Multi-task Gaussian Processes

Abstract

Modelling and forecasting functional data (time series, spatial measurements, …), even with a probabilistic flavour, is a common and well-handled problem nowadays. However, suppose that one is collecting data from hundreds of individuals, each of them gathering thousands of biological measurements, all evolving continuously over time. Such a context, frequently arising in biological or medical studies, quickly leads to highly correlated datasets where dependencies come from different sources (for instance, temporal trends or individual similarities). Explicit modelling of overly large covariance matrices accounting for these underlying correlations is generally unreachable due to theoretical and computational limitations. Therefore, practitioners often need to restrict their analysis by working on subsets of data or making arguable assumptions (fixing time, studying genes or individuals independently, …). To tackle these issues, we proposed a framework for multi-task Gaussian processes, tailored to handle multiple functional data simultaneously. By sharing information between tasks through a mean process instead of an explicit covariance structure, this method leads to a learning and forecasting procedure with linear complexity in the number of tasks. The resulting predictions remain Gaussian distributions and thus offer an elegant probabilistic approach to deal with correlated measurements. Group structures can also be exploited through clustering within the learning procedure to enhance prediction performances. We will finally present an extended framework in which as many sources of correlation as desired can be considered while maintaining linear complexity scaling. Several applicative examples are explored, coming from various fields like epidemiology, biology, or sports sciences.