00 x 000 ı 0 0000 a > i k m j l 6 i k l j m 7 => Cxk00 xl000 xm0000 ıij 0 a8C 0 000 0000 00 0 00 0000 000 > Cxj xl xm ıik a9 C xj xk xm ıil a10 > > Cxj 0 xk00 xl000 ıim0000 a11C > > Cxi ıj 0k00 ıl000m0000 a12 C xj 0 ıik00 ıl000m0000 a13 C xk00 ıij 0 ıl000m0000 a14C > > Cx 000 ı 0 ı 00 0000 a C x 0000 ı 0 ı 00 000 a C x ı 0 000 ı 00 0000 a C > l ij k m 15 m ij k l 16 i j l k m 17 > Cx 0 ı 000 ı 00 0000 a C x 00 ı 000 ı 0 0000 a C x 000 ı 00 ı 0 0000 a C > j il k m 18 k il j m 19 l ik j m 20 > 0000 00 0 000 00 000 0 0000 0 0000 00 000 > Cxm ıik ıj l a21 C xi ık l ıj m a22 C xj ıim ık l a23C ;> Cxk00 ıim0000 ıj 0l000 a24 C xl000 ıim0000 ıj 0k00 a25 C xm0000 ıil000 ıj 0k00 a26g (C503)

End of Table: Scalar products of basic invariants C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 877

Table C.18: Isotropic and homogenous correlation functions, two- and three- point functions, normal and longitudinal components 9 0 2 ˚11.r; r / D a1.r/r C a2.r/ D ˚pp = 0 0 ˚12.r; r / D ˚12.r; r / D ˚pn D ˚np D 0 ; (C 504) 0 0 ˚22.r; r / D ˚33.r; r / D ˚nn 9 2 00 > ˚111 D r1 .x1 x1/a1.r1;r2/ C r1Œa2.r1;r2/ C a3.r1;r2/ > 00 > C.x1 x1/a4.r1;r2/ D ˚ppp > 00 00 > 2 > ˚112 D r1 .x3 x2/a1.r1;r2/ C .x2 x2/a4.r1;r2/ > 00 00 > ˚ D r2.x x /a .r ;r / C .x x /a .r ;r / > 113 1 3 3 1 1 2 3 3 4 1 2 > ˚ D 0; ˚ D r a D ˚ ;˚ D 0 > 121 122 1 2 pnn 123 => ˚ D 0; ˚ D 0; ˚ D r a D ˚ 131 132 133 1 2 pnn (C505) ˚ D 0; ˚ D r a D ˚ ;˚ D 0 > 211 212 1 3 np n 213 > ˚ D ˚ D ˚ D 0 > 231 232 233 > > ˚311 D ˚312 D ˚313 D r1a3 D ˚np n > > ˚321 D ˚322 D ˚323 D 0 > 00 00 > ˚331 D .x1 x1/a4 D ˚nnp;˚332 D .x2 x2/a4; > 00 ;> ˚333 D .x3 x3/a4 ˚pp ˚nn a1.r/ D r2 ;a2.r/ D ˚nn 9 ˚ppp ˚pnn˚np n ˚nnp > a1.r2;r2/ D 2 00 > r1 .x1 x1/ > ˚pnn = a2.r2;r2/ D r1 (C506) ˚np n > a3.r2;r2/ D > r1 > ˚nnp ; a4.r2;r2/ D 00 .x1 x1/

0 xi xj ˚ij .r; r / D ˚nn.r/ıij C Œ˚pp .r/ ˚nn.r/ r2 (C507)

00 0 00 xi xj .xk xk / ˚ij k .r; r ; r / D ıjk˚nnn C ıik˚np n C 00 ıij ˚nnp r1 r1 .x x1/ 00 1 xi xj .xk xk / C 2 00 .˚ppp ˚pnn ˚np n ˚nnp/ r1 .x1 x1/ (C508) 0 00 ˚ij 0k0 .r; r D r / D xi xj 0 xk0 a1.r1/ C xi ıj 0k0 a2.r1/ C.xj 0 ıik0 C xk0 ıij 0 /a3.r1/ ˚ .r /˚ .r /2˚ .r / ppp 1 pnn 1 nnp 1 0 0 D 3 xi xj xk r1 .x 0 ı 0 Cx 0 ı 0 / xi 0 0 j ik k ij C˚pnn.r1/ ıj k C ˚nnp.r1/ r1 r1 (C509)

End of Table: Isotropy and homogenity 878 C Statistical Notions, Random Events and Stochastic Processes

Table C.19:Number of characteristic functions for multi-point functions based on vector-valued signals

multi-point correlation functions number of characteristic functions

22 34 410 526

End of Table: characteristic functions

Ex 5: Nonlocal Time Series Analysis

Example: Nonlocal time series analysis

Classical time series analysis, for instance Earth tide measurements, is based on local harmonic and disharmonic effects. Here we present a nonlocal concept to analyze on a global scale measurements at different locations. The tool of the generalized harmonic analysis base on correlation functions of higher order between vector-valued signals at different points and at different times. Its polyspectrum is divided into poly-coherence and poly-phase. A 3d-ﬁlder of Kolmogorov-Wiener type is designed for an array analysis. The optional distance between stations and the optimal registration time- Nyquist frequency- is given to avoid aliasing.

Standard procedures of signal analysis treat a registered time series of one station only. It is our target here to combine various observation stations: we are interested in the global picture handled by array analysis. We like to refer to applications in seismology and tidal analysis, the great success of tomography. A 3d- or multichannel ﬁlter makes possible to arrange an optimal signal-to-noise ratio in terms of an optimal design of the station distance and an optimal design of the registration time, namely statistical accuracy of numbers which decide upon a linear or nonlinear ﬁlter.

