Spectral Mixture Kernels for Multi-Output Gaussian Processes

Abstract

Early approaches to multiple-output Gaussian processes (MOGPs) relied on linear combinations of independent, latent, single-output Gaussian processes (GPs). This resulted in cross-covariance functions with limited parametric interpretation, thus conflicting with the ability of single-output GPs to understand lengthscales, frequencies and magnitudes to name a few. On the contrary, current approaches to MOGP are able to better interpret the relationship between different channels by directly modelling the cross-covariances as a spectral mixture kernel with a phase shift. We extend this rationale and propose a parametric family of complex-valued cross-spectral densities and then build on Cramér's Theorem (the multivariate version of Bochner's Theorem) to provide a principled approach to design multivariate covariance functions. The so-constructed kernels are able to model delays among channels in addition to phase differences and are thus more expressive than previous methods, while also providing full parametric interpretation of the relationship across channels. The proposed method is first validated on synthetic data and then compared to existing MOGP methods on two real-world examples.

Synergy of Sentinel-1 and Sentinel-2 Time Series for Cloud-Free Vegetation Water Content Mapping with Multi-Output Gaussian Processes

“…Aiming to improve training results each data channel was linearly detrended and normalized to have zero mean and unit variance utilizing the transformations provided by the MOGPTK. We analyzed four MOGP spectral kernels: (i) the Spectral Mixture Kernels for Multi-Output Gaussian Processes (MOSM) [70], (ii) the Cross-Spectral Mixture (CSM) [71], (iii) the Linear Model of Coregionalization (LMC) [39,40], and (iv) the Convolutional Model (CONV) [72,73] with four spectral components per channel (Q = 4). The MOGP kernels' parameters were randomly instantiated to establish an initial set of reasonable values for the S1 & S2 dataset using Independent Spectral Mixture (SM) as an estimation method.…”