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.     

ACCOUNTING FOR NEIGHBORING EFFECTS IN MEASURES OF SPATIAL CONCENTRATION*

ABSTRACT A common problem with spatial economic concentration measures based on areal data (e.g., Gini, Herfindhal, entropy, and Ellison‐Glaeser indices) is accounting for the position of regions in space. While they purport to measure spatial clustering, these statistics are confined to calculations within individual areal units. They are insensitive to the proximity of regions or to neighboring effects. Clearly, since spillovers do not recognize areal units, economic clusters may cross regional boundaries. Yet with current measures, any industrial agglomeration that traverses boundaries will be chopped into two or more pieces. Activity in adjacent spatial units is treated in exactly the same way as activity in far‐flung, nonadjacent areas. This paper shows how popular measures of spatial concentration relying on areal data can be modified to account for neighboring effects. With a U.S. application, we also demonstrate that the new instruments we propose are easy to implement and can be valuable in regional analysis.     

Local Indicators of Spatial Association—LISA

The capabilities for visualization, rapid data retrieval, and manipulation in geographic information systems (GIS) have created the need for new techniques of exploratory data analysis that focus on the “spatial” aspects of the data. The identification of local patterns of spatial association is an important concern in this respect. In this paper, I outline a new general class of local indicators of spatial association (LISA) and show how they allow for the decomposition of global indicators, such as Moran's I, into the contribution of each observation. The LISA statistics serve two purposes. On one hand, they may be interpreted as indicators of local pockets of nonstationarity, or hot spots, similar to the G_i and G*_i statistics of Getis and Ord (1992). On the other hand, they may be used to assess the influence of individual locations on the magnitude of the global statistic and to identify “outliers,” as in Anselin's Moran scatterplot (1993a). An initial evaluation of the properties of a LISA statistic is carried out for the local Moran, which is applied in a study of the spatial pattern of conflict for African countries and in a number of Monte Carlo simulations.     

On Geocomputational Determinants of Entropic Variations for Urban Dynamics Studies

Monitoring population characteristics and their patterns of spatial evolution are fundamental components for urban management and policy decision‐making. Societal issues such as health, transport, or crime are often explored using a range of models describing the urban dynamics of population attributes at specific scales that can be seen as complementary. Using and simulating data at different scales of aggregation asks for the need to analyze and compare spatiotemporal variations in order to better understand the model behaviors and emerging properties of the geosimulation. This article analyzes the uses of the entropy measure in the literature and constraining factors needed for its potential extension to explore the variations in geographic and time scales. In particular, the article discusses the need for a truly spatial entropy that takes into account the spatial contiguities of the observations usually aggregated within a zoning system of areal units. Two generic solutions are exposed for the various geometries and attribute structures used for census‐related analyses; they are based on existing measures for point data using (i) co‐occurrences of observations and (ii) discriminant ratios of distances between groups of observations. Their extensions to areal compositional data are articulated around their conceptual changes and geocomputational challenges. A revisited and new version of the entropy decomposition theorem, encompassing a spatiality concept semantically related to correlation, is also presented as efficiently reusing the constrained hierarchical zoning system of administrative units to enable discovery of emerging spatial pattern features from the geosimulation. A comparison of the results between the classical use of entropy and the spatial entropy framework devised shows the flexibility and added capabilities of the approach for new types of analyses, thus allowing new insight into studies of population dynamics.     

滇黔桂省际边界区域城镇经济联系的网络结构特征与演化分析

以滇黔桂省际边界区域28个城镇为例,构建城镇中心性评价指标体系,利用改进的熵值法测度各城镇的中心性得分,评价滇黔桂省际边界区域经济空间结构演变,利用修正的引力模型测度了1995年、2004年、2014年3个年份各城镇之间经济联系强度;借助GIS技术手段分析了城镇经济联系强度的空间格局,评价城镇等级体系演化特征;最后利用社会网络分析方法考察了城镇经济空间联系和网络关系的演化规律,得出如下结论：滇黔桂省际边界区域中心城市逐渐形成,经济重心由南向北转移,由右江区单中心逐步发展到兴义市、右江区、文山市三中心,再到兴义市发展成为强中心,平果县与靖西市具备发展为滇黔桂次中心城镇的潜力;城镇间的经济联系总体较弱,区域整体网络密度低,但呈现出不断增强的趋势,低等级城镇对高等级城镇的引力与高等级城镇对低等级城市的引力不同,形成两种不同的城镇网络体系,经济联系紧密地区主要集中在中心城市之间及中心城市与经济强县之间;三角结构的外围经济联系密度明显较低,对于大多数城镇来说,尚不具备经济辐射能力,辐射效应极弱。     

Spatial Entropy

A major problem in information theory concerns the derivation of a continuous measure of entropy from the discrete measure. Many analysts have shown that Shannon's treatment of this problem is incomplete, but few have gone on to rework his analysis. In this paper, it is suggested that a new measure of discrete entropy which incorporates interval size explicitly is required; such a measure is fundamental to geography and this statistic has been called spatial entropy. The use of the measure is first illustrated by application to one-and two-dimensional aggregation problems, and then the implications of this statistic for Wilson's entropy-maximizing method are traced. Theil's aggregation statistic is reinterpreted in spatial terms, and finally, some heuristics are suggested for the design of real and idealized spatial systems in which entropy is at a maximum.     

