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.     

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.     

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.     

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.     

Geography, Spatial Data Analysis, and Geostatistics: An Overview

Geostatistics is a distinctive methodology within the field of spatial statistics. In the past, it has been linked to particular problems (e.g., spatial interpolation by kriging) and types of spatial data (attributes defined on continuous space). It has been used more by physical than human geographers because of the nature of their types of data. The approach taken by geostatisticians has several features that distinguish it from the methods typically used by human geographers for analyzing spatial variation associated with regional data, and we discuss these. Geostatisticians attach much importance to estimating and modeling the variogram to explore and analyze spatial variation because of the insight it provides. This article identifies the benefits of geostatistics, reviews its uses, and examines some of the recent developments that make it valuable for the analysis of data on areal supports across a wide range of problems.     

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.     

Target-Density Weighting Interpolation and Uncertainty Evaluation for Temporal Analysis of Census Data

Conducting temporal analysis of census data often requires applying areal interpolation to integrate data that have been spatially aggregated using incompatible zoning systems. This article introduces a method of areal interpolation, target-density weighting (TDW), that is useful for long-term temporal analysis because it requires only readily available historical data and basic geographic information system operations. Then, through regression analysis of a large sample of U.S. census tract data, a model is produced that relates the error in TDW estimates of tract population to four basic properties of tracts. An analysis of model residuals combined with theorized absolute limits on interpolation error yields formulas with which we can compute upper and lower prediction bounds on the population in a tract of one census at the time of a different census. These prediction intervals enable the interpretation of different interpolated estimates with appropriately varying degrees of uncertainty.     

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.     

The Place of Population Geography in the System of Population Sciences

The area of research that is considered the proper domain of population geography is defined. In addition to an areal approach to the analysis of various aspects of population, such as growth, structure, migration and settlement patterns, there is great need for synthetic regional population studies, integrating all aspects of population within a particular territory. Causal relationships must be explained and the study of population must be related to the study of economic location.     

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.     

Kriging the Local Risk of a Rare Disease from a Register of Diagnoses

A substantial problem in studying the geographical epidemiology of rare noncontagious diseases is to estimate the risks of their development within populations. A geostatistical solution is described and illustrated by a case study of cancer among children in the West Midlands of England for the years 1980 to 1984 inclusive. Data consist of the numbers of diagnosed cases of cancer and of healthy children in each of 838 electoral wards, the centroids of which are known accurately. The rate of incidence or frequency, equal to the number of cases divided by the number of children, is a binomial variable and is treated as a realization of the underlying risk of a child's developing the disease that varies from place to place. The experimental variogram of the frequency was computed using the standard formulation. The variogram of the risk was obtained from it taking into account the numbers of children at risk and the error associated with each observed frequency. The variogram of the risk increased monotonically from 0 at zero lag to about 50 kilometers, and it was modeled as Whittle's two-dimensional elementary correlation function. The covariances of frequency and cross-covariances between the frequency and the risk were derived from it, and these were then used together with the data to krige the risk and map it. The risk of developing the disease is shown to have a patchy distribution, strongly autocorrelated at the regional scale of the investigation.     

Examining Land-Use through GIS-Based Kernel Density Estimation: A Re-Evaluation of Legacy Data from the Berbati-Limnes Survey

The use of archaeological survey data for evaluation of landscape dynamics has commonly been concerned with the distribution of settlements and changes in number of recorded sites over time. Here we present a new quantitative approach to survey-based legacy data, which allows further assessments of the spatial configuration of possible land-use areas. Utilizing data from an intensive archaeological survey in the Berbati-Limnes area, Greece, we demonstrate how GIS-based kernel density estimations (KDE) can be used to produce cluster-based density surfaces that may be linked to past land-use strategies. By relating density surfaces to elevation and slope, it is also possible to quantify shifts in the use of specific environments on a regional scale, allowing us to model and visualize land-use dynamics over time. In this respect, the approach provides more multifaceted information to be drawn from archaeological legacy data, providing an extended platform for research on human-environment interactions.     

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.     

Aggregation of Sampling Units: An Analytical Solution to Predict Variance

Geographical variables generally show spatially structured patterns corresponding to intrinsic characteristics of the environment. The size of the sampling unit has a critical effect on our perception of phenomena and is closely related to the variance and correlation structure of the data. Geostatistical theory uses analytical relationships for change of support (change of sampling unit size), allowing prediction of the variance and autocorrelation structure that would be observed if a survey was conducted using different sampling unit sizes. To check the geostatistical predictions, we use a test case about tree density in the tropical rain forest of the Pasoh Reserve, Malaysia. This data set contains exhaustive information about individual tree locations, so it allows us to simulate and compare various sampling designs. The original data set was reorganized to compute tree densities for 5 times 5-, 10 times 10-, and 20 times 20-meter quadrat sizes. Based upon the 5 times 5-meter data set, the spatial structure is modeled using a nugget effect (white noise) plus an exponential model. The change of support relationships, using within-quadrat variances inferred from the variogram model, predict the spatial autocorrelation structure and new variances corresponding to 10 times 10-meter and 20 times 20-meter quadrats. The theoretical and empirical results agreed closely, whereas neglecting the autocorrelation structure would have led to largely underestimating the variance. As the quadrat size increases, the range of autocorrelation increases, while the variance and the proportion of noise in the data decrease.     

Transitions in Social Organization: A Predictive Model from Southwestern Archaeology

A number of studies from experimental psychology suggest the existence of information-processing constraints that place limits on the number of people with which an individual may simultaneously interact. The existence of such constraints means that increases in the size of a human group will push that population toward “scalar thresholds,” at which point a transformation of the social order must take place to reorganize patterns of group interaction. The model developed here incorporates ethnographic data into the scalar theory of social change in an attempt to refine the precision with which scalar thresholds may be predicted. This scalar model contributes to an understanding of social change in small- to midsized sedentary populations and provides insight into processes by which social differentiation can emerge in these societies.     

Modelling of hydrology and potential population levels at Bronze Age Jawa,Northern Jordan: a Monte Carlo approach to cope with uncertainty

An annual water balance model for Wadi Rajil, in Northern Jordan, is used to simulate the ancient water supply system for the Early Bronze Age site of Jawa. The model includes: water delivery from the catchment; local pond storage; and water demand for people, animals and irrigation. A Monte Carlo approach is used to incorporate the uncertainty associated with a range of factors including rainfall, evaporation, water losses and use. The stochastic simulation provides estimates of potential population levels sustainable by the water supply system. Historical precipitation estimates from a Global Circulation Model, with uncertainty bounds, are used to reconstruct the climate at Jawa in the Early Bronze Age (EBA). Model results indicate that the population levels in the predicted wetter conditions during parts of the EBA could have risen to ∼6000 and may have been higher in wet years. However, palaeoclimatic proxies also suggest prolonged droughts in the EBA; and during these periods the water management system was unable to provide adequate supply for a population of 6000. The utility of Monte Carlo based hydrological modelling as a tool within archaeological science is discussed.     

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.     

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.