Efficient Parameter Estimation of Sampled Random Fields Using the Debiased Spatial Whittle Likelihood
Authors: Simons Frederik J , Guillaumin Arthur P. , Sykulski Adam M. , Olhede Sofia C. 
Published: March 2022

&lt;p&gt;We establish a theoretical framework, an algorithmic basis, and a&amp;#160;computational workflow for the statistical analysis of multi-variate&amp;#160;multi-dimensional random fields - sampled (possibly irregularly, with missing data) and finite (possibly bounded irregularly). Our research is practically motivated by geodetic and scientific problems of topography and gravity&amp;#160;analysis in geophysics and planetary physics, but our solutions fulfill the more general need for sophisticated methods of&amp;#160;inference that can be applied to massive remote-sensing data sets, and as such, our mathematical, statistical, and computational solutions transcend any particular&amp;#160;application.&amp;#160;The generic problem that we are addressing is: two (or more) spatial&amp;#160;fields are observed, e.g., by passive or active sensing, and we desire&amp;#160;a parsimonious statistical description of them, individually and in&amp;#160;their relation to one another. We consider the fields to be&amp;#160;realizations of a random process, parameterized as a Matern covariance&amp;#160;structure, a very flexible description that includes, as special&amp;#160;cases, many of the known models in popular use (e.g. exponential,&amp;#160;autoregressive, von Karman, Gaussian, Whittle, ...) Our&amp;#160;fundamental question is how to find estimates of the parameters of a&amp;#160;Matern process, and the distribution of those estimates for&amp;#160;uncertainty quantification. Our answer is, fundamentally: via&amp;#160;maximum-likelihood estimation. &amp;#160;We now provide a computationally and statistically efficient method for&amp;#160;estimating the parameters of a stochastic covariance model observed on a&amp;#160;regular spatial grid in any number of dimensions. Our proposed method, which we call the Debiased Spatial Whittle likelihood, makes important corrections&amp;#160;to the well-known Whittle likelihood to account for large sources of bias&amp;#160;caused by boundary effects and aliasing. We generalise the approach to&amp;#160;flexibly allow for significant volumes of missing data including those with&amp;#160;lower-dimensional substructure, and for irregular sampling boundaries. We&amp;#160;build a theoretical framework under relatively weak assumptions which&amp;#160;ensures consistency and asymptotic normality in numerous practical settings&amp;#160;including missing data and non-Gaussian processes. We also&amp;#160;extend our consistency results to multivariate processes. We provide&amp;#160;detailed implementation guidelines which ensure the estimation procedure can&amp;#160;still be conducted in O(n log n) operations, where n is the&amp;#160;number of points of the encapsulating rectangular grid, thus keeping the&amp;#160;computational scalability of Fourier and Whittle-based methods for large&amp;#160;data sets. We validate our procedure over a range of simulated and real&amp;#160;world settings, and compare with state-of-the-art alternatives,&amp;#160;demonstrating the enduring practical appeal of Fourier-based methods,&amp;#160;provided they are corrected and augmented by the procedures that we developed.&lt;/p&gt;