Figure 1.
Examples of area-proportional 3-Venn diagrams drawn with circles (A–C) and polygons (D–F) in literature.
(A) Comparing the cell-type of differentially regulated genes after an anti-cancer drug treatment [11]. The method used to draw the diagram has not been noted. This diagram is a reprint of Figure 3B in [11], previously published under a CC BY license. (B) Summarizing prognostic indicators of severe malaria [12]. The method used to draw the diagram has not been noted. This diagram is a reprint of Figure 3 in [12] (with the N value and the percentages in parenthesis removed), previously published under a CC BY license. (C) Analysing differences and similarities between identified chicken egg white proteome in three different studies [13]. Drawn using Venn Diagram Plotter [http://omics.pnl.gov/software/VennDiagramPlotter.php]. This diagram is a reprint of Figure 2A in [13], previously published under a CC BY license. (D) Analysing overlaps between gene libraries [14]. Drawn using DrawVenn [5]. This diagram is a reprint of Figure 4B in [14] under a CC BY license, with permission from John Wiley and Sons, original copyright 2009. (E) Studying transcriptome variation of different tissue types of the male field cricket, namely testis, accessory gland and the rest of the body [15]. Drawn using Convex Venn-3 [51]. This diagram is a reprint of Figure 1 (left) in [15] under a CC BY license, with permission from Nathan Bailey, original copyright 2013. (F) Summarizing genes affecting Top1-induced DNA damage identified in three different studies [16]. Drawn using DrawEuler [55]. This diagram is a reprint of Figure 3A in [16] (with added numeric labels indicating the quantitative data that according to the article each region in the diagram should represent), previously published under a CC BY license.
Figure 2.
Accurate area-proportional 3-Venn diagrams drawn with ellipses and eulerAPE for the data in Figure 1.
Each of these diagrams depicts the sets and the quantitative data indicated by the numeric labels in the regions of the corresponding diagram in Figure 1. These diagrams have been drawn with eulerAPE, but the labels have been added manually.
Figure 3.
Diagrams generated by various drawing methods for the same medical data from a journal article.
All the diagrams are meant to depict ω = {A = 0.25, B = 0.01, C = 0.11, AB = 0.10, AC = 0.29, BC = 0.03, ABC = 0.15}, which represents the findings of a medical survey from a journal article [37] that also included diagram D for ω. The diagrams generated for ω using circle-based drawing methods are marked as C, those of polygon-based methods are marked as P, and the only diagram with ellipses, that by eulerAPE, is E. Green indicates accurate diagrams with diagError ≤10−6. Red indicates diagrams with inaccurate or missing regions. D is a redrawing of Figure 5 (bottom) in [37], previously published under a CC BY license.
Figure 4.
A method for constructing an area-proportional 3-Venn diagram using circles.
(A) The quantitative values in each region indicate the required region areas, for which an area-proportional 3-Venn diagram should be drawn. (B) The first step of the construction whereby the three accurate 2-Venn diagrams are drawn. (C) The second step of the construction whereby the identical copies of the circle labelled c are rotated such that they overlap completely and only one circle labelled c is visible. (D) The instance when only three circles are visible, such that the regions of the 3-Venn diagram are obtained. (E) The actual area of the regions in the constructed diagram D, which, as in most cases when these diagrams are drawn with circles, do not correspond to the desired values in A. The numeric label in each region of this diagram indicates the regions' actual area.
Figure 5.
The starting diagram generator bisecting the interval along bisector line to position the third ellipse.
The centre of ellipse e3 is a point on the line L bisecting the angle ψ between the two tangents T1 and T2. The bisection method is applied in the interval indicated by the faded blue circles along L. The obtained centre should minimize the discrepancy of the required and the actual area of the region in exactly the three ellipses.
Figure 6.
The different ways the ellipses' properties are modified during the optimization search process.
At every iteration of the optimization algorithm, the (A) centre, (B) semi-axes and (C) angle of rotation of every ellipse are respectively modified by parameters pγ, pαβ and pθ in search for other solutions. (A) The grey points indicate the new centres that are obtained when one or both coordinates of the centre of an ellipse (solid black) are increasing or decreasing by pγ. (B) A label +pαβ means that that semi-axis was increased by the pαβ percentage, while -pαβ means that that semi-axis was decreased by the pαβ percentage. The dashed ellipses indicate how an ellipse (solid black) is changed when: (top, left) only the semi-major axis is increased or decreased by pαβ; (top, right) only the semi-minor is increased or decreased by pαβ; (bottom, left) the semi-axes are both increased or both decreased by pαβ; (bottom, right) one of the semi-axes is increased and the other is decreased by pαβ. (C) The dashed ellipses indicate how an ellipse (solid black) is changed when its angle of rotation is increased or decreased by pθ.
Figure 7.
The number of reruns to generate a good diagram for 61 data items in L1.
The number of reruns (1–10) that were required for eulerAPE to generate a good diagram for the 61 data items in L1 for which a non-good diagram was generated during the first run.
