Spatial models with explanatory variables in the dependence structure

Abstract

Please cite this article as: Ingebrigtsen, R., Lindgren, F., Steinsland, I., Spatial models with explanatory variables in the dependence structure. Spatial Statistics (2013), http://dx.doi.org/10. 1016/j.spasta.2013.06.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. AbstractGeostatistical models have traditionally been stationary. However, physical knowledge about underlying spatial processes often require models with nonstationary dependence structures. Thus, there has been an interest in the literature to provide flexible models and computationally efficient methods for non-stationary phenomena. In this work, we demonstrate that the stochastic partial differential equation (SPDE) approach to spatial modelling provides a flexible class of non-stationary models where explanatory variables easily can be included in the dependence structure. In addition, the SPDE approach enables computationally efficient Bayesian inference with integrated nested Laplace approximations (INLA) available through the R-package r-inla. We illustrate the suggested modelling framework with a case study of annual precipitation in southern Norway, and compare a non-stationary model with dependence structure governed by elevation to a stationary model. Further, we use a simulation study to explore the annual precipitation models. We investigate identifiability of model parameters and whether the deviance information criterion (DIC) is able to distinguish datasets from the non-stationary and stationary models.

Second-order non-stationary modeling approaches for univariate geostatistical data

“…Thus, a possible option for parameterizing the model without explicit construction of the deformation function is to control the major axis of the local deformation given by FðÁÞ through a vector field. Ingebrigtsen et al (2014) point out that, in general, nonstatioary modeling approaches based on space deformation and SPDE are not equivalent. For deformations into a higher dimensional space, the two approaches differ considerably.…”

“…For deformations into a higher dimensional space, the two approaches differ considerably. As highlighted by Ingebrigtsen et al (2014), the non-stationary SPDE defined in Eq. (36) with log j 2 ðxÞ ð Þ¼ The SPDE based non-stationary modeling approach also has a connection with the convolution based non-stationary modeling approach.…”

“…Bolin (2014) presents non-stationary SPDE models in a non-Gaussian framework. Ingebrigtsen et al (2014) model the non-stationarity in the SPDE using spatially varying explanatory variables. Work on parameterising and performing inference on non-stationary SPDE models with varying local anisotropy has been undertaken by Fuglstad et al (2015a).…”

Estimating unconsolidated sediment cover thickness by using the horizontal distance to a bedrock outcrop as secondary information

“…They approximate the estimation problem to stochastic partial differential equations. In this method non-stationarity of statistical moments are taken into account, but at the same time less computer resources are spent on matrix inversions which is a challenge for applications with a large number of observations (Lindgren et al, 2011;Ingebrigtsen et al, 2014;Hu and Steinsland, 2016).…”

Local Likelihood Estimation for Covariance Functions with Spatially-Varying Parameters: The convoSPAT Package for R

“…As noted, there are several other (albeit non convolution-based) methods for nonstationary spatial modeling that are available in software, namely the basis function approach in the fields package (Nychka, Furrer, and Sain 2017) and the Gaussian Markov random field approach in the INLA package (Lindgren, Rue, and Lindstrom 2011;Ingebrigtsen, Lindgren, and Steinsland 2014;Fuglstad, Lindgren, Simpson, and Rue 2015;Lindgren and Rue 2015), both available for R. However, as these methods arise from significantly different modeling approaches, the convoSPAT package represents a novel contribution to the set of available software for nonstationary spatial modeling. Software comparison across these various packages is beyond the scope of this paper and will be reserved for future work.…”