Empirical Zoning Distributions for Small Area Data

It is well established that using data summaries for a set of geographic areas or zones to estimate the parameters of a statistical model, commonly called ecological inference, frequently leads to the modifiable area unit problem (MAUP). In this article, the zoning effect of the MAUP is investigated for a range of scales. A zoning distribution is defined, and then used to characterize the zoning effect for parameter estimates from ecological analyses. Zone‐independent parameter estimates are obtained using the mean of the zoning distribution, and assessed using the variance of the zoning distribution. Zoning distributions are illustrated for parameter estimates from two ecological regression models at multiple scales using Australian National Health Survey data. For both a continuous response variable and a binary response variable, the empirical zoning distributions are unimodal, relatively symmetrical with appreciable variation, even when based on a large number of zones. The “ecological mean,” or expected value of the empirical zoning distribution at each scale, displays systematic variation with scale and the zoning distribution variance also depends on scale. The results demonstrate that the zoning effect should not be ignored, and the sensitivity of ecological analysis results to the analysis zones should be assessed.     

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.     

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.     

Geostatistical Prediction and Simulation of Point Values from Areal Data

The spatial prediction and simulation of point values from areal data are addressed within the general geostatistical framework of change of support (the term support referring to the domain informed by each measurement or unknown value). It is shown that the geostatistical framework (i) can explicitly and consistently account for the support differences between the available areal data and the sought-after point predictions, (ii) yields coherent (mass-preserving or pycnophylactic) predictions, and (iii) provides a measure of reliability (standard error) associated with each prediction. In the case of stochastic simulation, alternative point-support simulated realizations of a spatial attribute reproduce (i) a point-support histogram (Gaussian in this work), (ii) a point-support semivariogram model (possibly including anisotropic nested structures), and (iii) when upscaled, the available areal data. Such point-support-simulated realizations can be used in a Monte Carlo framework to assess the uncertainty in spatially distributed model outputs operating at a fine spatial resolution because of uncertain input parameters inferred from coarser spatial resolution data. Alternatively, such simulated realizations can be used in a model-based hypothesis-testing context to approximate the sampling distribution of, say, the correlation coefficient between two spatial data sets, when one is available at a point support and the other at an areal support. A case study using synthetic data illustrates the application of the proposed methodology in a remote sensing context, whereby areal data are available on a regular pixel support. It is demonstrated that point-support (sub-pixel scale) predictions and simulated realizations can be readily obtained, and that such predictions and realizations are consistent with the available information at the coarser (pixel-level) spatial resolution.     

flowAMOEBA: Identifying Regions of Anomalous Spatial Interactions

This study aims at developing a data‐driven and bottom‐up spatial statistic method for identifying regions of anomalous spatial interactions (clusters of extremely high‐ or low‐value spatial flows), based on which it creates a spatial flow weights matrix. The method, dubbed flowAMOEBA, upgrades a multidirectional optimum ecotope‐based algorithm (AMOEBA) from areal data to spatial flow data through a proper spatial flow neighborhood definition. The method has the potential to dramatically change the way we study spatial interactions. First, it breaks the convention that spatial interaction data are always collected and modeled between spatial entities of the same granularity, as it delineates the OD region of anomalous spatial interactions, regardless of the size, shape, scale, or administrative level. Second, the method creates an empirical spatial flow weights matrix that can handle network autocorrelation embedded in spatial interaction modeling, thus improving related policy‐making or problem‐solving strategies. flowAMOEBA is tested and demonstrated on a synthetic data set as well as a county‐to‐county migration data set.     

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.     

SPATIAL DEPENDENCY OF SEGREGATION INDICES

A few researchers have mentioned the scale sensitivity of segregation index, D. In this paper, I discuss analytically and empirically why using large enumeration areal units usually results in low segregation measures, and using small areal units produces relatively high segregation measures. The discussion is also applicable to the multi-group variant of D. A major finding is that if people of the same ethnic groups are positively spatially auto-correlated, increasing the size of areal units of analysis may not lower D initially, because only people of the same group are added. But enlarging the areal units subsequently may include population of other ethnic groups, and therefore could lower D. However, if the boundaries of the larger enumeration units are drawn to include only population of the same group, then D will not change significantly. Both the spatial autocorrelation of ethnic group population and zonal pattern are critical factors in determining the scale sensitivity of D.     

