Remarks on multivariate Gaussian Process

TL;DR

This paper clarifies the fundamentals of multivariate Gaussian processes, including their definitions, properties, and a special case of multivariate Brownian motion, while also introducing their application in multi-output regression.

Contribution

It provides a rigorous definition of multivariate Gaussian processes, proves their existence, and discusses key properties and applications in statistical learning.

Findings

Defined multivariate Gaussian processes via Gaussian measures.

Proved existence of multivariate Gaussian processes.

Introduced multivariate Brownian motion and Gaussian process regression.

Abstract

Gaussian processes occupy one of the leading places in modern statistics and probability theory due to their importance and a wealth of strong results. The common use of Gaussian processes is in connection with problems related to estimation, detection, and many statistical or machine learning models. With the fast development of Gaussian process applications, it is necessary to consolidate the fundamentals of vector-valued stochastic processes, in particular multivariate Gaussian processes, which is the essential theory for many applied problems with multiple correlated responses. In this paper, we propose a precise definition of multivariate Gaussian processes based on Gaussian measures on vector-valued function spaces, and provide an existence proof. In addition, several fundamental properties of multivariate Gaussian processes, such as strict stationarity and independence, are introduced. We further derive multivariate Brownian motion including It\^o lemma as a special case of a multivariate Gaussian process, and present a brief introduction to multivariate Gaussian process regression as a useful statistical learning method for multi-output prediction problems.