In the ﬁrst section we introduce the classical nonlocal analysis for N D 2 stations for a vector-valued signal. We assume no correlation between the signal components. The mono-spectrum, namely the Fourier-transform of scalar-valued correlation functions is separated by coherence- and phase-graphs.Thesecond section is a straight forward approach to registered vector-valued signals between N D 2 stations. Support has the origin in the analysis of coherence surface and of phase surface of Fourier-transformed correlation-tensors. Poly-spectra of signal space- time functions form the basis of N stations analysis in the third section.We C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 879 separate real- and imaginary-parts in terms of a poly-coherence-graph and of a poly-phase-graph, both matrix forms. An extended array analysis is based on a Kolmogorov-Wiener ﬁlter in the fourth section. We give a simple analysis to predict an optimal registration and an optimal station distance. At the end we give references to (a) array analysis,(b)mono-spectrum and (c) bi- and poly-spectra.

Scalar signals, “classical analysis”

In terms of the “classical time series analysis” registrations at two different places are to be combined: Which are the frequency bands in which we can detect coherences, which is the spectral phase-dislocation between two registrations. We are to keep in mind the Fundamental Lemma of N. Wiener that a frequency assured by coherence has the property of maximal probability, characterized by signal time- functions at two stations. We sum up the following operations: (i) Calculation of auto- and cross-correlation functions of scalar-valued signals, (ii) Calculation of Fourier-transforms of auto- and cross-correlation functions, separation of mono-spectra in terms of active and passive (cos versus sin terms) components, 2 2 (iii) Plots of coherence graphs H .k1;!1/ and phase graphs .k1;!1/ Here we depart from a scalar space-time representation of a signal, namely by a signal s.r;t/at a placement r and at the time t and a signal s.r0;t0/ at the placement r0 and at the time t 0 by correlation, in detail by (C512 - C 519). fŒs.r;t/;s.r0;t0/ denotes the probability density, to meet the signals s and s0 at the placements r; r0 and the times t;t0. we intend to transform the probability integral into a space- time integral assuming homogeneity and ergodicity by (c 513) and (C514). The e space-time complex mono-spectrum ˚.ri ;t1/ ! ˚.ki ;!1/ is defined by (C515), e direct and inverse, (C516), ˚.ki ;!1/ decomposed into a real-valuedp part and an imaginary part.“e” denotes the Fourier transform, i W 1, n indicates the dimension. In analogy to wave propagation in electrodynamics, we define e e e e Rcf˚gDW.ki ;!1/ active spectrum or co-spectrum, but Jmf˚gDW quadratic or “blind spectrum”. Backward transformation ˚.ri ;1/ cos.!11 < k1jr1 >/ e e and ˚.ri ;1/ sin.!11 < k1jr1 >/ lead to f.ki ;!1/; .ki ;!1/g Let us 2 e2 e2 finally H WD .ki ;!1/ C .ki ;!1/g as the space-time coherence graph,and e e tan .ki ;!1/gWD.ki ;!1/=.ki ;!1/; .ki ;!1/ the phase graph, both very important in defining “coherent light”. Figure C39 illustrates the phase graph, e2 C e2 the length of the complex vector in the polar coordinates. The auto correlation functions .r; r/ and .r0; r0/ clearly lead us to “angle” and “length” in the complex plane Fig. C40 illustrates the fundamental coherence- and phase- functions.

Finally the contributions of lackman (1965); Granger and Hatanaka (1964); Harris (1967); Kertz (1969, 1971); Lee (1960); Pugachev (1965); Robinson (1967), Robinson and Treitel (1964). 880 C Statistical Notions, Random Events and Stochastic Processes

Table C.20: Variance-covariance function of a scalar signal, direct and inverse Fourier transform, space-time representation

EŒs.r;t/s.r0;t0/ WD ˚.r; r0;t;t0/ (C510)

C1R C1R WD ds.r;t/ ds.r0;t0/s.r;t/s.r0;t0/f Œs.r;t/s.r0;t/ (C511) 1 1

CRT CRx CRy 0 0 1 ˚.r r;t t/ D lim 8T xy dt dx dy (C512) T;x;y!1 T x y

s.x;y;t/sŒx C .x0 C x/;y C .y0 C y/;t C .t 0 C t/ (C513)

0 0 r r WD r1;t t WD 1 (C514)

nC1 C1R C1R e n ˚.k1;!1/ D .2/ 2 d r1 d1˚.r1;1/ exp Ci.!11 k1r1/ 1 1 (C515)

nC1 C1R C1R n e ˚.r1;1/ D .2/ 2 d k1 d!1˚.k1;!1/ exp i.!11 k1r1/ 1 1 (C516)

e e e e e ˚.k1;!1/ D ReŒ˚C iImŒ˚ D .k1;!1/ C i .k1;!1/ (C 517)

nC1 C1R C1R e n .k1;!1/ WD .2/ 2 d r1 d1˚.r1;1/ cos.!11 k1r1/ (C518) 1 1

nC1 C1R C1R e n .k1;!1/ WD .2/ 2 d r1 d1˚.r1;1/ sin.!11 k1r1/ (C519) 1 1

2 e2 e2 H WD .k1;!1/ C .k1;!1/ (C520) e2 .k1;!1/ tg.k1;!1/ WD (C521) e2 .k1;!1/

End of Table: variance-covariance functions C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 881

Fig. C39 Decomposition into real and imaginary component Gauss representation fe; e g

Fig. C40 Scalar coherence and phase graph

C.9.52 First generalization: vector-valued classical analysis

At ﬁrst, we deal with separate single components of a general three-component tidal recording. We develop a joint analysis of all three components observed at two points on the surface of the Earth, for example. Its signal functions are correlated with each other, in the special case here the vertical component as well as the horizontal component. We receive a correlation tensor of second degree, namely in the form of the coherence matrix and the phase matrix. In detail, the analysis technique is based on the following operations: (i) Computation of auto- and cross-correlations of the vector-valued signal, (ii) Computation of the Fourier transform of the functions separation within the mono spectral matrix in “active”and“blind” spectra cosine-andsine-Fourier transform, (iii) Part of the coherenec matrix Hij .k1;!1/ as well as the phase matrix ij .k1;!1/.