Measuring the Scale of Segregation Using k‐Nearest Neighbor Aggregates

Nearly all segregation measures use some form of administrative unit (usually tracts in the United States) as the base for the calculation of segregation indices, and most of the commonly used measures are aspatial. The spatial measures that have been proposed are often not easily computed, although there have been significant advances in the past decade. We provide a measure that is individually based (either persons or very small administrative units) and a technique for constructing neighborhoods that does not require administrative units. We show that the spatial distribution of different population groups within an urban area can be efficiently analyzed with segregation measures that use population count‐based definitions of neighborhood scale. We provide a variant of a k‐nearest neighbor approach and a statistic spatial isolation and a methodology (EquiPop) to map, graph, and evaluate the likelihood of individuals meeting other similar race individuals or of meeting individuals of a different ethnicity. The usefulness of this approach is demonstrated in an application of the method to data for Los Angeles and three metropolitan areas in Sweden. This comparative approach is important as we wish to show how the technique can be used across different cultural contexts. The analysis shows how the scale (very small neighborhoods, larger communities, or cities) influences the segregation outcomes. Even if microscale segregation is strong, there may still be much more mixing at macroscales.     

Spatial and environmental impacts on adverse birth outcomes in Ontario

This study assesses the overall spatial variations and neighbourhood‐level “hot spots” of low birth weight and preterm birth incidence within three public health units in Ontario, Canada. The analysis uses a stepwise approach of intra‐class correlation analysis, a spatial scan statistic, and multilevel spatial modeling. Results show that neighbourhood level variation accounts for only 2–3 percent of the total variation of adverse birth outcomes in the study area. However, strong spatial autocorrelation is observed at the neighbourhood level, and spatial clusters of relatively high adverse birth outcome rates exist in areas that are associated with environmental risks, including pollution sources and proximity to highways. Thus, although estimated neighbourhood impacts on adverse birth outcomes are small compared with those of individual‐level risks, local high potential environmental risk areas are identifiable. Environmental surveillance and spatial statistical analysis should be conducted regularly by local health authorities to identify and monitor the impact of environmental changes on health in general and on birth outcomes in particular. Specific community‐oriented health interventions may be required to reduce observed local health impacts.     

Parameterization and Visualization of the Errors in Areal Interpolation

Areal interpolation involves the transfer of data from one zonation of a region to another, where the two zonations of space are geographically incompatible. By its very nature this process is fraught with errors. However, only recently have there been specific attempts to quantify these errors. Fisher and Langford (1995) employed Monte Carlo simulation methods, based on modifiable areal units, to compare the errors resulting from selected areal interpolation techniques. This paper builds on their work by parameterizing and visualizing the errors resulting from the areal weighting and dasymetric methods of areal interpolation. It provides the basis for further research by developing the methodology to produce predictive models of the errors in areal interpolation. Random aggregation techniques are employed to generate multiple sets of source zones and interpolation takes place from these units onto a fixed set of randomly generated target zones. Analysis takes place at the polygon, or target zone level, which enables detailed analysis of the error distributions, basic visualization of the spatial nature of the errors and predictive modeling of the errors based on parameters of the target zones. Correlation and regression analysis revealed that errors from the areal weighting technique were related to the geometric parameters of the target zones. The dasymetric errors, however, demonstrated more association with the population or attribute characteristics of the zones. The perimeter, total population, and population density of the target zones were shown to be the strongest predictive parameters.     

Nonlinear Multivariate Spatial Modeling Using NLPCA and Pair‐Copulas

A novel geostatistical modeling approach is developed to model nonlinear multivariate spatial dependence using nonlinear principal component analysis (NLPCA) and pair‐copulas. In spatial studies, multivariate measurements are frequently collected at each location. The dependence between such measurements can be complex. In this article, a multivariate geostatistical model is developed that can capture both nonlinear spatial dependence across locations and nonlinear dependence between measurements at a particular location. Nonlinear multivariate dependence between spatial variables is removed using NLPCA. Subsequently, a pair‐copula based model is fitted to each transformed variable to model the univariate nonlinear spatial dependencies. NLPCA and pair‐copulas, within the proposed model, are compared with stepwise conditional transformation (SCT) and conventional kriging. The results show that, for the two case studies presented, the proposed model that utilizes NLPCA and pair‐copulas reproduces nonlinear multivariate structures and univariate distributions better than existing methods based on SCT and kriging.     

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.     

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.     

