Explicit Link Between Periodic Covariance Functions and State Space Models

This paper shows how periodic covariance functions in Gaussian process regression can be reformulated as state space models, which can be solved with classical Kalman filtering theory. This reduces the problematic cubic complexity of Gaussian process regression in the number of time steps into linear time complexity. The representation is based on expanding periodic covariance functions into a series of stochastic resonators. The explicit representation of the canonical periodic covariance function is written out and the expansion is shown to uniformly converge to the exact covariance function with a known convergence rate. The framework is generalized to quasi-periodic covariance functions by introducing damping terms in the system and applied to two sets of real data. The approach could be easily extended to nonstationary and spatio-temporal variants.