Entropy in Spatial Aggregation

This paper introduces an approach to the measurement of locational phenomena in a spatial hierarchy using entropy statistics. A number of such statistics suitable for the study of spatial aggregation are derived, and each of these statistics is decomposed at different levels of the spatial hierarchy using principles of decomposition first applied by Theil. These decomposition statistics are compared with the variance analysis applied by Moellering and Tobler and with the spatial entropy measure suggested by Curry. The use of these statistics is then illustrated by data from the Reading subregion and New York City, and the paper is concluded with an analysis of a possible role for entropy and information in problems involving equal-area zoning.     

Spatial Autoregression

The paper by Cliff and Ord (1969) caused us to think more about the W matrix and the concept of spatial autoregression. This article reviews some applications of W , reinforcing the point that autoregression in time series is similar to, but not the same as, autoregression in spatial series.     

Substitutability,Spatial Structure,and Spatial Interaction

This paper attempts to distill the relationship among spatial structure, consumer preferences, and spatial interaction. It introduces the notion of substitutability. Destinations are substitutable in both economic-behavioral and physical-locational senses. Inadequate treatment of this can lead to variations in the estimated parameters of conventional spatial interaction models.     

Spurios Spatial Regression: Some Monte Carlo Results with a Spatial Unit Root and Spatial Cointegration

In this paper I introduce the concepts of spatial unit roots and spatial cointegration, and via Monte-Carlo simulation I illustrate their implications for spatial regression. It is shown that spatial unit roots lead to spurious (spatial) regression, as in the well-known case involving time-series. Spatial cointegration is similar to its time-series counterpart, although I demonstrate that OLS estimation of spatial error-correction models is not consistent.     

Exploring Spatial Patterns Using an Expanded Spatial Autocorrelation Framework

Spatial autocorrelation (SA) is regarded as an important dimension of spatial pattern. SA measures usually consist of two components: measuring the similarity of attribute values and defining the spatial relationships among observations. The latter component is often represented by a spatial weights matrix that predefines spatial relationship between observations in most measures. Therefore, SA measures, in essence, are measures of attribute similarity, conditioned by spatial relationship. Another dimension of spatial pattern can be explored by controlling observations to be compared based upon the degree of attribute similarity. The resulting measures are spatial proximity measures of observations, meeting predefined attribute similarity criteria. Proposed measures reflect degrees of clustering or dispersion for observations meeting certain levels of attribute similarity. An existing spatial autocorrelation framework is expanded to a general framework to evaluate spatial patterns and can accommodate the proposed approach measuring proximity. Analogous to the concept of variogram, clustergram is proposed to show the levels of spatial clustering over a range of attribute similarity, or attribute lags. Specific measures based on the proposed approach are formulated and applied to a hypothetical landscape and an empirical example, showing that these new measures capture spatial pattern information not reflected by traditional spatial autocorrelation measures.     

Interregional Migration Modeling via Entropy and Information Theory

In attempting to model gross migration flows, as distinct from net flows, entropy and information theory models supplement those from the demographic tradition. However, clear differences exist between these two classes, including the type of origin and destination information which needs to be supplied, the relationship between the overall decision to migrate and the conditional choice of destination, the role of behavioral variables other than distance, and the expected predictive performance. In this paper, an enhanced interregional migration model is introduced which attempts a further integration of both the above classes. A linear version of the model is used for exploratory data analysis on a large Australian population census data set. Several alternative hypotheses are then tested with the general model. Some guidelines are also indicated toward developing a fully dynamic version of the model as well as a formal hierarchical framework.     

Lagrange Multiplier Test Diagnostics for Spatial Dependence and Spatial Heterogeneity

Several diagnostics for the assessment of model misspecification due to spatial dependence and spatial heterogeneity are developed as an application of the Lagrange Multiplier principle. The starting point is a general model which incorporates spatially lagged dependent variables, spatial residual autocorrelation and heteroskedasticity. Particular attention is given to tests for spatial residual autocorrelation in the presence of spatially lagged dependent variables and in the presence of heteroskedasticity. The tests are formally derived and illustrated in a number of simple empirical examples.     

Interpreting Spatial Economic Structure and Spatial Multipliers: Three Perspectives

The measurement and interpretation of spatial economic structure have presented many problems for analysts. The notion of a spatial multiplier has received even less formal attention; part of the motivation for this paper is the examination of the degree to which sectoral decomposition analysis might prove useful in illuminating the paths of interaction across space that are associated with changes in exogenous activity levels. In this paper, we present three perspectives that examine spatial structure in the context of an aggregated interregional social accounting matrix for Bangladesh. These perspectives examine structure at the micro-, meso-, and macrolevels and are interpreted with the assistance of several alternative methods, including structural path analysis, the field of influence, the superposition principle, multiregional feedback loop analysis, and the Matrioshka Principle; the results are interpreted analytically and cartographically. Some attention is also devoted to the formulation of a general approach to spatial multipliers. It is shown that the structure of flows and the structure of influence reveal patterns that, taken together, provide important insights into an understanding of structure and may prove to form the basis for the eventual formal presentation of the spatial multiplier.     

Spatial Weights Matrices

The impetus for Cliff and Ord's landmark article, "The Problem of Spatial Autocorrelation," was to avoid the problem of topological invariance in the spatial weights matrix. The problem and three points of view on it – theoretical, topological, and empirical – are discussed.