Gaussian process regression with multiple response variables

Abstract

Gaussian process regression (GPR) is a Bayesian non-parametric technology that has gained extensive application in data-based modelling of various systems, including those of interest to chemometrics. However, most GPR implementations model only a single response variable, due to the difficulty in the formulation of covariance function for correlated multiple response variables, which describes not only the correlation between data points, but also the correlation between responses. In the paper we propose a direct formulation of the covariance function for multi-response GPR, based on the idea that its covariance function is assumed to be the "nominal" uni-output covariance multiplied by the covariances between different outputs. The effectiveness of the proposed multi-response GPR method is illustrated through numerical examples and response surface modelling of a catalytic reaction process.

Online Sparse Multi-Output Gaussian Process Regression and Learning

“…This is mainly due to the need of inverting a large covariance matrix. To reduce complexity, [19], [34] constrained the covariance structure to establish a more parsimonious multioutput model and speed up the computations. In [28], [29], the sparse approximation [35] was integrated into the convolved multi-output GP regression and two approximate inference techniques were developed.…”

“…First, we consider simultaneously modeling the following two well-correlated bivariate functions [19]…”

“…The standard GP model assumes a single output variable only. However, in practice, multi-output functions arise widely in geostatistics [16], robotics [17], sensor networks [18], chemometrics [19], traffic flow forecasting [20] and computational physics [21], [22], just to name a few. A typical example of multi-output regression is the robot inverse dynamics problem [17], [23].…”

Nonparametric regression method with functional covariates and multivariate response

“…The problem with multivariate response variables and functional covariates based on the L 1 -median regression estimation is described by Chaouch and Laïb (2013). Wang and Chen (2015) proposes the formulation of the covariance function for the multi-response Gaussian process regression. Xiang et al (2013) studies the multivariate nonparametric regression analysis in the context of longitudinal data.…”

Gaussian process regression with functional covariates and multivariate response

“…For example, [5,6] treat each response variable as a Gaussian process and multiple responses are modelled independently; [19] treats Gaussian processes as the outputs of stable linear filters; [20] proposes a direct formulation of the covariance function for multi-response GPR, based on the idea that its covariance function is assumed to be the "nominal" uni-output covariance multiplied by the covariances between different outputs. In this paper a new method is proposed to deal with correlated multivariate response variables with multivariate and functional covariates.…”