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.     

OLS ESTIMATION AND THE t TEST REVISITED IN RANK‐SIZE RULE REGRESSION*

ABSTRACT The rank‐size rule and Zipf's law for city sizes have been traditionally examined by means of OLS estimation and the t test. This paper studies the accurate and approximate properties of the OLS estimator and obtains the distribution of the t statistic under the assumption of Zipf's law (i.e., Pareto distribution). Indeed, we show that the t statistic explodes asymptotically even under the null, indicating that a mechanical application of the t test yields a serious type I error. To overcome this problem, critical regions of the t test are constructed to test the Zipf's law. Using these corrected critical regions, we can conclude that our results are in favor of the Zipf's law for many more countries than in the previous researches such as Rosen and Resnick (1980) or Soo (2005) . By using the same database as that used in Soo (2005) , we demonstrate that the Zipf law is rejected for only one of 24 countries under our test whereas it is rejected for 23 of 24 countries under the usual t test. We also propose a more efficient estimation procedure and provide empirical applications of the theory for some countries.     

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.     

A Note on the Extremities of Local Moran's Iis and Their Impact on Global Moran's I

By defining local Moran's I_i as a ratio of quadratic forms and making use of its overall additivity to match global Moran's I, we can identify spatial objects with a strong impact on global Moran's I. First, we concentrate on the spatial properties of local Moran's I_i expressed by the local linkage degree. Depending on whether we use the W- or C-coding of the spatial connectivity matrix, the variance of local Moran's I_i for a small local linkage degree will be either large or small. Note that spatial objects associated with a local Moran's I_i with a large variance affect the global statistic much more than spatial objects associated with a local Moran's I_i with a small variance. Counterintuitively, global Moran's I defined in the W-coding is most influenced by spatial objects with a small number of spatial neighbors. In contrast, spatial objects with a large number of spatial neighbors exert more impact on global Moran's I setup in the C-coding. Second, we investigate the impact of the empirical data on local Moran's I_i and show that local Moran's I_i will only be significant for extreme absolute residuals at and around the reference location. Clusters of average regression residuals cannot be detected by local Moran's I_i. Consequently, spatial cliques of extreme residuals contribute more to significance tests on global autocorrelation.     

Power of the Rank Adjacency Statistic to Detect Spatial Clustering in a Small Number of Regions

The rank adjacency statistic D is a statistical method for assessing spatial autocorrelation or clustering of geographical data. It was originally proposed for summarizing the geographical patterns of cancer data in Scotland (IARC 1985). In this paper, we investigate the power of the rank adjacency statistic to detect spatial clustering when a small number of regions is involved. The investigations were carried out using Monte Carlo simulations, which involved generating patterned/clustered values and computing the power with which the D statistic would detect it. To investigate the effects of region shapes, structure of the regions, and definition of weights, simulations were carried out using two different region shapes, binary and nonhinary weights, and three different lattice structures. The results indicate that in the typical example of considering Canadian total mortality at the electoral district level, the D statistic had adequate power to detect general spatial autocorrelation in twenty‐five or more regions. There was an inverse relationship between power and the level of connectedness of the regions, which depends on the weighting function, shape, and arrangement of the regions. The power of the D statistic was also found to compare favorably with that of Moran's I statistic.     

Analysis of a Distribution of Point Events Using the Network‐Based Quadrat Method

This study proposes a new quadrat method that can be applied to the study of point distributions in a network space. While the conventional planar quadrat method remains one of the most fundamental spatial analytical methods on a two‐dimensional plane, its quadrats are usually identified by regular, square grids. However, assuming that they are observed along a network, points in a single quadrat are not necessarily close to each other in terms of their network distance. Using planar quadrats in such cases may distort the representation of the distribution pattern of points on a network. The network‐based units used in this article, on the other hand, consist of subsets of the actual network, providing more accurate aggregation of the data points along the network. The performance of the network‐based quadrat method is compared with that of the conventional quadrat method through a case study on a point distribution on a network. The χ² statistic and Moran's I statistic of the two quadrat types indicate that (1) the conventional planar quadrat method tends to overestimate the overall degree of dispersion and (2) the network‐based quadrat method derives a more accurate estimate on the local similarity. The article also performs sensitivity analysis on network and planar quadrats across different scales and with different spatial arrangements, in which the abovementioned statistical tendencies are also confirmed.     

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.     

Redundancy as an Index of Change in Point Pattern Analysis

Point pattern analysis based on concepts from information theory can go beyond existing techniques. Direct measurement of spatial form is achieved when Thiessen polygons are constructed around the points; in this scheme, each point's proportion of total area may be treated like a probability. Three information-theoretic indices are available for analysis of a distribution of such probabilities. Entropy is density dependent. Redundancy, defined as the difference between observed and maximum entropy, seems to avoid this problem when the number of individuals in a pattern exceeds twenty. Comparisons of prior and posterior redundancy provide an indication of change in overall pattern form. An information gain expectation reflects changes for each individual in a pattern. Here, point-area redundancy parameters are determined for Poisson-generated patterns, using a gamma distribution of polygon areas and a computer-generated set. An application to an urban crime problem illustrates the use of these parameters in the analysis of pattern change.     

