The periodic zeta covariance function for Gaussian process regression

TL;DR

This paper introduces the periodic zeta covariance function for Gaussian process regression, which allows flexible modeling of seasonal and power-law spectral components, with detailed implementation guidance.

Contribution

It proposes the periodic zeta as a new covariance function for Gaussian processes, enabling better modeling of seasonal and spectral properties, and provides implementation details.

Findings

The periodic zeta covariance function effectively models seasonal processes.

It can be integrated with Matérn covariance for enhanced flexibility.

Implementation details facilitate practical adoption.

Abstract

I consider the Lerch-Hurwitz or periodic zeta function as covariance function of a periodic continuous-time stationary stochastic process. The function can be parametrized with a continuous index $\nu$ which regulates the continuity and differentiability properties of the process in a way completely analogous to the parameter $\nu$ of the Mat\'ern class of covariance functions. This makes the periodic zeta a good companion to add a power-law prior spectrum seasonal component to a Mat\'ern prior for Gaussian process regression. It is also a close relative of the circular Mat\'ern covariance, and likewise can be used on spheres up to dimension three. Since this special function is not generally available in standard libraries, I explain in detail the numerical implementation.