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.     

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.     

The Colocation Quotient: A New Measure of Spatial Association Between Categorical Subsets of Points. 协同区位商：点集分类子集间空间关联性的新度量标准

This article presents a new metric we label the colocation quotient (CLQ), a measurement designed to quantify (potentially asymmetrical) spatial association between categories of a population that may itself exhibit spatial autocorrelation. We begin by explaining why most metrics of categorical spatial association are inadequate for many common situations. Our focus is on where a single categorical data variable is measured at point locations that constitute a population of interest. We then develop our new metric, the CLQ, as a point‐based association metric most similar to the cross‐k‐function and join count statistic. However, it differs from the former in that it is based on distance ranks rather than on raw distances and differs from the latter in that it is asymmetric. After introducing the statistical calculation and underlying rationale, a random labeling technique is described to test for significance. The new metric is applied to economic and ecological point data to demonstrate its broad utility. The method expands upon explanatory powers present in current point‐based colocation statistics.     

Constructing the Spatial Weights Matrix Using a Local Statistic

Spatial weights matrices are necessary elements in most regression models where a representation of spatial structure is needed. We construct a spatial weights matrix, W , based on the principle that spatial structure should be considered in a two‐part framework, those units that evoke a distance effect, and those that do not. Our two‐variable local statistics model (LSM) is based on the G_i* local statistic. The local statistic concept depends on the designation of a critical distance, d_c, defined as the distance beyond which no discernible increase in clustering of high or low values exists. In a series of simulation experiments LSM is compared to well‐known spatial weights matrix specifications—two different contiguity configurations, three different inverse distance formulations, and three semi‐variance models. The simulation experiments are carried out on a random spatial pattern and two types of spatial clustering patterns. The LSM performed best according to the Akaike Information Criterion, a spatial autoregressive coefficient evaluation, and Moran's I tests on residuals. The flexibility inherent in the LSM allows for its favorable performance when compared to the rigidity of the global models.     

Categorical Wombling: Detecting Regions of Significant Change in Spatially Located Categorical Variables

Wombling is a method for discovering boundaries in a collection of continuous variables observed at the same geographic localities. We extend this method to categorical data. A categorical wombling statistic C_i, which identifies areas of rapid change, is defined for every pair i = 1,…, n of adjacent localities, and is equal to the average number of category mismatches at i. We use both simulation and theory to consider the order statistics of C_i under null hypotheses of randomness, and of spatial autocorrelation for each variable, but independence between variables. Graph-theoretical statistics derived from C_i investigate whether areas of rapid change resemble boundaries. Computer simulation is used to study the distributions of these under the two null hypotheses. The methods are applied to linguistic data in three European areas. Other potential applications exist in biology, linguistics, anthropology, and other social sciences.     

Beyond Moran's I: Testing for Spatial Dependence Based on the Spatial Autoregressive Model

The statistic known as Moran's I is widely used to test for the presence of spatial dependence in observations taken on a lattice. Under the null hypothesis that the data are independent and identically distributed normal random variates, the distribution of Moran's I is known, and hypothesis tests based on this statistic have been shown in the literature to have various optimality properties. Given its simplicity, Moran's I is also frequently used outside of the formal hypothesis-testing setting in exploratory analyses of spatially referenced data; however, its limitations are not very well understood. To illustrate these limitations, we show that, for data generated according to the spatial autoregressive (SAR) model, Moran's I is only a good estimator of the SAR model's spatial-dependence parameter when the parameter is close to 0. In this research, we develop an alternative closed-form measure of spatial autocorrelation, which we call APLE , because it is an approximate profile-likelihood estimator (APLE) of the SAR model's spatial-dependence parameter. We show that APLE can be used as a test statistic for, and an estimator of, the strength of spatial autocorrelation. We include both theoretical and simulation-based motivations (including comparison with the maximum-likelihood estimator), for using APLE as an estimator. In conjunction, we propose the APLE scatterplot, an exploratory graphical tool that is analogous to the Moran scatterplot, and we demonstrate that the APLE scatterplot is a better visual tool for assessing the strength of spatial autocorrelation in the data than the Moran scatterplot. In addition, Monte Carlo tests based on both APLE and Moran's I are introduced and compared. Finally, we include an analysis of the well-known Mercer and Hall wheat-yield data to illustrate the difference between APLE and Moran's I when they are used in exploratory spatial data 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.     

