Effects of Confidentiality-Preserving Geo-Masking on the Estimation of Semivariogram and of the Kriging Variance

Geostatistical methods, such as semivariograms and kriging are well-known spatial tools commonly employed in many disciplines such as health, mining, forestry, meteorology to name only few. They are based essentially on point-referenced data on a continuous space and on the calculation of distances between them. In many practical instances, however, the exact point location, even if exactly known, is geo-masked to preserve confidentiality. This typically happens when dealing with confidential data related to individuals-health and their biometric parameters. In these situations, the estimation of the semivariogram and, hence, the spatial prediction can become biased and highly inefficient. This paper examines the extent of the bias in the particular case when the geo-masking mechanism is known (called “intentional locational error”) and lays the ground to a full understanding of the phenomenon in more general cases. We also examine how the geo-masking affects the estimation of the kriging variance thus reducing the efficiency of spatial prediction. We pursue our aims by developing some theoretical results and by making use of simulated and real data analysis.     

Mapping the possible occurrence of archaeological sites by Bayesian inference

Regional archaeological prospections are often done by field walking, where the location of the sampled fields is often determined by factors like feared disturbance or recent plowing. The resulting data configuration can be suboptimal for spatial prediction of the archaeological potential by geostatistical methods like kriging. As an alternative, we propose a Bayesian method to map the possible occurrence of archaeological finds and compare this to indicator regression kriging. Three types of predictive models were implemented in the Bayesian context following deductive, inductive and mixed approaches to use auxiliary geographical information in the mapping. After prediction of a validation set, it was concluded that the mixed approach gave the best results in terms of map quality, and that the kriging method performed poorly. Usage of data on the presence and the absence of archaeological finds is to be preferred above usage of presence data only. Furthermore, a method is presented that filters those parts of a predictive map that are not strongly supported by evidence.     

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.     

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.     

Exploring Spatial Patterns Using an Expanded Spatial Autocorrelation Framework

Spatial autocorrelation (SA) is regarded as an important dimension of spatial pattern. SA measures usually consist of two components: measuring the similarity of attribute values and defining the spatial relationships among observations. The latter component is often represented by a spatial weights matrix that predefines spatial relationship between observations in most measures. Therefore, SA measures, in essence, are measures of attribute similarity, conditioned by spatial relationship. Another dimension of spatial pattern can be explored by controlling observations to be compared based upon the degree of attribute similarity. The resulting measures are spatial proximity measures of observations, meeting predefined attribute similarity criteria. Proposed measures reflect degrees of clustering or dispersion for observations meeting certain levels of attribute similarity. An existing spatial autocorrelation framework is expanded to a general framework to evaluate spatial patterns and can accommodate the proposed approach measuring proximity. Analogous to the concept of variogram, clustergram is proposed to show the levels of spatial clustering over a range of attribute similarity, or attribute lags. Specific measures based on the proposed approach are formulated and applied to a hypothetical landscape and an empirical example, showing that these new measures capture spatial pattern information not reflected by traditional spatial autocorrelation measures.     

Properties of Tests for Spatial Dependence in Linear Regression Models

Based on a large number of Monte Carlo simulation experiments on a regular lattice, we compare the properties of Moran's I and Lagrange multiplier tests for spatial dependence, that is, for both spatial error autocorrelation and for a spatially lagged dependent variable. We consider both bias and power of the tests for six sample sizes, ranging from twenty-five to 225 observations, for different structures of the spatial weights matrix, for several underlying error distributions, for misspecified weights matrices, and for the situation where boundary effects are present. The results provide an indication of the sample sizes for which the asymptotic properties of the tests can be considered to hold. They also illustrate the power of the Lagrange multiplier tests to distinguish between substantive spatial dependence (spatial lag) and spatial dependence as a nuisance (error autocorrelation).     

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.     

良渚遗址群考古地理信息系统构建探索

运用GIS软件对良渚遗址群多源数据进行数字化处理,构建良渚遗址GIS应用模型,为进一步考古研究与文物保护提供更为科学的方法和便捷的管理手段。本文阐述良渚遗址群考古信息的空间数据库与属性信息库的建设,并简单介绍了数据采集与处理过程。     

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.     

An Introduction to the Network Weight Matrix

This study introduces the network weight matrix as a replacement for the spatial weight matrix to measure the spatial dependence between links of a network. This matrix stems from the concepts of betweenness centrality and vulnerability in network science. The elements of the matrix are a function not simply of proximity, but of network topology, network structure, and demand configuration. The network weight matrix has distinctive characteristics, which are capable of reflecting spatial dependence between traffic links: (1) elements are allowed to have negative and positive values capturing the competitive and complementary nature of links, (2) diagonal elements are not fixed to zero, which takes the self‐dependence of a link upon itself into consideration, and (3) elements not only reflect the spatial dependence based on the network structure, but they acknowledge the demand configuration as well. We verify the network weight matrix by modeling traffic flows in a 3 × 3 grid test network with 9 nodes and 24 directed links connecting 72 origin‐destination (OD) pairs. Models encompassing the network weight matrix outperform both models without spatial components and models with the spatial weight matrix. The network weight matrix represents a more accurate and defensible spatial dependency between traffic links, and offers the potential to augment traffic flow prediction.     

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.     