A Local Model for Multivariate Analysis: Extending Wartenberg’s Multivariate Spatial Correlation

Local analysis can provide specific information about individual observations that is often useful in understanding nonstationary interactions among variables. This paper extends the application of Wartenberg’s Multivariate Spatial Correlation (MSC) method to a local setting. The original MSC can be considered as an adaptation of Principal Component Analysis for spatial effects with respect to spatial autocorrelation. The extended MSC method described in this paper, however, further incorporates another spatial effect, spatial heterogeneity, by the addition of geographic weights in standardizing the data and in calculating the spatial association weight matrix. The extension allows more local analysis and facilitates additional visualization of the results. The geographically weighted MSC is illustrated and justified using the classic dataset collected by André-Michel Guerry on moral statistics in 1830s France.     

城市和区域产业发展分析的增长商测度

区域产业分析的常用测度之一是区位商,该测度用于一个地理单元（如北京市）中某种行业在全部行业中的比重与特定区域（如全中国）的同种行业比重做比较。然而,区位商是一个静态概念,不能反映一种行业的相对增长幅度或者变化速度。基于数学中"商"的概念和理论地理学中测度构造的一般方法,本文定义了一个行业增长商测度,用于比较一个子区域中某种行业的增长幅度与特定地理区域中同一行业增长幅度,数值越大表明一个子区域的某种行业相对于特定区域的同种行业的增速越快、增幅越大。进一步地,定义了一个区域增长商,用于比较某种行业的相对增幅与全部行业的平均增幅,数值越大表明某种行业在全部行业中的增速越快。采用信阳实例说明了增长商分析的应用方法。作为参照,定义、对比并讨论了区位熵和增长熵测度。增长商是一个动态的测度,简明易懂,在区域产业分析中可以与区位商功能互补,在未来的主导产业选择中可能发挥一定作用。     

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.     

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.     

A History of the Concept of Spatial Autocorrelation: A Geographer's Perspective

Spatial autocorrelation is a concept that helps to define the field of spatial analysis. It is central to studies using spatial statistics and spatial econometrics. In this paper, we trace the early development of the concept and explain the academic links that brought the concept to the fore in the late 1960s. In geography, the importance of the work of Michael F. Dacey, Andrew D. Cliff, and J. Keith Ord is emphasized. Later, with the publication of a volume on spatial econometrics by Luc Anselin, spatial research and the use of the concept of spatial autocorrelation received a considerable boost. These developments are outlined together with comments about recent and possible future trends in spatial autocorrelation-based research.     

An Equivalence Theorem to Integrate Entropy-Maximizing Submodels within Overall Mathematical-Programming Frameworks

This paper presents a lemma and a theorem which demonstrate equivalent ways of formulating certain mathematical-programming problems. The first shows, in general terms, how constraints can be “absorbed” into the objective function and vice versa. The second shows how such ideas can be applied to spatial analysis problems involving the minimization of development costs subject to interactions being determined by (suboptimal) entropy-maximizing models. This provides both a new perspective on the relationship between programming and entropy models, and formulations that are computationally convenient.     

西方旅游者在华旅游活动研究——基于西方旅游批发商的中国游产品分析

有关西方旅游者在华旅游活动的现有研究缺乏系统性,且几乎都借助于中国官方机构的统计信息。由于统计方法存在明显的缺陷,这些统计信息往往无法有效地反映旅游者的实际活动轨迹和活动内容。本文以西方旅游批发商的产品样册为基础,运用自创的分析方法研究西方旅游者的在华旅游活动。研究表明,西方旅游者主要是受中华文明吸引而来华旅游,西方各国旅游批发商推出的中国游产品没有明显的差异性。     

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.     

基于区域联系和IPA的节事游客地方认同空间特征——以2011西安世界园艺博览会为例

本研究选取地方理论的地方认同维度,将其与节事研究领域相结合,以西安世界园艺博览会为案例,运用区域综合联系强度、IPA以及Taylor的距离衰减模型,探究参与该节事的游客对举办地西安的地方认同的空间特征。研究发现:世园会游客的地方认同在总体上表现为等级递减和跳跃式增长两种空间特征,且仅在国内各省(市、区)这个较大尺度上体现出明显的距离衰减效应。文章最后通过讨论提出了值得进一步研究的节事游客地方认同空间特征与空间极化、介入机会及空间扩散的关系问题。     

INDUSTRIAL DISPERSION IN THE SOVIET UNION: AN APPLICATION OF ENTROPY MEASURES

Industrial dispersion across the republics of the USSR between 1926 and 1974 is described through the use of entropy measures. Ostensibly, a set of location principles would guide such dispersion, and several of them stress regional equalization as an important consideration in locating industry. Therefore, trends in regional inequality (in industrialization) are investigated through the use of inequality statistics derived from entropy measures at three different spatial scales. Between 1926 and 1960, inequality among the republics decreased, but has increased between 1960 and 1974. However, inequality has decreased between the economic regions of the USSR between 1940 and 1975, and at the oblast level, there has been a tendency toward greater equality (homogeneity) within each economic region or republic since 1960. In attempting to account for these diverse trends, it is suggested that they are largely the result of the priority of efficiency considerations in location decisions rather than those of regional equity per se.     

The Detection of Clusters Using a Spatial Version of the Chi-Square Goodness-of-Fit Statistic

A test statistic for the detection of spatial clusters is developed by generalizing the common chi-square goodness-of-fit test. The paper includes a discussion of the relationship between the statistic and other associated statistics, and provides an analysis of both its null distribution and power. The paper concludes with the development of a local version of the statistic and an application to leukemia clustering in central New York.     

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.