All pairs
Azenplot can show the same information as apairs plot but with two important display differences.
First, the matrix organization of thepairs layout isreplaced by the “zig-zag” layout ofzenplot. Second, thenumber of plots produced is about half that of apairs plotallowing each plot in azenplot to be given more visualspace.
Producing all pairs withPairViz
A convenient function to produce all pairs can be found in thePairViz package found oncran and installed inR viainstall.packages("PairViz").
if (!has_pairviz) {cat("> **Optional dependency**: Examples using the **PairViz** package are skipped if it’s not installed. Install with `install.packages(\"PairViz\")`.\n\n")}else {library(PairViz)}We will illustrate this functionality and the difference between apairs plot and azenplot by first consideringa small dataset on earthquakes having only a few variates. Thedifference between the two plots becomes much more important for datahaving larger numbers of variates – we illustrate the difference againusing German data on voting patterns in two elections.
Example: Ground acceleration of earthquakes
The built-inR data set calledattenucontains measurements to estimate the attenuating effect of distance onthe ground acceleration of earthquakes in California.
There are 5 different variates used to describe the peak accelerationof 23 California earthquakes measured at different observation stations.The data set contains 182 different peak acceleration measurements andhas some missing data. The first few cases of the data set look like
## event mag station dist accel## 1 1 7.0 117 12 0.359## 2 2 7.4 1083 148 0.014## 3 2 7.4 1095 42 0.196## 4 2 7.4 283 85 0.135## 5 2 7.4 135 107 0.062## 6 2 7.4 475 109 0.054Its variates are
## [1] "event" "mag" "station" "dist" "accel"and we are interested in all pairs of these variates.
To get these, first imagine a graph having as its nodes the variatesof the data. An edge of this graph connects two nodes and hencerepresents a pair of variates. If interest lies in all pairs of varates,then the graph is a complete graph – it will have an edge between everypair of nodes. An ordering of variate pairs corresponds to any path onthe graph. To have an ordering of all pairs of variates, the path mustvisit all edges and is called an Eulerian, or Euler path. Such a pathalways exists for complete graphs on an odd number of nodes; when thenumber of nodes is even, extra edges must be added to the graph beforean Eulerian can exist.
For a complete graph with n nodes, the functioneseq(for Euler sequence) function from thePairViz packagereturns an order in which the nodes (numbered 1 to n) can be visited toproduce an Euler path. It works as follows.
## Since attenu has 5 variates, the complete graph has n=5 nodes## and an Euler sequence is given asPairViz::eseq(5)## [1] 1 2 3 1 4 2 5 3 4 5 1In terms of the variate names ofattenu, this is:
## [1] "event" "mag" "station" "event" "dist" "mag" "accel" ## [8] "station" "dist" "accel" "event"Precomputed complete graph.
this sequence traces an Eulerian path on the complete graph and sopresents every variate next to every other variate somewhere in theorder.
Optional dependency: This graph, and any othergraph, in this vignette can be displayed as a Graphviz plot using theplot() method fromRgraphviz. To install locally:
if (!requireNamespace("BiocManager",quietly=TRUE))install.packages("BiocManager")BiocManager::install("Rgraphviz")If theRgraphviz package is installed, then the abovegraph could be produced as follows:
Euler sequences viazenpath
This functionality (and more) fromPairViz has beenbundled together in thezenplots package as a singlefunctionzenpath. For example,
## [1] 5 1 2 3 1 4 2 5 3 4 5This sequence, while still Eulerian, is slightly different than thatreturned byeseq(5). The sequence is chosen so that allpairs involving the first index appear earliest in the sequence, thenall pairs involving the second index, and so on. We call this a “frontloaded” sequence and identify it with thezenpath argumentmethod = "front.loaded". Other possibilities aremethod = "back.loaded" andmethod = "balanced"giving the following sequences:
## Back loading ensures all pairs appear latest (back) for## high values of the indices.zenpath(5,method ="back.loaded")## [1] 1 2 3 1 4 2 5 3 4 5 1## Frot loading ensures all pairs appear earliest (front) for## low values of the indices.zenpath(5,method ="front.loaded")## [1] 5 1 2 3 1 4 2 5 3 4 5## Balanced loading ensures all pairs appear in groups of all## indices (Hamiltonian paths -> a Hamiltonian decomposition of the Eulerian)zenpath(5,method ="balanced")## [1] 1 2 3 5 4 1 3 4 2 5 1The differences are easier to see when there are more nodes. Below,we show the index ordering (top to bottom) for each of these threemethods when the graph has 15 nodes, here labelled”a” to “o” (to makeplotting easier).
