Stochastic Processes and Random Fields

Home , Random field, Stochastic process

Chapter 2

Sto chastic Pro cesses and Random

Fields

A topic of intrinsic interest in this course is sto chastic partial di erential equations (SPDE).

Although a rigorous treatment of this topic go es b eyond our goals, in order to intro duce

the fundamental ideas of the basic theory of SPDE, it is necessary to discuss the basic

concepts of sto chastic pro cesses and random elds. These two concepts are nothing more

than extensions of the random variable concept, treated in the previous lecture, for cases

in which these variables have temp oral or spatial dep endence, resp ectively. As a matter of

fact there is much similaritybetween the two concepts at a fundamental level, with some

particular nomenclature di erences.

2.1 De nition and Probabilistic Concepts

In the previous lecture, the symbol x(! ) referred to the value of a vector random variable

x, resulting from the realization of an exp eriment ! . Sto chastic pro cesses are those in which

the random variable is also a function of time, that is, the random variables are de ned on

the pro duct space T , where T represents the real time line. In this case, we denote by

x(!; t) the result of a sto chastic pro cess x(t). In what follows, we utilize the most common

abbreviation of a sto chastic pro cess, denoting the sto chastic variables as x(t), where !

will b e implicit in the notation. Sto chastic pro cesses are referred to as discrete{time or

continuous{time dep ending whether the time domain is discrete or continuous, resp ectively.

In sto chastic pro cesses, an event B in the probability space is denoted by

B = f! 2 : x(!; t) xg : (2.1)

The distribution function for a discrete{time pro cess with N random n{vectors x(t ); ; x(t ),

1 N

is de ned as:

F [x(t ); ; x(t )] P (f! 2 :x(!; t ) x ; ; x(!; t ) x g) : (2.2)

1 N 1 1 N N

x(t )x(t )

N 19

with probabilitydensity function (if it exits):

@ F [x(t ); ; x(t )]

1 N

x(t )x(t )

(2.3) p [x(t ); ; x(t )] =

1 N

x(t );;x(t )

@ x(t ) @ x(t )

1 N

n n

where we recall that @ =@ x = @ =@ x @x . Consequently,we can write

1 n

F [x(t ); ; x(t )] =

1 N

x(t )x(t )

Z Z

x(t ) x(t )

0 0 0 0

p (x ; ; x ) dx dx : (2.4)

x(t )x(t )

1 N 1 N

1 1

In case of a continuous{time pro cess, the probability distribution and probability density

functions are de ned for all times t, and can b e symb olically written as

F (x;t) P (f! 2 :x(!; t) xg) (2.5a)

@ F (x;t)

p (x;t) = (2.5b)

@ x

resp ectively.

The concepts of mean, variance and correlation intro duced in the previous lecture can b e

extended directly to the case of sto chastic pro cesses. Therefore, we de ne concisely these

quantities for this case:

Mean vector:

(t) Efx(t)g = xp (x) dx ; (2.6)

x(t)

Stationary mean value vector: de ned when the mean is indep endent of time, that is

x lim x(t) dt : (2.7)

t !1

For the case in which the stationary mean value coincides with the ensemble mean ,

the pro cess is called ergo dic in the mean.

Mean for discrete{time processes:

x lim x(kT ) (2.8)

K !1

2K +1

k =K

where T is the sampling p erio d.

Quadratic mean value matrix:

T T

(t) Efx(t)x(t) g = xx p (x) dx : (2.9)

x(t)

We can still de ne the stationary quadratic mean value based on the de nition of

stationary mean value, as we can de ne the stationary quadratic mean value for a

discrete{time pro cess utilizing the corresp onding de nition given ab ove. 20

Auto{correlation matrix:

(t; ) Efx(t)x ( )g ; (2.10)

or explicitly written,

Z Z

1 1

(t; )= zy p (z ; y ) dz dy : (2.11)

x(t)x( )

1 1

where the word auto refers to the sto chastic pro cess x(t).

