Modelling Non-Smooth Signals with Complex Spectral Structure

TL;DR

This paper introduces the CGPCM and RGPCM models that extend the GPCM to handle non-smooth signals with complex spectral structures, along with a new inference method that improves calibration and efficiency.

Contribution

It proposes two novel models, CGPCM and RGPCM, that relax smoothness assumptions and incorporate causality and nonparametric spectral modeling, plus an improved inference scheme.

Findings

Models effectively capture non-smooth signals.

The Gibbs sampler improves inference efficiency.

Experimental results show promising performance.

Abstract

The Gaussian Process Convolution Model (GPCM; Tobar et al., 2015a) is a model for signals with complex spectral structure. A significant limitation of the GPCM is that it assumes a rapidly decaying spectrum: it can only model smooth signals. Moreover, inference in the GPCM currently requires (1) a mean-field assumption, resulting in poorly calibrated uncertainties, and (2) a tedious variational optimisation of large covariance matrices. We redesign the GPCM model to induce a richer distribution over the spectrum with relaxed assumptions about smoothness: the Causal Gaussian Process Convolution Model (CGPCM) introduces a causality assumption into the GPCM, and the Rough Gaussian Process Convolution Model (RGPCM) can be interpreted as a Bayesian nonparametric generalisation of the fractional Ornstein-Uhlenbeck process. We also propose a more effective variational inference scheme, going beyond the mean-field assumption: we design a Gibbs sampler which directly samples from the optimal variational solution, circumventing any variational optimisation entirely. The proposed variations of the GPCM are validated in experiments on synthetic and real-world data, showing promising results.