1: Introduction.- 2: The Scope of Spatial Econometrics.- 3: The Formal Expression of Spatial Effects.- 4: A Typology of Spatial Econometric Models.- 5: Spatial Stochastic Processes: Terminology and General Properties.- 6: The Maximum Likelihood Approach to Spatial Process Models.- 7: Alternative Approaches to Inference in Spatial Process Models.- 8: Spatial Dependence in Regression Error Terms.- 9: Spatial Heterogeneity.- 10: Models in Space and Time.- 11: Problem Areas in Estimation and Testing for Spatial Process Models.- 12: Operational Issues and Empirical Applications.- 13: Model Validation and Specification Tests in Spatial Econometric Models.- 14: Model Selection in Spatial Econometric Models.- 15: Conclusions.- References.

Spatial Econometrics: Methods and Models

An overview of model-based geostatistics.- Gaussian models for geostatistical data.- Generalized linear models for geostatistical data.- Classical parameter estimation.- Spatial prediction.- Bayesian inference.- Geostatistical design.

/pdf/model-based-geostatistics-3h5fgto422.pdf

Model-based Geostatistics

Spatial Dependence in linear Regression Models with an Introduction to Spatial Econometrics : Regression Models with an Anselin Bera I. INTRODUCTION

The use of kriging models for approximation and metamodel-based design and optimization has been steadily on the rise in the past decade. The widespread usage of kriging models appears to be hampered by (1) the lack of guidance in selecting the appropriate form of the kriging model, (2) computationally efficient algorithms for estimating the model’s parameters, and (3) an effective method to assess the resulting model’s quality. In this paper, we compare (1) Maximum Likelihood Estimation (MLE) and Cross-Validation (CV) parameter estimation methods for selecting a kriging model’s parameters given its form and (2) and an R 2 of prediction and the corrected Akaike Information Criterion for assessing the quality of the created kriging model, permitting the comparison of different forms of a kriging model. These methods are demonstrated with six test problems. Finally, different forms of kriging models are examined to determine if more complex forms are more accurate and easier to fit than simple forms of kriging models for approximating computer models.

http://www2.mae.ufl.edu/haftka/eoed/protected/ASME04.kriging.pdf

Use of Kriging Models to Approximate Deterministic Computer Models

SUMMARY Maximum likelihood has frequently been suggested as a way to estimate covariance parameters in spatial Gaussian processes. The results of some simple examples based on simulated data are discussed. It is shown by an example with real data that the profile likelihood can be multimodal, and that the global maximum may not correspond to a sensible value of the parameters. Mardia & Marshall (1984) worked in the following framework. We have a Gaussian process {Z(x) I x E X} with mean given by a regression equation, E{Z(x)} =f(x)T', for a p-dimensional vector ,/. The covariance function cov {Z(x), Z(y)} = c(x, y; 0) is also specified by a q-dimensional vector 0. This process is observed at n points xI, .. , xn and from these observations we wish to estimate /3 and 0. Let Zn = {Z(xI), .. ., Z(xn)}T be the vector of observations, and K = [c(xi, xj; 0)] be its covariance matrix. Finally, let F be the matrix with rows f(Xi)T. Then Zn has a multivariate normal distribution with mean F18, dispersion matrix K and the log likelihood is, up to an additive constant,

Problems with likelihood estimation of covariance functions of spatial Gaussian processes

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Problems with likelihood estimation of covariance functions of spatial Gaussian processes