  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 Cram\'er'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.
