Basis-Function Models in Spatial Statistics

Abstract

Spatial statistics is concerned with the analysis of data that have spatial locations associated with them, and those locations are used to model statistical dependence between the data. The spatial data are treated as a single realisation from a probability model that encodes the dependence through both fixed effects and random effects, where randomness is manifest in the underlying spatial process and in the noisy, incomplete, measurement process. The focus of this review article is on the use of basis functions to provide an extremely flexible and computationally efficient way to model spatial processes that are possibly highly non-stationary. Several examples of basis-function models are provided to illustrate how they are used in Gaussian, non-Gaussian, multivariate, and spatio-temporal settings, with applications in geophysics. Our aim is to emphasise the versatility of these spatial statistical models and to demonstrate that they are now centre-stage in a number of application domains. The review concludes with a discussion and illustration of software currently available to fit spatial-basis-function models and implement spatial-statistical prediction.

Dynamic Data Assimilation of MPAS-O and the Global Drifter Dataset

“…Basis function models, which use linear combinations of continuous functions to estimate a state, are a flexible and computationally efficient method for solving non-stationary systems [2]. In these models, a set of basis functions are chosen or discovered, and the associated coefficient weights for an unknown function in the space are learned.…”

“…Note that since the dynamic information αj(t) has already been captured, the added complexity compared to the static method is O(1) per observation2 While we focus on Jan. 1st, 2008 to Jan 1st, 2009, the results presented here are representative of the general year-long phenomenon. The following discussion and analysis can be applied to any time slice, since the fully dynamic models have temporal information…”