Optimization-Based Construction of Quadrilateral Table Cartograms

A quadrilateral table cartogram is a rectangle-shaped figure that visualizes table-form data; quadrilateral cells in a table cartogram are transformed to express the magnitude of positive weights by their areas, while maintaining the adjacency of cells in the original table. However, the previous construction method is difficult to implement because it consists of multiple operations that do not have a unique solution and require complex settings to obtain the desired outputs. In this article, we propose a new construction for quadrilateral table cartograms by recasting the construction as an optimization problem. The proposed method is formulated as a simple minimization problem to achieve mathematical clarity. It can generate quadrilateral table cartograms with smaller deformation of rows and columns, thereby aiding readers to recognize the correspondence between table cartograms and original tables. In addition, we also propose a means of sorting rows and/or columns prior to the construction of table cartograms to reduce excess shape deformation. Applications of the proposed method confirm its capability to output table cartograms that clearly visualize the characteristics of datasets

Computing Fast and Scalable Table Cartograms for Large Tables

Given an m x n table T of positive weights and a rectangle R with an area equal to the sum of the weights, a table cartogram computes a partition of R into m x n convex quadrilateral faces such that each face has the same adjacencies as its corresponding cell in T, and has an area equal to the cell's weight. In this thesis, we explored different table cartogram algorithms for a large table with thousands of cells and investigated the potential applications of large table cartograms. We implemented Evans et al.'s table cartogram algorithm that guarantees zero area error and adapted a diffusion-based cartographic transformation approach, FastFlow, to produce large table cartograms. We introduced a constraint optimization-based table cartogram generation technique, TCarto, leveraging the concept of force-directed layout. We implemented TCarto with column-based and quadtree-based parallelization to compute table cartograms for table with thousands of cells. We presented several potential applications of large table cartograms to create the diagrammatic representations in various real-life scenarios, e.g., for analyzing spatial correlations between geospatial variables, understanding clusters and densities in scatterplots, and creating visual effects in images (i.e., expanding illumination, mosaic art effect). We presented an empirical comparison among these three table cartogram techniques with two different real-life datasets: a meteorological weather dataset and a US State-to-State migration flow dataset. FastFlow and TCarto both performed well on the weather data table. However, for US State-to-State migration flow data, where the table contained many local optima with high value differences among adjacent cells, FastFlow generated concave quadrilateral faces. We also investigated some potential relationships among different measurement metrics such as cartographic error (accuracy), the average aspect ratio (the readability of the visualization), computational speed, and the grid size of the table. Furthermore, we augmented our proposed TCarto with angle constraint to enhance the readability of the visualization, conceding some cartographic error, and also inspected the potential relationship of the restricted angles with the accuracy and the readability of the visualization. In the output of the angle constrained TCarto algorithm on US State-to-State migration dataset, it was difficult to identify the rows and columns for a cell upto 20 degree angle constraint, but appeared to be identifiable for more than 40 degree angle constraint

The State of the Art in Cartograms

Cartograms combine statistical and geographical information in thematic maps, where areas of geographical regions (e.g., countries, states) are scaled in proportion to some statistic (e.g., population, income). Cartograms make it possible to gain insight into patterns and trends in the world around us and have been very popular visualizations for geo-referenced data for over a century. This work surveys cartogram research in visualization, cartography and geometry, covering a broad spectrum of different cartogram types: from the traditional rectangular and table cartograms, to Dorling and diffusion cartograms. A particular focus is the study of the major cartogram dimensions: statistical accuracy, geographical accuracy, and topological accuracy. We review the history of cartograms, describe the algorithms for generating them, and consider task taxonomies. We also review quantitative and qualitative evaluations, and we use these to arrive at design guidelines and research challenges

