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.     

The Role of Spatial Scale in Geomorphic Systems

Geomorphic systems are characterized by numerous, complex interrelationships between system components, and by processes and controls which may operate over different spatial scales. Factors operating at any given spatial scale can be viewed as an abstracted subset of all relationships operating at all scales. The theory that relationships which operate over spatial scales an order of magnitude different are effectively independent of each other is formally stated in terms of abstracted systems. An example is given to illustrate the use of spatial statistics to determine what constitutes a significant spatial scale difference in controls over hydraulic geometry of a desert wash.     

A Simulation Study to Explore Inference about Global Moran's I with Random Spatial Indexes

Inference procedures for spatial autocorrelation statistics assume that the underlying configurations of spatial units are fixed. However, sometimes this assumption can be disadvantageous, for example, when analyzing social media posts or moving objects. This article examines for the case of point geometries how a change from fixed to random spatial indexes affects inferences about global Moran's I, a popular spatial autocorrelation measure. Homogeneous and inhomogeneous Matérn and Thomas cluster processes are studied and for each of these processes, 10,000 random point patterns are simulated for investigating three aspects that are key in an inferential context: the null distributions of I when the underlying geometries are varied; the effect of the latter on critical values used to reject null hypotheses; and how the presence of point processes affects the statistical power of Moran's I. The results show that point processes affect all three characteristics. Inferences about spatial structure in relevant application contexts may therefore be different from conventional inferences when this additional source of randomness is taken into account.     

THE FUTURE OF SPATIAL ECONOMETRICS*

ABSTRACT The purpose of this paper is threefold. First, we give an overview of the general direction the spatial econometrics literature has taken without attempting to provide a representative survey of all interesting work that has appeared. Second, we identify a number of problems in spatial econometrics that are as yet unresolved. Finally, we provide advocacy for the notion that new spatial econometric theory should be inspired by actual empirical applications as opposed to being directed by what appears to be the most obvious extension of what is currently available.     

现代服务业集聚形成机理空间计量分析

在纳入空间效应前提下,构建现代服务业集聚形成机理空间面板计量模型,对我国28个省域相关数据实证研究表明:我国现代服务业集聚在省域之间有较强的空间依赖性和正的空间溢出效应。技术差异在时间维度上对现代服务业集聚促进作用显著,在空间维度上并不显著;交易费用与现代服务业集聚有显著的负相关性;知识溢出、规模经济、政府行为对现代服务业集聚促进作用显著。     

What Were We Thinking?

This article outlines the context in geography and statistics in the mid 1960s, at the height of geography's so-called "quantitative revolution," that led us into a long-term collaboration about spatial statistics, which has continued in surges and lulls for some 40 years. We focus upon problems in spatial autocorrelation, including the measurement of autocorrelation, distribution theory, and variable geographical lattices. This narrative may not describe how it was, but it does describe how we remember the events of the time.     

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.     

How many bones make five? The art and science of guesstimation in archaeozoology

Six faunal statistics are described and assessed by means of simulation. Of these, the ‘probable number of individuals’ (PNI) methods, which include the Lincoln and Krantz indices, are shown to perform remarkably well. The assumptions of these PNI procedures are given and circumstances under which PNI methods can be employed are described. Palaeoeconomic methods occupy a middle ground between classical statistical theory and experimental archaeology, hence the use of the term ‘guesstimation’: this ground is explored and it is shown that these methods can have a considerable bearing on the interpretation of taphonomic and ‘spatial’ (intra-stratum) regularities. Promising avenues for future research are indicated.     

Maximum Getis–Ord Statistic Adjusted for Spatially Autocorrelated Data

Local statistics test the null hypothesis of no spatial association or clustering around the vicinity of a location. To carry out statistical tests, it is assumed that the observations are independent and that they exhibit no global spatial autocorrelation. In this article, approaches to account for global spatial autocorrelation are described and illustrated for the case of the Getis–Ord statistic with binary weights. Although the majority of current applications of local statistics assume that the spatial scale of the local spatial association (as specified via weights) is known, it is more often the case that it is unknown. The approaches described here cover the cases of testing local statistics for the cases of both known and unknown weights, and they are based upon methods that have been used with aspatial data, where the objective is to find changepoints in temporal data. After a review of the Getis–Ord statistic, the article provides a review of its extension to the case where the objective is to choose the best set of binary weights to estimate the spatial scale of the local association and assess statistical significance. Modified approaches that account for spatially autocorrelated data are then introduced and discussed. Finally, the method is illustrated using data on leukemia in central New York, and some concluding comments are made.     

Geography, Spatial Data Analysis, and Geostatistics: An Overview

Geostatistics is a distinctive methodology within the field of spatial statistics. In the past, it has been linked to particular problems (e.g., spatial interpolation by kriging) and types of spatial data (attributes defined on continuous space). It has been used more by physical than human geographers because of the nature of their types of data. The approach taken by geostatisticians has several features that distinguish it from the methods typically used by human geographers for analyzing spatial variation associated with regional data, and we discuss these. Geostatisticians attach much importance to estimating and modeling the variogram to explore and analyze spatial variation because of the insight it provides. This article identifies the benefits of geostatistics, reviews its uses, and examines some of the recent developments that make it valuable for the analysis of data on areal supports across a wide range of problems.     

