Analytic Long-Term Forecasting with Periodic Gaussian Processes

Gaussian processes are a state-of-the-art method for learning models from data. Data with an underlying periodic structure appears in many areas, e.g., in climatology or robotics. It is often important to predict the long-term evolution of such a time series, and to take the inherent periodicity explicitly into account. In a Gaussian process, periodicity can be accounted for by an appropriate kernel choice. However, the standard periodic kernel does not allow for analytic long-term forecasting, which requires to map distributions through the Gaussian process. To address this shortcoming, we re-parametrize the periodic kernel, which, in combination with a double approximation, allows for analytic long-term forecasting of a periodic state evolution with Gaussian processes. Our model allows for probabilistic long-term forecasting of periodic processes, which can be valuable in Bayesian decision making, optimal control, reinforcement learning, and robotics.