(2/5) Multi-Output Gaussian Processes: representation; code for realizations

Antonio Sala, UPV

Difficulty: **** ,       Relevance: ,      Duration: 21:58

Summary:

This video, a continuation of the video [mimoGP1EN], discusses the representation of multi-output stochastic processes in two ways:

• as a process that, indeed, has an auto-covariance function $k(x_1,x_2)$ whose result is a matrix of size according to the number of outputs

• as well as a stochastic process with one more input, $k({\xi }_1,{\xi }_2)$, with $\xi =(x,j)$ being an augmented input, a pair of abscissa + index (say, 1 or 2 if we are in a two-output case) indicating which output we are referring to, henceforth with $k({\xi }_1,{\xi }_2)$ being a scalar.

Actually, both representations are equivalent, as one would intuitively expect, and it is just a matter of preference when writing code in which one is used.

Although it is not the only option, generating these multi-output covariances is convenient and simple if one considers that the process is $y=Cu$, with $C$ being a matrix and $u$ a vector of “components”. If we assume that they are independent, they will each have an autocovariance independent of the rest, and $E[u(x_1)u{(x_2)}^T]$ will be a diagonal matrix.

The last part of the video presents a Matlab example of said processes (2 outputs) and generates realizations of it.

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