Information Entropy of Non‐Probabilistic Processes

We derive an expression for the entropy of non‐probabilistic distributions encountered in spatial and mathematical mappings. The entropy of non‐probabilistic distributions can be formulated using probabilistic notions of the hypothetical random redistribution of finite information. We show that the discrete approximation to the information content of spatial maps can be based on the discrete hypergeometric distribution. The resultant “associative” entropy is distinct from the Shannon entropy for probability distributions and addresses several shortcomings of the current entropy paradigm as applied to spatial analysis. The associative entropy statistic is distributed approximately as a chi‐squared random variable under limitations of variation. We formulate a univariate logical equivalent of the associative entropy statistic, freeing the paradigm from the degrees of freedom constraint to which it has been traditionally shackled. This entropy has application in spatial analysis and fuzzy set theory. The associative entropy is based on the concept of proportional information and is related to the Getis G‐statistics of spatial association and the Chi‐squared statistics of sample means. We explore the utility of the theory when applied to spatial distribution of vegetation in New Brunswick, Canada. The limitations and implications of the entropy expression are discussed and suggestions are made for future applications of the theory. This work is part of the development of an information theory framework for the analysis of landscape patterns of animal habitat.     

Estimation of adult skeletal age‐at‐death: statistical assumptions and applications

We examine the statistical assumptions underlying different techniques of estimating the age‐at‐death of a skeleton from one or more age indicators. The preferred method depends on which property of the distribution of the data in the reference sample is preserved in the skeleton to be aged. In cases where the conditional distribution of age given indicator is preserved, we provide ‘look‐up’ tables giving essentially unbiased age estimates and prediction intervals, using a large reference sample and the auricular surface and pubic symphysis age indicators. Where this assumption is violated, but the conditional distribution of indicator given age is preserved, we find that an alternative model which attempts to capture the biological process of development of an individual has some attractive features, which may make it suitable for further study. Copyright © 2006 John Wiley & Sons, Ltd.     

Determinants of Variation in Population–Employment Interaction Findings: A Quasi‐Experimental Meta‐Analysis. 人口‐就业相互作用的决定因素研究：一种准实验元分析法

This article reports about a metaregression analysis of empirical results generated using data for the northern Netherlands (1988–2002) in order to investigate the ambiguity in results in the population–employment interaction literature. Specifically, the analysis deals with the issue whether “jobs follow people” or “people follow jobs.” The article starts with introducing the basics of quasi‐experimental meta‐analysis and with identifying some advantages of using quasi‐experimental meta‐analysis as compared with the standard meta‐analysis approach. Two subsequent sections document the selection of the population–employment interaction model and salient characteristics of the data set as well as the setup of the primary analyses. A total of 4,050 quasi‐experimental empirical results for the jobs–people direction of causality are generated using different specifications and estimators for a spatial econometric interaction model. The subsequent metaregression analysis reveals that the empirical results are largely shaped by the spatial, temporal, and employment characteristics of the data sampling. The results also appear much more sensitive to different measurements of the model's key variables when compared with alternative specifications of the spatial weights matrix. The main determinant driving empirical results about jobs–people causality are differences in model specification and estimation, as revealed by an inherent bias in parameter estimates and misguided inferences for some of the commonly used specifications. Finally, suggestions for future research are identified.     

ART‐P‐MAP Neural Networks Modeling of Land‐Use Change: Accounting for Spatial Heterogeneity and Uncertainty

Spatial land‐use models over large geographic areas and at fine spatial resolutions face the challenges of spatial heterogeneity, model predictability, data quality, and of the ensuing uncertainty. We propose an improved neural network model, ART‐Probability‐Map (ART‐P‐MAP), tailored to address these issues in the context of spatial modeling of land‐use change. First, it adaptively forms its own network structure to account for spatial heterogeneity. Second, it explicitly infers posterior probabilities of land conversion that facilitates the quantification of prediction uncertainty. Extensive calibration under various test settings is conducted on the proposed model to optimize its utility in seeking useful information within a spatially heterogeneous environment. The calibration strategy involves building a bagging ensemble for training and stratified sampling with varying category proportions for experimentation. Through a temporal validation approach, we examine models’ performance within a systematic assessment framework consisting of global metrics and cell‐level uncertainty measurement. Compared with two baselines, ART‐P‐MAP achieves consistently good and stable performance across experiments and exhibits superior capability to handle the spatial heterogeneity and uncertainty involved in the land‐use change problem. Finally, we conclude that, as a general probabilistic regression model, ART‐P‐MAP is applicable to a broad range of land‐use change modeling approaches, which deserves future research.     