Algorithms for cartographic visualization

Maps are effective tools for communicating information to the general public and help people to make decisions in, for example, navigation, spatial planning and politics. The mapmaker chooses the details to put on a map and the symbols to represent them. Not all details need to be geographic: thematic maps, which depict a single theme or attribute, such as population, income, crime rate, or migration, can very effectively communicate the spatial distribution of the visualized attribute. The vast amount of data currently available makes it infeasible to design all maps manually, and calls for automated cartography. In this thesis we presented efficient algorithms for the automated construction of various types of thematic maps. In Chapter 2 we studied the problem of drawing schematic maps. Schematic maps are a well-known cartographic tool; they visualize a set of nodes and edges (for example, highway or metro networks) in simplified form to communicate connectivity information as effectively as possible. Many schematic maps deviate substantially from the underlying geography since edges and vertices of the original network are moved in the simplification process. This can be a problem if we want to integrate the schematized network with a geographic map. In this scenario the schematized network has to be drawn with few orientations and links, while critical features (cities, lakes, etc.) of the base map are not obscured and retain their correct topological position with respect to the network. We developed an efficient algorithm to compute a collection of non-crossing paths with fixed orientations using as few links as possible. This algorithm approximates the optimal solution to within a factor that depends only on the number of allowed orientations. We can also draw the roads with different thicknesses, allowing us to visualize additional data related to the roads such as trafic volume. In Chapter 3 we studied methods to visualize quantitative data related to geographic regions. We first considered rectangular cartograms. Rectangular cartograms represent regions by rectangles; the positioning and adjacencies of these rectangles are chosen to suggest their geographic locations to the viewer, while their areas are chosen to represent the numeric values being communicated by the cartogram. One drawback of rectangular cartograms is that not every rectangular layout can be used to visualize all possible area assignments. Rectangular layouts that do have this property are called area-universal. We show that area-universal layouts are always one-sided, and we present algorithms to find one-sided layouts given a set of adjacencies. Rectangular cartograms often provide a nice visualization of quantitative data, but cartograms deform the underlying regions according to the data, which can make the map virtually unrecognizable if the data value differs greatly from the original area of a region or if data is not available at all for a particular region. A more direct method to visualize the data is to place circular symbols on the corresponding region, where the areas of the symbols correspond to the data. However, these maps, so-called symbol maps, can appear very cluttered with many overlapping symbols if large data values are associated with small regions. In Chapter 4 we proposed a novel type of quantitative thematic map, called necklace map, which overcomes these limitations. Instead of placing the symbols directly on a region, we place the symbols on a closed curve, the necklace, which surrounds the map. The location of a symbol on the necklace should be chosen in such a way that the relation between symbol and region is as clear as possible. Necklace maps appear clear and uncluttered and allow for comparatively large symbol sizes. We developed algorithms to compute necklace maps and demonstrated our method with experiments using various data sets and maps. In Chapter 5 and 6 we studied the automated creation of ow maps. Flow maps are thematic maps that visualize the movement of objects, such as people or goods, between geographic regions. One or more sources are connected to several targets by lines whose thickness corresponds to the amount of ow between a source and a target. Good ow maps reduce visual clutter by merging (bundling) lines smoothly and by avoiding self-intersections. We developed a new algorithm for drawing ow trees, ow maps with a single source. Unlike existing methods, our method merges lines smoothly and avoids self-intersections. Our method is based on spiral trees, a new type of Steiner trees that we introduced. Spiral trees have an angle restriction which makes them appear smooth and hence suitable for drawing ow maps. We study the properties of spiral trees and give an approximation algorithm to compute them. We also show how to compute ow trees from spiral trees and we demonstrate our approach with extensive experiments

Interactive visualization for missing values, time series, and areal data

Visualization is widely used to explore data, examine variation, reveal trends, and diagnose models. Furthermore, interactive plots can re-focus the view to features of interest, drill down into a fine resolution, query or lookup elements, look at data from various directions, and connect plots with model analysis. However, for specific data types and specific exploratory purposes, the general interactions like brushing, panning, zooming, and querying can be insufficient. The lack of a grammar for interactive graphics makes differences between the user interactions on data and on the view of data difficult to delineate. This thesis partially addresses these issues and fills gaps in methodology from three application areas: missing values, temporal/longitudinal data, and areal data. Interactive graphics plays different roles in three areas. In missing data analysis, many imputation methods have been developed but little has been done for exploring the missing value structure to determine the missingness pattern, or to evaluate the imputations. This research addresses this gap, focusing on an interactive tool to explore missings, check the missingness assumptions, and compare imputation methods. For temporal and longitudinal data, using static plots is inadequate for exploring the trends, seasonality or unusual individuals, especially when the data set is large. This research develops special interactions and discusses the elements and pipeline in the interactivity construction. It is implemented in the R package, cranvastime, with details on how to use for a number of datasets. For the areal data, cartograms are widely used but there is no universally good algorithm for cartogram construction or evaluation. This research proposes an evaluation criterion and utilizes an interactive interface to optimize the visualization between the original shape-reserved map and area-reserved cartogram

Kahden muuttujan sävyjen sekoitus – väriasteikoiden suunnittelu kahden muuttujan koropleettikartoille räätälöidyllä työkalulla