A Spatial Price Determination Model for Multi‐Market Heterogeneously Distributed Resources: An Application to Forestry Markets in Sweden

The spatial dimension is a key paradigm in price determination, as attested by recent studies in the literature that highlighted the differential in market behavior between spatial and non‐spatial pricing settings. In this paper, we develop a model of spatial pricing for multi‐market heterogeneously distributed resources, with an application to the Swedish forestry sector. The focus of the model is to estimate the impact of spatial interaction on the demand for resources in terms of resource allocation, competition, and pricing. In its core, the pricing mechanism relies on a supply–demand framework. Using disaggregated data at the gridcell level for forest feedstock supply and harvesting costs in Sweden, we construct regional supply curves for each gridcell assuming a maximum transportation distance to delimit the potential market. Demand nodes are exogenously determined and are adjusted using a distance‐decay model to assess demand pressure across locations. We apply the model empirically to assess the impact on forest feedstock prices of a 20 TWh increase in biofuel production.     

Toward a Cultural Landscape Atlas: Representing all the Landscape as Cultural

Many government and non‐statutory registers utilise point datasets to represent cultural heritage places. An effect of this approach is to emphasise that cultural heritage comprises a series of spatially discrete material remains or ‘sites’, suggesting discrete locations which are somehow disconnected from their broader historical and landscape contexts. We advocate an alternative in which spatial representation of heritage is set within a cultural landscape framework, acknowledging that all parts of the landscape have inter‐connected cultural histories, associations and meanings resulting from long‐term and ongoing human–environmental interactions. Results from a collaborative cultural heritage research project undertaken at Culgoa National Park in Australia demonstrate the advantage of this approach. The mapping products produced by the work comprise an interactive electronic DVD Atlas and hard copy maps. Both focus on meeting the management needs of field‐based park staff.     

Toward a Theory of Spatial Statistics: Another Step Forward

An update is presented to an earlier piece outlining some steps that needed to be taken toward the establishment of a theory of spatial statistics. Findings that have appeared since this first paper are summarized and interpreted, and extensions and suggestions are offered for the further establishment of a basis for a theory of spatial statistics. Topics include boundary considerations, the role of latent spatial dependencies, and small-sample-size issues. These topics embrace the problems of data transformations, edge effect bias, reference sampling distributions, multivariate autocorrelation models, conditional expectations, and higher-order autoregressive structures. In part, a course is charted for the next step to be taken.     

Refining the definition of cultural levels at Karabi Tamchin: a quantitative approach to vertical intra-site spatial analysis

The visual identification of archaeological levels can be difficult when stratigraphy is complex. This study emphasizes the importance of three-dimensional intra-site spatial analysis as a means of testing the integrity of archaeological levels, including the identification of palimpsest deposits. A geographical information system (GIS) is applied to a three-dimensional spatial analysis of lithic and bone distributions from Karabi Tamchin, a Middle Palaeolithic site from the Crimea, Ukraine. K-means statistical clustering is combined with a series of data transformations to identify and interpret the vertical and horizontal spatial organization of the site. The results indicate that K-means cluster analysis, used in conjunction with GIS, provides an exceptional method of identifying discrete clusters of archaeological materials in three dimensions. Through an analysis of cluster contents within levels, it is possible to reconstruct and compare patterns of spatial organization at Karabi Tamchin, contributing to current debates regarding the cognitive complexity of Neanderthal populations.     