Cross{correlation matrix:

(t; ) Efx(t)y ( )g : (2.12)

where the word cross refers to the twostochastic pro cesses x(t)and y (t).

Auto{covariance matrix of a stochastic process:

C (t; )= cov fx(t); x( )g Ef[x(t) (t)][x( ) ( )] g ; (2.13)

x x

where the designation auto refers to the sto chastic pro cess in question, in this case,

only x(t). It is simple to show that

C (t; )= (t; ) (t) ( ) (2.14)

x x x

When t = ,we de ne the covariance matrix as:

P (t) C (t; t) ; (2.15)

x x

which is sometimes referred to as the variance matrix.

Cross{covariance matrix of a stochastic process:

C (t; ) Ef[x(t) (t)][y ( ) ( )] g ; (2.16)

x y

where the designation cross refers to the sto chastic pro cesses x(t)and y (t). We can

easily show that

C (t; )= (t; ) (t) ( ) (2.17)

xy xy x

Correlation matrices can b e de ned analogously to the de nitions given in the previous

lecture.

2.2 Indep endentProcess

Wesay that a sto chastic pro cess x(t) is indep endent when for all t and the probability

density p (x(t); x( )) = p p . In this way, according to (2.11) it follows that

x(t);x( ) x(t) x( )

Z Z

1 1

zy p (z )p (y ) dz dy : (t; ) =

x(t) x( )

1 1

Z Z

1 1

= dzz p (z ) dyy p (y )

x(t) x( )

1 1

= Efx(t)gEfx ( )g (2.18) 21

Therefore, from the de nition of auto{covariance matrix it follows that C (t; )= 0, for

t 6= , that is, an indep endent sto chastic pro cess is uncorrelated in time. In an entirely

analogous way,we can showthatiftwo sto chastic pro cesses x(t)andy (t) are indep endent,

they are also uncorrelated, that is, C (t; )= 0,forany t and . As in the case of random

variables, the contrary of this relation is not necessarily true, that is, two uncorrelated

pro cesses are not necessarily indep endent.

2.3 Markov Pro cess

As in sto chastic pro cesses in general, a Markov pro cess can b e continuous or discrete

dep ending on whether the time parameter is continuous or discrete, resp ectively. A discrete

sto chastic pro cess (i.e., sto chastic sequence) fx(t )g,fort >t ,oracontinuous sto chastic

k k 0

pro cess x(t), for t>t , is said to b e a Markov pro cess if, for all t,

p [x(t)j ( )] = p [x(t)jx( )] (2.19)

x(t)j( ) x(t)jx( )

where ( )= fx(s);t s tg, and analogously ( )= fx(s);t s tg.

0 0

More sp eci cally for the discrete case, a rst{order Markov pro cess, also referred to as a

Markov{1 pro cess, is one for which

p (x jx ; ; x ; x )= p (x jx ) (2.20)

k k 1 1 0 k k 1

x jx x x x jx

1 0

k k 1 k k 1

That is to say, a Markov{1 pro cess is one for which the probability density at time t, given

all states up to t,intheinterval [t ;], dep ends only on the state at the nal time, ,ofthe

interval. This is nothing more than a way of stating the causality principle: the state of a

pro cess at a particular moment in time is sucient for us to determine the future states of

the pro cess, without us having to know its complete history.

In the discrete case, we can write for the joint probability density

p ( ) = p (x ; ; x ; x )

k x x x k 1 0

1 0

k k

= p (x jx ; ; x ; x ) p (x ; ; x ; x )

k k 1 1 0 x x x k 1 1 0

x jx x x

1 0

k 1

1 0

k k 1

(2.21)

where we utilize the prop erty (1.77). Assuming that the sto chastic pro cess is rst{order