Bivariate maps are a type of map visualization where two related data series are displayed at once for each data point. They can answer questions of how two variables interrelate in a geographical context using several kinds of encodings — visual variables — such as shape or color. The most common types are choropleth maps that use color hue and lightness to encode data and symbol-based maps that use shape size for both data series. Bivariate maps have seen a minor surge in popularity with new software tools but remain an understudied visualization type with a lack of clear usage recommendations. The thesis consists of a theoretical and a practical part. The purpose was to collate existing recommendations about the design of bivariate maps and determine whether they are considered a useful type of visualization. The theoretical part was a literature survey of relevant visualization and cartography literature, including empirical studies. I also sought to see whether bivariate choropleths are considered more effective than other types. The practical part was building a web tool prototype for bivariate color scale creation limited to choropleth maps, the Bivariate hue blender. The tool uses the Hue-Chroma-Lightness (HCL) color space for scheme design. By rotating the hue angle of an input color by a user-defined amount, a new color can be created. Intermediate colors are generated by blending these two with each other and a light secondary input color. The primary purpose of the tool was to improve color scheme creation and the building process used the framework of research-based design. It involved building the tool, using it to evaluate seven existing palettes, and creating three new palettes. These were applied to four different bivariate maps using statistical data from Finland in two different geographical divisions. Test data was selected using contingency table visualizations to ensure that all classes contain values. In addition to the color scales, a bivariate ordinal texture design was created. Bivariate maps were found to be grouped in categories using the concept of integral and separable dimensions. Bivariate choropleth maps were found to be a relevant visualization type, provided that the data is suitable, and that the number of classes is no larger than 9. An issue pertaining to color contrast was identified — accessibility guidelines stipulate a lightness difference between adjacent hues that require the use of strokes in most choropleth maps. Questions concerning effectiveness of other types, how bivariate symbols interact and how viewers can use bivariate maps for analytical tasks remain unresolved. The tool was subjectively found to enable better control over bivariate color scale creation than other similar software. The evaluated bivariate palettes had issues in lightness uniformity and separation of colors, which could be resolved in the three new palettes. These were found to be at least as practical as the seven initial palettes. This work has concluded that bivariate maps can be considered useful in special cases with the right data, which should encourage visualization designers to employ them. It has contributed a prototype tool that aids the creation of new perceptually uniform color scales for bivariate choropleth maps. Three new colorblind-safe 3×3 palettes are an addition to the limited set of schemes in active use. The method of selecting data using contingency tables can aid in creating bivariate maps.Kahden muuttujan tietokartat ovat visualisointityyppi, jossa kaksi toisiinsa liittyvää tietosarjaa näytetään kunkin datapisteen kohdalla. Niillä voidaan tutkia kuinka kaksi muuttujaa ovat yhteydessä toisiinsa maantieteellisessä kontekstissa, käyttämällä useita erilaisia visuaalisia muuttujia – kuten muotoa tai väriä. Yleisimpiä tyyppejä ovat koropleettikartat, joissa käytetään värin sävyä ja vaaleutta tietojen esittämiseen, sekä symbolikartat, joissa käytetään muodon kokoa molemmille datasarjoille. Kahden muuttujan karttojen suosio on kasvanut uusien ohjelmistotyökalujen myötä, mutta ne ovat edelleen vähän tutkittu visualisointityyppi, josta puuttuvat selkeät käyttösuositukset. Opinnäytetyöni koostuu teoreettisesta ja käytännön osasta. Tarkoituksena on ollut koota olemassa olevia suosituksia kahden muuttujan kartoista ja selvittää, pidetäänkö niitä hyödyllisenä visualisointityyppinä. Teoriaosuus on kirjallisuuskatsaus visualisointi- ja kartografiakirjallisuuteen, mukaan luettuna myös empiiriset tutkimukset. Pyrin myös selvittämään, pidetäänkö kahden muuttujan koropleettikarttoja tehokkaampina kuin muita kahden muuttujan karttatyyppejä. Käytännön osuus on verkkotyökalun prototyyppi, Bivariate hue blender, joka on tehty kahden muuttujan väriasteikkojen luomista varten. Työkalu käyttää Hue-Chroma-Lightness (HCL; sävy, kromaattisuus, vaaleus) -väriavaruutta. Kun syötetyn värin sävykulmaa kääntää, syntyy uusi väri. Alkuperäisestä ja uudesta väristä luodaan kaksi erillistä väriasteikkoa vaaleasta aloitussävystä ja näitä yhdistämällä muodostetaan asteikon välivärit. Työkalun ensisijaisena tarkoituksena on ollut helpottaa väriasteikkojen luomista. Sen kehittämisessä on sovellettu tutkimukseen perustuvaa suunnittelua. Työkalun avulla on arvioitu seitsemän palettia ja luotu kolme uutta. Näitä on sovellettu neljään erilaiseen kahden muuttujan karttaan, joissa on käytetty tilastotietoja Suomesta kahden eri maantieteellisen jaon mukaan. Väriasteikkojen lisäksi on luotu kuviotekstuuri. Tutkimuksessa todetaan, että kahden muuttujan kartat voidaan jakaa luokkiin käyttäen kokonaisten ja eroteltavien ulottuvuuksien käsitettä. Koropleettikarttojen todetaan olevan toimiva laji, kunhan aineisto on sopiva ja luokkia enintään yhdeksän. Työssä tunnistettiin värikontrastiin liittyvä ongelma – esteettömyysohjeissa määrätyt vierekkäisten sävyjen vaaleuserot edellyttävät ääriviivojen käyttöä useimmissa kartoissa. Tutkimuksessa auki jäävät kysymykset koskevat muiden tyyppien tehokkuutta, kaksimuuttujaisten symbolien vuorovaikutusta ja sitä, kuinka katsoja lukee ja käyttää näitä karttoja. Työkalun voidaan todeta subjektiivisesti mahdollistavan paremman hallinnan kaksimuuttujaväriasteikkojen luomisessa vastaaviin ohjelmiin verrattuna. Arvioiduissa paleteissa oli ongelmia vaaleuden tasaisuudessa ja värien erottelussa, jotka nyt voitiin ratkaista kolmessa uudessa paletissa. Näiden todetaan olevan ainakin yhtä käytännöllisiä kuin seitsemän alkuperäistä palettia. Työn loppupäätelmä on, että kaksimuuttujaisia karttoja voidaan pitää hyödyllisinä tietyissä tapauksissa ja niihin soveltuvalla datalla, mikä voi kannustaa visualisointisuunnittelijoita käyttämään niitä. Työn tuloksena on prototyyppityökalu, joka auttaa luomaan uusia tasajakoisia väriskaaloja kahden muuttujan koropleettikarttoja varten. Kolme uutta palettia on lisäys aktiivisessa käytössä olevien kaksimuuttujaisten palettien rajalliseen joukkoon. Kontingenssitaulukoihin perustuva aineiston valintamenetelmä voi auttaa suunnittelijoita kahden muuttujan karttojen luomisessa

