Barry J. Kronenfeld

A Classification Method for Choropleth Maps Incorporating Data Reliability Information

Observations assigned to any two classes in a choropleth map are expected to have attribute values that are different. Their values might not be statistically different, however, if the data are gathered from surveys, such as the American Community Survey, in which estimates have sampling error. This article presents an approach to determine class breaks using the class separability criterion, which refers to the levels of certainty that values in different classes are statistically different from each other. Our procedure determines class breaks that offer the highest levels of separability given the desired number of classes. The separability levels of all class breaks are included in a legend design to show the statistical likelihood that values on two sides of each class break are different. The legend and the associated separability information offer map readers crucial information about the reliability of the spatial patterns that could result from the chosen classification method.

Gradation as a Communication Device in Area-Class Maps

This paper proposes a methodology to assess gradation as a cartographic tool for communicating information in area-class maps. The communication model is used as a theoretical foundation, suggesting distinction between errors that occur in encoding and decoding of geographic information. The proposed methodology begins with the determination of a target level of encoding error. Map alternatives are constrained to achieve this target, with gradation considered as one variable in the map production process. The result is a series of maps of equal encoding accuracy but varying in the degree of gradation represented. The individual maps of the series can then be evaluated in terms of decoding accuracy. The methodology is demonstrated by producing a series of alternative forest region maps of New York, Pennsylvania, and New Jersey based on U.S. Forest Service data on tree genus distributions. The series ranges from a 4-class graded area-class map to a 13-class crisp map. The results show gradation to be a viable alternative to the proliferation of map classes as a means of cartographic communication.

A heuristic multi-criteria classification approach incorporating data quality information for choropleth mapping

ABSTRACT Despite conceptual and technology advancements in cartography over the decades, choropleth map design and classification fail to address a fundamental issue: estimates that are statistically indifferent may be assigned to different classes on maps or vice versa. Recently, the class separability concept was introduced as a map classification criterion to evaluate the likelihood that estimates in two classes are statistical different. Unfortunately, choropleth maps created according to the separability criterion usually have highly unbalanced classes. To produce reasonably separable but more balanced classes, we propose a heuristic classification approach to consider not just the class separability criterion but also other classification criteria such as evenness and intra-class variability. A geovisual-analytic package was developed to support the heuristic mapping process to evaluate the trade-off between relevant criteria and to select the most preferable classification. Class break values can be adjusted to improve the performance of a classification.

Beyond the epsilon band: polygonal modeling of gradation/uncertainty in area-class maps

A spatial modeling technique is proposed to represent boundary uncertainty or gradation on area-class maps using a simple polygon tessellation with designated zones of indeterminacy or transition zones. The transition zone can be conceptualized as a dual of the epsilon band, but is more flexible and allows for a wide range of polygonal configurations, including polygons with sinuous boundaries, spurs, three-way transition zones, and null polygons. The model is specified using the medial axis to capture the general shape characteristics of a transition zone. Graph theoretic representation of an extended version of the medial axis captures key junctions in both shape and classification and is used to identify well-formed transition zones that can be logically and unambiguously handled by the model. A multivariate classification surface is specified by first defining degrees or probabilities of membership at every point on the medial axis and transition zone boundary. Degrees or probabilities of membership at all other points are defined by linear interpolation. The technique is illustrated with an example of a complex transition zone, and a simple isoline representation that can be derived from the model is presented. The proposed modeling technique promises to facilitate expert characterization of soil formations, ecological systems, and other types of areal units where gradation and/or boundary uncertainty are prevalent.

Restricted random labeling: testing for between-group interaction after controlling for joint population and within-group spatial structure

Abstract Statistical measures of spatial interaction between multiple types of entities are commonly assessed against a null model of either toroidal shift (TS), which controls for spatial structure of individual subpopulations, or random labeling (RL), which controls for spatial structure of the joint population. Neither null model controls for both types of spatial structure simultaneously, although this may sometimes be desirable when more than two subpopulations are present. To address this, we propose a flexible framework for specifying null models that we refer to as restricted random labeling (rRL). Under rRL, a specified subset of individuals is restricted and other individuals are randomly relabeled. Within this framework, two specific null models are proposed for pairwise analysis within populations consisting of three or more subpopulations, to simultaneously control for spatial structure in the joint population and one or the other of the two subpopulations being analyzed. Formulas are presented for calculating expected nearest neighbor counts and co-location quotients within the proposed framework. Differences between TS, RL and rRL are illustrated by application to six types of generating processes in a simulation study, and to empirical datasets of tree species in a forest and crime locations in an urban setting. These examples show that rRL null models are typically stricter than either TS or RL, which often detect “interactions” that are an expected consequence either of the joint population pattern or of individual subpopulation patterns.