Markov, according to (2.20) wehave that

p ( )= p (x jx ) p (x ; ; x ; x ) (2.22)

k k k 1 x x x k 1 1 0

x jx

1 0

k k 1

and utilizing rep eatedly the de nition (2.20) we obtain

p ( )= p (x jx ) p (x jx ) p (x jx ) p (x ) (2.23)

k k k 1 k 1 k 2 1 0 x 0

x jx x jx x jx

1 0

k k 1 k 1 k 2

Therefore, the joint probabilitydensity ofaMarkov{1 pro cess can b e determined from the

initial marginal probability density p [x(0)], and from the probabilitydensity p [x(t)jx(s)],

x(0) x(t)jx(s)

for t s 2 [t ;), and t<.The quantity p [x(t)jx(s)] is known as the transition

x(t)jx(s)

probability density of a Markov pro cess. 22

The concept of Markovian pro cess can b e extended to de ne Markov pro cesses of di erent

orders. For example, a discrete sto chastic pro cess for which the probabilitydensity at time

t dep ends on the pro cess at times t and t can b e de ned as those for whichwehave:

k k 1 k 2

p (x jx ; x )= p (x jx ; x ) (2.24)

k k 1 0 k k 1 k 2

x jx x x x jx x

1 0

k k 1 k k 1 k 2

which in this case is called a second{order Markov pro cess, or Markov{2. Analogously,

we can de ne k {order Markov pro cesses. In those cases considered in this course, the

de nition given in (2.20) for rst{order Markov pro cesses is sucient.

2.4 Gaussian Pro cess

A sto chastic pro cess in n dimensions fx(t);t 2 T g,where T is an arbitrary interval of time,

is said to b e Gaussian if for any N instants of time t ;t ; ;t in T , its density function,

1 2 N

distribution function, or characteristic function, is normal. In other words, the pro cess is

Gaussian if the vectors x(t ); x(t ); ; x(t ) are jointly Gaussian distributed. According

1 2 N

to what was seen in the previous lecture we can write the density function of this pro cess

as:

1 1

T 1

exp (z ) P p (z )= (z ) ; (2.25)

z z z

Nn=2 1=2

(2 ) jP j

where the vector z , of dimension Nn = N n, is de ned as:

1 0

x(t )

C B

x(t )

C B

; (2.26) z

C B

A @ .

x(t )

the mean vectors (t ), of dimension Nn are given by

(t ) Efz(t )g ; (2.27)

i i

for i =1; 2; ;N, and the covariance P , of dimension (N n) = Nn Nn, has elements

which are the sub{matrices

P =[P ] Ef[x(t ) (t )] [x(t ) (t )] g (2.28)

z x ij i i j j

x x

for i; j =1; 2; ;N. In this way, a Gaussian pro cess is completely determined by its mean

and its auto covariance. A pro cess whichissimultaneously Gaussian and Markovian is said

to b e a Gauss{Markov pro cess.

2.5 Stationary Pro cess

A precise de nition of the concept of stationary pro cess can b e given by returning to the

concept of probability.However, for what interests us, it is sucient to utilize wide{sense 23

stationary pro cesses, which only requires that the rst two moments b e time{indep endent.

In this sense, a stationary pro cess is one for which the mean is indep endent of time:

(t)= ; (2.29)

x x

and for which the correlation only dep ends on the time interval between events:

(t; )= (t ) ; (2.30)

x x

which can b e written as:

( )= (t + ; t)=Efx(t + )x (t)g : (2.31)

x x

An even weaker concept of stationary pro cess is de ned when the covariance is stationary.

In this case,

C ( )=C (t + ; t)= (t + ; t) (t + ) (t) (2.32)

x x x

x x

These concepts apply similarly to \cross{quantities", that is, the cross{correlation and

cross{covariance

( )= (t + ; t)=Efx(t + )y (t)g : (2.33a)

xy xy

C ( )= C (t + ; t)= (t + ; t) (t + ) (t) (2.33b)

xy xy xy

x y

resp ectively, are stationary. In this case, it is simple to show that

( )= ( ) (2.34a)

xy yx

C ( )= C ( ) (2.34b)

xy yx

since stationary covariances and correlations are invariant under a time translation of .