Theoretical Background 882 C Statistical Notions, Random Events and Stochastic Processes

Let si .x;t/ D ıgi .x;t/ be a vector-valued space-time-signal consisting of vertical and horizontal components subject to i D 1 or x;i D 2 or y and i D 3 or z. The signal function si .x;t/ at the placement x and at the time instant t and the 0 0 0 0 signal function sj .x ;t / at the placement x and the time instant t are correlated by means of

0 0 0 0 Efsi .x;t/sj .x ;t /gDW˚ij .x; x ;t;t / for i;j 2f1; 2; 3g (C522) As an analogue of our previous formulae we deﬁne the coherence matrix, namely the coherence tensor Hij ,andthephase matrix ij

e2 e2 Hij .k1;!1/ WD ij .k1;!1/ C ij .k1;!1/ (C523) e2 .k1;!1/ tg .k ;! / D ij ij 1 1 e2 (C524) ij .k1;!1/

The coherence matrix as well as the phase matrix are represented by special surface for instance by coherence ellipses, coherence hyperbolae etc.

C.9.53 Second generalization: poly spectra Up to now we analyzed only signal registrations at two different points by correlation, namely by coherence and phase. Here, we introduce the concept of poly spectra for the purpose of registration of Earth tides at all points we have registrations. In detail, the analysis technique is based on the following operations: (i) Computation of multi-point correlation functions of a vector-valued tidal signal (ii) Computation of the Fourier transforms called polyspectra. The concept is able to separate mono-, bi-, tri-, in short polyspectra in “active” and “blind” spectra, cosine- and sine-Fourier transform. (iii) Part of generalized coherence graphs

Hi1i2iN 1iN .k1; k2; ; kN 1;!1;!2; ;!N 1/ (C525) and the generalized phase graphs

i1i2iN 1iN .k1; k2; ; kN 1;!1;!2; ;!N 1/ (C526) at N tidal stations being functions of wave number vectors k1; k2; ; kN 1 and the frequencies !1;!2; ;!N 1. Theoretical Background

Let si1 .x;t/ ıgi.x;t/ be a tensor-valued space-time tidal signal at the ﬁrst tidal 0 station, si2 .x ;t/ be a tensor-valued tidal signal at the second tidal station. If we C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 883 correlate all signal functions with each other, e receive a multi-point correlation function equal to the number of tidal stations.

0 0 Efsi1 .x;t/si2 .x ;t / sin .xN ;tn/gWD˚i1i2iN 1iN (C527)

In case of space like homogeneity and the time like stationarity we are able to represent the various correlation functions by

˚i1i2iN 1iN .x1; x2; ; xN 1;1;2; ;N 1/ (C528)

subject to

0 00 0 00 x1 WD x x; x2 WD x x; etc :1 D t t; 2 D t t; etc (C529) and their polyspectra e ˚ i1i2iN 1iN .k1; k2; ; kN 1;!1;!2; ;!N 1/ D ZC1 ZC1 C n.N 1/ .N 1/ .2/ 2 d n.N 1/r d n.N 1/ 1 1

˚i1i2iN 1iN .r1;r2; ;rN 1;1;2; ;N 1/

exp Ci.!11 C !22 CC!N 1N 1 k1r1 k2r2 kN 1rN 1/ (C530) n is the dimension number, N the number of tidal stations. For the case of N D 3 we use the term “bispectrum”, for the case N D 4 we use the term “trispectrum” in general “polyspectrum”. The common separation of ˚i1i2iN 1iN in real- and imaginary part leads us to the Polycoherence graph as well as the poly phase graph

2 e 2 e Hi1i2iN 1iN D Rc Œ˚ i1i2iN 1iN C Jm Œ˚ i1i2iN 1iN (C531) e JmŒ˚ i1i2iN 1iN tgi1i2iN 1iN D e (C532) RcŒ˚ i1i2iN 1iN

Numerical results are presented in Brillinger (1964), Grafarend (1972), Hasselmann et al. (1963), Rosenblatt (1965, 1966), Sinaj (1963 i, ii), Tukey (1959), and Van Nees (1965). 884 C Statistical Notions, Random Events and Stochastic Processes

Table C.9.54: Generalized array analysis on the basis of a Kolmogorov- Wiener filter (3D-filter) We do not attempt to review “filter theory”, instead we present array analysis for the purpose to optimally reduce noise in tidal measurements in optimal station density. The method of analysis is based in the following operations: (i) Computation of signal correlation functions from tidal registration in multi- stations and afterwards in noise correlation functions from an analysis of noise characteristic multi-dimensional stochastic process, for instance of type Markov, homogenous and isotropic (ii) Computation of spectra from diverse correlation functions (iii) Solving the linear matrix equation of type Wiener-Hopf equations of optimal filter functions. The optimal station density is achieved from the postulate <kjx > 1,the optimal registration time from the postulate !t 1. x is the distance of the tidal stations in position space, x the distance in the Fourier space, for instance kx D 2=x wavelength aliasing t the discretization parameter in the time domain and ! the corresponding frequency distance in the Fourier space,for instance ! D 2=T , ! the frequency aliasing. Example

1 r D k cos ^k; r 2 cos ^k; r ! 1=t

^k; r D 0; D 103 Km ;rŠ 140 Km ^k; r D 300;D 103 Km ;rŠ 330 Km t D 1 sec ;!D 1 Hz ;TŠ 6 sec t D 1 msec ;!D 103 Hz ;TŠ 6 msec

Theoretical Background

Lets us assume that the vector-valued registration signal consists of two parts,the pure signal part and the superimposed noise:

ıgi .x;t/D si .x;t/C ni .x;t/ (C533)

0 The ﬁltered signal ıgi .x;t/has the linear structure of type

0 0 0 ıgi .x;t/D si .x;t/C ni .x;t/ (C534) C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 885

0 0 The filter error "i .x;t/ D si .x;t/ si .x;t/ C ni .x;t/ ni .x;t/ is designed by I1trij .x;t/ D min or I1 det ij .x;t/ D min subject to the error tensor ij .x;t/ WD Ef"i .x;t/"j .x;t/gDmin. The filter fij .x;t/is defined by the linear Volterra setup

ZC1 Z 0 n ni .x;t/WD d x1 d1fij .x1;1/nj .x x1;t 1/ (C535) 1

JŒij .fkl.x; t// D min lead the “necessary” generalized Wiener-Hopf equations.

