Euler diagrams are common and user-friendly visualizations for set relationships. Most Euler diagrams use circles, but circles do not always yield accurate diagrams. A promising alternative is ellipses, which, in theory, enable accurate diagrams for a wider range of input. Elliptical diagrams, however, have not yet been implemented for more than three sets or three-set diagrams where there are disjoint or subset relationships. The aim of this paper is to present eulerr: a software package for elliptical Euler diagrams for, in theory, any number of sets. It ts Euler diagrams using numerical optimization and exact-area algorithms through a two-step procedure, rst generating an initial layout using pairwise relationships and then nalizing this layout using all set relationships.
The Euler diagram, rst described by Leonard Euler in 1802 [
Euler diagrams may be fashioned out of any closed shape, and have been
implemented for triangles [
With four or more sets that all intersect, for instance, exact Euler diagrams
are impossible with circles, given that we require 24 1 = 15 intersections but
with four circles can yield no more than 13 [
Fitting elliptical or circular Euler diagrams must be done numerically even in
the two-set case [
In the nal layout, we need to compute the areas of the overlaps in the
diagram. Frederickson [
The R-package eulerr was created as part of a bachelor's thesis [
eulerr allows input in the form of disjoint subsets, unions and identities, a matrix of binary or boolean indices, a list of sample spaces, or a two- or three-way table. The Euler diagram is t in two steps: rst, an initial layout is formed with circles using only the sets' pairwise relationships. Second, this layout is ne-tuned taking all intersections into consideration. 2.1
For our initial layout, we adopt a constrained version of multi-dimensional
scaling (MDS) that is used in venn.js [
We begin by placing circles representing each set uniformly at random in a square space with area Pin=1 ri2 , where ri is the radius of the ith circle. The circles are initialized so that their areas are proportional to the size of their respective sets. The algorithm then moves the circles so that the separation between each pair of circles matches their respective sets' intersection. If the two sets are disjoint, however, the algorithm is indi erent to the relative locations of those circles as long as they do not intersect. The equivalent applies to subset sets: as long as the circle representing the smaller set remains within the larger circle, their locations are free to vary. In all other cases, the loss function (1) is the residual sums of squares of the separation of circles, d, required to obtain accurate pairwise overlaps and the actual distance in the layout. if disjoint if subset otherwise, (1) L(h; k) =
X where h and k are the x and y coordinates of the centers of the circles respectively. The analytic gradient follows naturally after some algebra.
We optimize (1) using the nonlinear optimizer nlm() from the R core
package stats [
To nd the overlap areas, we rst locate all the points of intersection of the
ellipses [
It is trivial to nd the area of the polygon section since it is always convex [
Having computed the areas of all the intersections we now decompose them into
disjoint areas, wherein each area uniquely represents a subset of the input. This is
the form we need for our nal optimization. We feed the initial layout computed
in section 2.1 to the optimizer|once again we employ nlm() from stats but now
1 By perfect, we refer to solutions with diagError < 10 6 (see equation (3)).
also provide the option to use ellipses rather than circles, allowing the \circles"
to rotate and the relation between the semiaxes to vary, altogether rendering
ve parameters to optimize per set and ellipse (or three if we restrict ourselves
to circles). For each iteration of the optimizer, the areas of all intersections are
analyzed and a measure of loss returned. The loss we use is the same as that in
venneuler [
Pin=1(Ai
!i)2 Pin=1 Ai2 with =
Pin=1 Ai!i Pin=1 !i2 ; where Ai is the area representing !i, the size of the ith disjoint intersection, and n the number of intersections in the set con guration.
As an additional option, the user may activate a last optimization step2 that
uses a Generalized Simulated Annealing optimizer [
To measure the goodness of t of the resulting diagram, we adopt two widely
used measures: the previously covered stress [
max i=1;2;:::;n Pin=1 !i !i
Ai Pin=1 Ai : (2) (3) The complete algorithm is provided in Algorithm 1.
eulerr is primarily written in R but its backbone is implemented in C++ and make
heavy use of the linear algebra library Armadillo [
In this section, we will study set con gurations|and the diagrams t to them| from previous papers featuring software for Euler diagrams. The packages we will study are eulerr 4.1.0, eulerAPE 3.0.0, and venneuler 1.1-0.