Starting from the bottom (the back of the sequence), “back loading”has the last index, “o”, complete its pairing with every other indexbefore “n” completes all of its pairings. All of “n”’s pairings completebefore those of “m”, all of “m”’s before “l”, and so on until the lastpairing of “a” and “b” are completed. Note that the last indices stillappear at the end of the sequence (since the sequence begins at the topof the display and moves down). The term “back loading” is used here ina double sense - the later (back) indices have their pairings appear asclosely together as possible towards the back of the returned sequence.A simple reversal, that isrev(zenpath(15, method = "back.loaded")), would have themappear at the beginning of the sequence. In this case the “back loading”would only be in one sense, namely that the later indexed (back) nodesappear first in the reversed sequence.
Analogously, “front loading” has the first (front) indices appear atthe front of the sequence with their pairings appear as closely togetheras possible.
The “balanced” case ensures that all indices appear in each block ofpairings. In the figure there are 7 blocks.
All three sequences are Eulerian, meaning all pairs appear somewherein each sequence.
Pairs plots versus zenplots
Eulerian sequences can now be used to compare apairsplot with azenplot when all pairs of variates are to bedisplayed.
First a pairs plot:
## We remove the space between plots and suppress the axes## so as to give maximal space to the individual scatterplots.## We also choose a different plotting character and reduce## its size to better distinguish points.pairs(attenu,oma=rep(0,4),gap=0,xaxt="n",yaxt="n")We now effect a display of all pairs usingzenplot.
## Plotting character and size are chosen to match that## of the pairs plot.## zenpath ensures that all pairs of variates appear## in the zenplot.## The last argument, n2dcol, is chosen so that the zenplot## has the same number of plots across the page as does the## pairs plot.zenplot(attenu[,zenpath(ncol(attenu))],n2dcol=4)Each display shows scatterplot of allchoose(5,2) = 10pairs of variates for this data. Each display occupies the same totalarea.
Withpairs each plot is displayed twice and arranged ina symmetric matrix layout with the variate labels appearing along thediagonal. This makes for easy look-up but uses a lot of space.
Withzenplot, each plot appears only once with itscoordinate defining variates appearing as labels on horizontal (top orbottom) and vertical (left or right) axis positions. The layout followsthe order of the variates in which the variates appear in the call tozenplot beginning in the top left corner of the display andthen zig-zagging from top left to bottom right; when the rightmostboundary or the display is reached, the direction is reversedhorizontally and the zigzag moves from top right to bottom left. Thefollowing display illustrates the pattern (had by simply callingzenplot):
## Call zenplot exactly as before, except that each scatterplot is replaced## by an arrow that shows the direction of the layout.zenplot(attenu[,zenpath(ncol(attenu))],plot2d="arrow",n2dcol=4)The zig zag pattern of plots appears as follows.
- The top left plot (of either
zenplotdisplay) hashorizontal variateeventand vertical variatemag. - To its right is a plot sharing the same vertical variate
magbut now with horizontal variatestation.Note that the variatestationhas some missing values andthis is recorded on its label asstation (some NA). - Below this a plot appears with the same horizontal variate
stationbut now with vertical variateevent.Since this is the first repeat appearance ofeventitappears with a suffix asevent.1. - To its right is a plot with shared vertical variate
eventand new horizontal variatedist. - Below this is a plot having shared horizontal variate
distand as vertical variate the first repeat of thevariatemag. - To its right is a plot having shared vertical variate
magand new horizontal variateaccel. - The right edge of the display is reached and the zigzag changeshorizontal direction repeating the pattern until either the left edge isreached (whereupon the horizontal direction is reversed) or the variatesare exhausted.
Like thepairs plot, thezenplot lays itsplots out on a two dimensional grid – the argumentn2dcol=4specifies the number of columns for the 2d plots (e.g. scatterplots). Asshown, this can lead to a lot of unused space in the display.
