Exact Integration of Stationary Gaussian Process Kernels

In applications such as object tracking and financial time series, it is often desirable to model stochastic processes that are the integral of another stationary process, leading potentially to non-stationarity and infinite variance. In tracking, for example, an object's velocity may be modeled by a stationary Gaussian Process (GP), while the position over time is an integrated GP (IGP). Such ‘integrated’ processes are also GPs, and we derive closed-form expressions for their kernels when the base kernel is from the Matérn, squared-exponential, rational quadratic, or Laplace class. Particular attention is given to the Matérn kernel, as it has special cases that allow for exact integration in terms of elementary functions. We show that mixtures of these special Matérn kernels also include the family of discrete state-space models (SSMs), thereby allowing integration of these SSMs without a discrete accumulation approximation. We provide an optimized filtering implementation for SSMs with integrated GP dynamics and compare our method with integrated tracking models from the literature on a real-world dataset of agile soccer/football player tracking.