2.6 Wiener{Khintchi ne Relation

A de nition that follows from the concept of stationary pro cess intro duced ab ove is given by

the Wiener{Khintchine relation. This relation de nes the sp ectral density of the stationary

covariance as b eing the Fourier transform of the covariance. For a continuous sto chastic

pro cess the p ower sp ectrum of the covariance can b e written as:

C (! ) C ( )e d ; (2.35)

x x

and consequently,by the inverse Fourier transform wehave that

C (! )e d! : (2.36) C ( )=

x x

For discrete stationary pro cesses the discrete Fourier transform de nes the corresp onding

power sp ectrum. 24

2.7 White Noise Pro cess

The simplest p ower sp ectrum that we can think of is the one given by a constant, that is,

^ ^

one for which C(! ) Q (! )= Q , where the sto chastic pro cess w is called white noise.

w w

In this case, the covariance b ecomes a Dirac delta:

Q e d! Q ( ) =

w w

= Q ( ) : (2.37)

Even if this noise is completely non{physical, b ecause it is in nite at the origin, it is of

great imp ortance in the development of sto chastic di erential equations.

2.8 Wiener Pro cess

A Wiener pro cess, also called Brownian motion, denoted by b(t), is de ned as the integral

of a stationary, Gaussian white noise pro cess w (t)withzeromean:

b(t)= w (t) dt ; (2.38)

where

cov fw (t); w ( )g = Q (t ) ; (2.39)

as wesawabove. Some of the prop erties of this pro cess are listed b elow:

1. b(t) is normally distributed.

2. Efb(t)g = 0, for all t 0.

3. P fb(0) = 0g =1.

4. b(t) has indep endent and stationary increments, that is, indep endent of time. We

refer to increments as b eing the di erences b(t ) b(t ); ; b(t ) b(t ), where

1 2 n1 n

i+1 i i

5. b(t) is a Markov pro cess.

Moreover the variance of a Wiener pro cess increases linearly in time:

Z Z

t t

T T

varfb(t)g = Efb(t)b (t)g = Efw (t )w (t )g dt dt

1 2 1 2

0 0

Z Z

t t

= Q (t t ) dt dt

w 1 2 1 2

0 0

= Q dt = Q t; (2.40)

w 1 w

0 25

where we used the following de nition of a delta function:

f (s) (t s) ds =;f(t) : (2.41)

It is imp ortant to notice that the diculty encountered in the description of a Gaussian

white noise pro cess, the problem of in nite variance, do es not exist for the Wiener pro cess.

That means, the latter is a well{b ehaved pro cess.

2.9 Spatial Random Fields

The literature on random sto chastic elds is relatively smaller than that on sto chastic

pro cesses. Still, there are several treatments, such as those of Vanmarcke[132] and Yaglom

[140]. A more recent treatment, directed toward earth science applications is the one of

Christakos [24]. In what follows, we will b e as concise as p ossible, keeping in mind that the

main purp ose of this section is to intro duce the concepts of homogeneity and isotropy for

random elds.

The concept of random elds can b e intro duced similarly to the wayweintro duced sto chastic

pro cesses. In this case, we asso ciate with each random variable x ; x ; ; x the p oints

1 2 n

r ; r ; ; r in the space R . A random spatial eld can b e considered a function of

1 2 n

events ! 2 , where is the sample space intro duced in the previous lecture, and also a

function of the spatial p osition r 2 R ,thatis,x(r)= x(!; r). When we write x(r)we

are simplifying the notation in a manner entirely analogous to what we did in the previous

section, when the variable was the time. This concept can b e extended to several random

variables dep ending on space in order to motivate the intro duction to vector random spatial

elds. Wedenoteby x(r) the vector random eld which represents the set of random spatial

elds x (r); x (r); ; x (r), that is,

1 2 m

x(r)= [x (r); x (r); ; x (r)] (2.42)

1 2 m

The distribution function of a vector random spatial eld is then de ned as:

F (x; r)= P (f! : x(r) x; r 2 R g) : (2.43)

We emphasize once more that the concept of random elds is an extension of the concept

of sto chastic pro cess. A sto chastic pro cess is a random eld for which the spatial argument

r 2 R ,isintro duced for n = 1 and r ! r ! t so that the random variable b ecomes x(t),

as b efore.

