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Karhunen-Loeve Expansion

Module by: Don Johnson. E-mail the author

The representation of the process, Xt X t , is the sequence of random variables X i X i . The choice basis of φ i t φ i t is unrestricted. Of particular interest is to restrict the basis functions to those which make the X i X i uncorrelated random variables. When this requirement is satisfied, the resulting representation of Xt X t is termed the Karhunen-Loève expansion. Mathematically, we require i,j,ij:E X i X j =E X i E X j i j i j X i X j X i X j . This requirement can be expressed in terms of the correlation function of Xt X t . E X i X j =E0TXα φ i αdα0TXβ φ j βdβ=0T0T φ i α φ j β R X αβdαdβ X i X j α 0 T X α φ i α β 0 T X β φ j β β 0 T α 0 T φ i α φ j β R X α β As E X i X i is given by E X i =0T m X α φ i αdα X i α 0 T m X α φ i α our requirement becomes

i,j,ij:0T0T φ i α φ j β R X αβdαdβ=0T m X α φ i αdα0T m X β φ j βdβ i j i j β 0 T α 0 T φ i α φ j β R X α β α 0 T m X α φ i α β 0 T m X β φ j β
(1)
Simple manipulations result in the expression
i,j,ij:0T φ i α0T K X αβ φ j βdβdα=0 i j i j α 0 T φ i α β 0 T K X α β φ j β 0
(2)
When i=j i j , the quantity E X i 2E X i 2 X i 2 X i 2 is just the variance of X i X i . Our requirement is obtained by satisfying 0T φ i α0T K X αβ φ j βdβdα= λ i δ i j α 0 T φ i α β 0 T K X α β φ j β λ i δ i j or
i,j,ij:0T φ i α g j αdα=0 i j i j α 0 T φ i α g j α 0
(3)
where g j α=0T K X αβ φ j βdβ g j α β 0 T K X α β φ j β Furthermore, this requirement must hold for each jj which differs from the choice of ii. A choice of a function g j α g j α satisfying this requirement is a function which is proportional to φ j α φ j α : g j α= λ j φ j α g j α λ j φ j α . Therefore,
0T K X αβ φ j βdβ= λ j φ j α β 0 T K X α β φ j β λ j φ j α
(4)
The φ i φ i which allow the representation of Xt X t to be a sequence of uncorrelated random variables must satisfy this integral equation. This type of equation occurs often in applied mathematics; it is termed the eigenequation. The sequences φ i φ i and λ i λ i are the eigenfunctions and eigenvalues of K X αβ K X α β , the covariance function of Xt X t . It is easily verified that: K X tu=i=1 λ i φ i t φ i u K X t u i 1 λ i φ i t φ i u This result is termed Mercer's Theorem.

The approach to solving for the eigenfunction and eigenvalues of K X tu K X t u is to convert the integral equation into an ordinary differential equation which can be solved. This approach is best illustrated by an example.

Example 1

K X tu=σ2mintu K X t u σ 2 t u . The eigenequation can be written in this case as σ2(0tuφudu+ttTφudu)=λφt σ 2 u 0 t u φ u t u t T φ u λ φ t Evaluating the first derivative of this expression, σ2tφt+σ2tTφuduσ2tφt=λdφtdt σ 2 t φ t σ 2 u t T φ u σ 2 t φ t λ t φ t or σ2tTφudu=λdφdt σ 2 u t T φ u λ t φ Evaluating the derivative of the last expression yields the simple equation (σ2φt)=λd2φd t 2 σ 2 φ t λ 2 t φ This equation has a general solution of the form φt=Asinσλt+Bcosσλt φ t A σ λ t B σ λ t . It is easily seen that BB must be zero. The amplitude AA is found by requiring φ=1 φ 1 . To find λλ, one must return to the original integral equation. Substituting, we have σ2A0tusinσλudu+σ2tAtTsinσλudu=λAsinσλt σ 2 A u 0 t u σ λ u σ 2 t A u t T σ λ u λ A σ λ t After some manipulation, we find that t,t 0 T :AλsinσλtAσtλcosσλT=λAsinσλt t t 0 T A λ σ λ t A σ t λ σ λ T λ A σ λ t or t,t 0 T :AσtλcosσλT=0 t t 0 T A σ t λ σ λ T 0 Therefore, n,n=12:σλT=(n1/2)π n n 1 2 σ λ T n 1/2 and we have λ n =σ2T2n+1/22π2 λ n σ 2 T 2 n 1/2 2 2 φ n t=2T1/2sin(n+1/2)πtT φ n t 2 T 1/2 n 1/2 t T

The Karhunen-Loève expansion has several important properties.

  • The eigenfunctions of a positive-definite covariance function constitute a complete set. One can easily show that these eigenfunctions are also mutually orthogonal with respect to both the usual inner product and with respect to the inner product derived from the covariance function.
  • If Xt X t Gaussian, X i X i are Gaussian random variables. As the random variables X i X i are uncorrelated and Gaussian, the X i X i comprise a sequence of statistically independent random variables.
  • Assume K X tu= N 0 2δtu K X t u N 0 2 δ t u : the stochastic process Xt X t is white. Then N 0 2δtuφudu=λφt u N 0 2 δ t u φ u λ φ t for all φt φ t . Consequently, if λ i = N 0 2 λ i N 0 2 , this constraint equation is satisfied no matter what choice is made for the orthonormal set φ i t φ i t . Therefore, the representation of white, Gaussian processes consists of a sequence of statistically independent, identically-distributed (mean zero and variance N 0 2 N 0 2 ) Gaussian random variables. This example constitutes the simplest case of the Karhunen-Loève expansion.

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