Geometrical aspects of linear prediction algorithms

Franc¸ois Desbouvries D´epartement Signal et Image Institut National des T´el´ecommunications 9 rue Charles Fourier, 91011 Evry, France [email protected]

Keywords: Schur and Levinson-Szeg¨o algorithms, Schur On the other hand, Toeplitz forms were studied indepen- complements, spherical trigonometry, analytic interpolation dently by Szeg¨o, who introduced a set of orthogonal poly- theory, Lobachevski geometry nomials with respect to an (absolutely continuous) positive measure on the unit circle. These polynomials obey a two- terms recursion [2] involving a set of parameters of modulus Abstract bounded by one, which later on were recognized to be equal to the Schur parameters [3]. In the 1940s, Toeplitz forms re- In this paper, an old identity of G. U. Yule among partial cor- ceived a revived interest in view of their natural occurrence relation coefficients is recognized as being equal to the cosine in the Kolmogorov-Wiener prediction and interpolation the- law of spherical trigonometry. Exploiting this connection en- ory of stationary processes (see e.g., [4, ch. 10], as well as ables us to derive some new (and potentially useful) relations the survey paper [5] and the references therein). Working on among partial correlation coefficients. Moreover, this obser- Wiener’s solution of the continuous time prediction problem, vation provides new (dual) non-Euclidean geometrical inter- Levinson proposed a fast algorithm for solving Toeplitz sys- pretations of the Schur and Levinson-Szeg¨o algorithms. tems; later on, the Levinson recursions were recognized as being the recurrence relations of Szeg¨o. 1 Introduction Finally, there was an intense activity in these fields beginning in the late 70s, mainly towards the development Linear prediction and interpolation is a major tool in time se- of fast algorithms for numerical linear algebra, on the one ries analysis and in signal processing. In this context, the hand, and in the domain of analytic interpolation theory, Schur and Levinson-Szeg¨o algorithms compute the partial on the other hand. Through these new developments and autocorrelation function of a wide-sense stationary process. extensions, new connections with other mathematical topics

As such, they have found a large variety of electrical engi- and disciplines were developed, including among others neering applications. displacement rank theory, J lossless transfer functions, Let us briefly recall the history of these algorithms. At the modern analytic function theory and operator theory. The beginning of the century, Schur, Carath´eodory and Toeplitz literature on these connections and extensions is vast; the were active in such fields as analytic function theory, Toeplitz reader may refer for instance to the papers [6] [7] [8] [9] and forms and moment problems. In 1917, Schur developped books [10] [11].

a recursive algorithm for checking whether a given func-

P 1

k The mathematical environment of these algorithms is thus

s(z )= s z

tion k is analytic and bounded by one

k =0 in the unit disk [1]. Such functions are characterized by very rich, and these various interactions have already been a sequence of parameters of modulus less than one (the thoroughly investigated by many researchers. In this wealthy Schur parameters) which are computed recursively from the context, our contribution in this paper consists in exhibiting

new unnoticed connections with spherical trigonometry. s

power series coefficients k by an elegant algorithm. On the

(z )=

other hand, Carath´eodory and Toeplitz showed that c

P 1

k As far as geometry is concerned, the Lobachevski geom-

c +2 c z 0

k is analytic and has positive real part for

k =1

P n

 etry was already known to be a natural environment of the

jz j < 1 a b c

j i

if and only if the Toeplitz forms i ,

j

i;j =0

 Schur and Levinson-Szeg¨o algorithms, since the core of these

c = c n n with , are positive for all . Let

n algorithms mainly consists in a linear fractional transforma-

c(z ) c 1+s(z )

0 tion leaving the unit circle invariant. However, a new point of

s(z )= () c(z )=c ;

0 (1)

(z )+c 1 s(z ) c view is obtained when considering the algorithms (via posi- 0

tive definite Toeplitz forms) in the particular context of their

s(z )j 1 c(z ) since j if and only if has positive real part, the application to linear prediction. Then, up to an appropriate Schur algorithm implicitely enables to test whether a Toeplitz normalization, the Schur and Levinson-Szeg¨o algorithms be- form is positive. come trigonometric identities in a spherical triangle. Since

2 P the real projective 2-space I is the quotient space obtained from the sphere by identifying antipodal points, we see that the alternate non-Euclidean geometry with constant curva- ture (i.e., the elliptic one) is indeed another natural geometri- AAAAAA cal environment of the Schur and Levinson-Szeg¨o algorithms as well. A AAAAAAA A

Let us briefly outline the underlying mechanisms lead-

X g

f AAAAAAAAAAAAAAAAAAAAAAAA

ing to this new interpretation. Let i be zero-mean A

1:n ^ square-integrable random variables, X the best lin- j b

n AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

X fX g i

ear mean-square estimate of j in terms of ,

i=1

1:n 1:n ^

~ c

X = X X

and j the corresponding estima- O

j j

tion error. The partial correlation coefficient (or par- AAAAA AAA AA

n

X fX g

X C

n+1 i

cor) of 0 and , given , is defined as =1

i AAAAAA

1:n 1:n 2 1=2 1:n 1:n 1:n 2 1=2

~ ~ ~

~ C

=[E (X ) ] E (X X )[E (X ) ]

 .