Thezenplot layout can be made more compact by differentchoices of the argumentn2dcol (odd values provide a morecompact layout). For example,
By default,zenplot tries to determine a value forn2dcol that minimizes the space unused by its zigzaglayout.
with layout directions as
As the direction arrows show, the default layout is to zigzaghorizontally first as much as possible.
This is clearly a much more compact display. Again, axes are sharedwherever a label appears between plots.
Unless explicitly specified, the value ofn2dcol isdetermined by the aregumentscaling which can either be anumerical value specifying the ratio of the height to the width of thezenplot layout or be a string describing a page whose ratioof height to width will be used. The possible strings areletter'' (the default),square’‘,A4'',golden’’(for the golden ratio), or ``legal’’.
Visual search
The display arrangement of a scatterplot matrix facilitates thelookup of the scatterplot for any particular pair of variates by simplyidentifying the corresponding row and columns.
The scatterplot matrix also simplifies the visual comparison of theone variate to each of several others by scanning along any single row(or column). Note however that this single row scan does come at theprice of doubling the number of scatterplots in the display.
These two visual search facilities are diminished by the layout of azenplot. Although the same information is available in azenplot the layout does not lend itself to easy lookup fromvariates to plots. If thezenplot layout is used in aninteractive graphical system, other means of interaction could beimplemented to have, for example, all plots containing a particularvariate (or pair of variates) distinguish themselves visually by havingtheir background colour change temporarily.
On the other hand, the reverse lookup from plot to variates issimpler in azenplot than in a scatterplot matrix,particularly for large numbers of variates.
Both layouts allow a visual search for patterns in the pointconfigurations. Having many plots be presented at once enables a quickvisual search over a large space for the existence of interesting pointconfigurations (e.g. correlations, outliers, grouping in data, lines,etc.).
When the number of plots is very large, an efficient compact layoutcan dramatically increase the size of the visual search space. This iswherezenplot’s zigzag layout outperforms the scatterplotmatrix.
This could be illustrated this on the following example.
Example: German election data
In thezenplots package the data setde_elect contains the district results of two Germanfederal elections (2002 and 2005) as well as a number of socio-economicvariates as well.
There are 299 districts and 68 variates yielding a possiblechoose(68,2) = 2278 different scatterplots.
This many scatterplots will overwhelm apairs plot. Inits most compact form, the pairs plot for the first 34 variates alreadyoccupies a fair bit of space:
(N.B. We do not execute any of these large plots simply to keep thestorage needs of this vignette to a minimum. We do encourage the readerto execute the code however on their own.)
If you execute the above code you will see interesting pointconfigurations including: some very strong positive correlations, somepositive and negative correlations, non-linear relations, the existenceof some outlying points, clustering, striation, etc.
Because this scatterplot matrix is for only half of the variates itshowschoose(34,2) = 561 different scatterplots, each onetwice. For a display of 1122 plots, only about one quarter of all 2278pairwise variate scatterplots available in the data set appear in thisdisplay.
A second scatterplot matrix on the remaining 34 variates would alsoshow only a quarter of the plots. The remaining half,\(34 \times 34 = 1156\) plots, are missingfrom both plots.
In contrast, thezenplot shows all 2278 plots at once.In fact, because an Eulerian sequence requires a graph to be even(i.e. each node has an even number of edges), whenever the number ofvariates,\(p\), is evenzenpath(...) will repeat exactly\(p/2\) pairs somewhere in the sequence itreturns.
To produce thezenplot of all pairs of variates on theGerman election data we callzenplot(de_elect[,zenpath(68)], pch="."). (Again, we don’tproduce it here so as to minimize the storage footprint of thisvignette.)
## Try invoking the plot with the following## zenplot(de_elect[,zenpath(68)], pch=".", n2dcol="square",col=adjustcolor("black",0.5))In approximately the same visual space as the scatterplot matrix(showing only 561 unique plots), thezenplot hasefficiently and compactly laid out all 2278 different plots plusr ncol(de_elect) duplicate plots. This efficient layoutmeans thatzenplot can facilitate visual search forinteresting point configurations over much larger collections of variatepairs – in the case of the German election data, this all possible pairsof variates are presented simultaneously.
In contrast, all pairs loses most of the detail