Nonparametric Significance Test for Discovery of Network‐Constrained Spatial Co‐Location Patterns

Spatial co‐location patterns are useful for understanding positive spatial interactions among different geographical phenomena. Existing methods for detecting spatial co‐location patterns are mostly developed based on planar space assumption; however, geographical phenomena related to human activities are strongly constrained by road networks. Although these methods can be simply modified to consider the constraints of networks by using the network distance or network partitioning scheme, user‐specified parameters or priori assumptions for determining prevalent co‐location patterns are still subjective. As a result, some co‐location patterns may be wrongly reported or omitted. Therefore, a nonparametric significance test without priori assumptions about the distributions of the spatial features is proposed in this article. Both point‐dependent and location‐dependent network‐constrained summary statistics are first utilized to model the distribution characteristics of the spatial features. Then, by using these summary statistics, a network‐constrained pattern reconstruction method is developed to construct the null model of the test, and the prevalence degree of co‐location patterns is modeled as the significance level. The significance test is evaluated using the facility points‐of‐interest data sets. Experiments and comparisons show that the significance test can effectively detect network‐constrained spatial co‐location patterns with less priori knowledge and outperforms two state‐of‐the‐art methods in excluding spurious patterns.     

Spatial equilibrium in deviations: An application to skill‐premium and skill‐mix heterogeneity

In this paper, we propose a novel—and general‐purpose—modeling approach. We give a linear representation of the spatial general equilibrium, expressed in terms of local percentage deviations from the benchmark case of symmetry, where all the areas in the economy are taken to be initially identical. To illustrate the flexibility of our approach, we revisit the literature on the spatial heterogeneity of local skill premia and local skill mix. We show that our approach is able to encompass a variety of alternative explanations in a simple “unifying framework.” Finally, we exploit a graphical version of the model to show how to implement empirical tests.     

Multi‐Type,Multi‐Zone Facility Location

The placement of facilities according to spatial and/or geographic requirements is a popular problem within the domain of location science. Objectives that are typically considered in this class of problems include dispersion, median, center, and covering objectives—and are generally defined in terms of distance or service‐related criteria. With few exceptions, the existing models in the literature for these problems only accommodate one type of facility. Furthermore, the literature on these problems does not allow for the possibility of multiple placement zones within which facilities may be placed. Due to the unique placement requirements of different facility types—such as suitable terrain that may be considered for placement and specific placement objectives for each facility type—it is expected that different suitable placement zones for each facility type, or groups of facility types, may differ. In this article, we introduce a novel mathematical treatment for multi‐type, multi‐zone facility location problems. We derive multi‐type, multi‐zone extensions to the classical integer‐linear programming formulations involving dispersion, centering and maximal covering. The complexity of these formulations leads us to follow a heuristic solution approach, for which a novel multi‐type, multi‐zone variation of the non‐dominated sorting genetic algorithm‐II algorithm is proposed and employed to solve practical examples of multi‐type, multi‐zone facility location problems.     

Gibrat Revisited: An Urban Growth Model Incorporating Spatial Interaction and Innovation Cycles. 再评吉尔布瑞特：顾及空间相互作用和创新周期的城市增长模型

We designed a geographical model for simulating the distribution of urban growth in systems of cities. The model incorporates the hierarchical and spatial diffusion of innovation cycles through gravitational interactions within a set of cities. Using theoretical simulations, we demonstrate that this model is able to reproduce the observed properties of urban systems for the log‐normal distribution of city sizes as well as the observed distribution of growth rates. Our experimentation was performed on a large harmonized historical database that includes a few hundred French urban agglomerations between 1831 and 1999 (Pumain‐INED database). Both spatial interaction and innovation cycles are necessary ingredients to explain the evolution of urban hierarchies. We suggest that Gibrat's generic stochastic growth model based on independent entities should be replaced by a more relevant model of spatially and temporally interdependent geographical entities.     

SKILL POLARIZATION IN LOCAL LABOR MARKETS UNDER SHARE‐ALTERING TECHNICAL CHANGE

This paper considers the “share‐altering” technical change hypothesis in a spatial general equilibrium model where individuals have different levels of skills. Building on a simple Cobb‐Douglas production function, our model shows that the implementation of skill‐biased technologies requires a sufficient proportion of highly educated individuals. Moreover, when technical progress disproportionately replaces middle‐skill jobs, the local distribution of skills will exhibit “fat‐tails,” where the proportion of both highly skilled and low‐skilled workers increases. These and several other predictions of the model are consistent with recent existing evidence, and avoid some major criticism against the “canonical” CES framework.     

SPATIAL ECONOMETRIC MODELING OF ORIGIN‐DESTINATION FLOWS*

ABSTRACT Standard spatial autoregressive models rely on spatial weight structures constructed to model dependence among n regions. Ways of parsimoniously modeling the connectivity among the sample of N=n² origin‐destination (OD) pairs that arise in a closed system of interregional flows has remained a stumbling block. We overcome this problem by proposing spatial weight structures that model dependence among the N OD pairs in a fashion consistent with standard spatial autoregressive models. This results in a family of spatial OD models introduced here that represent an extension of the spatial regression models described in Anselin (1988) .     