;n+1 0 0 n+1 n+1 0 B

It is bounded by 1 in magnitude and is classicaly interpreted AAAAAA B a

X X

n+1

as the correlation coefficient of 0 and , once the in-

n

fX g

fluence of i has been removed. In 1907, G. U. Yule =1 i AAAAAA

[12] showed that the parcors could be computed recursively : AAAAAAAAAAA

1:n1 1:n1 1:n1

  

0;n+1 0;n n;n+1

1:n

q q

 :

= (2)

0;n+1

1:n1 1:n1

2 2

1 ( ) 1 ( )

0;n n;n+1

It happens that this well known formula is formally equal to

the fundamental cosine law of spherical trigonometry : Figure 1: The spherical triangle ABC.

cos a cos b cos c

cos A =

; (3)

b sin c sin Let us now turn to the organization of this paper. Non- which gives an angle of a spherical triangle, in terms of its Euclidean hyperbolic aspects of the Schur algorithm are im- three sides (see figure 1). This observation establishes an plicit in [15] but do not seem otherwise to be well known. unexpected link between statistics and time-series analysis, Yet the Lobachevski geometry is, by construction, an essen- on the one hand, and spherical trigonometry (a branch of tial feature of the algorithm, which deserves to be better ap- trigonometry), on the other hand. preciated. More precisely, we show in section 2 that Schur’s In former papers [13] [14], spherical trigonometry was layer-peeling type solution to the Carath´eodory problem nec- shown also to admit a close connection with the electrical essarily makes use of automorphisms of the unit disk which, engineering topic of recursive least-squares adaptive filtering on the other hand, happen to be the direct isometries of the which, as linear regression analysis, is a mean square ap- Lobachevski plane. proximation problem. Now, the Schur and Levinson-Szeg¨o The last two sections are devoted to the new geometrical

algorithms can be written as algebraic recursions within a co- interpretations in terms of spherical trigonometry. So in sec-

n +1) variance matrix or its inverse; due to the identification (2) = tion 3, we relate recursive regressions within a set of ( (3), they admit a connection with spherical trigonometry as random variables, algebraic manipulations in a covariance well. matrix or in its inverse, and spherical trigonometry. We show Indeed, the source of such analogies is that (time- or that adding (resp. removing) a new variable in the regres- order-) recursive least-squares algorithms can be devel- sion problem which, in terms of Schur complements on the oped from projection identities. In linear regression, one covariance matrix (resp. on its inverse), amounts to using recognizes that the mean-square error to be minimized the quotient property [16, p. 279], corresponds in terms of is a distance, so the projection theorem can be applied spherical trigonometry to applying the law of cosines (resp. in the Hilbert space generated by the random variables. the polar law of cosines). Introducing a new variable in the regression problem Lastly in section 4, we further assume that the random amounts to updating a projection operator, and the problem variables are taken out of a stationary time series, and we use can indeed be described in terms of projections in a space the results of section 3 to interpret in parallel the Schur and generated by three vectors. But three unit-length vectors Levinson-Szeg¨o algorithms in terms of spherical trigonom- form a tetrahedron in 3D-space, and deriving projective etry. The Schur (resp. Levinson-Szeg¨o) relations consist in identities in a normalized tetrahedron results in deriving two Schur complement recursions (in the forward and back- trigonometric relations in the spherical triangle determined ward sense) in the original covariance matrix (resp. in its by this tetrahedron (see figure 1, and [14] for details). inverse), and can indeed be interpreted in dual spherical tri- angles. Up to an appropriate normalization, the Schur (resp. order decreasing Levinson-Szeg¨o) recursions coincide with 2.2 The Schur mechanism and automorphisms of the two coupled occurrences of the law of cosines (resp. of unit circle the polar law of cosines), and the Levinson-Szeg¨o recursions with two coupled occurrences of the polar five elements for- Let us now analyse the design of the Schur algorithm in terms

mula. of automorphisms of the unit disk. Automorphisms of a do-

def

G Aut(G) =

main G are bi-holomorphic mappings of :

1 1

f 2H(G; G); f f 2H(G; G)g

2 Non-Euclidean (hyperbolic) geometrical as- f s.t. exists, and , where

H(G ;G ) G

2 1 1 is the set of holomorphic functions of onto

pects of the Schur and Levinson-Szego¨ algo-

G S H(ID ; ID) ID

2 . The Schur class coincides with where is rithms the open unit disk. The mapping (4) can be decomposed into two steps : In this section, we first briefly recall the mechanisms under-

lying the Schur algorithm. We next show that the choice by 0

s(z ) a s (z )+ a

0 0

0 1

s (z )= ) s(z )= ;

( (7)

1

 0

Schur of a recursive solution to the Carath´eodory problem 

1 a s(z ) 1+a s (z )

0 1

naturally sets the algorithm in a non-Euclidean hyperbolic 0

1 0

environment. 0

s (z )= s (z ) s 7! s

and 1 . Since the transform in (7)

1 1 z is a linear fractional transformation (LFT), we begin with

2.1 The Schur algorithm recalling elementary (Euclidean) properties of LFTs [17]

def

ob M ob = fz 7!

The Carath´eodory analytic interpolation problem consists in [18]. Let M denote the M¨obius group

P

az +b

n

k

; ad bc 6=0g ad bc =0

a z + s(z )=

s with ( corresponds to the

cz +d

finding all functions such that 1. k

k =0

def

+1 n trivial situation where the mapping is either constant or un-

O (z ) 2S = ff (z ) jz j < 1; , and 2. s analytic in and

defined). Then the mapping

f (z )j1 jz j < 1g j for . In 1917, Schur proposed a ”layer-

peeling” type algorithm [1] (i.e., in which the interpolation

 : GL(2; C) ! M ob 

data are processed recursively) which we briefly recall. 

a b

az +b

M = 7!  ;  (z )= M

Let us first consider the case where there is only one inter- M with

cz +d

c d a

polation point 0 . Due to the maximum principle, the prob-

ja j > 1 ?

lem has no solution if 0 , and admits the unique solu-

 =   2 C M

is a group homomorphism. Since M for all ,

s(z )=a ja j =1 ja j < 1

0 0

tion 0 if .If , let

(M )=1

there is no loss of generality in supposing that det .