The distribution function is related to the probability densityby means of the expression

@ F (x; r)

(2.44) p (x; r)=

@ x

and consequently

0 0

F (x; r)= p (x ; r) dx : (2.45)

x x

1 26

The concepts of mean, variance and correlation can b e extended directly for the case of

random spatial elds. Therefore we de ne concisely these quantities in this case:

Mean value of a random eld:

xp (x; r) dx ; (2.46) (r) Efx(r)g =

x(r)

Auto{covariance matrix of a random spatial eld:

C (r ; r )= cov fx(r ); x(r )g Ef[x(r ) (r )][x(r ) (r )] g ; (2.47)

x i j i j i i j j

x x

for two spatial p oints r and r , where we made an analogy with what wesawin

i j

sto chastic pro cesses; auto refers to the random eld in question, in this case x(r).

Then, wehave that

T T

C (r ; r )= Efx(r )x (r )g (r ) (r ) (2.48)

x i j i j x i j

When r = r ,wehave the variance matrix which describ es the lo cal b ehavior of the

i j

random eld.

In order to simplify and more easily demonstrate the notation, consider the case of a scalar

random eld x(r). The mean intro duced ab ove b ecomes a scalar quantity (r), that is, a

function of \one" spatial variable r 2 R .Thecovariance b ecomes a function (no longer a

matrix) of \two" spatial variables r ; r . The variance is a function given by

i j

(r)= C (r; r = r) (2.49)

x j

for r = r .We can still intro duce the spatial correlation function (r ; r )between two

i x i j

p oints as:

C (r ; r )

x i j

(r ; r )= (2.50)

x i j

(r ) (r )

x i x j

A scalar random spatial eld is said to b e uncorrelated when

(

(r) ; for r = r = r

i j

C (r ; r )= (2.51)

x i j

0 ; otherwise

and in fact, such a random eld is said to b e a white eld (analogously to the white pro cess

seen previously).

Basically all the concepts de ned for random pro cesses can b e generalized for spatial random

elds:

Markovian process ! Markovian eld

Gaussian process ! Gaussian eld

white process ! white eld 27

as for the concepts of characteristic function, conditional probability function, conditional

mean, conditional covariance, etc.

Avery imp ortant generalization is that of the concept of stationarity of a sto chastic pro cess,

which for spatial random elds translates into the concept of homogeneity. In the wide sense,

a spatial random eld is said to b e homogeneous when its mean value is indep endentof

the spatial variable, and its covariance dep ends only on the distance b etween two p oints in

space. For the scalar case, this can b e written as:

(r) = (2.52a)

C (r ; r ) = C (r = r r ) (2.52b)

x i j x i j

Another fundamental concept is that of the isotropic spatial random eld, which is de ned

as a eld for which

C (r ; r )=C (r = jr r j) (2.53)

x i j x i j

is satis ed. That is, a spatial random eld is said to b e isotropic when its covariance dep ends

only on the magnitude of the distance b etween twopoints in space.

It is p ossible to show (e.g., Christakos [24]) that for a homogeneous random eld, not

necessarily isotropic, we can write the covariance function as

exp(iw r) C (w ) dw (2.54) C (r)=

x x

where C (w ) is the sp ectral density function that, by the inverse Fourier transform, can b e

written as:

C (w )= exp(iw r) C (r) dr (2.55)

x x

(2 )

This result can b e generalized for the case of vector random elds. Notice that for real

random elds, the covariance and sp ectral density can in fact b e expressed in terms of

Fourier cosine integrals.

