Remarks on multi-output Gaussian process regression

Multi-output regression problems have extensively arisen in modern engineering community. This article investigates the state-of-the-art multi-output Gaussian processes (MOGPs) that can transfer the knowledge across related outputs in order to improve prediction quality. We classify existing MOGPs into two main categories as (1) symmetric MOGPs that improve the predictions for all the outputs, and (2) asymmetric MOGPs, particularly the multi-fidelity MOGPs, that focus on the improvement of high fidelity output via the useful information transferred from related low fidelity outputs. We review existing symmetric/asymmetric MOGPs and analyze their characteristics, e.g., the covariance functions (separable or non-separable), the modeling process (integrated or decomposed), the information transfer (bidirectional or unidirectional), and the hyperparameter inference (joint or separate). Besides, we assess the performance of ten representative MOGPs thoroughly on eight examples in symmetric/asymmetric scenarios by considering, e.g., different training data (heterotopic or isotopic), different training sizes (small, moderate and large), different output correlations (low or high), and different output sizes (up to four outputs). Based on the qualitative and quantitative analysis, we give some recommendations regarding the usage of MOGPs and highlight potential research directions.