Dynamic covariance estimation via predictive Wishart process with an application on brain connectivity estimation

Modeling the complex dependence in multivariate time series data is a fundamental problem in statistics and machine learning. Traditionally, the task has been approached with methods such as multivariate autoregressive models and multivariate generalized autoregressive conditional heteroskedasticity models, and Gaussian process based methods are recently becoming popular by leveraging the flexibility of non-parametric learning. However, few methods exist that directly model the dynamics of the covariance matrices except generalized Wishart process (GWP), and even the generalized Wishart process is limited with applications on small dataset due to the extremely high computational capacity induced by multiple Gaussian processes. In this regard, a novel stochastic process named as Predictive Wishart Process (PWP) is proposed, which provides a collection of positive semi-definite random matrices indexed by input variables. The PWP projects process realizations of GWP to a lower dimensional subspace to efficiently estimate every GWP. The theoretical properties of it are examined, and both Bayesian inference and efficient variational expectation maximization are explored in relation to it. Moreover, the PWP is empirically tested on synthetically generated time-series data to validate competitive reconstructive performance and efficient predictive performance, and applied on a large-scale real functional magnetic resonance imaging (fMRI) dataset from Human Connectome Project (HCP) to demonstrate its practicality. A thorough statistical analysis with visualizations is conducted on the brain connectivity, and also a PWP-based multi-task learning framework is proposed to extract meaningful features from individual fMRIs.