Procedure: Haar wavelet expansion

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

The eigenfunction \(\phi_{i}(t)\) is expressed as a linear combination of Haar orthonormal basis functions

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

The Haar wavelet, the simplest wavelet basis function, is defined as

\[\begin{split}\psi(x)=\left\{ \begin{array}{cc} 1 & 0<x<\frac{1}{2} \\ -1 & \frac{1}{2} \leq x <1 \\ 0 & \mbox{otherwise} \end{array}\right..\end{split}\]

Inputs

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

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

  3. \(M=2^n\) orthogonal basis functions on \([0,1]\) constructed in the following way

    \[\begin{split}\psi_1 &= 1 \\ \psi_i &= \psi_{j,k}(x) \\ i &= 2^j+k+1 \\ j &= 0,1, \cdots, n-1 \\ k &= 0,1, \cdots, 2^j-1\end{split}\]

    where

    \[\begin{split}\psi(x)=\left\{ \begin{array}{cc} 1 & k2^{-j}<x<2^{-j-1}+k2^{-j} \\ -1 & 2^{-j-1}+k2^{-j} \leq x <2^{-j}+k2^{-j} \\ 0 & \mbox{otherwise} \end{array}\right.\end{split}\]

Output

  1. The eigenvalues \(\lambda_i\) , \(i=1\cdots p\).
  2. The matrix of unknown coefficients \(D\).
  3. An approximated covariance function; \(R(s,t)=\Psi(s)^T D^T \Lambda D \Psi(t)\).

Procedure

  1. Write the eigenfunction as \(\phi_i(t)=\sum_{k=1}^M d_{ik} \psi_{k}(t)=\Psi^T(t) D_i\) so that, \(R(s,t)=\sum_{m=1}^M \sum_{n=1}^M a_{mn}\psi_m(s) \psi_n(t)=\Psi(s)^T A \Psi(t)\).
  2. Choose \(\strut{M}\) time points such that \(t_j=\frac{2i-1}{2M}, 1 \leq i \leq M\).
  3. Compute the covariance function \(C\) for those \(M\) points.
  4. Apply the 2D wavelet transform (discrete wavelet transform) on \(C\) to obtain the matrix \(A\).
  5. Substitute \(R(s,t)=\Psi(s)^T A \Psi(t)\) in \(\lambda_i\phi_i(t)=\int_{0}^{1} R(s,t)\phi_i(s)\), then we have \(\lambda_i\Psi^T(t)D_i=\Psi^T(t)AHD_i\).
  6. Define \(H\) as a diagonal matrix with diagonal elements \(h_{11}=1\), \(h_{ii}=2^{-j}\) \(i=2^{j}+k+1, \quad j=0,1, \cdots, n-1\) and \(k=0,1, \cdots, 2^j-1\).
  7. Define the whole problem as \(\Lambda_{p \times p} D_{p \times M} \Psi(t)_{M \times 1}= D_{p \times M} H_{M \times M } A_{M \times M} \Psi(t)_{M \times 1}\).
  8. From (7), we have \(\Lambda D=D H A\).
  9. Multiply both sides of (8) by \(H^{\frac{1}{2}}\)
  10. Express the eigen-problem as \(\Lambda D H^{\frac{1}{2}}=D H^{\frac{1}{2}} H^{\frac{1}{2}} A H^{\frac{1}{2}}\) or \(\Lambda \hat{D}=\hat{D}. \hat{A}\) where \(\hat{D}=DH^{\frac{1}{2}}\) and \(\hat{A}=H^{\frac{1}{2}} A H^{\frac{1}{2}}\).
  11. Solve the eigen-problem in (10), then \(\Phi(t)=D\Psi(t)=\hat{D} H^{-\frac{1}{2}}\Psi(t)\).