(2; C)

From now on we shall thus restrict to SL . Then the

1 s(z ) a zs (z )+ a

0 1 0

s (z )= , s(z )= :

1 (4)

 I I M ob 

 kernel of reduces to and , and is isomorphic to

z 1 a s(z ) 1+a zs (z )

1

0 0



M ob (SL(2; C)=I )

the associated quotient group : = .

a az +b a bcad 1 b

s 2

= c 6=0 + z + Then the key property of this transformation is that =

Now, 2 (resp. )if

cz +d c c z +d=c d d

S , s 2S

1 , so that =0 (resp. c ), so any LFT is a succession of translations,

inversions, rotations, and/or homotheties. Since all these ge-

zs (z )+a

1 0

(s 2S s(0) = a ) , (s(z )= s 2S): 1

and 0 and ometrical transformations preserve circles, a LFT maps any



1+ a zs (z ) 1

0 (5) circle in the complex plane (possibly of infinite radius) into 

another circle (possibly of infinite radius). In particular, M a

In the case of a single interpolation point 0 , equa-

T M

tion (5) provides a parametrization of all solutions to the maps the unit circle T onto itself if and only if belongs to

(1; 1) SL(2; C) 

Carath´eodory problem in terms of an arbitrary Schur func- the subgroup SU of consisting of unitary

diag (+1; 1)

matrices, with = :

s a 1

tion 1 . A new interpolation point can be accomodated

  s

by further restricting this set of possible functions 1 . From

a b

2 2

s 2S

(1; 1) = f jaj jbj =1g

equation (5), we see that ( and has interpolation con- SU with

 

b a

(a


a )) s 2S

n 1

straints 0 if and only if ( and has inter-

1 1 1

(a a


a

polation constraints )), in which the depend

   

0 i n1

0

1 0 1 0

s (z )

H

a = a

= fM 2 SL(2; C); M M = g: 1

on the data i . In particular, if and only if s.t.

1!

0 1 0 1

z =0

1

s (0) = a

1 : we are thus led back to the same metric con-

0

N N

1 strained interpolationproblem, but now of order n . These Let be the set of such M¨obius transforms; is a subgroup

1

ob

considerations lead by induction to the Schur algorithm : of M , and

az + b

1 s (z ) s (0)

def

2 2 p p

N = fz 7! jaj jbj =1g ;

: s (z )=s(z ); js (0)j < 1;s (z )=

p p+1

0 if s.t.

  

(0)s (z ) z 1 s b z + a

p

p



(SU (1; 1)=I ): (6) =

1

In x2.3 we will deal with geometrical aspects of the recursion (6). This

 2N ID is why the regular case only is considered here; all details of the general case Finally, observe that any M also maps the interior can be found in [1]. of the unit-circle onto itself (and similarly for its exterior).

ja j < 1

We now turn back to the Schur algorithm. Since 0 , This geometry is thus, by construction, the natural geometri- we have cal environment of the Schur algorithm.

Finally, let us briefly consider the Levinson-Szeg¨o algo-

(s(z ) 2H(ID; ID) s(0) = a )

and 0 (8)

rithm. It is not a solution to an analytic interpolation prob-

() (s(z ) 2H(ID; ID) s(0) = a )

and 0 (9) lem, but can nevertheless be rephrased (via the Schur-Cohn

0 0

(z ) 2H(ID ; ID) (0) = 0) ) (s s

( and (10) stability test) in the framework of section 2.1 [24], and thus

1 1

(z )=

shares the same geometrical environment. For let a

() s (z ) 2H(ID ; ID) ; 1

(11) P

n

n i n  

a z (z )=z (a(1=z )) f (z )=b(z )=a(z )

, b and .

i =0

whence (5). Equivalence (8)/(9) is due to the maximum prin- i

(z ) TT Then f is rational and has modulus 1 on . So, by

ciple and (10)/(11) to the Schwarz lemma. As for (9)/(10), it

(z ) ID

the maximum modulus theorem, (f is analytic in and

0

s =   s  : z 7! M

holds because M , where the mapping

1

jf (z )j =1 TT jf (z )j1 ID

p on ) if and only if ( in and

2

(z a )= 1ja j

0 0

z a

0

jf (z )j =1 TT f

p

 (z )= N  = on ), i.e., if and only if is a rational bounded 

M belongs to

 1a z

2

(1a z )= 1ja j 0 0

0 function of the lossless type (a Blashke product). But this

(ID)

Aut . In fact, it is interesting to notice that (7) was the

f = f

can be checked via the Schur algorithm, because 0 is

only possible choice, because, as is well known [19] [20],

n jf (0)j < 1

a Blashke product of order if and only if p for D

the automorphisms of I indeed coincide with the group of

0  p  n 1 jf (0)j =1

and n ; the recursions coincide with

LFTs which leave the unit-disk invariant : the order-decreasing Levinson-Szeg¨o algorithm.

N = Aut(ID) :

1

R =R n

3 Schur Complements in 0: and spher- 0:n 2.3 Hyperbolic geometry of the Schur and Levinson- ical trigonometry Szeg¨o algorithms We now turn from analytical to geometrical considerations. From now on we shall deal with spherical trigonometry as- These are obtained naturally in the framework of the theory pects of the Schur and Levinson-Szeg¨o algorithms. In this

developped in [21], which aims at describing the holomor- intermediate section, we first recall some elementary projec-

n C

phic structure of a domain G of in terms of geometric tive identities. We next bring back from spherical trigonom-

G; d ) d

( etry some relations among parcors which we will refer to G properties of the space G . If the distance is cho-

sen such that any holomorphic mapping of G onto itself is in sections 3.3 and 4. In the view of section 4, we even-

a contraction, then automorphisms of G are isometries with tually consider Schur complementation in a covariance ma-

d trix or in its inverse, because Schur complements provide respect to G and can thus be interpreted (in the spirit of F. Klein) as rigid motions with respect to the geometry speci- the connection between the Schur and Levinson-Szeg¨o al-

d gorithms and spherical trigonometry. The reason why is

fied by G . t

The Schwarz-Pick lemma [17] [20] [21] [22] provides a that algebraically the elementary recursion with pivot k;k :

1

0

t ! t = t t t t

i;j i;k k;j

i;j , reduces to a cosine law when i;j

nice illustration of this general methodology to the present k;k

p p

t t j;j

situation; as expected, we shall meet non-Euclidean hyper- normalized by i;i .