Implementing Spatial Data Analysis Software Tools in R

This article reports on work in progress on the implementation of functions for spatial statistical analysis, in particular of lattice/area data in the R language environment. The underlying spatial weights matrix classes, as well as methods for deriving them from data from commonly used geographical information systems are presented, handled using other contributed R packages. Since the initial release of some functions in 2001, and the release of the spdep package in 2002, experience has been gained in the use of various functions. The topics covered are the ingestion of positional data, exploratory data analysis of positional, attribute, and neighborhood data, and hypothesis testing of autocorrelation for univariate data. It also provides information about community building in using R for analyzing spatial data.     

Multiscalar Settlement Ecology Study of Piedmont Village Tradition Communities,A.D. 1000–1600

This research analyzes the spatial patterning of settlement sites in relation to landscape features to determine the factors that influenced settlement location choices for Late Precontact (A.D. 1000–1600) Piedmont Village Tradition (PVT) communities in the Yadkin, Dan, Haw, and Eno river valleys of the Piedmont Southeast. We employ geographic information systems to estimate characteristics of past landscapes, nearest neighbor analysis to describe basic settlement patterns, and discriminant function analysis to determine spatial correlations between settlements and landscape features. We examine the data on three scales and also assess potential changes over time. Results indicate that settlement location choices were broadly similar on the regional scale, but specific influences varied between and within valleys and over time. When examined with current archaeological, ethnohistoric, and linguistic information, the results suggest that PVT communities engaged in regional interaction networks in highly variable ways and that the relationship between subsistence and settlement varied according to settlement size. Using these results, we explore the roles PVT communities played in the formation and maintenance of natural and cultural landscapes in the Late Precontact Southeast.     

Adaptive Cell Tower Location Using Geostatistics. 基于地统计学的适应性基站发射塔选址研究

In this article, we address the problem of allocating an additional cell tower (or a set of towers) to an existing cellular network, maximizing the call completion probability. Our approach is derived from the adaptive spatial sampling problem using kriging, capitalizing on spatial correlation between cell phone signal strength data points and accounting for terrain morphology. Cell phone demand is reflected by population counts in the form of weights. The objective function, which is the weighted call completion probability, is highly nonlinear and complex (nondifferentiable and discontinuous). Sequential and simultaneous discrete optimization techniques are presented, and heuristics such as simulated annealing and Nelder–Mead are suggested to solve our problem. The adaptive spatial sampling problem is defined and related to the additional facility location problem. The approach is illustrated using data on cell phone call completion probability in a rural region of Erie County in western New York, and accounts for terrain variation using a line‐of‐sight approach. Finally, the computational results of sequential and simultaneous approaches are compared. Our model is also applicable to other facility location problems that aim to minimize the uncertainty associated with a customer visiting a new facility that has been added to an existing set of facilities.     

An ethnoarchaeological inductive model for predicting archaeological site location: A case-study of pastoral settlement patterns in the Val di Fiemme and Val di Sole (Trentino,Italian Alps)

In this paper I propose a novel integration of inductive predictive modelling and ethnoarchaeology. The case study concerns seasonal upland pastoral settlement patterns in the eastern Italian Alps. A sample of modern pastoral sites has been selected, and their relationships with environmental variables have been analyzed in order to create a model to predict the location of archaeological upland pastoral sites. The model has been tested with modern and archaeological control samples. It has proved to be useful for predicting the location of specific site categories. Ethnoarchaeological fieldwork has been carried out in the same alpine sample area, which has enhanced the interpretative potential of the proposed model, suggesting that the spatial of the analyzed sites could well be related to their dairying function. The creation of ethnoarchaeological locational models with a predictive potential may be very important not only in helping to tackle some theoretical and methodological problems in predictive modelling, but also in enhancing the importance of ethnoarchaeology in landscape archaeology projects.     

Investigating Global and Local Categorical Map Configuration Comparisons Based on Coincidence Matrices

The simple and intuitive nature of the coincidence matrix has not only made it the current "gold standard" for accuracy assessment (based on a sample of map pixels), but also a common tool for describing difference between two categorical maps (when all pixels are enumerated). It is this latter case of map comparison that this article explores. Coincidence matrices, although providing significant information regarding thematic agreement between two categorical maps (composition), can lack significantly in terms of conveying information about differences or similarities in the spatial arrangement (configuration) of those map categories in geographic space. This article introduces means for distilling the available configuration information from a coincidence matrix while demonstrating some simple categorical map comparisons. Specifically, while the coincidence matrix summarizes per-pixel compositional persistence or change, the introduced technique further quantifies the global and local configurational uncertainty between compared maps. I demonstrate how this quantification of configurational uncertainty can be used to gauge which thematic mismatch types are most significant and how to measure/present local configurational uncertainty in a spatial context. Implementation is through a straightforward mathematical algorithm in R that is illustrated by several examples.     

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.     

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.     

Analyzing the Relationship Between Smoking and Coronary Heart Disease at the Small Area Level: A Bayesian Approach to Spatial Modeling

We model the relationship between coronary heart disease and smoking prevalence and deprivation at the small area level using the Poisson log-linear model with and without random effects. Extra-Poisson variability (overdispersion) is handled through the addition of spatially structured and unstructured random effects in a Bayesian framework. In addition, four different measures of smoking prevalence are assessed because the smoking data are obtained from a survey that resulted in quite large differences in the size of the sample across the census tracts. Two of the methods use Bayes adjustments of standardized smoking ratios (local and global adjustments), and one uses a nonparametric spatial averaging technique. A preferred model is identified based on the deviance information criterion. Both smoking and deprivation are found to be statistically significant risk factors, but the effect of the smoking variable is reduced once the confounding effects of deprivation are taken into account. Maps of the spatial variability in relative risk, and the importance of the underlying covariates and random effects terms, are produced. We also identify areas with excess relative risk.