Data-driven Warping of Gaussian Processes for Spatial Interpolation of Skewed Data
Authors: Hristopulos Dionissios , Agou Vasiliki , Pavlides Andrew 
Published: March 2022

&lt;p&gt;Gaussian processes are a flexible machine learning framework that can be used for spatial interpolation and space-time prediction as well. Gaussian process regression (GPR) is quite similar to the geostatistical kriging method.&amp;#160; It encompasses various types of kriging (e.g., simple, ordinary, universal and regression kriging).&amp;#160; In addition, it is formulated in an inherently Bayesian framework which allows taking into account a priori beliefs regarding the distribution of the model&amp;#8217;s hyper-parameters. Thus, it also incorporates Bayesian versions of kriging [1]. &amp;#160;GPR is based on the assumption that the stochastic component of the observations follows a Gaussian distribution.&amp;#160; However, this is not the case for various environmental variables (e.g., amount of precipitation, hydraulic conductivity, wind speed), which follow skewed probability distributions.&amp;#160; The skewness is handled within the geostatistical framework using nonlinear transforms such that the marginal distribution of the data in the latent space becomes normal.&amp;#160; This procedure is known as Gaussian anamorphosis in geostatistics.&amp;#160; In the context of GPR, the term warped Gaussian process is used to denote the nonlinear transformation of the observations [2].&amp;#160; &amp;#160;Gaussian anamorphosis (warping) is usually implemented using explicit, monotonically increasing nonlinear functions.&amp;#160; A different approach involves generating the warping function with the help of the empirically estimated cumulative probability distribution of the data.&amp;#160; This approach provides flexibility because the transformation is data-driven (non-parametric) and is thus not constrained by specific functional forms.&amp;#160; Furthermore, the cumulative distribution function of the data can be accurately estimated using smoothing kernels [3].&amp;#160; We investigate warped Gaussian process regression using synthetic datasets and precipitation reanalysis data from the Mediterranean island of Crete. Cross validation analysis is used to establish the advantages of non-parametric warping for the interpolation of incomplete data. We demonstrate that warped GPR equipped with data-driven warping provides enhanced flexibility compared to &quot;bare&quot; GPR and can lead to improved predictive accuracy for non-Gaussian data. &amp;#160;&lt;/p&gt;&lt;p&gt;Keywords: Gaussian processes, Mediterranean island, non-Gaussian, warping, precipitation&lt;/p&gt;&lt;p&gt;Funding: This research is co-financed by Greece and the European Union (European Social Fund- ESF) through the Operational Programme &amp;#171;Human Resources Development, Education and Lifelong Learning 2014-2020&amp;#187;in the context of the project &amp;#8220;Gaussian Anamorphosis with Kernel Estimators for Spatially Distributed Data and Time Series and Applications in the Analysis of Precipitation&amp;#8221; (MIS 5052133).&lt;/p&gt;&lt;p&gt;&lt;strong&gt;References&lt;/strong&gt;&lt;/p&gt;&lt;p&gt;[1] T. Hristopulos, 2020. Random Fields for Spatial Data Modeling. Springer Netherlands, http://dx.doi.org/10.1007/978-94-024-1918-4.&lt;/p&gt;&lt;p&gt;[2] Snelson, E., Rasmussen, C.E. and Ghahramani, Z., 2004. Warped Gaussian processes. &lt;em&gt;Advances in neural information processing systems&lt;/em&gt;, &lt;em&gt;16&lt;/em&gt;, pp.337-344.&lt;/p&gt;&lt;p&gt;[3] Pavlides, A., Agou, V., and Hristopulos, D. T., 2021. Non-parametric Kernel-Based Estimation of Probability Distributions for Precipitation Modeling. &lt;em&gt;arXiv preprint arXiv:2109.09961&lt;/em&gt;.&lt;/p&gt;