Daniel A. Griffith

SPATIAL MODELING IN ECOLOGY: THE FLEXIBILITY OF EIGENFUNCTION SPATIAL ANALYSES

Recently, analytical approaches based on the eigenfunctions of spatial configuration matrices have been proposed in order to consider explicitly spatial predictors. The present study demonstrates the usefulness of eigenfunctions in spatial modeling applied to ecological problems and shows equivalencies of and differences between the two current implementations of this methodology. The two approaches in this category are the distance-based (DB) eigenvector maps proposed by P. Legendre and his colleagues, and spatial filtering based upon geographic connectivity matrices (i.e., topology-based; CB) developed by D. A. Griffith and his colleagues. In both cases, the goal is to create spatial predictors that can be easily incorporated into conventional regression models. One important advantage of these two approaches over any other spatial approach is that they provide a flexible tool that allows the full range of general and generalized linear modeling theory to be applied to ecological and geographical problems in the presence of nonzero spatial autocorrelation.

A linear regression solution to the spatial autocorrelation problem

Abstract. The Moran Coefficient spatial autocorrelation index can be decomposed into orthogonal map pattern components. This decomposition relates it directly to standard linear regression, in which corresponding eigenvectors can be used as predictors. This paper reports comparative results between these linear regressions and their auto-Gaussian counterparts for the following georeferenced data sets: Columbus (Ohio) crime, Ottawa-Hull median family income, Toronto population density, southwest Ohio unemployment, Syracuse pediatric lead poisoning, and Glasgow standard mortality rates, and a small remotely sensed image of the High Peak district. This methodology is extended to auto-logistic and auto-Poisson situations, with selected data analyses including percentage of urban population across Puerto Rico, and the frequency of SIDs cases across North Carolina. These data analytic results suggest that this approach to georeferenced data analysis offers considerable promise.

Semiparametric Filtering of Spatial Autocorrelation: The Eigenvector Approach

In the context of spatial regression analysis, several methods can be used to control for the statistical effects of spatial dependencies among observations. Maximum likelihood or Bayesian approaches account for spatial dependencies in a parametric framework, whereas recent spatial filtering approaches focus on nonparametrically removing spatial autocorrelation. In this paper we propose a semiparametric spatial filtering approach that allows researchers to deal explicitly with (a) spatially lagged autoregressive models and (b) simultaneous autoregressive spatial models. As in one non-parametric spatial filtering approach, a specific subset of eigenvectors from a transformed spatial link matrix is used to capture dependencies among the disturbances of a spatial regression model. However, the optimal subset in the proposed filtering model is identified more intuitively by an objective function that minimizes spatial autocorrelation rather than maximizes a model fit. The proposed objective function has the advantage that it leads to a robust and smaller subset of selected eigenvectors. An application of the proposed eigenvector spatial filtering approach, which uses a cancer mortality dataset for the 508 US State Economic Areas, demonstrates its feasibility, flexibility, and simplicity.

Explorations into the Relationship between Spatial Structure and Spatial Interaction

This paper explores the relationship between spatial structure and spatial interaction at the intraurban level. To examine this relationship an experimental framework is designed based on the application of a doubly constrained entropy-type gravity model to journey-to-work data for twenty-four Canadian urban areas. The study demonstrates that distance-decay exponents are strongly influenced by geographic structure and the geometry of origins and destinations. As such, both the influence of map pattern and the friction of distance should be explicitly incorporated into spatial interaction models. The paper also explores the impact of city size and the nature of the economic base of the urban area upon distance-decay exponents.

Modelling urban population density in a multi-centered city

Modelling the geographic distribution of urban population densities has been attempted in several ways. Recently a controversy emerged in the Journal of Urban Economics regarding whether or not calibrations of these models render unbiased parameter estimates. Three sources of bias were dealt with in these discussions, namely (1) model specification error, (2) the estimation procedure employed, and (3) the definition of areal unit observation size. Additional sources of bias overlooked in this controversy include the presence of multiple centers in a city, and the existence of externalities. This paper explores these additional sources, from both a conceptual and an empirical point of view.

