Multivariate Gaussian and Student$-t$ Process Regression for Multi-output Prediction

Gaussian process for vector-valued function model has been shown to be a useful method for multi-output prediction. The existing method for this model is to re-formulate the matrix-variate Gaussian distribution as a multivariate normal distribution. Although it is effective in many cases, re-formulation is not always workable and difficult to extend because not all matrix-variate distributions can be transformed to related multivariate distributions, such as the case for matrix-variate Student$-t$ distribution. In this paper, we propose a new derivation of multivariate Gaussian process regression (MV-GPR), where the model settings, derivations and computations are all directly performed in matrix form, rather than vectorizing the matrices as done in the existing methods. Furthermore, we introduce the multivariate Student$-t$ process and then derive a new method, multivariate Student$-t$ process regression (MV-TPR) for multi-output prediction. Both MV-GPR and MV-TPR have closed-form expressions for the marginal likelihoods and predictive distributions. The usefulness of the proposed methods is illustrated through several simulated examples. In particular, we verify empirically that MV-TPR has superiority for the datasets considered, including air quality prediction and bike rent prediction. At last, the proposed methods are shown to produce profitable investment strategies in the stock markets.