ID bolic geometry, since G reduces to the unit disk , which

is a Euclidean model of the Lobachevski plane (see e.g. 3.1 Partial correlation coefficients, recursive projec-

d (:; :) C

[23]). Let M denote the M¨obius distance in : for all

tions and Yule’s parcor identitity

0

z z

0 0

z; z 2 C;d (z; z )=

M

0 . The Schwarz-Pick lemma

z z

1 Our geometrical results are based on the properties of orthog-

H(ID; ID) states that any function f belonging to is a con- onal projectors and can thus be formalized in any Hilbert

traction with respect to the M¨obius distance : space. However, the natural framework in this paper is

2

IL ( ; A;P)

2H(ID; ID) ) f the space of complex, zero-mean, square-

( ; A;P) 0

0 integrable random variables defined on , endowed

f (z )f (z )

0

z z

 ; z; z 2 ID;



 0

for all 0

1(f (z )) f (z ) 1z z

X; Y )= E (XY )

with the inner product ( . P

Let M denote the orthogonal projector on the Hilbert

Aut(ID)

equality holds if and only if f belongs to .

M ?

^

A H(M) M P = I P

space generated by , M , the pro-

Let us now turn back to the discussion at the end of section M

M M

~ ^

A M A = A A X

0 jection of onto , and . denotes nor-

7! s

2.2. Schur had to chose the functional s (or, in general,

1

1=2

= X (X; X )

malization to unit norm : X . The (some-

s 7! zs Aut(ID) ID

p+1

p ) within , and automorphisms of pre- 

times called total) correlation coefficient A;B (resp. partial d

serve the M¨obius distance M , and thus the Poincar´e distance

M

0

 A B A B 1+d (z;z ) M

0 correlation coefficient )of and (resp. of and ,

A;B

d (:; :) d (z; z ) = log

P 0

P , with . More precisely,

1d (z;z )

M

M  =

with respect to a commun subspace ) is defined as A;B

they are well known to coincide with the direct isometries of





M M

M M

e e

2

(A; B )=( )  =(  ; B )= 2 A

B;A

H ;d ) ( (resp. ).

A;B B;A the Lobachevski plane P [17] [18] :

Let us now consider recursive projections. It is well known

2



f N = Aut(ID) (H ;d )g : = direct isometries of P that

2 2

(H ;d )

The full isometry group of P is obtained by including the map

? ?



P z 7! z P = P + P = P ;P

M M

~ ~

;A M

as a generator. M (12)

M M;A

A A

P ;A

where M , say, is the orthogonal projector onto the closed enables us to hint that some spherical trigonometry relations A subspace generated by M and . These identities are of might hold as well when transposed to the parcors frame- utmost importance in RLS adaptive filtering as well as in work, and indeed they do. Similarly, they can all be derived Kalman filtering. From (12), it is easy to show that from (17); however, though purely algebraic, these relations

are not necessarily intuitive.

M;A M M M M M 1 M

~ ~ ~ ~ ~ ~ ~

= B (B ; A )(A ; A ) A ; B (13) Among any 4 elements there exists one and only one re- lation. These 15 relations are the 3 cosine laws, the 3 co-

which gives the useful relations sine laws in the polar triangle, the 3 self-dual sine formu-

;A M;A M M M 2

M las and the 6 self-dual cotangent formulas. They all have a

~ ~ ~ ~

B ; B )=(B ; B )=1j j ( (14) A;B parcor equivalent. However, there are many different rela-

and tions among any 5 elements (or between the 6), and it always

M

M;B M;A ~

~ seems possible to find new ones. Thus we give only one of

A ; B )= : ( (15) A;B them, the five elements formula. For sake of brevity, proofs In order to get spherical trigonometry relations, we need are omitted.

consider three projection residuals. From (13) we have

M;A M;A M M

~ ~ ~ ~

C ; B ) = (C ; B )

( (16) 3.2.1 The cosine law in the polar triangle

M M M M 1 M M

~ ~ ~ ~ ~ ~

(C ; A )(A ; A ) (A ; B )

In the polar triangle, the cosine law reads :

M M 1=2 M M 1=2

~ ~ ~ ~

C ; C ) (B ; B )

Dividing by ( , and using (14),

cos A +cosB cos C

cos a =

we get : (18)

sin B sin C

M M M

  

C;B C;A A;B ;A

M Similarly, (17) admits the polar version :

q q

; 

= (17)

C;B

M M

2 2

1 j 1 j j j

C;A A;B

M;A M;B M;C

 +  

C;B C;A A;B

M

q q

= ;

 (19)

C;B

;B M;C

which is formally equal to (3) (at least in the real case), up to M

2 2

1 j j 1 j j A;B a straightforward identification of variables. C;A which was already known to Yule [12, (19) p. 93]. 3.2 New relations among parcors induced by spherical trigonometry 3.2.2 The sine law Let us first briefly recall some facts from spherical trigonom-

etry (and in particular the duality principle). Three points The spherical triangle self-dual sine law is the following for-

; B C (0; 1)

A and on the sphere determine the spherical tri- mula :

ABC AB

A sin B sin C

angle , which consists of the 3 arcs of great circles , sin

= :

= (20) BC

AC and obtained by intersecting the sphere and the planes

sin a sin b sin c

OAC OBC OAB, and (see figure 1). A spherical triangle has 6

Similarly, the following relation holds among parcors :

c A; B

elements : the 3 sides a; b and , and the 3 angles and

d

v v v

a BOC

C . The side , say, is defined as the angle and is equal

u u u

M;A M;B M;C

2 2 2

u u u

BC A

j j j j 1 j 1 j

to the length of the arc . The angle , say, is defined as 1

B;C C;A A;B

t t t

= = :

OAB OAC

M M M

2 2

the dihedral angle between the planes and , and is 2

1 j 1 j 1 j j j j

C;A A;B also equal to the angle made by the tangents to the spherical B;C

(21) A triangle ABC at point .