We begin with a set relationship from Wilkinson [
Data: N sets, n intersections, !: the required disjoint areas for an optimal diagram, Fi: the size of the ith set, : a prede ned tolerance threshold. Result: N ellipses with parameters = fh; k; a; b; g. for i 1 to N do
Ai !i ri pAi= xi; yi
U 0; qPiN=1 ri2 foreach i < j N do nd dij that minimizes Oij (Fi \ Fj) 2, where Oij is the overlap of the circles representing Fi and Fj for i 1 to 10 do obtain h(i), k(i) by minimizing (1) using a local optimizer j i 2 f1; 10g that minimizes (1) obtain nal by minimizing (2) using a local optimizer with h = h(j), k = k(j), a = 0, b = 0, = 0 as starting values if diagError( nal) > then obtain last-ditch using a global optimizer if diagError( nal) > diagError( last-ditch) then return last-ditch else return nal else return
nal speci ed as disjoint subsets using the &-operator such that "A&B", for instance, are the items unique to the intersection between A and B. We t this speci cation with venneuler and eulerr, in the latter case with both circles and ellipses, using euler(): the workhorse of eulerr. f1 <- venneuler::venneuler(wilkinson) f2 <- euler(wilkinson) f3 <- euler(wilkinson, shape = "ellipse") # fit with venneuler # fit with eulerr (circles) # fit with eulerr (ellipses) eulerr manages to t this con guration perfectly using ellipses in addition to producing a marginally better circular diagram. The stress values are 0.007, 0.004, and 4:687 10 12 for venneuler, eulerr (with circles), and eulerr (with ellipses) respectively.
Diagrams in eulerr are plotted using plot(), which allows considerable customization of the resulting diagram. In the following code, we plot the circular diagram using the default options and the elliptical one with a few modi cations (Fig. 2). plot(f2) plot(f3, fills = rainbow(6, s = 0.5, v = 1), # change fills edges = list(lex = 3, col = "white"), # white, broader edges labels = list(font = 3)) # italic labels
B F
D A
C
A
B
C
Micallef and Rodgers feature a diagram from Lenz et al. [
"A&B" = 0.41, "A&C" = 0.04, "B&C" = 0, "A&B&C" = 0.11)
Because eulerAPE cannot t set con gurations with empty intersections, the
authors used 0.00001 as a proxy for ;. Using eulerr, however, we can t the
diagram using the original data.
plot(euler(lenz, shape = "ellipse"), legend = TRUE) # add a legend
The ts from both packages are exact (Fig. 3). Although we instructed eulerr
to allow ellipses in the t, the algorithm stuck to circles, which, given that the
t is exact, is the appropriate choice since circles are easier to interpret [
E
B
Finally, we provide a benchmark of the accuracy of venneuler, eulerAPE, and eulerr in reproducing 1,000 random three-set combinations sampled from U (10 6; 1) (Fig. 4). We use three-set combinations with all intersections present in order to enable all tested packages to t the diagrams.
In terms of both stress and diagError, the elliptical diagrams from eulerr and eulerAPE perform the best, with the former coming out marginally ahead with median stress and diagError at 3:123 10 13 and 1:638 10 7 respectively, whilst the equivalent gures for eulerAPE are 7:834 10 12 and 8:219 10 7.
For the circular diagrams, eulerr achieves the lowest median stress at 0.022, followed by venneuler and eulerAPE at 0.035 and 0.08 respectively. In terms of diagError, venneuler performs best followed by eulerr and eulerAPE with respective median diagErrors of 0.048, 0.055, and 0.067.
stress 0.0 0.2 0.6
0.8 0.4 diagError eulerAPE (ellipses) eulerAPE (circles)
venneuler eulerr (ellipses) eulerr (circles) 0.0 0.2 0.4 0.6 0.8 In this paper, we have presented an R-based software package, eulerr, for generating elliptical Euler diagrams for any number of sets. We have examined its performance for set relationships from previous publications of software for Euler diagrams and shown that eulerr performs adequately for our examples and for random three-set combinations. In general, we have also shown that elliptical Euler diagrams have the potential to outperform circular diagrams. The reason for this is simple: elliptical Euler diagrams feature two additional degrees of freedom for each shape in the diagram, provided by stretch and rotation.
eulerr is the rst software to feature area-proportional elliptical Euler
diagrams for more than three sets. The only other software for elliptical Euler
diagrams, eulerAPE, is restricted to three sets. This limitation is discussed by
the authors of the package, who argue that Euler diagrams with more than three
sets often lack well-formed solutions and that their complexity make
implementations di cult [
The results of this paper are limited to a few cases and it is not known whether they generalize to other set combinations. This is a topic for future research in the eld, which should examine di erent software for Euler diagrams in large-scale simulation studies.