Evaluating the Comprehension of Euler Diagrams

We describe an empirical investigation into layout criteria that can help with the comprehension of Euler diagrams. Euler diagrams are used to represent set inclusion in applications such as teaching set theory, database querying, software engineering, filing system organisation and bio-informatics. Research in automatically laying out Euler diagrams for use with these applications is at an early stage, and our work attempts to aid this research by informing layout designers about the importance of various Euler diagram aesthetic criteria. The three criteria under investigation were: contour jaggedness, zone area inequality and edge closeness. Subjects were asked to interpret diagrams with different combinations of levels for each of the criteria. Results for this investigation indicate that, within the parameters of the study, all three criteria are important for understanding Euler diagrams and we have a preliminary indication of the ordering of their importance

How Should We Use Colour in Euler Diagrams?

This paper addresses the problem of how best to use colour in Euler diagrams. The choice of using coloured curves, rather than black curves, possibly with coloured fill is often made in tools that automatically draw Euler diagrams for information visualization as well as when they are drawn manually. We address the problem by empirically evaluating various different colour treatments: coloured or black curves combined with either no fill or coloured fill. By collecting performance data, we conclude that Euler diagrams with coloured curves and no fill significantly outperform all other colour treatments. Most automated layout algorithms adopt colour fill and are, thus, reducing the effectiveness of the Euler diagrams produced. As Euler diagrams can be used in a multitude of areas, ranging from crime control to social network analysis, our results stand to increase the ability of users to accurately and quickly extract information from their visualizations

The Impact of Shape on the Perception of Euler Diagrams

Euler diagrams are often used for visualizing data collected into sets. However, there is a significant lack of guidance regarding graphical choices for Euler diagram layout. To address this deficiency, this paper asks the question `does the shape of a closed curve affect a user's comprehension of an Euler diagram?' By empirical study, we establish that curve shape does indeed impact on understandability. Our analysis of performance data indicates that circles perform best, followed by squares, with ellipses and rectangles jointly performing worst. We conclude that, where possible, circles should be used to draw effective Euler diagrams. Further, the ability to discriminate curves from zones and the symmetry of the curve shapes is argued to be important. We utilize perceptual theory to explain these results. As a consequence of this research, improved diagram layout decisions can be made for Euler diagrams whether they are manually or automatically drawn

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

eulerForce: Force-directed Layout for Euler Diagrams

Euler diagrams use closed curves to represent sets and their relationships. They facilitate set analysis, as humans tend to perceive distinct regions when closed curves are drawn on a plane. However, current automatic methods often produce diagrams with irregular, non-smooth curves that are not easily distinguishable. Other methods restrict the shape of the curve to for instance a circle, but such methods cannot draw an Euler diagram with exactly the required curve intersections for any set relations. In this paper, we present eulerForce, as the first method to adopt a force-directed approach to improve the layout and the curves of Euler diagrams generated by current methods. The layouts are improved in quick time. Our evaluation of eulerForce indicates the benefits of a force-directed approach to generate comprehensible Euler diagrams for any set relations in relatively fast time

The impact of topological and graphical choices on the perception of Euler diagrams

This paper establishes the impact of topological and graphical properties on the comprehension of Euler diagrams. To-date, various studies have examined the impact of individual properties of Euler diagrams, such as curve shape and orientation. This has allowed us to establish guides for using these properties such as ‘draw Euler diagrams with circles’ and ‘draw Euler diagrams without regard to orientation’. However, until the work described here, questions still remain, for example ‘do these guides, when combined, make a significant difference to real-world Euler diagrams?’, and if so, ‘should they be used by those visualising set data with Euler diagrams?’ To answer these questions an empirical study was conducted to compare Euler diagrams that have been drawn by others for their real-world data, against versions that adhere to all of the guides in combination. The study establishes that both the accuracy and the speed with which information is derived from Euler diagrams is significantly improved when Euler diagrams adhere, where possible, to all the guides. The improvement is considerable when using the guided diagrams, with on average, the error rate being more than halved from 21.4% to 10.3%, and a 9 s improvement in the average time taken, from 34.2s to 24.9s. As Euler diagrams are regularly used to visualise information in a multitude of areas, ranging from crime control to social network analysis, our results indicate that applying the guides to these diagrams will improve the ability of users to accurately and quickly extract information

Evaluating Visualizations of Sets and Networks that Use Euler Diagrams and Graphs

This paper presents an empirical evaluation of state-of-the-art visualization techniques that combine Euler diagrams and graphs to visualize sets and networks. Focusing on SetNet, Bubble Sets and WebCola – techniques for which there is freely available software – our evaluation reveals that they can inaccurately and ineffectively visualize the data. Inaccuracies include placing vertices in incorrect zones, thus incorrectly conveying the sets in which the represented data items lie. Ineffective properties, which are known to hinder cognition, include drawing Euler diagrams with extra zones or graphs with large numbers of edge crossings. The results demonstrate the need for improved techniques that are more accurate and more effective for end users.The Leverhulme Trus

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

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