An extension of Matérn covariance function to multi-ouput case The Matérn class is used to chose covariance functions for univariate Gaussian Processes. Is there an known extension of this class to the multi-output/dimensional Gaussian Process case?
 A: The simplest multivariate extension of a Matérn covariance kernel
or of any kernel $K(t,\,s)$ for continuous input takes the form
$$
   \text{Cov}[Y_i(t), \, Y_j(s)] = \Sigma_{i,j}\, K(t - s) 
$$
where $\mathbf{Y}(t)= [Y_i(t)]_{1 \leq i \leq m}$ is the multivariate
process and $\boldsymbol{\Sigma} = [\Sigma_{i,j}]_{i,j}$
is a $m \times m$ positive definite matrix.
This is the continuous-time version of "Seemingly Unrelated"
models in Time Series: SU Regression Equations (SURE)
or SU Time Series Equations (SUTSE). With this covariance
kernel the Gaussian r.vs $Y_i(t)$ and $Y_j(s)$ are uncorrelated
conditional on $Y_j(t)$ or conditional on $Y_i(s)$.
This is a special case of a tensor product $K_{\text{cat}}
\otimes K_{\text{cont}}$ of a kernel with categorical input $i \in \{1,\ \dots,\,m\}$  and a
kernel with continuous input $t$
$$
K_{\text{prod}}\{ [i,\,t], \, [j,\,s] \} = K_{\text{cat}}(i,\,j)\, K_{\text{cont}} (t,
\,s)
$$
This representation may provide a tip to use the product kernel with software
accepting only univariate output.           