Spatial variation in cervical cancer screening participation and outcomes among Indigenous and non‐Indigenous Australians in Queensland

Indigenous women continue to experience a disproportionately higher burden of cervical cancer than non‐Indigenous women in Australia. The National Indigenous Cervical Screening Project used probabilistic record linkage to combine population‐based administrative databases and identify Indigenous women on Pap Smear Registers. This study aimed to quantify the spatial variation by local government areas (LGAs) for Indigenous and non‐Indigenous women in Queensland in cervical screening participation rates and related outcomes. Empirical Bayes local geostatistical smoothing was performed to reduce the likelihood of spurious variation between small areas. The cohort included 1,091,260 women (2 per cent Indigenous) aged 20 to 69 with 2,393,708 Pap smears between 2006 and 2011. Indigenous women had smoothed LGA‐specific 5‐year participation rates (interquartile range (IQR) 38.9–53.3 per 100 eligible women) consistently lower than non‐Indigenous women (IQR 80.7–85.3). The non‐overlapping confidence intervals of these rates suggest that the Indigenous differential was significant. Compared with Indigenous women, non‐Indigenous women had consistently lower and more stable prevalence rates of histologically confirmed high grade abnormalities (IQR 8.0–10.1 versus 15.0–21.3 per 1,000 screened women). Although the LGA‐specific rates also suggest that a higher proportion of non‐Indigenous women were followed‐up within two months of an abnormal screening result, the wide confidence intervals for these estimates limit our ability to draw definitive conclusions about spatial patterns for this outcome. These findings highlight the importance of continued monitoring and ongoing efforts to identify drivers of these patterns and develop effective strategies to improve participation and potentially reduce the cervical cancer burden among Indigenous women.     

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 Effect on Attribute Prediction of Location Uncertainty in Spatial Data

A datum is considered spatial if it contains location information. Typically, there is also attribute information, whose distribution depends on its location. Thus, error in location information can lead to error in attribute information, which is reflected ultimately in the inference drawn from the data. We propose a statistical model for incorporating location error into spatial data analysis. We investigate the effect of location error on the spatial lag, the covariance function, and optimal spatial linear prediction (that is, kriging). We show that the form of kriging after adjusting for location error is the same as that of kriging without adjusting for location error. However, location error changes entries in the matrix of explanatory variables, the matrix of co‐variances between the sample sites, and the vector of covariances between the sample sites and the prediction location. We investigate, through simulation, the effect that varying trend, measurement error, location error, range of spatial dependence, sample size, and prediction location have on kriging after and without adjusting for location error. When the location error is large, kriging after adjusting for location error performs markedly better than kriging without adjusting for location error, in terms of both the prediction bias and the mean squared prediction error.     

A Descriptive Analysis of Discrete U.S. Industrial Complexes*

Abstract. We use the Getis/Ord local G statistic and detailed county‐level industry employment data from the U.S. Bureau of Labor Statistics to isolate discrete industrial complexes—or groups of nominally linked industries clustered in particular locations—for two recent years: 1989 and 1997. We describe the characteristics of the complexes in terms of their number, spatial extent, broad regional distribution, and other factors. Data from the two periods help illustrate key shifts in industrial locations, including the continuing concentration of the apparel industry in the Southeast and the ongoing southern shift in U.S. vehicle production.     

Exploring Goodness-of-Fit and Spatial Correlation Using Components of Tango's Index of Spatial Clustering

The ability to detect anomalies such as spatial clustering in data sets plays an important role in spatial data analysis, leading to interest in test statistics identifying patterns exhibiting significant levels of clustering. Toward this end, Tango (1995) proposed a statistic (and its associated distribution under a null hypothesis of no clustering) assessing overall patterns of spatial clustering in a set of observed regional counts. Rogerson (1999) observed that Tango's index may be decomposed into the summation of two distinct statistics, the first mirroring standard tests of goodness-of-fit (GOF), and the second an index of spatial association (SA) similar to Moran's I . In this article, we investigate the effectiveness of Rogerson's expression of Tango's statistic in separating GOF from SA in data sets containing clusters. We simulate data under the null hypothesis of no clustering as well as two alternative hypotheses. The first alternative hypothesis induces a poor fit from the null hypothesis while maintaining independent observations and the second alternative hypothesis induces spatial dependence while maintaining fit. Using Rogerson's decomposition and leukemia incidence data from upstate New York, we show graphically that one is unable to statistically distinguish poor fit from autocorrelation.     

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.     

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.     

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.