We now turn to the duality principle of spherical trigonom- 0

etry. Let A be the pole (with respect to the equator passing 3.2.3 The cotangent formulas

0

C A B

through B and ) which is in the same hemisphere as ;

0 0 0 0

A B C

and C are defined similarly. The spherical triangle is These are the 6 self-dual formulas obtained by permuting

0 0 0 0 0

A B C a A

the polar triangle of ABC.In , the elements and , variables into the equation

A  a

say, are equal respectively to  and (see e.g., [25]

b sin a = cos C cos a +sinC cot B:

[26]). So, for any spherical trigonometry formula there ex- cot (22)

a; b; c; A; B ; C )

ists a dual relation, obtained by replacing (

 A;  B;  C;  a;  b;  c)

by ( , respectively. Similarly, the following relation among parcors holds :

q There are three degrees of freedom in a spherical triangle, q

 

M;C

M

M

M M

2 2

 = 1 j 1 j j j = 

so there cannot be more than three distinct relations among 

A;C

B;C

A;C B;C A;B 

the six elements. All the spherical trigonometry relations can 

q q

M;B M;C M;B 2

thus be derived from the three cosine laws obtained by per- 2

= j  j 1 j 1 j

+ (23)

A;C A;C muting variables into (3). Now, the identification (3) = (17) A;B

1

n

R =(r ) P = R =

n i;j 0:n

3.2.4 The five elements formula Lemma 3.1 Let 0: and

i;j =0

0:n

n

( p ) i; j 2 [0


n]

i;j . Then for all , =0 These are the 6 formulas obtained by permuting variables i;j

into

r = (X ;X) ;

i j

i;j (28)

[0:n]nj

[0:n]ni

~

~

X

X

j cos b sin c =sinb cos A cos c +sina cos B:

(24) i

p = ( ; ):

i;j (29)

[0:n]ni [0:n]ni [0:n]nj [0:n]nj

~ ~ ~ ~

(X ;X ) (X ;X )

i j j

The dual equations are i

1

n

R =(r ) P = R =

n i;j 0:n

Lemma 3.2 Let 0: and

i;j =0 0:n

cos B sin C = sin B cos a cos C + sin A cos b:

(25) n

( p ) i; j 2 [0


n]

i;j . Then for all ,

i;j =0

Similarly, the following relations among parcors hold : r

i;j

= ( X ; X )= ;

p

j i;j

i (30)

q

r r

i;i j;j

M

M

2

1 j j =

 (26)

p C;A

B;A

i;j [0:n]ni [0:n]nj [0:n]ni;j

~ ~

= ( X ; X )= :

p q

q (31)

i j i;j

p p

i;i j;j

M;A M;B

M

M M

2 2

 + 1 j j  1 j j 

B;A

C;B C;A C;A C;B

q =

We are now ready to extend lemma 3.2. Let M

M;B M;C

 

2

 j j =

1 (27)

C;A B;A

A B

q q

with A invertible. Then the Schur complement

C D

M;B M;C M;A

M M

2 2

1 j j   + 1 j j 

C;B C;A

1 C;A B;A C;B

M=A) A M (M=A)=D CA B ( of in is defined as .

The following theorem encompasses and generalizes lemma

1

p = 1 =R

R 3.2 (which corresponds to the particular case ): n

3.3 Schur Complements in 0: and spherical

0:n

trigonometry

1

n

= R =(r ) P = R

n i;j 0:n

Theorem 3.1 Let 0: and

i;j =0 0:n

0:p 0:p n

We now consider covariance matrices and their inverses. n

R = (r ) (p )

i;j . Let moreover (resp.

i;j =p+1 0:n i;j =0

i;j

n

fX g p  q

0:p 0:p

Let i be scalar random variables. For , let

n i=0

=(p ) P R

0:p

T ) be the Schur complement of

i;j =p+1 0:n

i;j

X =[X


X ]

:q p q

p . In all this section, we will assume

R (p +1) (p +1)

2 0:n

n in (of the top left corner

fX g IL ( ; A;P)

that : i belong to , and that the covari-

i=0 T

[I 0]P [I 0] P P p = 1

p+1 0:n p+1 0:n 0:n

H of in ). For ,weset

R = E (X X ) X

n 0:n 0:n

ance matrix 0: of is invertible.

0:n

0:p 0:p 0:p 0:p

R P r p = R = P = r = p

0:n 0:n i;j i;j

(X ;


;X )

H , , , , and

l l 0:n 0:n i;j i;j

1

k

f

X

0:p

In the sequel, the general notation i of sec-

=   p 2 [1; 0;


;n 1]

i;j

;


;l l . Then for all , and for all

i;j

1 k

~

X  r;s  q

tion 3.1 is simplified to . Let p . Since

i

2 [p +1


n] i; j ,

we will essentially use contiguous sets of indices (with-

q

H(fX g )

m

pmq

~

0:p 0:p 0:p 0:p X

out loss of generality), we also replace i ,

r r = r =  ;

6=r m6=r;m6=s

m (32)

i;j i;i j;j i;j

H(fX g ) H(fX g )

m m

pmq pmq

~ ~

q

X X i

i and respectively by

[p+1:n]ni;j

0:p 0:p 0:p

[p:q ]nr [p:q ]nr;s

p:q

: =  = p p p

~ ~

~ (33)

i;j j;j i;j i;i

X X

X , and . Similar notations are adopted

i i i

for the correlation coefficients, so that the partial correlation

m6=r

(fX g )

H We now turn to the connection with the spherical trigonome-

m

pmq

p 

coefficient (of order q ) , say, is denoted

X ;X j

i try cosine laws :

[p:q ]nr

simply by  . In our conventions, the order of the sec-

i;j

p:q p:q

~ Corollary 3.1 Up to normalization, an elementary (i.e. rank

p q 

ondary (upper) indices and is meaningful : X ,

0:p 0:p

i;j i

p:q p:q p:q p:q P

1) Schur complement step on R (resp. on ) performs

0:n 0:n

r P p

(and later on R , , and ) reduce respectively

i;j i;j i;j

i;j the law of cosines (3) (resp. the polar law of cosines (18)) :

X  R r P p p>q

i;j i;j i;j i;j i;j

to i , , , , and if . In this way the

2 [1; 0;


;n 2] i; j 2 [p +2


n]

notation changes continuously from the total to the partial For all p , and for all ,

p 0:p 0:p

situation. For instance, there is no conceptual need to distin- 0:

  

i;j i;p+1 p+1;j

0:p+1

q guish between total and partial correlation coefficients since q

= ;

