  The use of covariance kernels is ubiquitous in the field of spatial
statistics. Kernels allow data to be mapped into high-dimensional feature
spaces and can thus extend simple linear additive methods to nonlinear methods
with higher order interactions. However, until recently, there has been a
strong reliance on a limited class of stationary kernels such as the Matern or
squared exponential, limiting the expressiveness of these modelling approaches.
Recent machine learning research has focused on spectral representations to
model arbitrary stationary kernels and introduced more general representations
that include classes of nonstationary kernels. In this paper, we exploit the
connections between Fourier feature representations, Gaussian processes and
neural networks to generalise previous approaches and develop a simple and
efficient framework to learn arbitrarily complex nonstationary kernel functions
directly from the data, while taking care to avoid overfitting using
state-of-the-art methods from deep learning. We highlight the very broad array
of kernel classes that could be created within this framework. We apply this to
a time series dataset and a remote sensing problem involving land surface
temperature in Eastern Africa. We show that without increasing the
computational or storage complexity, nonstationary kernels can be used to
improve generalisation performance and provide more interpretable results.