Geographic information system‐based edge effect correction for Ripley's K‐function under irregular boundaries

Ripley's K‐function is a test to detect geographically distributed patterns occurring across spatial scales. Initially, it assumed infinitely continuous planar space, but in reality, any geographic distribution occurs in a bounded region. Hence, the edge problem must be solved in the application of Ripley's K‐function. Traditionally, three basic edge correction methods were designed for regular study plots because of simplified geometric computation: the Ripley circumference, buffer zone, and toroidal methods. For an irregular‐shaped study region, a geographic information system (GIS) is needed to support geometric calculation of complex shapes. The Ripley circumference method was originally implemented by Haase and has been modified into a Python program in a GIS environment via Monte Carlo simulation (hereafter, the Ripley–Haase and Ripley–GIS methods). The results show that in terms of the statistical powers of clustering detection for irregular boundaries, the Ripley–GIS method is the most stable, followed by the buffer zone, toroidal, and Ripley–Haase methods. After edge effects of irregular boundaries have been eliminated, Ripley's K‐function is used to estimate the degree of spatial clustering of cities in a given territory, and in this paper, we demonstrate that by reference to the relationship between urban spatial structure and economic growth in China.     

The Assessment of Spatial Autocorrelation through Constrained Multiple Regression

Most published measures of spatial autocorrelation (SA) can be recast as a (normalized) cross-product statistic that indexes the degree of relationship between corresponding entries from two matrices—one specifying the spatial connections among a set of n locations, and the other reflecting a very explicit definition of similarity between the set of values on some variable x realized over the n locations. We first give a very brief sketch of the basic cross-product approach to the evaluation of SA, and then generalize this strategy to include less restrictive specifications for the notion of similarity between the values on x. Using constrained multiple regression, the characterization of variate similarity basic to any assessment of SA can itself be framed according to the information present in the measure of spatial separation. These extensions obviate the inherent arbitrariness in how SA is usually evaluated, which now results from the requirement of a very restrictive definition of variate similarity before a cross-product index can be obtained.     

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.     

Globalising Economic Governance,Political Projects,and Spatial Imaginaries: Insights from Four Australasian1 Cities

This paper explores how territorial economic governance is discursively constituted in a globalising and neoliberalising world. It acknowledges both the increasingly recognised formative role of spatial imaginaries in economic interventions and the workings of co‐constitutive political projects that link particular imaginaries with political ambitions and policy strategies. Embracing recent calls for comparative ethnographic urban research at the local‐global interface, it explores currently dominant spatial imaginaries across the four Australasian cities of Auckland, Sydney, Melbourne, and Perth. Based on multiple qualitative methods, this study claims that a considerable number of actor's spatial associations and reference points can be related to particular city‐specific governmental projects; Auckland's Super‐City, Sydney's Global and Green City, Melbourne's Liveable City, and Perth's Vibrant City. It is demonstrated how discursive governance techniques such as ‘story‐telling’, benchmarking, and policy transfer have been pivotal in the activation, circulation, and performance of those spatial imaginaries and their transformation into temporarily dominant visions for strategic urban interventions aimed at repositioning urban actors, spaces, and activities. While spatial imaginaries can be related to differently framed global aspirations, the effects of spatial political projects on urban governance and investment trajectories differ significantly across space. Theoretically, the paper stresses the importance of particular conceptions of territorial relations and time‐ and place‐specific institutional mediation in shaping context‐dependent discursive material governance alignments.     

The contribution of Sir Alan Wilson to spatial interaction and transport modelling

ABSTRACT

In a series of papers in the mid-1960s and early 70s, building on his key 1967 paper, Alan Wilson made a series of fundamental contributions to the specification and application of land-use transport models. In this paper the basis of his entropy-maximizing approach to spatial distribution models is outlined with a demonstration that the methods have applications very much beyond its transport roots. The entropy maximization method laid the foundations for the development of a range of multinomial logit share models. He expanded the core transport ideas to the whole range of comprehensive models, initially building on, and extending, Lowry's iconic Pittsburgh model. The factors that have sustained the longevity of Wilson's models are explored and articulated.     