 (34)

i;j

0:p 0:p

2 2

0

j j 1 j j

a total correlation coefficient is simply a parcor of order . 1

+1 p+1;j

In this section, we shall first recall (and slightly extend) i;p

p+1:n]ni;j [p+1:n]ni;p+1 [p+1:n]np+1;j

some results [27] [28] [29] [30] giving the covariance ma- [

 +  

i;j i;p+1 p+1;j

[p+2:n]ni;j

X

q 0:n q

 =

trix of (resp. its inverse) in terms of covariances of :

i;j

[p+1:n]ni;p+1 [p+1:n]np+1;j n

2 2

fX g

i

j j 1 j j

the random variables (resp. of the random vari- 1

i=0

i;p+1 p+1;j

[0:n]ni

n

~

g X

ables f ). We thus get lemmas 3.1 and 3.2, which

i=0

i (35)

are generalized to theorem 3.1 by considering Schur comple-

1

R R n

ments in 0: and in . Lastly these recursions receive Proof :

0:n

0:p R

a spherical trigonometry interpretation. We begin with the The Schur complementation step !

0:n

0:p 0:p 0:p+1

(R =r = R

following elementary results. ) reads componentwise :

0:n p+1;p+1 0:n

0:p 0:p 0:p 0:p 0:p 0:p 0:p+1

1

~ ~ ~ ~ ~ ~

) ; X )(X ; X ) (X ; X =(X r

p+1 p+1 p+1 i j i i;j

0:p 0:p 0:p+1 0:p+1

~ ~ ~ ~

(X p  1 s (0) ; X )=(X ; X )

, due to (16). Normaliz- In this case, for all the Schur parameter p is equal

p+1 j i i

1:p1

ing as in section 3.1 we get (34). to the (partial) correlation coefficient  . It is convenient

0;p

 1 We next consider (35). Similarly, from the quotient to write the algorithm in vector form [31] [32] : for p ,

property of Schur complements (see e.g. [16]), we have

2 3 2 3

0 1 0

0:p 0:p 0:p+1

u v u 0

p1 p1

p

(P =p P

)= . But this equality reads com-

0:n p+1;p+1 0:n

1 2 1 1

6 7 6 7

u v u v

p1 p1 p p

7 6 7

ponentwise 6

 

6 7 6 7

s (0) . . 1 . .

p

6 7 6 7

p 0:p 0:p 0:p 0:p+1

0: . . . .

1 ;

. . =

 . . (36)

p p (p ) p = p :

6 7 6 7

i;j i;p+1 p+1;p+1 p+1;j i;j

s (0) 1

k +1

k k k

p

6 7 6 7

u v u v

p1

p1 p p

q 4 5 4 5 q

p 0:p

0: . . . . p

Dividing by p , and using (33), we get . . j;j

i;i . . . .

q

[p+1:n]ni;j [p+1:n]ni;p+1 [p+1:n]np+1;j

0:p+1 0:p

 +   = p =p

0 i i

i;j i;p+1 p+1;j i;i i;i

u = r u = v = r i>0

0 i

q with initialization , and for .

0 0 0

[p+2:n]ni;j

0:p+1 0:p

 p

=p . Remarking from (29) and From the point of view of analytic interpolation theory,

i;j j;j j;j

th

p

p 0:p

0: which was that of section 2.1, this step of the algo-

P p = P =

+1:n

from the equality p that

0:n i;i r

rithm incorporates the new data p in the covariance exten-

[p+1:n]ni [p+1:n]ni

1

~ ~

(X ; X

) , and using (14), we see that

i i

q sion problem. This problem is recursive and “hierarchical”

q

[p+1:n]ni;p+1

0:p+1 0:p

2

(r ;


;r ) R

p1 p1

by nature : given 0 such that is positive

p 1 j =p =

j . We thus get (35),

i;i i;i i;p+1

> 0 R > 0 r p definite ( ), p if and only if belongs to a disk

which is the polar cosine law (19) = (18). Q

p1

1:i1

2

(1 j j ) r

(of decreasing radius 0 ), the center of

0;i

i=1

(r ;


;r ) k  0

p1 which depends on 0 . So for all , the row 4 Non-Euclidean (spherical) geometrical as-

number k of (36) integrates the contributionof the correlation r

pects of the Schur and Levinson-Szego¨ algo- r

p+k

lag p in the subsequent (possible) compatibibility of

(r ;


;r )

p+k 1

rithms with 0 . In particular, the row number zero

r (r ;


;r )

0 p1

tells whether p is compatible with the data

[X X


X ]=

0 1 n > 0 R > 0

From now on, we shall further assume that R

1 p

via the following test : assuming that p , if

1 0

[X X


X ]; fX ;t 2 ZZ g

t t1 tn t

s (0) = v =u j < 1 where is a zero-mean, dis- j

and only if p .

p1 p1

crete time, wide sense stationary time series. As a conse- This progressive incorporation of the constraints

R

r ;


;r ;


n

quence, 0: is a Toeplitz matrix. For simplicity, let us de- p

2 in the analytic interpolation problem

R R r r

n n i;j j i

note 0: by and by . The parcors satisfy a shift- corresponds to the progressive incorporation of the random

i; j; k ; p; q 2 ZZ ;p  q; i; j 62

;


;X ;

invariance property : for all X

1 tp+1

variables t in the linear prediction

p:q p+k :q +k

 =  p;


;qg

f , . Among all correlation coeffi-

i;j

+k;j +k

i problem, and thus to the progressive updating of the asso-

1:p1

?

