  Kernel design for Multi-output Gaussian Processes (MOGP) has received
increased attention recently. In particular, the Multi-Output Spectral Mixture
kernel (MOSM) arXiv:1709.01298 approach has been praised as a general model in
the sense that it extends other approaches such as Linear Model of
Corregionalization, Intrinsic Corregionalization Model and Cross-Spectral
Mixture. MOSM relies on Cram\'er's theorem to parametrise the power spectral
densities (PSD) as a Gaussian mixture, thus, having a structural restriction:
by assuming the existence of a PSD, the method is only suited for multi-output
stationary applications. We develop a nonstationary extension of MOSM by
proposing the family of harmonizable kernels for MOGPs, a class of kernels that
contains both stationary and a vast majority of non-stationary processes. A
main contribution of the proposed harmonizable kernels is that they
automatically identify a possible nonstationary behaviour meaning that
practitioners do not need to choose between stationary or non-stationary
kernels. The proposed method is first validated on synthetic data with the
purpose of illustrating the key properties of our approach, and then compared
to existing MOGP methods on two real-world settings from finance and
electroencephalography.