Accuracy of Count Data Estimated by the Point‐in‐Polygon Method

This paper analyzes the accuracy of count data estimated by the point‐in‐polygon method. A point‐in‐polygon interpolation model is proposed, based on a stochastic distribution of points and the target zone, in order to represent a variety of situations. The accuracy of estimates is numerically investigated in relation to the size of the target zone and the distribution of points, and the optimal location of representative points is discussed. The major findings of this paper are as follows: (1) though the relative accuracy of estimates generally increases monotonously with the size of the target zone, the monotoneity is often disturbed by the periodicity in the spatial configuration of source zones and the point distribution; (2) the point‐in‐polygon and the areal weighting interpolation methods have the same accuracy of estimates when points are concentrated in less than 12–15 percent area around the representative point in source zones; (3) the point‐in‐polygon method is not so robust against the locational gap between points and the representative point; (4) the optimal location of representative points is given by the spatial median of points.     

A Bayesian Approach to Modeling Binary Data: The Case of High‐Intensity Crime Areas

This paper reports the fitting of a number of Bayesian logistic models with spatially structured or/and unstructured random effects to binary data with the purpose of explaining the distribution of high‐intensity crime areas (HIAs) in the city of Sheffield, England. Bayesian approaches to spatial modeling are attracting considerable interest at the present time. This is because of the availability of rigorously tested software for fitting a certain class of spatial models. This paper considers issues associated with the specification, estimation, and validation, including sensitivity analysis, of spatial models using the WinBUGS software. It pays particular attention to the visualization of results. We discuss a map decomposition strategy and an approach that examines properties of the full posterior distribution. The Bayesian spatial model reported provides some interesting insights into the different factors underlying the existence of the three police‐defined HIAs in Sheffield.     

A Geostatistical Linear Regression Model for Small Area Data. 一种适用于小区域数据的地统计线性回归模型

We present a new linear regression model for use with aggregated, small area data that are spatially autocorrelated. Because these data are aggregates of individual‐level data, we choose to model the spatial autocorrelation using a geostatistical model specified at the scale of the individual. The autocovariance of observed small area data is determined via the natural aggregation over the population. Unlike lattice‐based autoregressive approaches, the geostatistical approach is invariant to the scale of data aggregation. We establish that this geostatistical approach also is a valid autoregressive model; thus, we call this approach the geostatistical autoregressive (GAR) model. An asymptotically consistent and efficient maximum likelihood estimator is derived for the GAR model. Finite sample evidence from simulation experiments demonstrates the relative efficiency properties of the GAR model. Furthermore, while aggregation results in less efficient estimates than disaggregated data, the GAR model provides the most efficient estimates from the data that are available. These results suggest that the GAR model should be considered as part of a spatial analyst's toolbox when aggregated, small area data are analyzed. More important, we believe that the GAR model's attention to the individual‐level scale allows for a more flexible and theory‐informed specification than the existing autoregressive approaches based on an area‐level spatial weights matrix. Because many spatial process models, both in geography and in other disciplines, are specified at the individual level, we hope that the GAR covariance specification will provide a vehicle for a better informed and more interdisciplinary use of spatial regression models with area‐aggregated data.     

LOCAL FOOD,URBANIZATION, AND TRANSPORT‐RELATED GREENHOUSE GAS EMISSIONS

We argue that “buying local” does not necessarily reduce transport‐related greenhouse gas emissions, even if production technologies and yields are homogeneous in space. We develop a partial‐equilibrium model of rural‐urban systems where the spatial distribution of food production within and between regions is endogenous. We exhibit cases where locating some food production in the least‐urbanized regions results in lower emissions and higher welfare than if all regions are self‐sufficient. The optimal spatial allocation of food production does not exclude the possibility that some regions should be self‐sufficient, provided that their urban population sizes are neither too large nor too small.     

Area‐to‐Point Prediction Under Boundary Conditions

This article proposes a geostatistical solution for area‐to‐point spatial prediction (downscaling) taking into account boundary effects. Such effects are often poorly considered in downscaling, even though they often have significant impact on the results. The geostatistical approach proposed in this article considers two types of boundary conditions (BC), that is, a Dirichlet‐type condition and a Neumann‐type condition, while satisfying several critical issues in downscaling: the coherence of predictions, the explicit consideration of support differences, and the assessment of uncertainty regarding the point predictions. An updating algorithm is used to reduce the computational cost of area‐to‐point prediction under a given BC. In a case study, area‐to‐point prediction under a Dirichlet‐type BC and a Neumann‐type BC is illustrated using simulated data, and the resulting predictions and error variances are compared with those obtained without considering such conditions.     