(p)= g

f ciated projection operator (this, of course, is nothing but

2IN

cients (total or partial), the function p

0;p

(1) = 

 the classical lattice or Gram-Schmidt interpretation of the

;1 (with 0 , as in theorem 3.1) is the partial autocor- relation function of the process. Schur algorithm [33]). To see this, let us rewrite the Schur

Let us now turn back to the Schur and Levinson-Szeg¨o al- algorithm in terms of projective identities. It is easily seen

 0 gorithms. In this final section, we shall write the commun (by induction) that for k , the two recursions of the row

(lattice) recursions of both algorithms as two Schur comple- number k of (36) are two coupled occurrences of the same

ment recursions (in the forward and backward directions), identity (16) : h

but acting on the covariance matrix (in the Schur case) or on i

tp+1:t1 tp+1:t1 tp+1:t1 tp+1:t1

~ ~ ~ ~

(X ; X )(X ; X )

tp t

tpk pk its inverse, i.e., on the covariance matrix of the normalized t

interpolation process (in the Levinson-Szeg¨o case). From

3 2

tp+1:t1 tp+1:t1

~ ~

) ;X (X

t

p

section 3.3, the link with spherical trigonometry will fol- t

1

tp+1:t1 tp+1:t1

~ ~

7 6

(X ;X )

tp tp

low immediately : up to normalization, the Schur (resp. in- 

5 4

tp+1:t1 tp+1:t1

~ ~

) (X ;X

t

tp

verse Levinson-Szeg¨o) algorithm performs the law of cosines 1

tp+1:t1 tp+1:t1

~ ~

(X ;X ) t

(resp. the polar law of cosines). This is a new feature of t

h i

tp+1:t tp+1:t tp:t1 tp:t1

~ ~ ~

the classical duality of the Schur and Levinson-Szeg¨o algo- ~

(X ; X )(X ; X ) :

= (37)

tp t

pk tpk rithms. As for the Levinson-Szeg¨o algorithm, it is an imple- t mentation of the polar five elements formula. Since all these quantities are covariances of estimation er- rors, they reduce to parcors when appropriately normalized; 4.1 Spherical geometry of the Schur algorithm so a connection of the recursive equations (37) with spherical trigonometry is expected.

The new (spherical) geometrical interpretation of the algo- In fact, both equations are easily seen to

def

p1

rithm stems from the connection between the Schur algo- 1: =

be Schur complement recursions in R

0:n

tp+1:t1 tp+1:t1

n ~

rithm and linear regression. Let us thus initialize (6), via ~

((X ; X

)) . These two Schur

i;j =0

ti tj

(1), with complementation steps correspond to augmenting the set

p1

2

fX g

ti

z + r z +

r of variables in the projective space in its two

i=1

1 2

s (z )= :

0

X 2

(contiguous) opposite directions : the forward one t

r + r z + r z +

0 1 2

2

X

[tp+1:t]ntk

tp+1:t1

p

and the backward one t . Because of stationarity,

~ ~

(X ; X )

6

t

tk

the resulting quantities still are covariances of estimation ;

4

[tp+1:t]ntk [tp+1:t]ntk

~ ~

(X ; X )

tk k

residuals with respect to the same subspace, because the t

| {z }

tp:t1 tp:t1

~ ~

[( X X )

right hand side of (37) also reads ;

p1

tp1

tpk 1

a

tp:t1 tp:t1

k

~ ~

(X ; X )]

, and the two coefficients in the 3

t

tpk

1:p1 1:p1



[tp:t1]ntk

tp+1:t1

 ( ) ~

transformation matrix reduce to and . ~

0;p 0;p

(X ; X )

7

tp

tk

From the discussion in section 3, the link with spherical 5

[tp:t1]ntk [tp:t1]ntk

~ ~

) ; X X

( (39)

tk k

trigonometry is immediate : t

{z } |

p1



) (a

k

Theorem 4.1 p

 k  p p  2

Up to normalization, an elementary step of the Schur al- (the first equation is valid for 1 , with , and the

 k  p 1 p  2

gorithm performs two coupled occurrences of the law of second for 0 with ).

1

R

 1 k  0

p These equations are Schur complement recursions in ; p

cosines : for all , and for all , 0:

p

fX g

i

they correspond to reducing the set of variables t

i=0

3 2

1:p1



0;p

1 in the projective space in its two (extremum) opposite di-

p p

h i

1:p1 1:p1

7 2 2 6

X X

1j j 1j j

t tp

p1 1:p1

1: rections : the forward one and the backward one .

p;0 0;p

7 6

=  ;

1:p1

p;p+k 0;p+k

5 4



1 0

p; From the discussion in section 3.3, we thus expect that, after

p p

1:p1 1:p1

2 2

1j j 1j j

0 0;p p; appropriate normalization of the covariances of the estima-

tion errors, (39) will reduce to some spherical trigonometry

h i

q q

0:p1 1:p p1 1:p1

1: polar law.

2 2

:

j j 1 j 1 j 

; (38)

p;p+k 0;p+k ;p+k p;p+k 0 This hint is enforced when looking at the ran- dom variables in the left hand side of (37) and

Proof.

Z = (fX g ;X ) (A; B ; C ) =

tp+1:t1 tp+1:t1 tp+1:t1

i 0ip tm

(39). Let t ,

1=2

~ ~ ~

; ( X ; X (X

Divide (37) by )

t t

tpk

(X ;X ;X ) M = H(ZnfX ;X ;X g)

tp tm t tp tm

t , and .

tp+1:t1 tp+1:t1

1=2

~ ~

( X ; X

) , which is equal to

tp

tpk

M = H(fX g ) m  p M =

i 1ip1

So t if , and

tp+1:t1 tp+1:t1 tp+1:t1

1=2 1=2

~ ~ ~ i6=m

) ; X X ) (X

1  m  p 1 (fX g ) , and use H

tp

tpk tpk

i

t if . Then (37)

1ip1

(14). can be visualized as projective identities within the tetra-

M M M

~ ~ ~

A ; B ; C )

hedron ( , and (39) as projective identities

M;B ;C M;C;A M;A;B

~ ~ ~

A ; B ; C ) 4.2 Spherical geometry of the Levinson-Szeg¨o algo- within the tetrahedron ( , which

rithm can easily be shown [14] to be the polar tetrahedron of

M M M

~ ~ ~

( ; B ; C ) We now turn to the spherical geometry of the Levinson- A .

