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.     

Reconstructing Population Density Surfaces from Areal Data: A Comparison of Tobler's Pycnophylactic Interpolation Method and Area-to-Point Kriging

We compare Tobler's pycnophylactic interpolation method with the geostatistical approach of area-to-point kriging for distributing population data collected by areal unit in 18 census tracts in Ann Arbor for 1970 to reconstruct a population density surface. In both methods, (1) the areal data are reproduced when the predicted population density is upscaled; (2) physical boundary conditions are accounted for, if they exist; and (3) inequality constraints, such as the requirement of non-negative point predictions, are satisfied. The results show that when a certain variogram model, that is, the de Wijsian model corresponding to the free-space Green's function of Laplace's equation, is used in the geostatistical approach under the same boundary condition and constraints with Tobler's approach, the predicted population density surfaces are almost identical (up to numerical errors and discretization discrepancies). The implications of these findings are twofold: (1) multiple attribute surfaces can be constructed from areal data using the geostatistical approach, depending on the particular point variogram model adopted—that variogram model need not be the one associated with Tobler's solution and (2) it is the analyst's responsibility to justify whether the smoothness criterion employed in Tobler's approach is relevant to the particular application at hand. A notable advantage of the geostatistical approach over Tobler's is that it allows reporting the uncertainty or reliability of the interpolated values, with critical implications for uncertainty propagation in spatial analysis operations.     

Area‐to‐Point Prediction Under Boundary Conditions

This article proposes a geostatistical solution for area‐to‐point spatial prediction (downscaling) taking into account boundary effects. Such effects are often poorly considered in downscaling, even though they often have significant impact on the results. The geostatistical approach proposed in this article considers two types of boundary conditions (BC), that is, a Dirichlet‐type condition and a Neumann‐type condition, while satisfying several critical issues in downscaling: the coherence of predictions, the explicit consideration of support differences, and the assessment of uncertainty regarding the point predictions. An updating algorithm is used to reduce the computational cost of area‐to‐point prediction under a given BC. In a case study, area‐to‐point prediction under a Dirichlet‐type BC and a Neumann‐type BC is illustrated using simulated data, and the resulting predictions and error variances are compared with those obtained without considering such conditions.     

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.     

Bayesian Areal Wombling for Geographical Boundary Analysis

In the analysis of spatially referenced data, interest often focuses not on prediction of the spatially indexed variable itself, but on boundary analysis , that is, the determination of boundaries on the map that separate areas of higher and lower values. Existing boundary analysis methods are sometimes generically referred to as wombling , after a foundational article by Womble (1951). When data are available at point level (e.g., exact latitude and longitude of disease cases), such boundaries are most naturally obtained by locating the points of steepest ascent or descent on the fitted spatial surface (Banerjee, Gelfand, and Sirmans 2003). In this article, we propose related methods for areal data (i.e., data which consist only of sums or averages over geopolitical regions). Such methods are valuable in determining boundaries for data sets that, perhaps due to confidentiality concerns, are available only in ecological (aggregated) format, or are only collected this way (e.g., delivery of health-care or cost information). After a brief review of existing algorithmic techniques (including that implemented in the commercial software BoundarySeer), we propose a fully model-based framework for areal wombling, using Bayesian hierarchical models with posterior summaries computed using Markov chain Monte Carlo methods. We explore the suitability of various existing hierarchical and spatial software packages (notably S-plus and WinBUGS) to the task, and show the approach's superiority over existing nonstochastic alternatives, both in terms of utility and average mean square error behavior. We also illustrate our methods (as well as the solution of advanced modeling issues such as simultaneous inference) using colorectal cancer late detection data collected at the county level in the state of Minnesota.     

A Geostatistical Linear Regression Model for Small Area Data. 一种适用于小区域数据的地统计线性回归模型

We present a new linear regression model for use with aggregated, small area data that are spatially autocorrelated. Because these data are aggregates of individual‐level data, we choose to model the spatial autocorrelation using a geostatistical model specified at the scale of the individual. The autocovariance of observed small area data is determined via the natural aggregation over the population. Unlike lattice‐based autoregressive approaches, the geostatistical approach is invariant to the scale of data aggregation. We establish that this geostatistical approach also is a valid autoregressive model; thus, we call this approach the geostatistical autoregressive (GAR) model. An asymptotically consistent and efficient maximum likelihood estimator is derived for the GAR model. Finite sample evidence from simulation experiments demonstrates the relative efficiency properties of the GAR model. Furthermore, while aggregation results in less efficient estimates than disaggregated data, the GAR model provides the most efficient estimates from the data that are available. These results suggest that the GAR model should be considered as part of a spatial analyst's toolbox when aggregated, small area data are analyzed. More important, we believe that the GAR model's attention to the individual‐level scale allows for a more flexible and theory‐informed specification than the existing autoregressive approaches based on an area‐level spatial weights matrix. Because many spatial process models, both in geography and in other disciplines, are specified at the individual level, we hope that the GAR covariance specification will provide a vehicle for a better informed and more interdisciplinary use of spatial regression models with area‐aggregated data.     

