Gaussian Process Regression Models

Gaussian process regression (GPR) models are nonparametric kernel-based probabilistic models. You can train a GPR model using the fitrgp function.

Consider the training set {(xi,yi);i=1,2,...,n}, where xi∈?d and yi∈?, drawn from an unknown distribution. A GPR model addresses the question of predicting the value of a response variable ynew, given the new input vector xnew, and the training data. A linear regression model is of the form

where ε∼N(0,σ2). The error variance σ² and the coefficients β are estimated from the data. A GPR model explains the response by introducing latent variables, f(xi), i=1,2,...,n, from a Gaussian process (GP), and explicit basis functions, h. The covariance function of the latent variables captures the smoothness of the response and basis functions project the inputs x into a p-dimensional feature space.

A GP is a set of random variables, such that any finite number of them have a joint Gaussian distribution. If {f(x),x∈?d} is a GP, then given n observations x1,x2,...,xn, the joint distribution of the random variables f(x1),f(x2),...,f(xn) is Gaussian. A GP is defined by its mean function m(x) and covariance function, k(x,x′). That is, if {f(x),x∈?d} is a Gaussian process, then E(f(x))=m(x) and Cov[f(x),f(x′)]=E[{f(x)−m(x)}{f(x′)−m(x′)}]=k(x,x′).

Now consider the following model.

where f(x)~GP(0,k(x,x′)), that is f(x) are from a zero mean GP with covariance function, k(x,x′). h(x) are a set of basis functions that transform the original feature vector x in R^d into a new feature vector h(x) in R^p. β is a p-by-1 vector of basis function coefficients. This model represents a GPR model. An instance of response y can be modeled as

Hence, a GPR model is a probabilistic model. There is a latent variable f(x_i) introduced for each observation xi, which makes the GPR model nonparametric. In vector form, this model is equivalent to

where

The joint distribution of latent variables f(x1), f(x2), ..., f(xn) in the GPR model is as follows:

close to a linear regression model, where K(X,X) looks as follows:

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