Abstract: This article optimizes a continuous area cartogram algorithm published in The Professional Geographer by Dougenik, Chrisman, and Niemeyer (DCN) in 1985. The DCN algorithm simulates a rubber sheet and is an iterative and approximate solution of cartogram construction. Although it remains popular because of its conceptual simplicity and cartographic quality, the DCN algorithm cannot completely preserve topology and its mathematical properties are inadequately explained. This article presents an optimization to the DCN algorithm, named Opti-DCN, with three improvements. First, it provides a mathematical condition for topology preservation. Second, new transformation equations that meet this condition are deduced from mathematics, which simultaneously optimize the global elasticity coefficient, a key parameter that greatly impacts the convergence rate of the rubber-sheet algorithm and the topological integrity of its generated cartograms. Last, the new algorithm simplifies the way of generating transforming f...

Abstract: We present a new circular-arc cartogram model in which countries are drawn as polygons with circular arcs instead of straight-line segments. Given a political map and values associated with each country in the map, a cartogram is a distorted map in which the areas of the countries are proportional to the corresponding values. In the circular-arc cartogram model straight-line segments can be replaced by circular arcs in order to modify the areas of the polygons, while the corners of the polygons remain fixed. The countries in circular-arc cartograms have the aesthetically pleasing appearance of clouds or snowflakes, depending on whether their edges are bent outwards or inwards. This makes it easy to determine whether a country has grown or shrunk, just by its overall shape. We show that determining whether a given map and given area-values can be realized as a circular-arc cartogram is an NP-hard problem. Next we describe a heuristic method for constructing circular-arc cartograms, which uses a max-flow computation on the dual graph of the map, along with a computation of the straight skeleton of the underlying polygonal decomposition. Our method is implemented and produces cartograms that, while not yet perfectly accurate, achieve many of the desired areas in our real-world examples.

Abstract: The World Map of Organic Agriculture presents countries as proportional in size to their share of the total of world organic hectares. Such a map can be referred to as an equal-area cartogram or a density-equalising map. Equal-density cartograms are a tool for presenting a fresh view of tabulated geographic data sets. The World Map of Organic Agriculture accounts for all of the hectares of organically managed agricultural land (certified organic and in-conversion organic) reported by FiBL and IFOAM in 'The World of Organic Agriculture - Statistics & Emerging Trends 2013'. The map visually reveals relationships between the territories of the world and highlights the regional strengths and weaknesses of the global diffusion of organic agriculture. The World Map of Organic Agriculture is generated by the Worldmapper GIS algorithm developed at the University of Sheffield as a cartographic visualisation tool.

Abstract: Most real data sets contain atypical observations, often referred to as outliers. Their presence may have a negative impact in data modeling using machine learning. This is particularly the case in data density estimation approaches. Manifold learning techniques pro- vide low-dimensional data representations, often oriented towards visual- ization. The visualization provided by density estimation manifold learn- ing methods can be compromised by the presence of outliers. Recently, a cartogram-based representation of model-generated distortion was pre- sented for nonlinear dimensionality reduction. Here, we investigate the impact of outliers on this visualization when using manifold learning tech- niques that behave robustly in their presence.

Abstract: Despite the rapid development of geoinformation technology and GIS - a classic cartogram is still widely used method for presenting geographic features and phenomena, especially with regard to the relative values connected with the basic fields. The aim of this article was to investigate how the size and shape of the different basic fields influence the results of the phenomenon presentation (in this case anthropogenic line forms). In the experiment were used fields in the shape of: square, hexagon, circle and triangle with different sizes: 1 km, 2 km, 4 km, 8 km and 10 km. Different field areas with the same height, but of a different shape affected to varied quantitative characteristics within them. However, different field heights have caused an obvious increase or decrease the detail of the results. To take a look at the image of the spatial distribution of line forms compared cartograms with another, independent method - kernel density analysis. After setting kernel density image with cartograms one turned out that basic fields shape did not change the image of spatial relations significantly and wellcharacterized them in general. For this study area the best results obtained after the application fields with heights of 2 km and 4 km in the shape of squares and hexagons. It appears that the hexagons better than squares reflect the spatial image of the forms (hexagons allow better representation of the directions and shapes of the studied phenomena), however, they are less common in a geostatistical researches, and that's why they are more difficult to use, especially for comparative analysis.

Abstract: Telecommunication companies compete in increasingly aggr essive markets. Avoiding customer defection, or churn, should be at the core of successful management in such conte xt. These companies store and manage abundant customer usage data. Their analysis using advance d techniques can be a source of valuable insight into customers’ behavior over time. Exploratory data visua lization can help in this task. Many important contributions to multivariate data visualization using no nlinear techniques have recently been made. In this paper, we analyze a database of customer landline telephone usage in Brazil. These data are first visualized using a nonlinear manifold learning model, Generative Topo graphic Mapping (GTM). This visualization is enhanced using a cartogram technique, inspired in geograph ical representation methods, that reintroduces the local nonlinear distortion into the representation space. Y t another geographical information visualization technique, namely the Flow Maps, is then used to visualize cu stomer migrations over time periods in the GTM data representation space. The experimental results sh own in this paper provide evidence to support that the use of these methods can assist experts in the process of u eful knowledge extraction, with an impact on customer retention management strategies.