Can Dasymetric Mapping Significantly Improve Population Data Reallocation in a Dense Urban Area?

The issue of reallocating population figures from a set of geographical units onto another set of units has received a great deal of attention in the literature. Every other day, a new algorithm is proposed, claiming that it outperforms competitor procedures. Unfortunately, when the new (usually more complex) methods are applied to a new data set, the improvements attained are sometimes just marginal. The relationship cost‐effectiveness of the solutions is case‐dependent. The majority of studies have focused on large areas with heterogeneous population density distributions. The general conclusion is that as a rule more sophisticated methods are worth the effort. It could be argued, however, that when we work with a variable that varies gradually in relatively homogeneous small units, simple areal weighting methods could be sufficient and that ancillary variables would produce marginal improvements. For the case of reallocating census data, our study shows that, even under the above conditions, the most sophisticated approaches clearly yield the better results. After testing fourteen methods in Barcelona (Spain), the best results are attained using as ancillary variable the total dwelling area in each residential building. Our study shows the 3‐D methods as generating the better outcomes followed by multiclass 2‐D procedures, binary 2‐D approaches and areal weighting and 1‐D algorithms. The point‐based interpolation procedures are by far the ones producing the worst estimates.     

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.     

A Geostatistical Framework for Area‐to‐Point Spatial Interpolation

The spatial prediction of point values from areal data of the same attribute is addressed within the general geostatistical framework of change of support; the term support refers to the domain informed by each datum or unknown value. It is demonstrated that the proposed geostatistical framework can explicitly and consistently account for the support differences between the available areal data and the sought‐after point predictions. In particular, it is proved that appropriate modeling of all area‐to‐area and area‐to‐point covariances required by the geostatistical frame‐work yields coherent (mass‐preserving or pycnophylactic) predictions. In other words, the areal average (or areal total) of point predictions within any arbitrary area informed by an areal‐average (or areal‐total) datum is equal to that particular datum. In addition, the proposed geostatistical framework offers the unique advantage of providing a measure of the reliability (standard error) of each point prediction. It is also demonstrated that several existing approaches for area‐to‐point interpolation can be viewed within this geostatistical framework. More precisely, it is shown that (i) the choropleth map case corresponds to the geostatistical solution under the assumption of spatial independence at the point support level; (ii) several forms of kernel smoothing can be regarded as alternative (albeit sometimes incoherent) implementations of the geostatistical approach; and (iii) Tobler's smooth pycnophylactic interpolation, on a quasi‐infinite domain without non‐negativity constraints, corresponds to the geostatistical solution when the semivariogram model adopted at the point support level is identified to the free‐space Green's functions (linear in 1‐D or logarithmic in 2‐D) of Poisson's partial differential equation. In lieu of a formal case study, several 1‐D examples are given to illustrate pertinent concepts.     

A Scale‐Sensitive Test of Attraction and Repulsion Between Spatial Point Patterns

There exist a variety of tests for attraction and repulsion effects between spatial point populations, most notably those involving either nearest‐neighbor or cell‐count statistics. Diggle and Cox (1981) showed that for the nearest‐neighbor approach, a powerful test could be constructed using Kendall's rank correlation coefficient. In the present paper, this approach is extended to cell‐count statistics in a manner paralleling the K‐function approach of Lotwick and Silverman (1982). The advantage of the present test is that, unlike nearest‐neighbors, one can identify the spatial scales at which repulsion or attraction are most significant. In addition, it avoids the torus‐wrapping restrictions implicit in the Monte Carlo testing procedure of Lotwick and Silverman. Examples are developed to show that this testing procedure can in fact identify both attraction and repulsion between the same pair of point populations at different scales of analysis.     

Spatial Decomposition of Segregation Indices: A Framework Toward Measuring Segregation at Multiple Levels

Segregation measures based upon data gathered at different geographical levels cannot provide consistent results because of the scale effect under the Modifiable Areal Unit Problem (MAUP) umbrella. This paper proposes a framework, which decomposes segregation attributable to different geographical levels, to conceptually link segregation values obtained from multiple geographical levels together such that differences in segregation values among levels are accounted for. Using two different indices, the dissimilarity index D and the diversity index H, this paper illustrates the decomposition methods specific to these indices. When these indices are decomposed, local measures of segregation pertaining to multiple geographical levels are computed. These local segregation measures indicate the levels of segregation contributed by the local units and the regional units to the entire study area.