Reducing Errors in Rainfall Estimates through Rain Gauge Location

Facility location models are examined as a framework for generating rain gauge networks designed to reduce errors in mean areal precipitation (MAP) estimation. Errors in estimating MAP may be divided into two types: (i) capture error, not observing a storm which occurs in a gauged area, and (ii) extrapolation error, using a rain gauge measurement to represent a heterogeneous area. In this paper, five rain gauge location models are developed to minimize these errors. The models include adaptations of the maximal covering location problem, the p-median model, and three models derived from multicriteria cluster analysis. The models are tested using precipitation data from an experimental watershed maintained by the U.S. Department of Agriculture in Arizona. Analysis of the results reveals, for the particular watershed, that (1) in sparse networks, location of rain gauges can play a larger role than number of rain gauges in reducing errors in MAP estimates; (2) models based on mean hydrologic data provide nearly as good networks as models based on spatially correlated data; and (3) models yielding the best networks for estimating precipitation for flood predictions are different from the models providing the best precipitation estimates for low flow forecasts.     

Potential Contributions of Spatial Analysis to Geographic Information Systems for Transportation (GIS-T)

Geographic information systems (GIS) are having tremendous impacts on many scientific and application domains. The traditional subfield of spatial analysis is witnessing a major resurgence and enhancement due to GIS and geographical information science (GISci), an interdisciplinary field focusing on the theory and methodology underlying GIS software. The interdisciplinary field of geographic information systems for transportation (GIS-T) has emerged to focus on the role of GIS in transportaton analysis and planning. This paper suggests the benefits of closer linkages between spatial analysis, GISci, and transportation through a focused review of spatial analytical issues and their potential contributions to GIS-T. Specifically, this paper reviews the following issues: (i) modifiable areal units; (ii) boundary problems and spatial sampling; (iii) spatial dependence and spatial heterogeneity; and (iv) alternative representations of geographic environments. The discussion highlights the general issues as well as identifies their specific relevance to GIS-T. In addition, this paper identifies some emerging tools from GISci that can address these spatial analytical issues in GIS-T.     

A FURTHER EXPLORATION INTO THE ROBUSTNESS OF SPATIAL AUTOCORRELATION SPECIFICATIONS*

ABSTRACT In a recent study, the robustness of linear models with various spatial autocorrelation specifications was assessed through Monte Carlo experiments, and the geostatistical models were concluded to dominate the weight matrix models in the prediction. The present study tests the soundness of this conclusion with a different framework for prediction and presents some experimental results that can call into doubt the dominance of the geostatistical models over the weight matrix models.     

Global Climatic Change: the Difficulties of Assessing Impacts

The Intergovernmental Panel on Climate Change (IPCC) report, initiated in 1988, is complete, was debated at the Second World Climate Conference in November, 1990 and was subsequentty submitted to the United Nations General Assembly. IPCC: (i) asserted the reality of humanity's disturbance of the natural climate system; (ii) demanded studies to improve our knowledge of processes vulnerable to climatic changes: and (iii) called for policy responses to mitigate and adapt to these changes. Two fundamental issues are: how will global climatic change affect natural resources and human population and how will the impetus towards policy responses, particularly greenhouse gas emission reduction treaties, affect industry, the economy and trade? A necessary first step in the highly desirable and geographical aspiration of striving to link numerical climate modelling to the predictions of socioeconomic systems is increased awareness and improved understanding of current physical and social models. In this paper I review the status of numerical climatic modelling especially as it pertains to scenarios of the effects of human-enhanced greenhouse warming. These projections are of futures which are themselves the result of socio-economic predictions. Development of appropriate adaptive strategies depends crucially upon improved simulation of the continental near-surface climate and on improved spatial resolution of climate models by at least two orders of magnitude. Such increased resolution is likely to demand a thousandfold increase in computing power. The physical results of global climatic change are likely to be less significant than the social and economic effects resulting from international agreements on emission reduction Recent shifts in international research and policy responses place today's studies of global climatic change firmly at the focus of human-environment interactions and hence at the core of modern geography.     

Developing Geostatistical Space–Time Models to Predict Outpatient Treatment Burdens from Incomplete National Data

Basic health system data such as the number of patients utilizing different health facilities and the types of illness for which they are being treated are critical for managing service provision. These data requirements are generally addressed with some form of national Health Management Information System (HMIS), which coordinates the routine collection and compilation of data from national health facilities. HMIS in most developing countries are characterized by widespread underreporting. Here we present a method to adjust incomplete data to allow prediction of national outpatient treatment burdens. We demonstrate this method with the example of outpatient treatments for malaria within the Kenyan HMIS. Three alternative modeling frameworks were developed and tested in which space–time geostatistical prediction algorithms were used to predict the monthly tally of treatments for presumed malaria cases (MC) at facilities where such records were missing. Models were compared by a cross-validation exercise and the model found to most accurately predict MC incorporated available data on the total number of patients visiting each facility each month. A space–time stochastic simulation framework to accompany this model was developed and tested in order to provide estimates of both local and regional prediction uncertainty. The level of accuracy provided by the predictive model, and the accompanying estimates of uncertainty around the predictions, demonstrate how this tool can mitigate the uncertainties caused by missing data, substantially enhancing the utility of existing HMIS data to health-service decision makers.     