新国家空间理论视角下城市群的国家空间选择性研究

基于新国家空间理论,将城市群置于社会经济转型宏观背景下,分析其空间选择的国家空间项目与国家空间策略,并初步探讨了其空间效应。城市群作为目前国家空间选择性的具体形式,其产生背景、依据以及生产策略体现了国家空间选择性内涵与演变路径,是国家应对城市企业主义危机采取的一种新型城市与区域治理形式。在国家空间项目及其策略的支持下,城市群享有优先于其它地区发展的特权并实现经济快速发展,从而培育地方增长极并推动区域协调发展。然而,城市群的空间选择并未改变城市企业主义的逻辑而是将其转移到了区域尺度,并且存在城市间竞争激烈等一系列问题,亟待与其它空间项目与策略的配套实施以实现区域崛起并提高国家竞争力。     

Large Sample-Size Distribution of Statistics Used In Testing for Spatial Correlation

Test statistics for testing for spatial correlation in continuous variables have been given by both Moran and Geary and have subsequently been generalized. It has been conjectured for a long time that under the hypothesis of no spatial correlations all these statistics are normally distributed when the sample size is large. This paper proves a very general theorem on the large sample normality of quadratic forms. As corollaries to the theorem the asymptotic normality, under the hypothesis, of all the above-mentioned statistics is established. The necessary conditions are quite unrestrictive. It is also shown, by means of a counter example, that the conditions given in a similar theorem (Cliff and Ord) are inadequate to ensure normality.     

DIAGNOSTICS FOR REGRESSION MODELING IN SPATIAL ECONOMETRICS*

ABSTRACT. This paper describes statistics for model criticism in spatial econometrics. The purpose of these statistics is to evaluate how well a chosen model fits the data and to identify influential cases and how they affect the aggregate picture. The paper reviews results in Martin (1992) for the regression model with correlated errors where the coefficients of the variance matrix are assumed either known or fixed. The problems of applying the statistics in spatial econometric modeling are discussed. An application is reported which considers diagnostics for the mean function and highlights cases that might influence estimates of the parameter of the error model. Different ways of assessing the influence of cases are also described.     

LIBERATION OR RESTRICTION? THE JOB AS AN INFLUENCE ON URBAN ENVIRONMENTAL PERCEPTION AND BEHAVIOR

It is because social relations are so frequently and so inevitably correlated with spatial relations; because physical distances so frequently are, or seem to be, the indexes of social distance, that statistics have any significance whatever for sociology. And this is true, finally, because it is only as social and psychical facts can be reduced to, and correlated with, spatial facts that they can be measured at all.¹     

A Statistical Method for the Detection of Geographic Clustering

Kernel‐based, smoothed estimates of spatial variables are useful in exploratory analyses because they yield a clear visual image of geographic variability in the underlying variable. In this paper I suggest an approach for assessing the significance of peaks in the surface that result from the application of the smoothing kernel. The approach may also be thought of as a method for assessing the maximum among a set of suitably defined local statistics. Local statistics for data on a regular grid of cells are first defined by using a Gaussian kernel. Results from integral geometry are then used to find the probability that the maximum local statistic (M) exceeds a given critical value (M). Approximations are provided that make implementation of the approach straightforward. Future work will address several other issues associated with local statistics that have been defined in this way, including edge effects, and the effects of global spatial autocorrelation on the choice of critical value.     

LOCALIZATION AND INDUSTRY CLUSTERING ECONOMETRICS: AN ASSESSMENT OF GIBBS MODELS FOR SPATIAL POINT PROCESSES

The objective of this paper is to assess an approach to statistical modeling of point referenced establishment data that permit inclusion of “environmental” or establishment‐specific covariates and specific forms of interestablishment interaction. Gibbs models are used to decompose the conditional intensity of the spatial point process into trend and interaction components. The trend is composed of access measures (primarily different classes of roads) and three different interaction processes are tested: Geyer, area interaction, and Strauss hard core. While the models used have proved to be useful in ecology, we are unaware of any applications to establishment or firm data. In empirical application, the models yield intuitively appealing results for the trend component, and the ability to specify the interaction component gives deeper insights into interestablishment spatial dynamics than any previously published methods.     

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.     

The Analysis of Spatial Association by Use of Distance Statistics

Introduced in this paper is a family of statistics, G, that can be used as a measure of spatial association in a number of circumstances. The basic statistic is derived, its properties are identified, and its advantages explained. Several of the G statistics make it possible to evaluate the spatial association of a variable within a specified distance of a single point. A comparison is made between a general G statistic and Moran's I for similar hypothetical and empirical conditions. The empirical work includes studies of sudden infant death syndrome by county in North Carolina and dwelling unit prices in metropolitan San Diego by zip-code districts. Results indicate that G statistics should be used in conjunction with I in order to identify characteristics of patterns not revealed by the I statistic alone and, specifically, the G_i and G_i* statistics enable us to detect local “pockets” of dependence that may not show up when using global statistics.