Convolved Gaussian process regression models for multivariate non-Gaussian data

Abstract

Multivariate regression analysis has been developed rapidly in the last decade for dependent data. The most di cult part in multivariate cases is how to construct a crosscorrelation between response variables. We need to make sure that the covariance matrix is positive de nite which is not an easy task. Several approaches have been developed to overcome the issue. However, most of them have some limitations, such as it is hard to extend it to the case involving high dimensional variables or capture individual characteristics. It also should point out that the meaning of the cross-correlation structure for some methods is unclear. To address the issues, we propose to use convolved Gaussian process (CGP) priors (Boyle & Frean, 2005). In this dissertation, we propose a novel approach for multivariate regression using CGP priors. The approach provides a semiparametric model with multi-dimensional covariates and o ers a natural framework for modelling common mean structures and covariance structures simultaneously for multivariate dependent data. Information about observations is provided by the common mean structure while individual characteristics also can be captured by the covariance structure. At the same time, the covariance function is able to accommodate a large-dimensional covariate as well. We start to make a broader problem from a general framework of CGP proposed by Andriluka et al. (2006). We investigate some of the stationary covariance functions and the mixed forms for constructing multiple dependent Gaussian processes to solve a more complex issue. Then, we extend the idea to a multivariate non-linear regression model by using convolved Gaussian processes as priors. We then focus on an applying the idea to multivariate non-Gaussian data, i.e. multivariate Poisson, and other multivariate non-Gaussian distributions from the exponential family. We start our focus on multivariate Poisson data which are found in many problems relating to public health issues. Then nally, we provide a general framework for a multivariate binomial data and other multivariate non-Gaussian data. The de nition of the model, the inference, and the implementation, as well as its asymptotic properties, are discussed. Comprehensive numerical examples with both simulation studies and real data are presented.