Computing the Region Areas of Euler Diagrams Drawn with Three Ellipses

Ellipses generate accurate area-proportional Euler diagrams for more data than is possible with circles. However, computing the region areas is difficult as ellipses have various degrees of freedom. Numerical methods could be used, but approximation errors are introduced. Current analytic methods are limited to computing the area of only two overlapping ellipses, but area-proportional Euler diagrams in diverse application areas often have three curves. This paper provides an overview of different methods that could be used to compute the region areas of Euler diagrams drawn with ellipses. We also detail two novel analytic algorithms to instantaneously compute the exact region areas of three general overlapping ellipses. One of the algorithms decomposes the region of interest into ellipse segments, while the other uses integral calculus. Both methods perform equally well with respect to accuracy and time

The State-of-the-Art of Set Visualization

Sets comprise a generic data model that has been used in a variety of data analysis problems. Such problems involve analysing and visualizing set relations between multiple sets defined over the same collection of elements. However, visualizing sets is a non-trivial problem due to the large number of possible relations between them. We provide a systematic overview of state-of-the-art techniques for visualizing different kinds of set relations. We classify these techniques into six main categories according to the visual representations they use and the tasks they support. We compare the categories to provide guidance for choosing an appropriate technique for a given problem. Finally, we identify challenges in this area that need further research and propose possible directions to address these challenges. Further resources on set visualization are available at http://www.setviz.net

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

Drawing Area-Proportional Euler Diagrams Representing Up To Three Sets

Area-proportional Euler diagrams representing three sets are commonly used to visualize the results of medical experiments, business data, and information from other applications where statistical results are best shown using interlinking curves. Currently, there is no tool that will reliably visualize exact area-proportional diagrams for up to three sets. Limited success, in terms of diagram accuracy, has been achieved for a small number of cases, such as Venn-2 and Venn-3 where all intersections between the sets must be represented. Euler diagrams do not have to include all intersections and so permit the visualization of cases where some intersections have a zero value. This paper describes a general, implemented, method for visualizing all 40 Euler-3 diagrams in an area-proportional manner. We provide techniques for generating the curves with circles and convex polygons, analyze the drawability of data with these shapes, and give a mechanism for deciding whether such data can be drawn with circles. For the cases where non-convex curves are necessary, our method draws an appropriate diagram using non-convex polygons. Thus, we are now always able to automatically visualize data for up to three sets

VennPlex--a novel Venn diagram program for comparing and visualizing datasets with differentially regulated datapoints.

With the development of increasingly large and complex genomic and proteomic data sets, an enhancement in the complexity of available Venn diagram analytical programs is becoming increasingly important. Current freely available Venn diagram programs often fail to represent extra complexity among datasets, such as regulation pattern differences between different groups. Here we describe the development of VennPlex, a program that illustrates the often diverse numerical interactions among multiple, high-complexity datasets, using up to four data sets. VennPlex includes versatile output features, where grouped data points in specific regions can be easily exported into a spreadsheet. This program is able to facilitate the analysis of two to four gene sets and their corresponding expression values in a user-friendly manner. To demonstrate its unique experimental utility we applied VennPlex to a complex paradigm, i.e. a comparison of the effect of multiple oxygen tension environments (1–20% ambient oxygen) upon gene transcription of primary rat astrocytes. VennPlex accurately dissects complex data sets reliably into easily identifiable groups for straightforward analysis and data output. This program, which is an improvement over currently available Venn diagram programs, is able to rapidly extract important datasets that represent the variety of expression patterns available within the data sets, showing potential applications in fields like genomics, proteomics, and bioinformatics

Euler diagrams drawn with ellipses area‑proportionally (Edeap)

Background: Area-proportional Euler diagrams are frequently used to visualize data from Microarray experiments, but are also applied to a wide variety of other data from biosciences, social networks and other domains. Results: This paper details Edeap, a new simple, scalable method for drawing areaproportional Euler diagrams with ellipses. We use a search-based technique optimizing a multi-criteria objective function that includes measures for both area accuracy and usability, and which can be extended to further user-defned criteria. The Edeap software is available for use on the web, and the code is open source. In addition to describing our system, we present the frst extensive evaluation of software for producing area-proportional Euler diagrams, comparing Edeap to the current state-of-the-art; circle-based method, venneuler, and an alternative ellipse-based method, eulerr. Conclusions: Our evaluation—using data from the Gene Ontology database via GoMiner, Twitter data from the SNAP database, and randomly generated data sets—shows an ordering for accuracy (from best to worst) of eulerr, followed by Edeap and then venneuler. In terms of runtime, the results are reversed with venneuler being the fastest, followed by Edeap and fnally eulerr. Regarding scalability, eulerr cannot draw non-trivial diagrams beyond 11 sets, whereas no such limitation is present in Edeap or venneuler, both of which draw diagrams up to the tested limit of 20 sets