Venn diagram and UpSet plot maker

Counting. Each item is matched across lists (after trimming spaces; duplicates within a list count once; optionally ignoring case) and given the set of lists it appears in. The exclusive count of a combination, such as “A ∩ B only”, is the number of items in exactly those lists; these are the regions a Venn diagram draws, and they add up to the number of unique items. The inclusive count, $|A\cap B|$, also includes items that are in further lists. The two are linked by inclusion–exclusion: for three sets, A ∩ B only $=|A\cap B|-|A\cap B\cap C|$, and A only $=|A|-|A\cap B|-|A\cap C|+|A\cap B\cap C|$. When you type sizes and overlaps, the tool uses these to find the regions and rejects numbers that would make a region negative.

Two sets, exactly. Each circle’s area equals its set size, $\pi r_i^2=|S_i|$. The overlap of two circles whose centres are $d$ apart is the lens $A_{\cap }(d)=r_1^2\arccos \frac{d^2+r_1^2-r_2^2}{2d{r}_1}$ $+r_2^2\arccos \frac{d^2+r_2^2-r_1^2}{2d{r}_2}$ $-\frac{1}{2}\sqrt{k}$, where $k=(-d+r_1+r_2)$ $\times (d+r_1-r_2)$ $\times (d-r_1+r_2)$ $\times (d+r_1+r_2)$. The lens area falls steadily from the smaller circle’s area at $d=|r_1-r_2|$ to 0 at $d=r_1+r_2$. Bisection finds the $d$ where it equals the shared count, so all three regions have exactly the right areas.

Three sets, by fitting. Three circles have six free numbers once position and rotation are fixed (three radii and three distances) but seven regions to match, so most data cannot be drawn exactly with circles (Chen and Boutros 2011; Micallef and Rodgers 2014). Following eulerr (Larsson and Gustafsson 2018) and venneuler (Wilkinson 2012), the tool starts from the distances that give each pair its overlap, then adjusts the radii and centres to minimise the sum of squared differences between drawn and wanted region areas, $\sum _r({\hat{a}}_r-a_r{)}^2$, with Nelder–Mead from several starts. Region areas are computed exactly from the circle geometry (polygon plus circular segments). It then reports two measures: stress, $\sum _r(a_r-\beta {\hat{a}}_r{)}^2/\sum _r{a}_r^2$ with $\beta$ the least-squares scale (venneuler’s measure; 0 is perfect), and diagError, the largest difference between a region’s share of the drawing and its share of the items (eulerAPE and eulerr’s measure). A region that the circles cannot show at all is listed under the diagram, never silently dropped.

Classic layouts. The non-proportional option uses equal circles for 2–3 sets, Venn’s arrangement of four ellipses for 4 sets and Grünbaum’s (1975) five congruent ellipses with five-fold symmetry for 5 sets, so every one of the 3, 7, 15 or 31 regions exists. Each count is written at the point of its region farthest from any edge. With six or more sets no ellipse Venn diagram exists, and UpSet plots are clearer anyway.

UpSet plots (Lex et al. 2014; Conway et al. 2017) show each intersection as a column: a bar for its size above, and below it a matrix whose filled dots mark the sets involved, joined by a line. Horizontal bars give each set’s size. By default the bars are exclusive counts, as in UpSetR, so every item is counted once; inclusive counts are an option.

Sources: Lex A, Gehlenborg N, Strobelt H, Vuillemot R, Pfister H. UpSet: visualization of intersecting sets. IEEE Trans Vis Comput Graph 2014;20:1983–1992, doi:10.1109/TVCG.2014.2346248. Conway JR, Lex A, Gehlenborg N. UpSetR: an R package for the visualization of intersecting sets and their properties. Bioinformatics 2017;33:2938–2940, doi:10.1093/bioinformatics/btx364. Wilkinson L. Exact and approximate area-proportional circular Venn and Euler diagrams. IEEE Trans Vis Comput Graph 2012;18:321–331, doi:10.1109/TVCG.2011.56. Larsson J, Gustafsson P. A case study in fitting area-proportional Euler diagrams with ellipses using eulerr. Proceedings of the International Workshop on Set Visualization and Reasoning 2018, CEUR Workshop Proceedings 2116:84–91, ceur-ws.org/Vol-2116/paper7.pdf. Micallef L, Rodgers P. eulerAPE: drawing area-proportional 3-Venn diagrams using ellipses. PLoS ONE 2014;9:e101717, doi:10.1371/journal.pone.0101717. Chen H, Boutros PC. VennDiagram: a package for the generation of highly-customizable Venn and Euler diagrams in R. BMC Bioinformatics 2011;12:35, doi:10.1186/1471-2105-12-35. Grünbaum B. Venn diagrams and independent families of sets. Math Mag 1975;48:12–23, doi:10.1080/0025570X.1975.11976431.

The built-in example is three illustrative lists from a hypoxia study: 40 genes up after 24 h of hypoxia, 34 HIF1A ChIP-seq targets and 38 proteins up by proteomics. The gene names are real; the lists are made up for the example.

1. Count the regions. 64 genes are in at least one list. Exclusive counts: hypoxia only 10, ChIP only 5, proteomics only 15, hypoxia ∩ ChIP only 11, hypoxia ∩ proteomics only 5, ChIP ∩ proteomics only 4, all three 14. They add up to 64. Inclusive: hypoxia ∩ ChIP $=11+14=25$, hypoxia ∩ proteomics $=19$, ChIP ∩ proteomics $=18$.
2. Two circles exactly. For hypoxia and ChIP alone, the radii are $r_1=\sqrt{40/\pi }=3.568$ and $r_2=\sqrt{34/\pi }=3.290$ (in units where the area is the gene count). The lens area is 25 at $d=1.746$: solving the lens equation by bisection gives that distance, and putting it back gives an overlap of 25.000.
3. Three circles by fitting. The best circles give stress 0.0048 and diagError 2.0 percentage points. The largest error is ChIP ∩ proteomics only: 4 of 64 genes (6.3 %) but 4.3 % of the drawn area; hypoxia ∩ proteomics only gets 6.4 % for 7.8 %. All other regions are within 1.1 points. Fitting the same loss with SciPy gives the same stress; matplotlib-venn’s layouts give 0.0069 (cost-based) and 0.0057 (default).
4. Read the members. The 14 genes in all three lists are VEGFA, CA9, SLC2A1, PGK1, BNIP3, LDHA, NDRG1, P4HA1, ALDOA, ENO1, HK2, GPI, TPI1 and PGAM1, the core hypoxia-response genes. The exclusive table lists them once, under “all 3”; the inclusive row “Hypoxia ∩ HIF1A ChIP-seq” lists them again with the 11 genes that are only in those two lists.