ZC1 ZC1 0 0 n 00 00 00 00 ˚si sj .r; r ;t;t / D d r dt fik.r; r /˚jk.r; r / (C536) 1 1 e e e e C f ij D ˚ si sj .k1;!1/Œ˚ si sk C ˚ ni nk (C537)

ŒC identiﬁes the generalized inverse of type minimum norm least squares. e e e e F Œii D ˚ ssŒiiŒ˚ ssŒii C ˚ nnŒii (C538)

The result of 3D or linearized multi-channel leads to an easy solution. If we would care for a nonlinear ﬁlter subject to the multi-index J WD .j1;j2; ;j˛/ and K WD .k1;k2; ;kˇ/ we will receive the solution

˚ij D .fik;˚jk/ (C539) with respect to the scalar product .:; :/ of the corresponding Hilbert space. Numericals examples you can ﬁnd in C. B. Archambeau et. al. (1965), J. W. Britill and F. E. Whiteway (1965), H. W. Briscoe and P. L. Fleck (1965), J. P. Burg (1964), J. F. Claerbout (1964), P. Embree et. al. (1963), E. Grafarend (1972), A. N. Kolmogorov (1941), N. Wiener (1949, 1964, 1968).

Comments

In experimental sciences, namely geosciences like geophysics, geodesy, environmental sciences, namely, weather, air pollution, rainfall, oceanography, temperature affected sciences take action on a spatially and temporally large scale to very large scales. For this reason, they never can be considered stationary or homogenous- isotropic, but non-stationary inhomogeneous or anisotropic in the statistical sense Spock and Pilz (2008) if not Guttorp and Sampson (1994)as one of the attempts to model non-stationary, other examples are geodetic networks 886 C Statistical Notions, Random Events and Stochastic Processes in two- and three-dimensional Euclidean space and on two-, three- and four- dimensional manifolds including the sphere, the ellipsoid or General Relativity as an excellent example for inhomogeneity and anisotropy presented here. C. Calder and N. Cressie (2007) review different directions in spectral and convolution based on Gaussian non-stationary random modeling. Speck and Pilz (2008) present a spectral decomposition without any restriction to the Gaussian approach. They advice the reader to apply a special Generalized Linear Mixed Model (GLMM), the topic of our book, namely derived from the spectral decomposition of non-stationary random fields. Topics (i) Stationary and isotropic random field versus (ii) Non-stationary, locally isotropic random field (iii) Interpretation in the context of linear mixed models (LMMs)

Y D Fˇ C Ab C "0

a. Fˇ: design matrix of the first kind, result: regression function, trend behaviour F fixed design matrix b. Ab: non-deterministic random fluctuations around the trend functions, A fixed design matrix. c. "0: independent, homoscedastic error term centered at zero (iv) Relation to non-stationary, locally isotropic random fields (v) Generalized linear mixed models (GLMMs) (vi) Extending generalized linear mixed models (DHGLMMs) (vii) Limear and mixed models and Hermitic polynomials (viii) Anisotropic stochastic process axisymmetric, Chandrasekhar (1950), Grafarend (1972) Appendix D Basics of Groebner Basis Algebra

D-1 Deﬁnitions

To enhance the understanding of the theory of Groebner bases presented in Chap. 15, the following deﬁnitions supplemented with examples are presented. Three commonly used monomial ordering systems are; lexicographic ordering, graded lexicographic ordering and graded reverse lexicographic ordering.Firstwe deﬁne the monomial ordering before considering the three types.

Deﬁnition D.1. (Monomial ordering):

Zn A monomial ordering on kŒx1;:::;xn is any relation > on 0 or equivalently any ˛ Zn relation on the set x ;˛2 0 satisfying the following conditions: Zn (a) Is total (or linear) ordering on 0 Zn (b) If ˛>ˇand 2 0,then˛ C >ˇC Zn (c) < is a well ordering on 0. Zn This condition is satisﬁed if and only if every strictly decreasing sequence in 0 eventually terminates.

Deﬁnition D.2. (Lexicographic ordering):

This is akin to the ordering of words used in dictionaries. If we deﬁne a polynomial in three variables as P D kŒx;y;z and specify an ordering x>y>z, i.e., x comes before y and y comes before z, then any term with x will supersede that of y which in tern supersedes that of z. If the powers of the variables for respective monomials Zn are given as ˛ D .˛1; :::;˛n/ and ˇ D .ˇ1; :::;ˇn/; ˛; ˇ 2 0; then ˛>lex ˇ if in the vector difference ˛ ˇ 2 Zn; the most left non-zero entry is positive. For the ˛ ˇ same variable (e.g., x) this subsequently means x >lex x :

E. Grafarend and J. Awange, Applications of Linear and Nonlinear Models, 887 Springer Geophysics, DOI 10.1007/978-3-642-22241-2, © Springer-Verlag Berlin Heidelberg 2012 888 D Basics of Groebner Basis Algebra

Example D.1. x>y5z9 is an example of lexicographic ordering. As a second example, consider the polynomial f D 2x2y8 3x5yz4 C xyz3 xy4,wehavethelexicographic order; f D3x5yz4 C 2x2y8 xy4 C xyz3 jx>y>z.

Deﬁnition D.3. (Graded lexicographic ordering):

In this case, the total degree of the monomials is taken into account. First, one considers which monomial has the highest total degree before looking at the lexicographic ordering. This ordering looks at the left most (or largest) variable Zn of a monomial and favours the largest power. Let ˛; ˇ 2 0; then ˛>grlex ˇ if Pn Pn n j˛j D ˛i > jˇj D ˇi or j˛j D jˇj,and˛>lex ˇ; in ˛ ˇ 2 Z ; the most iD1 iD1 left non zero entry is positive.

Example D.2.

8 3 2 6 2 3 x y z >grlex x y z j.8;3;2/ >grlex .6;2;3/,sincej.8;3;2/j D 13 > j.6;2;3/j D 11 and ˛ ˇ D .2; 1; 1/. Since the left most term of the difference (2) is positive, the ordering is graded lexicographic. As a second example, consider the polynomial f D 2x2y8 3x5yz4 C xyz3 xy4; we have the graded lexicographic order; f D3x5yz4 C 2x2y8 xy4 C xyz3 jx>y>z:

Deﬁnition D.4. (Graded reverse lexicographic ordering):

In this case, the total degree of the monomials is taken into account as in the case of graded lexicographic ordering. First, one considers which monomial has the highest total degree before looking at the lexicographic ordering. In contrast to the graded lexicographic ordering, one looks at the right most (or largest) variable of Zn a monomial and favours the smallest power. Let ˛; ˇ 2 0; then ˛>grevlex ˇ if Pn Pn n j˛j D ˛i > jˇj D ˇi or j˛j D jˇj,and˛>grevlex ˇ,andin˛ ˇ 2 Z the iD1 iD1 right most non zero entry is negative.

