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The use of covariance kernels is ubiquitous in the field of spatialstatistics. Kernels allow data to be mapped into high-dimensional featurespaces and can thus extend simple linear additive methods to nonlinear methodswith higher order interactions. However, until recently, there has been astrong reliance on a limited class of stationary kernels such as the Matern orsquared exponential, limiting the expressiveness of these modelling approaches.Recent machine learning research has focused on spectral representations tomodel arbitrary stationary kernels and introduced more general representationsthat include classes of nonstationary kernels. In this paper, we exploit theconnections between Fourier feature representations, Gaussian processes andneural networks to generalise previous approaches and develop a simple andefficient framework to learn arbitrarily complex nonstationary kernel functionsdirectly from the data, while taking care to avoid overfitting usingstate-of-the-art methods from deep learning. We highlight the very broad arrayof kernel classes that could be created within this framework. We apply this toa time series dataset and a remote sensing problem involving land surfacetemperature in Eastern Africa. We show that without increasing thecomputational or storage complexity, nonstationary kernels can be used toimprove generalisation performance and provide more interpretable results.

Spatial Mapping with Gaussian Processes and Nonstationary Fourier Features