基于空间自相关和时空扫描统计量的聚集比较分析

聚集是区域经济研究中的重点问题之一,而聚集定位问题又是深入分析聚集需要解决的首要问题。由于聚集具有高度的尺度敏感性,采用空间自相关方法分析时,尺度选择容易受研究者主观判断的影响,而且空间自相关方法也未考虑聚集的时间特征。与之相比,Kulldorff等学者提出的扫描统计量方法表现出了明显的优势。研究探索性地选用浙江省各市、县工业从业人口的聚集问题,从尺度选择、尺度转换和时空融合三个方面,比较了空间自相关和时空扫描统计量方法在探测聚集问题上的差异性,进而证实了时空扫描统计量方法不仅有效解决了人为选择尺度的偏倚问题,实现了尺度推绎、转换的自动化,而且更加有利地融合了立体、动态、多尺度的时空分析优势。     

On the Comparative Effectiveness of the Economic Structure of Regions

An important problem in insuring optimal operation of the centrally planned and state-controlled economy of the Soviet Union is to measure the efficacy of the industrial structure and productive specialization of republics and economic regions. Several measures of the efficacy of regional economies have been proposed, involving various relationships between labor productivity, the value of capital plant and equipment and the value of output. All these measures are unsatisfactory, in the author's view, and an alternate approach is suggested, using the net concept of national income produced by regions. This approach has been made possible by the recent calculation of input-output tables for republics and economic regions. The idea of using national income as a measure of regional economic efficacy was first proposed in the Soviet literature by A. Ye. Probst, whose comments appear elsewhere in the issue of Soviet Geography.     

An Approximation for the Rank Adjacency Statistic for Spatial Clustering with Sparse Data

The rank adjacency statistic D provides a simple method to assess regional clustering. It is defined as the weighted average absolute difference in ranks of the data, taken over all possible pairs of adjacent regions. In this paper the usual normal approximation to the D statistic is found to give inaccurate results if the data are sparse and some regions have tied ranks. Adjusted formulae for the moments of D that allow for the existence of ties are derived. An example of analyses of sparse mortality data (with many regions having no deaths, and hence tied ranks) showed satisfactory agreement between the adjusted formulae and the empirical distribution of the D statistic. We conclude that the D statistic, when used with adjusted moments, provides a valid approximate method to evaluate spatial clustering, even in sparse data situations.     

Comparison of Moran's I and Geary's c in Multivariate Spatial Pattern Analysis

This article compares multivariate spatial analysis methods that include not only multivariate covariance, but also spatial dependence of the data explicitly and simultaneously in model design by extending two univariate autocorrelation measures, namely Moran's I and Geary's c. The results derived from the simulation datasets indicate that the standard Moran component analysis is preferable to Geary component analysis as a tool for summarizing multivariate spatial structures. However, the generalized Geary principal component analysis developed in this study by adding variance into the optimization criterion and solved as a trace ratio optimization problem performs as well as, if not better than its counterpart the Moran principal component analysis does. With respect to the sensitivity in detecting subtle spatial structures, the choice of the appropriate tool is dependent on the correlation and variance of the spatial multivariate data. Finally, the four techniques are applied to the Social Determinants of Health dataset to analyze its multivariate spatial pattern. The two generalized methods detect more urban areas and higher autocorrelation structures than the other two standard methods, and provide more obvious contrast between urban and rural areas due to the large variance of the spatial component.     

The Size-adjusted Critical Region of Moran's I Test Statistics for Spatial Autocorrelation and Its Application to Geographical Areas

In this article, we explore the expression of the asymptotic approximation of the distribution function of Moran's I test statistic for the check of spatial autocorrelation, and we derive a more accurate critical value, which gives the rejection region similar to significant level α to the order of N^?1 (N = sample size). We show that in some cases our procedure effectively finds the significance of positive spatial autocorrelation in the problem testing for the lack of the spatial autocorrelation. Compared with our method, the testing procedure of Cliff and Ord (1971) is clearly ad hoc and should not be applied blindly, as they pointed out. Our procedure is derived from the theory of asymptotic expansion. We numerically analyze four types of county systems with rectangular lattices and three regional areas with irregular lattices.     

Local Spatial Autocorrelation Statistics: Distributional Issues and an Application

The statistics G_i(d) and G_i*(d), introduced in Getis and Ord (1992) for the study of local pattern in spatial data, are extended and their properties further explored. In particular, nonbinary weights are allowed and the statistics are related to Moran's autocorrelation statistic, I. The correlations between nearby values of the statistics are derived and verified by simulation. A Bonferroni criterion is used to approximate significance levels when testing extreme values from the set of statistics. An example of the use of the statistics is given using spatial-temporal data on the AIDS epidemic centering on San Francisco. Results indicate that in recent years the disease is intensifying in the counties surrounding the city.     

Comparative Spatial Filtering in Regression Analysis

One approach to dealing with spatial autocorrelation in regression analysis involves the filtering of variables in order to separate spatial effects from the variables’ total effects. In this paper we compare two filtering approaches, both of which allow spatial statistical analysts to use conventional linear regression models. Getis’ filtering approach is based on the autocorrelation observed with the use of the G_i local statistic. Griffith's approach uses an eigenfunction decomposition based on the geographic connectivity matrix used to compute a Moran's I statistic. Economic data are used to compare the workings of the two approaches. A final comparison with an autoregressive model strengthens the conclusion that both techniques are effective filtering devices, and that they yield similar regression models. We do note, however, that each technique should be used in its appropriate context.