Gaussian process: Cholesky factor of covariance, spectral factor (Matlab example)

Antonio Sala, UPV

Difficulty: ***** ,       Relevance: ,      Duration: 19:34

Summary:

This video presents the interpretation of the Cholesky factorization (lower triangular) of the covariance matrix of a Gaussian stochastic process (in discrete time, since we approximate a continuous process through a finite number of test abscissa points). An initial part reviews the basic ideas from the video [gpkh2EN], about the square roots of a covariance matrix and the principal components. Then, it focuses on the main objective, which is to interpret the meaning of the triangular structure of the Cholesky factor... it is given a ‘causal’ interpretation: the latent variable ’$j$’ has an effect on the outputs in the $j+1$, …, $241$ positions. Thus, a realization of the process ’from left to right’ is constructed.

It is observed that the columns of $Q$ converge to a single sequence, which moves downwards (when the steady state is reached and the finite window effects have disappeared). The video justifies why this gives rise to a convolution formula to calculate the effect of the latent variables on the outputs; The convolution kernel can be interpreted as the impulse response of a time-invariant linear system that is called ’causal spectral factor’ in literature.

As square roots are not unique, there are other representations (anticausal, bilateral) of interest, which will be discussed in the video [gpanticaEN], a continuation of this one.

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