Figure 8.
Time and total number of iterations to generate good diagrams for data in L1.
The log10 (time in seconds) and log10(total number of iterations) taken to generate good diagrams for 9939 of the 10,000 data items in L1 during the first run (labelled as ‘Run 1’) and for 61 of the 10,000 data items in L1 during any of the one to a maximum of 10 reruns (labelled as ‘Reruns’).
Figure 9.
Examples of good diagrams generated after the first run for data in L1.
(A) and (B) illustrate (i) the good diagram that was found using (ii) the starting diagram generated for the data item in L1 ({a = 2273, b = 24458, c = 44454, ab = 7116, ac = 740, bc = 18807, abc = 12092} for A and {a = 17033, b = 6248, c = 16230, ab = 615, ac = 289, bc = 840, abc = 922} for B) that was equal to the set of region areas of (iii) a randomly generated 3-Venn diagram.
Figure 10.
Examples of good diagrams generated after the first rerun for data in L1.
(A) An example of (i) a non-good diagram with a very low diagError (6.51×10−4) generated during the first run and (ii) the good diagram generated during the first rerun for the data ({a = 10018, b = 27132, c = 39737, ab = 9567, ac = 11454, bc = 3, abc = 668}) in L1 obtained from (iii) a random diagram. The good diagram in ii was generated in 1.2 seconds and 86 iterations (including the first run and the one rerun). (B) An example of (i) a non-good diagram with a low diagError (8.38×10−3) generated during the first run and (ii) the good diagram generated during the first rerun for the data ({a = 53804, b = 39550, c = 1256, ab = 15606, ac = 15, bc = 29904, abc = 3597}) in L1 obtained from (iii) a random diagram. The good diagram in ii was generated in 2.9 seconds and 367 iterations (including the first run and the one rerun).
Figure 11.
Examples of diagrams generated by venneuler and eulerAPE (circles and ellipses) for data in L2.
Examples of diagrams generated with (i) circles by venneuler, (ii) circles by eulerAPE, and (iii) ellipses by eulerAPE for random 3-set data in L2. (A) Diagrams generated for data {a = 3491, b = 3409, c = 3503, ab = 120, ac = 114, bc = 132, abc = 126}. Ai is missing region abc and has stress = 5.69×10−4 and diagError = 1.16×10−2. Aii and Aiii have the required regions, one for every data set relation. Aii has stress = 8.36×10−3 and diagError = 2.63×10−2. Aiii has stress = 3.96×10−12 and diagError = 6.55×10−7. (B) Diagrams generated for data {a = 45910, b = 3261, c = 45467, ab = 58845, ac = 3028, bc = 16406, abc = 18496}. Bi is missing region ac and has stress = 3.17×10−3 and diagError = 2.07×10−2. There are two regions in Bi depicting only b. Bii and Biii have the required regions, one for every data set relation. Bii has stress = 2.13×10−2 and diagError = 4.36×10−2. Biii has stress = 3.43×10−12 and diagError = 6.85×10−7. (C) Diagrams generated for data {a = 3664, b = 46743, c = 59811, ab = 1742, ac = 2099, bc = 17210, abc = 24504}. Ci, Cii and Ciii have the required regions, one for every data set relation. Ci has stress = 4.27×10−3 and diagError = 2.30×10−2. Cii has stress = 8.31×10−3 and diagError = 2.44×10−2. Ciii has stress = 1.13×10−12 and diagError = 4.03×10−7.
Figure 12.
Stress and diagError of all the diagrams generated by venneuler and eulerAPE (circles and ellipses).
The (A) stress and (B) diagError of all the diagrams generated with circles by venneuler, with circles by eulerAPE and with ellipses by eulerAPE for the 10,000 3-set data in L2. The 10,000 diagrams generated with circles by venneuler had stress in [3.77×10−5, 6.14×10−1] with median 3.04×10−2 and mean 6.41×10−2, and diagError in [1.56×10−3, 2.46×10−1] with median 4.56×10−2 and mean 5.73×10−2. The 10,000 diagrams generated with circles by eulerAPE had stress in [1.91×10−10, 7.79×10−1] with median 7.00×10−2 and mean 1.13×10−1, and diagError in [3.30×10−6, 3.31×10−1] with median 6.28×10−2 and mean 6.73×10−2. The 10,000 diagrams generated with ellipses by eulerAPE had stress in [3.98×10−14, 2.24×10−1] with median 7.59×10−12 and mean 1.17×10−10, and diagError in [6.00×10−8, 1.39×10−1] with median 8.00×10−7 and mean 2.94×10−3.
Figure 13.
The figure in a medical journal article and the figure recreated with eulerAPE.
(A) The figure with two Venn diagrams drawn with circles in a medical journal article [37]. This is a redrawing of Figure 5 in [37], previously published under a CC BY license. (B) The figure as it would have looked like if the diagrams were drawn with ellipses using eulerAPE. Labels for eulerAPE's diagrams were added manually.