Szeg¨o algorithm. Remember from Theorem 3.1 that succes- Theorem 4.2

1

R R n

sive Schur complements in 0: (resp. in ) correspond n 0: Up to normalization, an elementary step of the order- to an increase (resp. a reduction) in the number of variables decreasing Levinson-Szeg¨o algorithm (resp. of the Levinson- in the regression problem. So, as was already the case at the Szeg¨o algorithm) performs two coupled occurrences of the end of section 2, the comparison with the Schur algorithm polar law of cosines (resp. of the polar five elements for- indeed proves easier when dealing with order-decreasing re- mula) :

cursions.

 2 1  k  p 1

For all p , and for all ,

th

Let us introduce the forward j order linear predic-

3 2

P

[0:p]n0;p

j

j j tj :t1 j



~

0;p

1

fa g X a X =

i

tion coefficients by t

p p t

i i=1 i

i=0

h i

[0:p]np;0 [0:p]n0;p

j 2 2 7 6

1j 1j j j

[0:p]nk;0 [0:p]nk;p

p;0 0;p

a = 1: 7

with From the Wiener Hopf equations, and 6

0

 ; =

[0:p]np;0

k;0 k;p

5 4

[tj :t]nti [tj :t]nt



j

p;0 ~ ~ 1

a =(X ; X )= p

Theorem 3.1, we get p

t

i ti

[0:p]np;0 [0:p]n0;p

2 2

1j j 1j j

[tj :t]nti [tj :t]nti

p;0 0;p

~ ~

X ; X )

( . So the order-decreasing

ti ti

 

q q

[0:p]nk;p [0:p]nk;0 [0:p]nk;p;0 p]nk;0;p Levinson-Szeg¨o recursions read : [0:

2 2

; 1 j 1 j j ; j 

k;p k;0 k;p k;0

2 3

[tp:t]ntk [tp:t]ntk

tp+1:t tp:t1

~ ~ ~

~ (40)

) ) (X ; X (X ; X

6 7

tp t

tk tk 5

4 and

[tp:t]ntk [tp:t]ntk [tp:t]ntk [tp:t]ntk

~ ~ ~ ~

 

(X ; X ) (X ; X )

q q

tk tk tk tk

| {z } | {z }

[0:p]nk;p;0 [0:p]nk;0;p [0:p]nk;p [0:p]nk;0

2 2

j ; j  1 j 1 j

p p



k;p k;0 k;p k;0

) (a a

pk k

2 3

[0:p]n0;p

2 3

1:p1



0;p

1



p p 0;p

1

h i

[0:p]np;0 [0:p]n0;p

1:p1 1:p1

6 7 2 2

2 2

1j j 1j j

1j 1j j j

[0:p]nk;0 [0:p]nk;p

6 7

p;0 0;p

p;0 0;p

6 7

=  ; :

 =

4 5

[0:p]np;0

1:p1

k;0 k;p

4 5

 

p;0

1 1 p;0

p p

1:p1 1:p1

2 2

[0:p]np;0 [0:p]n0;p

1j 1j j j

2 2

1j j 1j j

p;0 0;p

p;0 0;p (41) Proof : References Using (15), (39) can be rewritten as :

[1] I. Schur, Uber¨ Potenzreihen, die im Innern des Ein-

"

[tp:t]nt [tp:t]nt

=2

1 heitskreises beschr¨ankt sind, Journal f¨ur die Reine und

~ ~

(X ; X )

[0:p]nk;0

t t

) ; 

( Angewandte Mathematik, Vol. 147, pp. 205-32, 1917;

k;0

[tp:t]ntk [tp:t]ntk

1=2

~ ~

) ; X (X

tk k t and 148, pp. 122-45, 1918. English translation : On

power series which are bounded in the interior of the

#

[tp:t]ntp [tp:t]ntp

=2

1 unit circle, in Operator Theory, Advances and Applica-

~ ~

(X ; X )

tp tp [0:p]nk;p

 )

( tions 18 (I. Gohberg, ed.), part I, pp. 31-59; and part II,

k;p

[tp:t]ntk [tp:t]ntk

1=2

~ ~

(X ; X )

tk k

t pp. 61-88, Birkh¨auser Verlag, Basel, 1986.

3 2

p]n0;p

[0: [2] G. Szeg¨o, Orthogonal polynomials, American Mathe-



0;p

1

[0:p]np;0 [0:p]n0;p

2 2 matical Society, Providence, 1939.

1j j 1j j

7 6

p;0 0;p

= 

5 4

[0:p]np;0



p;0

1 [3] N. I. Akhiezer, The classical moment problem and

[0:p]np;0 [0:p]n0;p

2 2

1j j 1j j

0 0;p p; some related questions in analysis, Hafner publishing

" Company, New York, 1965.

[tp+1:t]nt [tp+1:t]nt

1=2

~ ~

(X ; X )

[0:p]nk;0;p

t t

 ) ;

( [4] U. Grenander and G. Szeg¨o, Toeplitz forms and their

k;0

[tp+1:t]ntk [tp+1:t]ntk

1=2

~ ~

(X ; X )

k tk t applications, University of California Press, Berkeley,

1958.

#

[tp:t1]ntp [tp:t1]ntp

1=2

~ ~

(X ; X )

p tp [0:p]nk;p;0

t [5] T. Kailath, A view of three decades of linear filtering

( ) :

k;p

[tp:t1]ntk [tp:t1]ntk

1=2

~ ~

(X ; X

) theory, IEEE Transactions on Information Theory, Vol.

tk tk

20, N. 2, pp. 146-81, 1974.

[tp:t]ntk

tp+1:t1 tp+1:t1

1=2

~ ~ ~

) =(X ; ; X X

Next divide by (

t t

k

t [6] P. Delsarte, Y. Genin, Y. Kamp and P. Van Dooren,

[tp:t]ntk

tp+1:t1

1=2

~ ~

( X ; X

) , which is equal to

tp

k

t Speech modeling and the trigonometric moment prob-

[tp:t]ntk [tp:t]ntk

tp+1:t1

1=2 1=2

~ ~ ~

X ) = (X ; X

) . Us- lem, Philips J. Res., Vol. 37, N. 5/6, pp. 277-92, 1982.

tp

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