C (r) = C (w ) dw (2.56a) r) cos(w

x x

cos(w r) C (r) dr (2.56b) C (w ) =

x x

(2 )

The imp ortance of these results lies in the fact that they provide a relatively simple criterion

to determine whether a continuous and symmetric function in R can b e a covariance

function. In fact, the necessary and sucient condition for a continuous function C x(r ; r )

i j

in R to b e a covariance function is that it b e a p ositive{semide nite function, that is,

Z Z

C (r ; r )f (r )f (r ) dr dr 0 (2.57)

x i j i j i j

n n

R R

for any function f (r). This criterion is generally very dicult to verify,even for homoge-

neous random elds. However, utilizing the sp ectral representation ab ove, Bo chner's [15] 28

theorem says that the criterion for a continuous and symmetric function in R to b e a

covariance function is that its sp ectral function b e p ositive{semide nite

C (w ) 0 (2.58)

for w 2 R .

A relevant result that app ears in atmospheric data assimilation concerns the isotropic case

with n = 2, that is, in R . In this case, C (r)= C (r ), where r = jrj. Intro ducing

x x

p olar co ordinates: r =(x; y )= (r cos ; r sin )and w =(w cos '; w sin '); and recalling

the change of variables in integrals means that we should calculate the determinant of the

Jacobian matrix that corresp onds to the transformation, that is

@x @x

cos r sin

@r @

jJac(r;)j = = r (2.59)

@y @y

sin r cos

@r @

where the notation j:j is used for the determinant. In this way, using the fact that the

integral over R for any function f (x; y ) is transformed into an integral over the circle C as

Z Z Z Z

1 1 1 2

f (x; y ) dx dy = f (r cos ; r sin ) rdrd (2.60)

0 0 0 0

(e.g., Ap ostol [4], pp. 479{485), the integral in (2.56b) b ecomes

Z Z

1 2

^ ^

C (w )= C (w ) = C (r ) r cos(w r) d

x x x

(2 )

0 0

Z Z

1 2

= C (r ) r cos[wr cos( ')] d (2.61)

(2 )

0 0

where the last equality is obtained by treating the inner pro duct explicitly:

w r = rw cos cos ' + rw sin sin '

= rw cos( ') (2.62)

where w = jw j. Now p erforming the transformation, ! + ' + =2, wehave that

cos( ') !sin , and therefore the integral of the expression ab ove is indep endentof

'. This means that the result of the integral is also indep endentof', as should b e the case

for isotropic covariances. Intro ducing the Bessel function of order zero:

cos(x sin ) d (2.63) J (x)=

(e.g., Arfken [5], pp. 579{580), wehavethatintwo dimensions

C (w )= J (wr)C (r )rdr (2.64)

x 0 x

Utilizing the orthogonality of the Bessel function of order zero:

0 0

(w w ) (2.65) rJ (wr)J (w r ) dr =

0 0

0 29

(e.g., Arfken [5], p. 594), we obtain for the isotropic covariance function in two dimensions

the formula:

C (r )= 2 J (wr)C (w )wdw (2.66)

x 0 x

It is interesting to mention that the concept of ergo dicity can also b e extended to spatial

random elds. In an entirely analogous way to what can b e done for sto chastic pro cesses,

a spatial random eld is said to b e ergo dic if its spatial mean and covariance coincide with

its ensemble mean and covariance, resp ectively.

Exercises

1. (Problem 4.3, Meditch[103]) Assuming that three scalar sto chastic pro cess fx(t);t 2

T g, fy(t);t 2 T g and fz(t);t 2 T g are pairwise indep endent, show that they are not

necessarily triplewise (simultaneously) indep endent.

2. Calculate the p ower sp ectrum for stationary pro cesses having the following auto cor-

relation functions:

2 2

2 =T

(a) Gaussian pulse: ( )= e

2 j j

(b) Damp ed cosine wave: ( )= e cos !

(

1 j j ; for j j 1

( )=

0 ; otherwise

3. (Problem 2.17, Brown [19]) The stationary pro cess x(t) has mean = const: and an

auto correlation function of the form

2 2

2 =T

(t; t + )= ( )= e

Another pro cess y (t) is related to x(t)by the deterministic equation

y (t)=ax(t)+ b

where a and b are known constants.