Cartographic techniques for communicating class separability: enhanced choropleth maps of median household income, Iowa

In mapping population characteristics, data are usually portrayed as accurate without error. However, many population datasets provide estimates derived from surveys or samples, and a certain level of uncertainty is associated with each estimate. Ignoring estimated uncertainty information in mapping may produce misleading maps and generate spurious spatial patterns. In this paper, we introduce a measure of separability to indicate the likelihood that units assigned to different classes are truly different statistically. A series of map symbolization techniques is proposed to communicate class separability to the cartographer or map reader, and presented in four series of maps of American Community Survey data on median household income for Iowa counties. These map series illustrate several different techniques: a legend designed to communicate separability between classes, graduated line symbols to communicate separability between individual map units, and a color scheme in which perceptual color differences are related to class separability. Each map series presents three alternative classifications to illustrating how the proposed symbolization techniques could assist a cartographer in choosing the more preferable classification scheme.

Manual construction of continuous cartograms through mesh transformation

ABSTRACT A computer-assisted framework is proposed to support the manual construction of cartograms. The framework employs a joint triangulation, similar to that used in rubber-sheeting, to define a piecewise affine transformation between map space and cartogram space. This guarantees preservation of all topological relations within and among transformed datasets with insertion of a finite number of points. To support intuitive user control of cartogram appearance, methods are developed to translate generically defined user adjustments of the cartogram into mesh vertex positions on either the source map mesh or cartogram mesh. The framework is implemented in a working prototype application and used to create sample cartograms of the USA and China. Results are compared with cartograms produced using diffusion and carto3f algorithms in terms of accuracy, aesthetic appearance, and approximate construction time. Qualitative aspects of the manual construction process are also discussed.

Visualizing statistical significance of disease clusters using cartograms

BackgroundHealth officials and epidemiological researchers often use maps of disease rates to identify potential disease clusters. Because these maps exaggerate the prominence of low-density districts and hide potential clusters in urban (high-density) areas, many researchers have used density-equalizing maps (cartograms) as a basis for epidemiological mapping. However, we do not have existing guidelines for visual assessment of statistical uncertainty. To address this shortcoming, we develop techniques for visual determination of statistical significance of clusters spanning one or more districts on a cartogram. We developed the techniques within a geovisual analytics framework that does not rely on automated significance testing, and can therefore facilitate visual analysis to detect clusters that automated techniques might miss.ResultsOn a cartogram of the at-risk population, the statistical significance of a disease cluster is determinate from the rate, area and shape of the cluster under standard hypothesis testing scenarios. We develop formulae to determine, for a given rate, the area required for statistical significance of a priori and a posteriori designated regions under certain test assumptions. Uniquely, our approach enables dynamic inference of aggregate regions formed by combining individual districts. The method is implemented in interactive tools that provide choropleth mapping, automated legend construction and dynamic search tools to facilitate cluster detection and assessment of the validity of tested assumptions. A case study of leukemia incidence analysis in California demonstrates the ability to visually distinguish between statistically significant and insignificant regions.ConclusionThe proposed geovisual analytics approach enables intuitive visual assessment of statistical significance of arbitrarily defined regions on a cartogram. Our research prompts a broader discussion of the role of geovisual exploratory analyses in disease mapping and the appropriate framework for visually assessing the statistical significance of spatial clusters.

An interactive mapping system incorporating data reliability information

Data quality should be considered in compiling maps in order to reveal reliable information about the spatial variation of a phenomenon. However, creating classes in a choropleth map by maximizing data reliability (i.e. the statistical differences of observed values between classes) often lead to useless maps with very uneven number of observations in different classes. An interactive mapping system is developed to allow users to adjust class breaks so that the resultant maps are relatively reliable and more effective in spatial pattern detection.