Gaussian process: anticausal and bilateral representations (Matlab example)

Antonio Sala, UPV

Difficulty: ***** ,       Relevance: ,      Duration: 13:48

Summary:

This video presents the ‘anticausal’ and ‘bilateral’ (mixture of causal and anticausal) representations of a Gaussian process, in a practical way via a Matlab example. The representations come from using a particular square root of the covariance matrix, since such matrix squre root is not unique. The first three minutes review the orthogonal square root that gives rise to the Karhunen-Loeve principal components (video [gpkh2EN]) and the lower triangular Cholesky factorization that gives rise to the causal representation (video [gpcholEN]).

By permuting rows and columns of the Cholesky decomposition, equivalent to a rearrangement of the random variables, we obtain an ’anti-causal’ representation where each latent variable influences the output of the process in the same place and on the previous ones (past), following the way we decided to order the abscissas from the Gaussian process (which we could interpret as ’time’). This gives rise to an animation where a realization of the Gaussian process is built ’from right to left’.

The second part of the video discusses the symmetric square root $Q=V\cdot \sqrt{D}\cdot V^T$. In that case $K=Q{Q}^T=Q^2$. The result is a ’mixture’ of causal and anticausal (symmetric) action of the latent variables on the output, except effects of initial conditions or finite window; this will give rise to what would be a bilateral convolution kernel. An intuitive interpretation is that each row of $Q$ (well, just the ‘stationary’ ones) is like a ‘hammer blow’ that deforms symmetrically on either side of the impact point, and that the intensity of the impact is the standard normally distributed latent variable.

*Link to my [ whole collection] of videos in English. Link to larger [ Colección completa] in Spanish.