(a) What is the auto{correlation function for y (t)?

(b) What is the cross{correlation function ( )?

4. (Problem 2.20, Brown [19]) Two random pro cesses are de ned by

x(t) = a sin(!t + )

y (t) = b sin (!t + )

where is a random variable with uniform distribution b etween0and2 ,and! is a

known constant. The co ecients a and b are b oth normal random variables Nf0; g

and are correlated with a correlation co ecient . What is the cross{correlation

function ( )? (Assume a and b are indep endentof .)

xy 30

5. Show that the following are admissible candidates for a covariance function:

(a) C (r)= a (r), for a>0andr 2 R

2 1

(b) C (x; y )= C (r = jx y j)= exp(r ), for x; y 2 R .(Hint: In this case,

x x

the pro of can b e obtained by either showing that (2.57) is true, or showing that

(2.58) is satis ed. Use (2.58) and expand exp(2xy )inTaylor series.)

6. Show that in R the isotropic sp ectral density function can b e expressed as

1 sin (wr)

C (w )= C (r )rdr

x x

2 w

and that consequently the corresp onding covariance function is given by

sin (wr)

C (r )= 4 C (w )wdw

x x

7. (Problem 7.14, Mayb eck [101]) In Monte Carlo analyses and other typ e of system sim-

ulations (e.g., non{identical twin exp eriments), it is often desired to generate samples

of a discrete{time white Gaussian noise vector pro cess, describ ed by mean zero and

covariance

Efw w g = Q

k k

with Q nondiagonal. Indep endent scalar white Gaussian noises can b e simulated

readily through use of pseudorandom co des (as wehave seen in our rst computer

assignment), but the question remains, how do es one prop erly provide for cross{

covariances of the scalar noises?

(a) Let v be a vector pro cess comp osed of indep endent scalar white Gaussian noises

of zero mean and unit variance:

Efv g =0 Efv g =1 for k =1; 2;:::

j;k

where v is the j {th comp onentofv ,attime t . Show that

j;k k k

w = L v for all k

k k k

prop erly mo dels the desired characteristics. The matrix L ab ove corresp onds

. Notice, to the Cholesky lower triangular square ro ot of Q , that is, Q = L L

k k k

that if the Cholesky upp er triangular square ro ot had b een used instead, the

corresp onding expression for generating w would b e

w = U v for all k

k k k

where, in this case, Q = U U .

k k

, where U (b) If U and D are the U{D factors of Q , that is, if Q = U D U

k k k k k k k

are upp er triangular and unitary matrices and D are diagonal matrices, show

that,

w = U u for all k

k k k 31

also provides the desired mo del if u isavector pro cess comp osed of indep endent

scalar white Gaussian noises of mean zero and variance:

Efu g = d ;

j;j ;k

j;k

where u is the j {th comp onentofu , at time t ,andd is the (j; j ) element

j;k k k j;j ;k

of D , at time t (i.e., the j {th element along its diagonal).

k k

8. Computer Assignment: Wewant to use the results of the previous problem to p erform

Monte Carlo exp eriments for a given correlation and/or covariance structure. As a

preparation for that, in this problem we are going to create a Matlab function that

generates a homogeneous correlation on a grid de ned over R and examine some of

its prop erties. Let us start by consider the interval (L ;L ], and let us divide it in

x x

a uniform grid of J points. Consider also the homogeneous and isotropic, Gaussian

1 1

correlation function in R R , that is,

2 2

Q(x; y )=Q(r = jx y j)= exp( (x y ) =L )

where r = jx y j is the distance b etween anytwo p oints in the domain, and L is the

(de)correlation length. Therefore the p oints in the discrete domain can b e de ned as

x = j x

where x =2L =J ,forj 2fJ=2+ 1;J=2g, and the elements of the homogeneous,

isotropic correlation matrix Q are given by

Q = Q(x ;y )

ij i j

(a) Construct a Matlab function that returns the covariance matrix Q,given the

half{length of the domain L ,thenumb er of grid p oints J , and the (de)correlation

length L .For (L ;L ;J)= (1; 0:2; 32), compute Q using this function. Makea

d x d

contour plot of the correlation array.(Note:Arealconvenientway of generating

meshgrid .) this matrix in Matlab is using the intrinsic function

(b) For the parameters of the previous item, plot the correlation function at the

following two sp eci c lo cations: x 2f0;L g.

j x

Check its eigenvalues.)

