Spectral Mixture Kernels with Time and Phase Delay Dependencies

Spectral Mixture (SM) kernels form a powerful class of kernels for Gaussian processes, capable to discover patterns, extrapolate, and model negative covariances. Being a linear superposition of quasi-periodical Gaussian components, an SM kernel does not explicitly model dependencies between components. In this paper we investigate the benefits of modeling explicitly time and phase delay dependencies between components in an AM kernel. We analyze the presence of statistical dependencies between components using Gaussian conditionals and posterior covariance and use this framework to motivate the proposed SM kernel extension, called Spectral Mixture kernel with time and phase delay Dependencies (SMD). SMD is constructed in two steps: first, time delay and phase delay are incorporated into each base component; next, cross-convolution between a base component and the reversed complex conjugate of another base component is performed which yields a complex-valued and positive definite kernel representing correlations between base components. The number of hyper-parameters of SMD, except the time and phase delay ones, remains equal to that of the SM kernel. We perform a thorough comparative experimental analysis of SMD on synthetic and real-life data sets. Results indicate the beneficial effect of modeling time and phase delay dependencies between base components, notably for natural phenomena involving little or no influence from human activity.