Abstract: In most applications cartograms are still used as unusual visualisations, so that this element of unconventional and sometimes provocative mapping remains in the focus of cartographic display. This may explain the reluctant use of cartograms as a basemap to show additional information. The potential of using gridded cartograms as both, a basemap, and an adaptable map projection for a multitude of applications has been demonstrated in the previous chapters. Gridded cartograms open a wide range of new applications that redraw the diverse geographies of the world. To show that potential requires further demonstrations of how this technique can be applied to uses that are of broader value to more than an enthusiastic group of peculiar geographers. While the previous chapters looked at conceptual issues and a general assessment of the cartographic values of the gridded cartogram technique, this chapter employs the technique to a broader range of application areas.

Abstract: The global competitiveness index (GCI) of a country is a comprehensive indicator to measure its social-economic development, sustainability and prospects reflecting national competitiveness by World Economic Forum (WEF, 2013). Thematic maps are commonly used to visualize such geo-distributed data. Cartogram resizes the area of polygons according to a specific variable providing a new perspective to understand the distribution of country's competitiveness on world. Essentially, cartogram presents a different cognitive space which differs from the well-known geographic space. In this paper, we generate the GCI cartogram with a combination of color scheme and compare it with the traditional thematic map to gain a better understanding of the distribution of country’s competitiveness.

Abstract: A visual demonstration of the benefits behind using cartograms, over the traditional choropleth map. Poster shows iGeology app activity in cartogram format.

Abstract: Model interpretability is a problem for multivariate data in general and, very specifically, for dimensionality reduction techniques as applied to data visualization. The problem is even bigger for nonlinear dimensionality reduction (NLDR) methods, to which interpretability limitations are consubstantial. Data visualization is a key process for knowledge extraction from data that helps us to gain insights into the observed data structure through graphical representations and metaphors. NLDR techniques provide flexible visual insight, but the locally varying representation distor- tion they generate makes interpretation far from intuitive. For some NLDR models, indirect quantitative measures of this mapping distortion can be calculated explicitly and used as part of an interpretative post-processing of the results. In this Master Thesis, we apply a cartogram method, inspired on techniques of geographic representation, to the purpose of data visualization using NLDR models. In particular, we show how this method allows reintroducing the distortion, measured in the visual maps of several self-organizing clustering methods. The main capabilities and limitations of the cartogram visualization of multivariate data using standard and hierarchical self-organizing models were investigated in some detail with artificial data as well as with real information stemming from a neuro-oncology problem that involves the discrimination of human brain tumor types, a problem for which knowledge dis- covery techniques in general, and data visualization in particular should be useful tools.

Abstract: The approach for the creation of gridded cartograms has been introduced as an alternative concept for a more accurate way of using cartogram techniques. The method introduced is not entirely new, but builds on and aims to improve the use of an existing diffusion-based density-equalising algorithm. The resulting maps created with this new approach need to be evaluated to assess the success and value of the methodology in comparison to other approaches. This chapter introduces the basic results for an alternative mapping of population distribution as the basic outcome of the methodology and evaluates, whether the outcomes constitute an improvement of the existing Worldmapper population cartogram.

Abstract: Gridded population cartograms are an alternative way of applying the principle of value-by-area maps with a higher geographical accuracy that provide novel analytical and visualisation capabilities. The previous chapter assessed these basic features of the new cartograms in relation to population at different scales and resolutions. The results led to the conclusion, that gridded cartograms embody a conformal reprojection where the connecting areas of the grid connect each grid cell seamlessly and at almost square angles and make gridded cartograms more similar to a (unusual) map projection rather than being a sketchy map like other cartograms. This characteristic requires further investigation into whether gridded cartograms can provide a reliable base for a map projection.

Abstract: In a rectilinear dual of a planar graph vertices are represented by simple rectilinear polygons, while edges are represented by side-contact between the corresponding polygons. A rectilinear dual is called a cartogram if the area of each region is equal to a pre-specified weight. The complexity of a cartogram is determined by the maximum number of corners (or sides) required for any polygon. In a series of papers the polygonal complexity of such representations for maximal planar graphs has been reduced from the initial 40 to 34, then to 12 and very recently to the currently best known 10. Here we describe a construction with 8-sided polygons, which is optimal in terms of polygonal complexity as 8-sided polygons are sometimes necessary. Specifically, we show how to compute the combinatorial structure and how to refine it into an area-universal rectangular layout in linear time. The exact cartogram can be computed from the area-universal layout with numerical iteration, or can be approximated with a hill-climbing heuristic. We also describe an alternative construction of cartograms for Hamiltonian maximal planar graphs, which allows us to directly compute the cartograms in linear time. Moreover, we prove that even for Hamiltonian graphs 8-sided rectilinear polygons are necessary, by constructing a non-trivial lower bound example. The complexity of the cartograms can be reduced to 6 if the Hamiltonian path has the extra property that it is one-legged, as in outer-planar graphs. Thus, we have optimal representations (in terms of both polygonal complexity and running time) for Hamiltonian maximal planar and maximal outer-planar graphs. Finally we address the problem of constructing small-complexity cartograms for 4-connected graphs (which are Hamiltonian). We first disprove a conjecture, posed by two set of authors, that any 4-connected maximal planar graph has a one-legged Hamiltonian cycle, thereby invalidating an attempt to achieve a polygonal complexity 6 in cartograms for this graph class. We also prove that it is NP-hard to decide whether a given 4-connected plane graph admits a cartogram with respect to a given weight function.