A sampling approach to estimate the log determinant used in spatial likelihood problems

Likelihood-based methods for modeling multivariate Gaussian spatial data have desirable statistical characteristics, but the practicality of these methods for massive georeferenced data sets is often questioned. A sampling algorithm is proposed that exploits a relationship involving log-pivots arising from matrix decompositions used to compute the log determinant term that appears in the model likelihood. We demonstrate that the method can be used to successfully estimate log-determinants for large numbers of observations. Specifically, we produce an log-determinant estimate for a 3,954,400 by 3,954,400 matrix in less than two minutes on a desktop computer. The proposed method involves computations that are independent, making it amenable to out-of-core computation as well as to coarse-grained parallel or distributed processing. The proposed technique yields an estimated log-determinant and associated confidence interval.

A spatial error model with continuous random effects and an application to growth convergence

We propose a spatial error model with continuous random effects based on Matérn covariance functions and apply this model for the analysis of income convergence processes (\(\beta \)-convergence). The use of a model with continuous random effects permits a clearer visualization and interpretation of the spatial dependency patterns, avoids the problems of defining neighborhoods in spatial econometrics models, and allows projecting the spatial effects for every possible location in the continuous space, circumventing the existing aggregations in discrete lattice representations. We apply this model approach to analyze the economic growth of Brazilian municipalities between 1991 and 2010 using unconditional and conditional formulations and a spatiotemporal model of convergence. The results indicate that the estimated spatial random effects are consistent with the existence of income convergence clubs for Brazilian municipalities in this period.     

A model for integrated spatial land use characteristics linking to surface nutrient concentration

Quantifying land use heterogeneity helps better understand how it influences biophysical systems. Land use area proportions have been used conventionally to predict water quality variables. Lacking an insight into the combined effect of various spatial characteristics could lead to the statistical bias and confused understanding in previous studies. In this study, using spatial techniques and mathematical models, a diagnostic model was developed and applied for quantifying and incorporating three spatial components, namely, slope, distance to sampling spots, and arrangement. The upper catchment of Miyun Reservoir was studied as the test area. Total nitrogen, total phosphorus, and chemical oxygen demand of water samples from field measurements were used to characterize the surface water quality in 52 sub-watersheds. Using parameter calibrations and determinations, combined spatial characteristics were explored and detected. Adjusted land use proportions were calculated by spatial weights of discriminating the relative contribution of each location to water quality and used to build the integrated models. Compared with traditional methods only using area proportions, our model increased the explanatory power of land use and quantified the effects of spatial information on water quality. This can guide the optimization of land use configuration to control water eutrophication.     

A spatial panel ordered-response model with application to the analysis of urban land-use development intensity patterns

This paper proposes and estimates a spatial panel ordered-response probit model with temporal autoregressive error terms to analyze changes in urban land development intensity levels over time. Such a model structure maintains a close linkage between the land owner’s decision (unobserved to the analyst) and the land development intensity level (observed by the analyst) and accommodates spatial interactions between land owners that lead to spatial spillover effects. In addition, the model structure incorporates spatial heterogeneity as well as spatial heteroscedasticity. The resulting model is estimated using a composite marginal likelihood (CML) approach that does not require any simulation machinery and that can be applied to data sets of any size. A simulation exercise indicates that the CML approach recovers the model parameters very well, even in the presence of high spatial and temporal dependence. In addition, the simulation results demonstrate that ignoring spatial dependency and spatial heterogeneity when both are actually present will lead to bias in parameter estimation. A demonstration exercise applies the proposed model to examine urban land development intensity levels using parcel-level data from Austin, Texas.