Regression analysis for multivariate process data of counts using convolved Gaussian processes

Abstract

Research on Poisson regression analysis for process data of counts has developed rapidly in the last decade. One of the difficult problems in the multivariate case is how to construct a cross-correlation structure whilst, at the same time, making sure that the covariance matrix is positive definite. To address this issue, we propose to use convolved Gaussian processes (CGP) in this paper. This approach provides a semi-parametric model and offers a natural framework for modelling the common mean structure and covariance structure simultaneously. The CGP enables the model to define a different covariance structure for each component of the response variables. This flexibility ensures the model can cope with data coming from different sources or having different structures, and thus to provide accurate estimation and prediction. In addition, the model is able to accommodate large-dimensional covariates. The definition 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.