POPULATION DISPERSION VS. CONCENTRATION IN A TWO‐REGION MIGRATION MODEL WITH ENDOGENOUS NATURAL AMENITIES*

ABSTRACT Contrary to the predictions of the basic spatial equilibrium model, the long‐run distribution of population across rural U.S. counties with high‐valued natural amenities has become relatively more concentrated versus dispersed. We provide an explanation by developing a two‐region model with mobile labor, production externalities and endogenous natural amenities. We find that strong preferences for natural amenities generally foster population dispersion. However, such preferences can also lead to population concentration when ecological degradation is low and man‐made capital is a relatively scarce input into natural amenity production. Investments that enhance natural amenities are found to reduce the divergence between the steady state and optimal outcomes.     

A NEW SPATIAL MULTIPLE DISCRETE‐CONTINUOUS MODELING APPROACH TO LAND USE CHANGE ANALYSIS

This paper formulates a multiple discrete‐continuous probit (MDCP) land use model within a spatially explicit economic structural framework for land use change decisions. The spatial MDCP model is capable of predicting both the type and intensity of urban development patterns over large geographic areas, while also explicitly acknowledging geographic proximity‐based spatial dependencies in these patterns. At a methodological level, the paper focuses on specifying and estimating a spatial MDCP model that allows the dependent variable to exist in multiple discrete states with an intensity associated with each discrete state. The formulation also accommodates spatial dependencies, as well as spatial heterogeneity and heteroskedasticity, in the dependent variable, and should be applicable in a wide variety of fields where social and spatial dependencies between decision agents (or observation units) lead to spillover effects in multiple discrete‐continuous choices (or states). A simulation exercise is undertaken to evaluate the ability of the proposed maximum approximate composite marginal likelihood (MACML) approach to recover parameters from a cross‐sectional spatial MDCP model. The results show that the MACML approach does well in recovering parameters. An empirical demonstration of the approach is undertaken using the city of Austin parcel level land use data.     

PANEL DATA MODELS WITH SPATIALLY DEPENDENT NESTED RANDOM EFFECTS

This paper focuses on panel data models combining spatial dependence with a nested (hierarchical) structure. We use a generalized moments estimator to estimate the spatial autoregressive parameter and the variance components of the disturbance process. A spatial counterpart of the Cochrane‐Orcutt transformation leads to a feasible generalized least squares procedure to estimate the regression parameters. Monte Carlo simulations show that our estimators perform well in terms of root mean square error compared to the maximum likelihood estimator. The approach is applied to English house price data for districts nested within counties.     

Spatiotemporal Modeling of Correlated Small‐Area Outcomes: Analyzing the Shared and Type‐Specific Patterns of Crime and Disorder

This research applies a Bayesian multivariate modeling approach to analyze the spatiotemporal patterns of physical disorder, social disorder, property crime, and violent crime at the small‐area scale. Despite crime and disorder exhibiting similar spatiotemporal patterns, as hypothesized by broken windows and collective efficacy theories, past studies often analyze a single outcome and overlook the correlation structures between multiple crime and disorder types. Accounting for five covariates, the best‐fitting model partitions the residual risk of each crime and disorder type into one spatial shared component, one temporal shared component, and type‐specific spatial, temporal, and space–time components. The shared components capture the underlying spatial pattern and time trend common to all types of crime and disorder. Results show that population size, residential mobility, and the central business district are positively associated with all outcomes. The spatial shared component is found to explain the largest proportion of residual variability for all types of crime and disorder. Spatiotemporal hotspots of crime and disorder are examined to contextualize broken windows theory. Applications of multivariate spatiotemporal modeling with shared components to ecological crime theories and crime prevention policy are discussed.     

Using Linear Integer Programming for Multi‐Site Land‐Use Allocation

Research in the area of spatial decision support (SDS) and resource allocation has recently generated increased attention for integrating optimization techniques with GIS. In this paper we address the use of spatial optimization techniques for solving multi‐site land‐use allocation (MLUA) problems, where MLUA refers to the optimal allocation of multiple sites of different land uses to an area. We solve an MLUA problem using four different integer programs (IP), of which three are linear integer programs. The IPs are formulated for a raster‐based GIS environment and are designed to minimize development costs and to maximize compactness of the allocated land use. The preference for either minimizing costs or maximizing compactness has been made operational by including a weighting factor. The IPs are evaluated on their speed and their efficacy for handling large databases. All four IPs yielded the optimal solution within a reasonable amount of time, for an area of 8 × 8 cells. The fastest model was successfully applied to a case study involving an area of 30 × 30 cells. The case study demonstrates the practical use of linear IPs for spatial decision support issues.