Example D.3.

8 3 2 6 2 3 x y z >grevlex x y z j.8;3;2/ >grevlex .6;2;3/ since j.8;3;2/j D 13 > j.6;2;3/j D 11 and ˛ ˇ D .2; 1; 1/. Since the right most term of the difference (-1) is negative, the ordering is graded reverse lexicographic. As a second example, D-2 Buchberger Algorithm 889 consider the polynomial f D 2x2y8 3x5yz4 C xyz3 xy4 ,wehavethegraded reverse lexicographic order: f D 2x2y8 3x5yz4 xy4 C xyz3 jx>y>z. P ˛ If we consider a non-zero polynomial f D ˛ a˛x in kŒx1; ::::;xn and ﬁx the monomial order, the following additional terms can be deﬁned:

Deﬁnition D.5.

Zn Multidegree of f : Multideg (f/ D max.˛ 2 0 ja˛ ¤ 0/ Leading Coefﬁcient multideg.f / of f :LC(f/D amultideg.f / 2 k Leading Monomial of f :LM(f/D x (with coefﬁcient 1) Leading Term of f :LT.f / D LC.f / LM .f /.

Example D.4.

Consider the polynomial f D 2x2y8 3x5yz4 C xyz3 xy4 with respect to lexicographic order fx>y>zg ,wehave Multideg (f )=(5,1,4) LC (f )= 3 LM (f )= x5yz4 LT (f )= 3x5yz4

The deﬁnitions of polynomial ordering above have been adopted from Cox et al. (1997), pp. 52–58.

D-2 Buchberger Algorithm computes the Groebner basis using the computer algebraic systems ((CAS) of Mathematica and Maple. With Groebner basis, most nonlinear equations that are encountered in geodesy and geoinformatics can be solved. All that is required of the user is to write algorithms that can easily run in Mathematica or Maple using steps that are discussed in Sects. D-21 and D-22 below.

D-21 Mathematica Computation of Groebner Basis

Groebner basis can be computed using algebraic softwares of Mathematica (e.g., version 7 onwards). In Mathematica, Groebner basis command is executed by writing

InŒ1 WD GroebnerBasisŒfpolynomialsg; fvariablesg; (D.1) 890 D Basics of Groebner Basis Algebra where In[1]:= is the Mathematica prompt which computes the Groebner basis for the Ideal generated by the polynomials with respect to the monomial order speciﬁed by monomial order options.

Example D.5. (Mathematica computation of Groebner basis):

In Example 15.3 on p. 536, the systems of polynomial equations were given as f1 D xy 2y 2 2 f2 D 2y x :

Groebner basis of this system would be computed by

InŒ1 WD GroebnerBasisŒff1;f2g; fx;yg; (D.2) leading to the same values as in (15.10) on p. 536. With this approach, however, one obtains too many elements of Groebner basis which may not be relevant to the task at hand. In a case where the solution of a speciﬁc variable is desired, one can avoid computing the undesired variables, and alleviate the need for back-substitution by simply computing the reduced Groebner basis. In this case (D.1) modiﬁes to

InŒ1 WD GroebnerBasisŒfpolynomialsg; fvariablesg; felimsg; (D.3) where elims is for elimination order. Whereas the term reduced Groebner basis is widely used in many Groebner basis literature, we point out that within Mathematica software, the concept of a reduced basis has a different technical meaning. In this book, as in its predecessor, and to keep with the tradition, we maintain the use of the term reduced Groebner basis.

Example D.6. (Mathematica computation of reduced Groebner basis):

In Example D.2.1, one would compute the reduced Groebner basis using (D.3)as

InŒ1 WD GroebnerBasisŒff1;f2g; fx;yg; fyg; (D.4) which will return only 2x2 C x3. Note that this form is correct only when the retained and eliminated variables are disjoint. If they are not, there is absolutely no guarantee as to which category a variable will be put in! As an example, consider the solution of three polynomials x2 C y2 C z2 1, xy z C 2,andz2 2x C 3y. In the approach presented in (D.4), the solution would be

InŒ1 WD GroebnerBasisŒfx2Cy2Cz21; xyzC2;z22xC3yg; fx;y;zg; fx;yg; (D.5) leading to D-2 Buchberger Algorithm 891

f1024 832z 215z2 C 156z3 25z4 C 24z5 C 13z6 C z8g:

Lichtblau (Priv. Comm.) however suggests that the retained variable, in this case .z/ and the eliminated variables .x; y/ be separated, i.e.,

InŒ1 WD GroebnerBasisŒfx2 C y2 C z2 1; xy z C 2;z2 2x C 3yg; fzg; fx;yg: (D.6) The results of (D.6)arethesameasthoseof(D.5), but with the advantage of a vastly better speed if one uses an ordering better suited for the task at hand, e.g., elimination of variables in this example. For the problems that are solved in our books, the condition of the retained and eliminated variables being disjoint is true, and hence the approach in (D.5) has been adopted. The univariate polynomial 2x2 C x3 is then solved for x using the roots command in Matlab (see e.g., Hanselman and Littleﬁeld (1997), p. 146) by

roots. Œ1 200/: (D.7)

The values of the row vector in (D.7) are the coefﬁcients of the cubic polynomial x3 2x2 C0xC0 D 0 obtained from (D.4). The values of y from Example (15.3) on p. 536 can equally be computed from (D.4) by replacing y in the option part with x and thus removing the need for back substitution. We leave it for the reader to compute the values of y from Example (15.3) and also those of z in Example (15.3) using reduced Groebner basis (D.3) as an exercise. The reader should conﬁrm that the solution of y leads to y3 2y with the roots y D 0 or y D˙1:4142.From experience, we recommend the use of reduced Groebner basis for applications in geodesy and geoinformatics. This will; fasten the computations, save on computer space, and alleviates the need for back-substitution.

