I am trying to create a spatiotemporal Gaussian Process Regression model where I am interested in prediction in both space and time.
So, I assume the models follow the form:
$$ Y(s; t) \sim GP (\mu(s; t), \Sigma(s; t)) $$
This is for all $(s; t)$ in our space-time domain where $\mu$ is the process mean and $\Sigma$ is the covariance function. So, my spatial data is denoted by 2-dimensional indexes and there is a time index.
I am wondering how such a covariance function can be specified. I am just starting out and wanted to just fit a GP with 0 mean and an RBF kernel which is smooth in both space and time. Also, currently $Y(s; t)$ is a scalar quantity. However, I am not sure how to specify this kernel. Should I create 3 kernels one for spatial x-dimension and one for spatial y-dimension and one for time and combine them by adding or multiplication? However, this does not seem right as I am not sure if I can use this to predict in $(x, y, t)$.
On more thinking, I think this should be a blocked covariance kernel where each diagonal block is the kernel for $x$, $y$ and $t$ respectively. This way I can even introduce correlations among the dimensions. Does that sound reasonable?