Karhunen-Loeve (PCA) components of a Gaussian Process: Matlab example (1)

Antonio Sala, UPV

Difficulty: **** ,       Relevance: ,      Duration: 18:47

Summary:

This video presents how to obtain the principal components (Karhunen-Loeve eigenfunctions in the continuous limit case) of a stochastic Gaussian process, with a ’quadratic exponential’ covariance kernel (said kernel choice was arbitrary). For simplicity and for graphical representations, a ’discrete’ set of 241 test points will be considered in order to analyze the structure of the covariance $K_{241\times 241}$.

The principal components, obtained from the diagonalization $K=VD{V}^T$, are one of the many ‘matrix square roots’ $K=Q{Q}^T$ that can be devised. Specifically, they are an ‘orthogonal’ square root in the sense that the columns of $Q=V\sqrt{D}$ are orthogonal to each other (uncorrelated components).

The video discusses various conceptual issues related to matrix square roots and principal components, and ends by graphically representing some of them.

The video [gpkh2pEN], a continuation of this one, will animate the reconstruction of a realization from standard normal ’latent’ principal components, and will also do so with a process for which observations are avaliable at certain points.

As the matrix square root $Q$ of the covariance matrix is not unique, there are other causal, anticausal and bilateral representations of the Gaussian process, which will be analysed in videos [gpcholEN] and [gpanticaEN].

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