(d) From the gures constructed in the previous items, we see that the correlation

decreases very quickly toward values that are nearly zero. (You can actually

printoutthevalues in Q to check it further). It could b e computationally

advantageous, particularly to reduce storage requirements, to approximate this

correlation \function" (matrix) by one that neglects correlation values b eyond

a certain cut{o length L . In this way, only the elements of the matrix cor-

resp onding to jr j L would need to b e stored. Without worrying ab out the

storages savings, mo dify the function of item (a), to construct a new matrix Q ,

by replacing the values of Q for which jr j >L by zero. Using the same param-

eters as in item (a), and a cut{o value of L =3L ,makea contour plot of the

c d

resulting correlation structure. Also, rep eat item (b). 32

(e) A visual comparison of the plots in items (a) and (b) with those of item (d) seem

to indicate that our approximation is fairly reasonable. Is the correlation of the

previous item, an acceptable correlation matrix?

9. Computer Assignment: The result obtained in the last item of the previous exercise

makes us wonder what is the correct way of constructing a correlation eld that has the

structure that wewant, but is zero b eyond a certain correlation length. The pro cedure

to generate what are called compact{supp ort correlation functions is through the use

of convolution of functions. (Note: see Gaspari and Cohn (1996) for the details on

2 3

constructing these correlation functions in R and R , that are of primary imp ortance

in mo deling covariances for data assimilation). Another way of lo oking at convolution

of functions is to think on the Hadamard pro duct for the case of matrices. Without

getting into the mathematical details, this problem has the intention to guide you

though the steps of building an actual correlation matrix for the function of the

previous problem. Consider then the compact{supp ort triangular correlation function

discussed earlier in the text:

(

1 jx y j=L ; for jx y jL

c c

T (x; y )==

0 otherwise

Then p erform the following tasks:

(a) Rep eat items (a){(c) of the previous exercise, but now for the compact{supp ort

function T (r )= T (jx y j). (Note: The matrix T has elements T = T (x ;y ).)

ij i j

(b) Construct a matrix Q as the Hadamard pro duct of the matrix Q, of item (a)

in the previous exercise, and T from the previous item, corresp onding to the

function T (r ). That is, let Q be given by

Q Q T =[Q T ]

ij ij

(Note: Matlab do es the Hadamard pro duct trivially). Make a contour plot Q,

and rep eat item (c) of the previous exercise.

Q are like, plot the correlation functions obtained from these three matrices at

point x = 0. (Please, have all three curves on the same frame).

10. Computer Assignment: Using the Matlab function created in item (a) of Exercise 8

(i.e., without a cut{o length), let us apply the results of Exercise 7 to understand

b etter what correlated noise actually is.

(a) Create a Matlab function that p erforms a Monte Carlo exp erimentgiven the

numberofsamples. Wewant the output of this function to b e the sampled

correlation matrix, obtained from a weighted sum of outer pro ducts of the vectors

w of Exercise 7. To obtain the Cholesky decomp osition of the correlation matrix

Q of Exercise 8, use the Matlab function chol .Make contour plots for the three

sampled correlation matrices obtained by using 100, 1000, 10000 samples.

(b) Now using the identity matrix, in the 32{dimensional space of Exercise 8, p erform

aMonte Carlo run, with 1000 samples, assuming this identity matrix is the

correlation matrix of interest. Makea contour plot of the resulting sampled

correlation matrix. Compare this result with those obtained in the previous

item. In particular, explain the meaning of using an identity correlation matrix. 33