D-22 Maple Computation of Groebner Basis

In Maple Version 10 the command is accessed by typing > with (Grobner); Basis (WL;T) is for Groebner, gbasis (WL;T) is for reduced Groebner basis. WL list or set of polynomials, T monomial order description, e.g. “plex” (variables) – pure lexicographic order and > is the Maple prompt and the semicolon ends the Maple command). Once the Groebner basis package has been loaded, the execution command then becomes > gbasis (polynomials, variables) which computes the Groebner basis for the ideal generated by the polynomials with respect to the monomial ordering speciﬁed by variables in the executable command. We should point out that at the time of writing this book, May 2012, version 16 of Maple has been released. 892 D Basics of Groebner Basis Algebra

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A. Bjerhammar’s left inverse, 121 Cauchy–Green tensor, 128 Additive rank partitioning, 268, 289 Cayley calculus, 236 Adjoint form, 151 Cayley inverse, 342, 395 Algebraic partitioning, 69 Centering matrix, 465, 468 Algorithms Central point, 362 Buchberger, 534, 536 Central statistical moment of second order, Euclidean, 534 189 ANOVA, 240 Centralized Lagrangean, 465 Antisymmetry operator, 152 Choleski decomposition, 617 Arithmetic mean, 146, 187, 192, 379 Choleski factorization, 136 Arnoldi method, 135 Circular normal distribution, 363, 374, 375 Associativity, 595 Clifford algebra, 593 Columns space, 141 Commutativity, 595 Balancing variance, 335 Complex algebra, 593 Basis, 168 Composition algebra, 592, 622 Bessel function, 371 Condition equation, 194 Best Linear Uniformly Unbiased Estimation, Condition equations, 122 207 Conformal, 380 Bias matrix, 308, 332, 386 Constrained Lagrangean, 108, 196, 237, 238, Bias norms, 309 284, 311 Bias vector, 308, 332, 340, 386, 387, 392, 394, Constraint Lagrangean, 208 395 Coordinate mapping, 265 Bias weight matrix, 340, 342, 394, 395 Coordinates, 152 BIQUUE, 426 Cartesian, 557 Bounded parameters, 3 Correlated observations, 185 Break points, 142 Covariance, 754 Buchberger algorithm, 592 Covariance components, 231 Covariance factor, 227 Covariance factorization, 215 Canonical base, 136 Covariances, 215 Canonical form, 673, 696 Criterion matrix, 118 Canonical LESS, 134 Cross product, 150, 153, 155, 158 Canonical MINOLESS, 297, 299, 300 Cumulative pdf, 679, 738 Canonical observation vector, 132 Cumulative probability, 723–725 Canonical unknown vector, 136 Curved surfaces, 102 Cartesian coordinates, 117, 642, 675 Cyclic ordering, 152

E. Grafarend and J. Awange, Applications of Linear and Nonlinear Models, 1011 Springer Geophysics, DOI 10.1007/978-3-642-22241-2, © Springer-Verlag Berlin Heidelberg 2012 1012 Index

D. Hilbert, 538 First derivatives, 96, 106, 191, 284, 303, 366, Datum defect, 265 413, 450 Datum problem, 495 First moment, 194, 754 Dedekind, 538 First moments, 422 Degrees of freedom, 94, 228, 267, 682, 729 First order moments, 205 Diagonal matrix, 467 1st Grassmann relation, 150 Differentiable manifold, 16, 102, 104 Fisher normal distributed, 378 Differential Geometry, 102 Fisher sphere, 361 Dilatation, 473, 674, 683 Fisher-von Mises, 362 Dilatation factor, 650 Fixed effects, 187, 307, 317, 326, 329, 333, Direct equations, 374 334, 339, 384 Direct map, 641 Fourier, 381 Dispersion matrix, 205, 207, 309, 394, 424, Fourier–Legendre linear model, 59 754 Fourier–Legendre space, 51 Dispersion vector, 217 Fourier-Legendre series, 381 Distances, 557 Fourth central moment, 755 Division algebra, 593, 602, 622 Fourth order moment, 196 Division algorithm, 539 Free parameters, 3 Frobenius, 311 E. Noether, 538 Frobenius error matrix, 462 EIGEN, 728 Frobenius matrix, 468 Eigencolumn, 300 Frobenius norm, 116, 311, 332, 387 Eigenspace, 302 Frobenius norms, 334, 387 Eigenspace analysis, 111, 130 Frobenius theorem, 593 Eigenspace synthesis, 130, 142 Fuzzy sets and systems, 128 Eigensystems, 135, 301 Eigenvalue, 300, 302 Eigenvalue Analysis, 728 G-inverse, 103 Eigenvalue decomposition, 130, 132, 136 Gain matrix, 520 Eigenvalues, 558 Gamma function, 645 positive eigenvalues, 113 Gauss elimination, 110, 209, 414, 451, 534, zero eigenvalues, 113 535, 545 Eigenvectors, 302 Gauss normal distribution, 362 Elliptic curve, 375 Gauss–Jacobi Combinatorial Algorithm, 172, Equations 176, 178 polynomial, 538 Gauss–Markov model, 208 Error propagation, 470 Gauss-Lapace normal distribution, 688 Error propagation law, 210, 402, 403, 407, 758 Gauss-Laplace pdf, 677 Error vector, 210 Gauß trapezoidal, 402 Error-in-variables, 384 General bases, 130 Euclidean, 113 General Gauss–Markov model Euclidean length, 95 with mixed effects, 426 Euclidean metric forms, 375 General inverse Helmert transformation, 673 Euclidean space, 112, 118 General reciprocal, 602 Euclidian observation space, 642 General Relativity, 113 Euler circles, 104 Generalized inverse, 103, 105, 129, 194, 315, Euler–Lagrange strain, 128 317 Euler–Lagrange tensor, 128 Geodesic, 362 Expectation operator, 757 Geodesy, 529 Grassmann column vector, 166 Grassmann coordinates, 141, 150, 152, 154, Fibering, 95, 104, 263, 268 158, 159, 166, 169 Finite matrices, 602 direct, 170 First central moment, 754 inverse, 170, 171 Index 1013

