Procedure: Fourier Expansion

Description and Background

This procedure is used to find the eigenvalues and the eigenfunctions of the covariance kernel when the analytical solution is not available. The orthonormal basis \(\{\theta_i\}\) is used to solve the problem. This procedure uses Fourier basis functions as the set of orthonormal basis functions.

In general, the eigenfunction \(\phi_{i}(t)\) is written as

\[\phi_{i}(t)=\sum_{k=1}^M d_{ik} \theta_k(t)=\theta(t)^T D_i=D_i^T\theta(t),\]

where \(\{\theta_i(t)\}\) is the set of orthonormal basis functions and \(\{d_{ik}\}\) is the set of unknown coefficients for the expansion.

Inputs

  1. The covariance function, see AltCorrelationFunction, and estimates of its parameters.

  2. \(p\), the number of eigenvalues and eigenfunctions required to truncate the Karhunen Loeve Expansion at.

  3. A set of \(M\) adequate basis functions, where \(M\) is chosen to be odd. The basis functions are written as

    \[\begin{split}\theta_1(t) &= 1, \\ \theta_2(t) &= \cos(2 \pi t), \\ \theta_3(t) &= \sin(2 \pi t), \\ \theta_{2i}(t) &= \cos(2\pi it), \\ \theta_{2i+1}(t) &= \sin(2\pi it), i=1, 2, \ldots \frac{M-1}{2}.\end{split}\]

Output

  1. The set of eigenvalues \(\{\lambda_i\}\), \(i=1\cdots p\).
  2. The matrix of unknown coefficients, \(D\).
  3. An approximated covariance kernel; \(R(s,t)=\sum_{i=1}^p \lambda_i \phi_i(s) \phi_i(t)=\phi(s)^T\Lambda \phi(t)=\theta(s)^T D^T \Lambda D \theta(t)\).

Procedure

  1. Replace \(\phi_i(s)\) in the Fredholm equation \(\int_0^1R(s,t)\phi_i(s)ds=\lambda_i\phi_i(t)\) with \(D_i^T\theta(t)\), then we have \(D_i^T\int_0^1 R(s,t) \theta(s)ds=D_i^T\lambda_i\theta(t)\).
  2. Multiply both sides of \(D_i^T\int_0^1 R(s,t) \theta(s)ds=D_i^T\lambda_i\theta(t)\) by \(\theta(t)\).
  3. Integrate both sides of \(D_i^T\int_0^1 R(s,t) \theta(s)\theta(t)^Tds=D_i^T\lambda_i\theta(t)\theta(t)^T\) with respect to \(\strut{t}\).
  4. Define \(A=\int_0^1\int_0^1R(s,t)\theta(s)\theta(t)dsdt, \quad B=\int_0^1\theta(t)\theta^T(t)dt\) where \(A\) is a symmetric positive definite matrix and \(B\) is a diagonal positive matrix.
  5. Matrix implemenation. Write the integration in (4) as \(D_{p \times M}A_{M \times M}=\Lambda_{p \times p} DB_{M \times M}\) which is equivalent to \(AD^T=BD^T\Lambda\).
  6. Express \(\strut{B}\) as \(B^{\frac{1}{2}}B^{\frac{1}{2}}\).
  7. Express the form in (5) as \(AD^T=B^{\frac{1}{2}}B^{\frac{1}{2}}D^T\Lambda\).
  8. Multilpy both sides of (7) by \(B^{-\frac{1}{2}}\), so that \(B^{-\frac{1}{2}}AB^{-\frac{1}{2}}B^{\frac{1}{2}}D^T=B^{-\frac{1}{2}}D^T\Lambda\).
  9. Assume \(E=B^{-\frac{1}{2}}D^T\) then \(B^{-\frac{1}{2}}AB^{-\frac{1}{2}}E=E\Lambda\).
  10. Solve the eigen-problem of \(B^{-\frac{1}{2}}AB^{-\frac{1}{2}}E=E\Lambda\).
  11. Compute \(D\) using \(D=E^T B^{-\frac{1}{2}}\).