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 Matérn 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.

协方差核的使用在空间统计领域中无处不在。核允许将数据映射到高维特征空间，因此可以将简单的线性加法方法扩展到具有更高阶相互作用的非线性方法。然而，直到最近，人们仍然强烈依赖有限类别的固定核，例如 Matérn 或平方指数，限制了这些建模方法的表达能力。最近的机器学习研究主要集中在频谱表示上，以对任意平稳核进行建模，并引入了包括非平稳核类别的更通用的表示。在本文中，我们利用傅立叶特征表示、高斯过程和神经网络之间的联系来概括以前的方法，并开发一个简单而有效的框架来直接从数据中学习任意复杂的非平稳核函数，同时注意避免使用状态过度拟合来自深度学习的最先进方法。我们重点介绍可以在此框架内创建的非常广泛的内核类。我们将其应用于时间序列数据集和涉及东非地表温度的遥感问题。我们证明，在不增加计算或存储复杂性的情况下，非平稳内核可以用来提高泛化性能并提供更可解释的结果。