Grassmann manifold, 102 Injective mapping, 638 Grassmann space, 141, 152, 154, 158, 164, Injectivity, 94 168, 169 Injectivity defect, 302 Grassmann vector, 159 Inner product, 111, 363 Grassmann–Plücker coordinates, 122 Interval calculus, 122 Greatest common divisors (gcd), 534 Invariant quadratic estimation, 187, 221, 237 Groebner basis, 527, 539, 592 Inverse equations, 374 definition, 540 Inverse Jacobian determinant, 372 Maple computation, 891 Inverse map, 102, 641 Mathematica computation, 889 Inverse partitioned matrices, 109 Inverse spectrum, 149 Inverse transformations, 132 Hankel matrix, 606, 610 IQUUE, 190 Hat matrix, 114, 122, 146, 149, 150 Helmert equation, 231 Helmert matrix, 231, 609 Jacobi determinant, 666, 673 augmented, 678 Jacobi map, 102 extended, 678 Jacobi matrix, 69, 102, 114, 403, 641, 700, 759 quadratic, 678 Jacobian, 641 Helmert random variable, 694 Jacobian determinant, 551, 647, 684 Helmert traces, 231 Join, 150 Helmert transformation, 679 Join and meet, 141 quadratic , 677 rectangular , 677 Helmert’s ansatz, 229, 230 Kalman filter, 493, 517, 519, 520 Helmert’s distribution, 681 Kernel, 94 Helmert’s polar coordinates, 733 Khatri–Rao product, 116 Helmert’s random variable, 708 Killing analysis, 598 Hesse matrix, 114, 204, 403, 474, 759 Kolmogorov–Wiener prediction, 440 High leverage points, 172 Krige’s prediction concept, 441 Hilbert basis invariants, 128 Kronecker, 538 Hilbert Basis Theorem, 538, 539 Kronecker matrix, 116 Hilbert space, 108 Kronecker–Zehfuss product, 116, 338, 391, Hodge dualizer, 151 495 Hodge star operator, 141, 150, 587, 640 Kummer, 538 Hodge star symbol, 158 KW prediction, 441 Holonomity condition, 120, 122 Homogeneous equations, 223 Homogeneous inconsistent equations, 411 Lagrange function, 204 Homogeneous linear prediction, 438 Lagrange multiplier, 105, 108, 110, 191, 238, Horizontal rank partitioning, 122, 132 414 Hurwitz theorem, 593 Lagrange multipliers, 284, 310, 413, 415, 422, Hypothesis testing, 339, 394 423 Lagrange parameter, 449 Lagrange polynomial, 124 Ideal, 536, 537 Lagrangean, 72, 336, 366, 390, 556, 557 Ideal membership, 540 multiplier, 536 Idempotent, 113, 213 Lanczos method, 136 Implicit Function Theorem, 263 Langevin distribution, 362 Inconsistent linear equation, 506 Langevin sphere, 361 Independent random variables, 640 Latent condition equations, 142, 168 Inhomogeneous inconsistent equations, 411 Latent conditions, 164 Inhomogeneous linear equations, 100 Latent restrictions, 141, 168 Injective, 265, 267 Laurent polynomials, 599 1014 Index

Leading Monomial (LM), 539 Moore-Penrose inverse, 602 Least Common Multiple (LCM), 541 Multideg, 540 Least squares, 95, 180, 185, 240, 563, 602 Multilinear algebra, 150 Historical background, 180 Multiplicative rank partitioning, 266, 272 Least squares solution, 98, 103, 108, 268, 272, Multipolynomial resultant, 547, 592 411, 414 Multipolynomial resultants, 529, 549 Left complementary index, 267 Multivariate Gauss–Markov model, 496 Left eigenspace, 133 Multivariate LESS, 495 Left Gauss metric, 128 Multivariate polynomials, 539 Left inverse, 129 LESS, 98 Leverage point, 157, 162 Necessary condition, 557 Lexicographic ordering, 536, 545, 887 Necessary conditions, 96, 106, 109, 191, 204, graded reverse, 888 208, 238, 270, 284, 304, 312, 336, Lie algebra, 592, 598 337, 367, 390, 413, 448, 451 Likelihood normal equations, 204 Non-symmetric matrices, 135 Linear estimation, 332 Non-vanishing eigenvalues, 111 Linear Gauss–Markov model, 421 Nondegenerate bilinear form, 111 Linear model, 108, 121, 150 Nonlinear error propagation, 758 Linear operator, 95, 265, 268, 757 Nonlinear model, 102 Linear space, 103, 265 Nonlinear models, 104, 525 Loss function, 361, 366 Nonlinear prediction, 438 LUUE, 190 Nonlinear problem, 461 Normal equation, 367 Normal equations, 110, 191, 310, 337, 390, Maple, 536 391, 412, 414–416, 511 Mathematica, 536 Normal models, 196 Matlab, 536 Normal space, 101 “roots” command, 536 Normalized column vectors, 166 Matrix algebra, 265, 593, 602 Normalized Grassmann vector, 159 Maximal rank, 103 Null space, 94, 265–267 Maximum, 536 Maximum Likelihood Estimation, 187, 202 Maximum norm solution, 113 Objective functions, 113, 114 Mean, 198 Oblique normal distribution, 374, 375 Mean matrix, 754 Observation equation, 518 Mean Square Estimation Error, 307, 339, 345, Observation point, 102 386, 392–395, 399 Observation space, 94, 102, 111, 113, 130, Measurement errors, 517 187, 214, 265, 266, 268, 364 Median, 187, 198 Observational equations, 511 Median absolute deviation, 187 Observed Fischer information, 204 Meet, 150, 153 Octonian algebra, 593 Minimum, 536 One variance component, 223, 226, 230, 236 Minimum bias, 310 Optimality conditions, 206 Minimum distance mapping, 101, 102 Optimization problem, 202 Minimum geodesic distance, 362, 366 Orthogonal, 51 Minimum norm, 268, 272, 618, 619 Orthogonal column space, 141 Minimum variance, 207, 422 Orthogonal complement, 101 MINOLESS, 266 Orthogonal matrix, 606 Mixed models, 240 Orthogonal projection , 104 Model manifold, 265 Orthogonality condition, 157, 229 Modiﬁed Bessel function, 363 Orthogonality conditions, 164 Monomial ordering, 887 Orthonormal, 51, 679 Moore–Penrose inverse, 274, 289 Orthonormal base vectors, 130, 150 Index 1015