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.     

Spatial/temporal mismatch: a conflation protocol for Canada Census spatial files

The Canada census is one of the chief sources of demographic and socio-economic data for researchers in this country. Census variables are linked to geography files that allow researchers using geographic information systems (GIS) to view and analyze spatial data. Some of the most useful analysis, however, is based on changes in attribute values over time and space. Analysis of spatio- temporal events such as shifting migration patterns or changes in the distribution of health status permits a more dimensioned perspective than the viewing of static spatial phenomena. The analysis of spatio-temporal phenomena is limited by major changes in the spatial framework (e.g., location of road networks and other spatial entities) between national censuses. This paper addresses this limitation by (i) illustrating the extent of spatial mismatch between the 1996 and the 2001 census; (ii) examining attempts to rectify this problem in other jurisdictions and (iii) presenting a 'made-in-Canada' solution for conflation of census geometries. We believe that this solution will enhance the ability of Canadian researchers to describe and analyze socio-economic, health and demographic shifts across time and space. The research is supported by an ftp site for downloading the census geography rectification software presented in this paper.     

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.     

Geostatistical Analysis of County-Level Lung Cancer Mortality Rates in the Southeastern United States

The analysis of health data and putative covariates, such as environmental, socioeconomic, demographic, behavioral, or occupational factors, is a promising application for geostatistics. Transferring methods originally developed for the analysis of earth properties to health science, however, presents several methodological and technical challenges. These arise because health data are typically aggregated over irregular spatial supports (e.g., counties) and consist of a numerator and a denominator (i.e., rates). This article provides an overview of geostatistical methods tailored specifically to the characteristics of areal health data, with an application to lung cancer mortality rates in 688 U.S. counties of the southeast (1970–1994). Factorial Poisson kriging can filter short-scale variation and noise, which can be large in sparsely populated counties, to reveal similar regional patterns for male and female cancer mortality that correlate well with proximity to shipyards. Rate uncertainty was transferred through local cluster analysis using stochastic simulation, allowing the computation of the likelihood of clusters of low or high cancer mortality. Accounting for population size and rate uncertainty led to the detection of new clusters of high mortality around Oak Ridge National Laboratory for both sexes, in counties with high concentrations of pig farms and paper mill industries for males (occupational exposure) and in the vicinity of Atlanta for females.     

Interpreting Intra-site Spatial Patterns in Seasonal Contexts: an Ethnoarchaeological Case Study from the Western Alps

This paper deals with the ethnoarchaeological analysis of the spatial pattern of artefacts and ecofacts within two traditional pastoral huts (a dwelling and a seasonal dairy) in the uplands of Val Maudagna (Cuneo province, Italian western Alps). The composition of the ethnoarchaeological assemblages of the two huts was studied and compared; point pattern analysis was applied to identify spatial processes mirrored in the interactions between objects; Moran’s I correlogram and empirical variogram were used to investigate the effects of trampling on the displacement of objects on the floor. The results were compared with information provided by the herder who still used the huts. The quantitative and ethnographical data enabled inferences to be made that can help in the interpretation of archaeological seasonal sites. The function of a seasonal site can be recognized, as can the impact of delayed curation on the composition of the assemblage and the importance of the intensity of occupation compared with the frequency of occupation. The spatial organization of activities is reflected in the spatial patterns of objects, with clearer identification of activity areas in intensively occupied sites, and there is evidence for the behaviour behind the spatial segregation of activities. Trampling is a crucial post-depositional factor in the displacement of artefacts and ecofacts, especially in non-intensively exploited sites. From a methodological point of view, this research is another example that highlights the importance of integrating quantitative methods (especially spatial analysis and geostatistical methods) and ethnoarchaeological data in order to improve the interpretation of archaeological sites and assemblages.     

Geostatistical Smoothing of Areal Data: Mapping Employment Density with Factorial Kriging

This article summarizes area-to-point (ATP) factorial kriging that allows the smoothing of aggregate, areal data into a continuous spatial surface. Unlike some other smoothing methods, ATP factorial kriging does not suppose that all of the data within an area are located at a centroid or other arbitrary point. Also, unlike some other smoothing methods, factorial kriging allows the user to utilize an autocovariance function to control the smoothness of the output. This is beneficial because the covariance function is a physically meaningful statement of spatial relationship, which is not the case when other spatial kernel functions are used for smoothing. Given a known covariance function, factorial kriging gives the smooth surface that is best in terms of minimizing the expected mean squared prediction error. I present an application of the factorial kriging methodology for visualizing the structure of employment density in the Denver metropolitan area.