Visualizing Set Relations and Cardinalities Using Venn and Euler Diagrams

In medicine, genetics, criminology and various other areas, Venn and Euler diagrams are used to visualize data set relations and their cardinalities. The data sets are represented by closed curves and the data set relationships are depicted by the overlaps between these curves. Both the sets and their intersections are easily visible as the closed curves are preattentively processed and form common regions that have a strong perceptual grouping effect. Besides set relations such as intersection, containment and disjointness, the cardinality of the sets and their intersections can also be depicted in the same diagram (referred to as area-proportional) through the size of the curves and their overlaps. Size is a preattentive feature and so similarities, differences and trends are easily identified. Thus, such diagrams facilitate data analysis and reasoning about the sets. However, drawing these diagrams manually is difficult, often impossible, and current automatic drawing methods do not always produce appropriate diagrams. This dissertation presents novel automatic drawing methods for different types of Euler diagrams and a user study of how such diagrams can help probabilistic judgement. The main drawing algorithms are: eulerForce, which uses a force-directed approach to lay out Euler diagrams; eulerAPE, which draws area-proportional Venn diagrams with ellipses. The user study evaluated the effectiveness of area- proportional Euler diagrams, glyph representations, Euler diagrams with glyphs and text+visualization formats for Bayesian reasoning, and a method eulerGlyphs was devised to automatically and accurately draw the assessed visualizations for any Bayesian problem. Additionally, analytic algorithms that instantaneously compute the overlapping areas of three general intersecting ellipses are provided, together with an evaluation of the effectiveness of ellipses in drawing accurate area-proportional Venn diagrams for 3-set data and the characteristics of the data that can be depicted accurately with ellipses

Efficient Algorithms for Graph-Theoretic and Geometric Problems

This thesis studies several different algorithmic problems in graph theory and in geometry. The applications of the problems studied range from circuit design optimization to fast matrix multiplication. First, we study a graph-theoretical model of the so called ''firefighter problem''. The objective is to save as much as possible of an area by appropriately placing firefighters. We provide both new exact algorithms for the case of general graphs as well as approximation algorithms for the case of planar graphs. Next, we study drawing graphs within a given polygon in the plane. We present asymptotically tight upper and lower bounds for this problem Further, we study the problem of Subgraph Isormorphism, which amounts to decide if an input graph (pattern) is isomorphic to a subgraph of another input graph (host graph). We show several new bounds on the time complexity of detecting small pattern graphs. Among other things, we provide a new framework for detection by testing polynomials for non-identity with zero. Finally, we study the problem of partitioning a 3D histogram into a minimum number of 3D boxes and it's applications to efficient computation of matrix products for positive integer matrices. We provide an efficient approximation algorithm for the partitioning problem and several algorithms for integer matrix multiplication. The multiplication algorithms are explicitly or implicitly based on an interpretation of positive integer matrices as 3D histograms and their partitions

Generalization of Central Place Theory Using Mathematical Programming

首都大学東京, 2015-03-25, 博士（理学）, 乙第101号首都大学東

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version