Orthonormal frame of reference, 154 Quadratic form, 189 Orthonormal matrix, 467, 474, 608 Quadratic Lagrangean, 337, 390 Orthonormality, 673 Quaternion algebra, 593 Outlier, 188, 198 Quotient set, 104 Outliers, 172, 332 Overdetermined, 95, 186 Overdetermined system of linear equations, Random, 428 136 Random effect, 329, 339, 384, 386, 392, 399, 426, 438 Random regression model, 474 Parameter space, 94, 102, 175, 187, 265, 266, Random variable, 519 268, 364, 450 Range, 267 Partial differential equations, 540 Range space, 94, 103, 266, 267 Partial inverse, 602 Rank factorization, 289, 617 Partial redundancies, 164 Rank identity, 284 PCA, 728 Rank partitioning, 12, 95, 132, 157, 164, 215, Permutation matrix, 198 263, 266, 298 Permutation operator, 151 Reduced Groebner basis, 537, 890 Plücker coordinate, 153 Reduced Lagrangean, 465 Plücker coordinates, 141, 150, 154, 158, 159, Redundancy, 147 166, 169 Redundancy matrix, 146, 147, 157, 164 Polar coordinates, 363, 642, 675 Regularization parameter, 307 Polar decompositions, 131 Relative index, 111 Polynomial, 528 Residual vector, 192, 316 Polynomial coefficients, 94 Ricci calculus, 236 Polynomial equations, 540 Ridge estimator, 342, 395 Polynomial of degree one, 94 Riemann manifold, 380 Polynomials Right complementary index, 267 equivalence, 540 Right eigenspace, 133 ordering Right eigenspace analysis, 146 lexicographic, 535 Right Gauss metric, 128 roots, 536 Right strain component, 128 univariate, 536, 559 Risk function, 449, 464 Positive definite, 106, 109, 113, 116, 130, 207, Robust, 332 208, 296, 308, 312, 314, 332, 338, Robust approximation, 122 391, 416, 427, 558, 617 Root mean square error, 187, 198, 679 Positive semidefinite, 106, 108, 112, 296, 617 Rotation, 131, 461, 674 Prediction equations, 520 Rotation matrix, 684 Prediction error, 520 Rotational invariances, 439 Prediction stage, 519 Principal Component Analysis, 728 Probability density function, 363, 638 Sample mean, 665, 667, 676 Probability distribution, 194 Sample median, 188, 198 Procrustes Algorithm, 461, 464, 472 Sample variance, 667, 676, 683 Projection lines, 556 Scale change, 674 Projection matrices, 213 Scale factor, 461 Pseudo-observations, 176 Schur complement, 615 Pseudoinvese, 602 Second central moment, 339, 393 Pullback operation, 175 Second central moments, 385 Second derivative, 106, 109, 271, 304, 367, 413, 450 Quadratic bias, 342, 396 Second moment, 194, 754 Quadratic estimation, 193 Second order design, 114, 115 Quadratic estimator, 193 Second order moment, 196 1016 Index

Second order moments, 205 Total degree, 888 Selfdual, 155 Total least squares, 108, 448 Semidefinite, 111 Trace, 390 Sequential optimization problem, 312 Transformation matrix, 521 Series inversion, 659 Translation, 461, 674 Set-theoretical partitioning, 104 Translation vector, 473 Similarity transformation, 521, 522 Translational invariant, 190, 193, 195, 228 Simple linear model, 200 Trend components, 441 Singular value decomposition, 467, 728 Tykhonov–Phillips regularization, 302, 307 Skew product symbol, 158 Tykhonov–Phillips regulator, 342, 395 Slices, 103 Slicing, 95, 263 Spatial redundancies, 157 Unbiased estimation, 677 Special Gauss–Markov model, 189, 205, 206, Unconstrained Lagrangean, 96, 106, 208, 220, 223, 225, 239, 315, 318, 271, 464 322, 326 Uniformly unbiased, 191, 193 Special linear Gauss–Markov model, 392 Unity, 595 with datum defect, 308 Univariate polynomials, 539 Special Relativity, 113 Unknown parameters, 115 Spherical coordinates, 361 Updating equations, 520 Standard deviation, 188, 681 Standard multivariate model, 497 State differential equation, 522 State equations, 519 Vandermonde matrix, 606, 611 Statistical moment of first order, 189 Variance, 215, 231, 754 Stochastic, 205 Variance factor, 227 Stochastic observations, 187, 517 Variance factorization, 215 Student’s random variable, 699 Variance-covariance, 215, 240, 403, 754 Subdeterminants, 157, 162, 169 Vector derivative, 106 Sufficiency condition, 96, 106, 109, 192, 284, Vector of curtosis, 755 413 Venn diagram, 104 Sufficiency conditions, 204, 208, 304, 312, Vertical rank partitioning, 95, 98 337, 390, 391 Von Mises, 362 Sufficient condition, 558 Von Mises circle, 361 Sufficient conditions, 451 Von Mises distribution, 755 Surjective, 94, 265, 267 Surjectivity defect, 94, 228, 302 SVD, 728 Weak isotropy, 439 Sylvester resultant, 548 Weight factor, 342, 395 Symmetric matrix, 129, 189, 224 Weight matrix, 115, 199, 472 System equations, 518 Weighted arithmetic mean, 175 Weighted arithmetic means, 178 Weighted Frobenius matrix norm, 207 Tangent space, 102 Weighted left inverse, 129 Taylor series, 399, 653, 655 Weighted Moore–Penrose inverse, 293 Taylor–Karman matrix, 404 Witt algebra, 592, 599 Taylor–Karman structure, 118, 402 Theory of Turbulence, 118 Third central moment, 754 Zehfuss product, 116 Three fundamental partitionings, 95 Zero correlation, 385