Mark D. Ecker

Bayesian Variogram Modeling for an Isotropic Spatial Process

Here we adopt a Bayesian approach to variogram modeling. In particular, we seek to analyze a recent dataset of scallop catches. We have the results of the analysis of an earlier dataset from the region to supply useful prior information. In addition, the Bayesian approach enables inference about any aspect of spatial dependence of interest rather than merely providing a fitted variogram. We utilize discrete mixtures of Bessel functions that allow a rich and flexible class of variogram models. To differentiate between models, we introduce a utility-based model choice criterion that encourages parsimony. We conclude with a fully Bayesian analysis of the scallop data.

Bayesian Modeling and Inference for Geometrically Anisotropic Spatial Data

A geometrically anisotropic spatial process can be viewed as being a linear transformation of an isotropic spatial process. Customary semivariogram estimation techniques often involve ad hoc selection of the linear transformation to reduce the region to isotropy and then fitting a valid parametric semivariogram to the data under the transformed coordinates. We propose a Bayesian methodology which simultaneously estimates the linear transformation and the other semivariogram parameters. In addition, the Bayesian paradigm allows full inference for any characteristic of the geometrically anisotropic model rather than merely providing a point estimate. Our work is motivated by a dataset of scallop catches in the Atlantic Ocean in 1990 and also in 1993. The 1990 data provide useful prior information about the nature of the anisotropy of the process. Exploratory data analysis (EDA) techniques such as directional empirical semivariograms and the rose diagram are widely used by practitioners. We recommend a suitable contour plot to detect departures from isotropy. We then present a fully Bayesian analysis of the 1993 scallop data, demonstrating the range of inferential possibilities.

An Analysis of the Influence of Location in the Market for Undeveloped Urban Fringe Land

This paper blends and extends the Colwell and Munneke (1997) and Isakson (1997) models of urban land values to include a spatiotemporal plattage component. By including a monocentric term in the Isakson model, we arrive at a Colwell and Munneke-type model with a temporal component included in the plattage term. We further extend this model to include a spatiotemporal effect in the plattage term and demonstrate the use of geostatistical plattage models. (R14)

Spatial modeling and prediction under stationary non-geometric range anisotropy

For modeling spatial processes, we propose a rich parametric class of stationary range anisotropic covariance structures that, when applied in R2, greatly increases the scope of variogram contors. Geometric anisotropy, which provides the most common generalization of isotropy within stationarity, is a special case. Our class is built from monotonic isotropic correlation functions and special cases include the Matérn and the general exponential functions. As a result, our range anisotropic correlation specification can be attached to a second order stationary spatial process model, unlike ad hoc approaches to range anisotropy in the literature. We adopt a Bayesian perspective to obtain full inference and demonstrate how to fit the resulting model using sampling-based methods. In the presence of measurement error/microscale effect, we can obtain both the usual predictive as well as the noiseless predictive distribution. We analyze a data set of scallop catches under the general exponential range anisotropic model, withholding ten sites to compare the accuracy and precision of the standard and noiseless predictive distributions.

Bayesian Spatial Modeling of Housing Prices Subject to a Localized Externality

This work proposes a non stationary random field model to describe the spatial variability of housing prices that are affected by a localized externality. The model allows for the effect of the localized externality on house prices to be represented in the mean function and/or the covariance function of the random field. The correlation function of the proposed model is a mixture of an isotropic correlation function and a correlation function that depends on the distances between home sales and the localized externality. The model is fit using a Bayesian approach via a Markov chain Monte Carlo algorithm. A dataset of 437 single family home sales during 2001 in the city of Cedar Falls, Iowa, is used to illustrate the model.

A note on a non-stationary point source spatial model

A point source, non-stationary covariance structure model is proposed, having only one additional parameter over a standard, stationary covariance structure, spatial model. Additionally, the proposed model is demonstrated to fit better than the three extra parameter, point source, non-stationary spatial model proposed by Ecker and De Oliveira (Commun Stat Theory Methods 37:2066–2078, 2008). The proposed model is fit from a Bayesian perspective and illustrated using a house sales dataset from Cedar Falls, Iowa.