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 filter-ing theory. This reduces the problematic cu-bic 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 co-variance function is written out and the ex-pansion is shown to uniformly converge to the exact covariance function with a known convergence rate. The framework is gener-alized 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 non-stationary and spatio-temporal variants.