(1/5) Multi-Output Gaussian Processes: Motivation

Antonio Sala, UPV

Difficulty: *** ,       Relevance: ,      Duration: 15:34

Summary:

This video introduces and motivates multi-output stochastic processes, based on two basic examples:

• Temperature and humidity readings at weather stations in different positions, from which you want to ’interpolate’

• The statistical relationship between the position and velocity of a mass subject to random accelerations.

In these examples, we will have a variable, say $y_1(x)$, and another one, $y_2(x)$, so that apart from the ‘marginal autocovariances’, say, $E[y_1(x_a)y_1(x_b)]$, $E[y_2(x_a)y_2(x_b)]$ (we assume zero mean for notational simplicity), we could have a covariance between the signals $E[y_1(x_a)y_2(x_b)]$ that would allow us to interpolate humidity with temperature data or estimate speed with position data, for example.

How are these covariances generated? Well, the theory of stochastic differential equations and filtering justifies the second case (position, velocity)... The former case (temperature, humidity) could be more ’descriptive’ or assuming certain components (underlying latent variables) to be present in the data-generating mechanism. The ‘intuitive’ details of this case are discussed in the video [mimogp2EN], a continuation of this one. Stochastic differential equations are a topic of greater complexity (but of greater theoretical interest, too), discussed in other materials.

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