Spatial-Filtering-Based Contributions to a Critique of Geographically Weighted Regression (GWR)

Interaction terms are constructed with georeferenced attribute variables and spatial filter eigenvectors, and then used to compute geographically varying regression coefficients. These coefficients, which are analogous to geographically weighted regression (GWR) coefficients, display preferable properties, and this specification is used to critique selected features of GWR. Comparisons are illustrated with the Georgia data appearing in the standard GWR tutorial.

Modeling Spatial Autocorrelation in Spatial Interaction Data: An Application to Patent Citation Data in the European Union

Spatial interaction models of the gravity type are widely used to model origin-destination flows. They draw attention to three types of variables to explain variation in spatial interactions across geographic space: variables that characterize an origin region of a flow, variables that characterize a destination region of a flow, and finally variables that measure the separation between origin and destination regions. This paper outlines and compares two approaches, the spatial econometric and the eigenfunction-based spatial filtering approach, to deal with the issue of spatial autocorrelation among flow residuals. An example using patent citation data that capture knowledge flows across 112 European regions serves to illustrate the application and the comparison of the two approaches. Copyright (c) 2008, Wiley Periodicals, Inc.

A Variance-Stabilizing Coding Scheme for Spatial Link Matrices

In spatial statistics and spatial econometrics two coding schemes are used predominately. Except for some initial work, the properties of both coding schemes have not been investigated systematically. In this paper we do so for significant spatial processes specified as either a simulta-neous autoregressive or a moving average process. Results show that the C-coding scheme emphasizes spatial objects with relatively large numbers of connections, such as those in the interior of a study region. In contrast, the W-coding scheme assigns higher leverage to spatial objects with few connections, such as those on the periphery of a study region. To address this topology-induced heterogeneity, we design a novel S-coding scheme whose properties lie in between those of the C-coding and the W-coding schemes. To compare these three coding schemes within and across the different spatial processes, we find a set of autocorrelation parameters that makes the processes stochastically homologous via a method based on the exact conditional expectation of Morans I. In the new S-coding scheme the topology induced heterogeneity can be removed in toto for Morans I as well as for moving average processes and it can be substantially alleviated for autoregressive processes.

Effective Geographic Sample Size in the Presence of Spatial Autocorrelation

Abstract As spatial autocorrelation latent in georeferenced data increases, the amount of duplicate information contained in these data also increases. This property suggests the research question asking what the number of independent observations, say , is that is equivalent to the sample size, n, of a data set. This is the notion of effective sample size. Intuitively speaking, when zero spatial autocorrelation prevails, ; when perfect positive spatial autocorrelation prevails in a univariate regional mean problem, . Equations are presented for estimating based on the sampling distribution of a sample mean or sample correlation coefficient with the goal of obtaining some predetermined level of precision, using the following spatial statistical model specifications: (1) simultaneous autoregressive, (2) geostatistical semivariogram, and (3) spatial filter. These equations are evaluated with simulation experiments and are illustrated with selected empirical examples found in the literature.

Error propagation modelling in raster GIS: overlay operations

Performing data manipulations on maps that possess error as a result of the process of data collection leads to error propagation. The errors that are present in maps are modified by such operations in ways that may undermine the purposeofanalysisand lead to increased uncertainty in thevalidity ofthe conclusions that are drawn. This paper analyses how source map error propagates as a result of overlay operations. Geman and Gemans corruption model for individual source map error is used for the analysis which allows for attribute measurement error and location error that can then interact with the (true) source map geography. This paper reports theoretical results on the univariate overlay problem and then extends these results through simulation. Throughout a set of source maps and error processes are used with specified properties in order to examine in detail the interactions that can take place between the different elements of the source map structure and the error process. The paper uses ANOVA metho...