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cvxhullR Allan Reese, University of Hull, UK ------------------------------------------------------------------------------- program cvxhull, sortpreserve version 8.0 syntax varlist(min=2 max=2) [if] [in] /// [ , Hulls(numlist int min=0 max=2 >0) MDPlot(int 8) /// GROup(varname) SELect(numlist int min=0 >0 sort) /// PREfix(string) noREPort noRETain /// MEAns SCATopt(string) noGRAph SAVing(string)]

Find convex hulls in 2-D unsorted data. Draws graph and/or generates variables

cvxhullyvar xvar[ifexp] [inrange] [,hulls(int[int])group(varname)select(numlist)mdplot(int)meansscatopt(string)prefix(string)noreportnoretainnographsaving(graph_filename[, replace])]

Description

cvxhullidentifies points defining the convex hulls of a group of points in two-dimensional space. Each hull is defined by two lines joining the bottom-left point to the top-right point. The data are sorted within the routine butsortpreservereturns the data to their original order after the call, whether or not it was completed. The routine may be called repeatedly to calculate a variety of hulls.Two new variables may be added to the dataset for each level of hull that is defined and retained. Default varnames start with _cvxh, but you can change this prefix. The "y" coordinates for each line are returned as _cvxh1l & _cvxh1r, _cvxh2l & _cvxh2r, etc.

Other returned variables are: _cvxhgrp if a group variable was defined, containing integers 1,2,3 etc corresponding to group membership; _cvxhhull containing the index number of the hull for ...; _cvxhcnt containing minimum and maximum counts of the number of points comprising that hull; _cvxhpts identifying all points lying on a hull at level k, by having k as its value for that observation. This may be useful for choosing points for labelling by cvxplot.

Two counts are returned for each hull: the minimum count is the number of sides or vertices, returned in the "bottom left" element and matched by a positive hull number; and the maximum count is the number of points lying on the hull, returned in the "top right" element and matched by a negative hull number. (Single-point hulls return only the first.)

In addition to the returned variables, the routine uses ten internal temporary variables (plus another for sortpreserve). All the variables named or implied in the previous paragraphs will be dropped at the start of the program. If the program is broken in on, some of the returned variables may have been created.

It has been suggested that counts of points in successive hulls can be used as analogues to the distribution function of a single variable. The counts can be inspected by

tab _cvxhgrp _cvxhhull [fw=_cvxhcnt]

or, if no group variable was specified (and selecting the minimum counts),

tab _cvxhhull [fw=_cvxhcnt] if _cvxhhull>0

These counts can be put in order by an auxiliary routine

cvxindexthat sorts the values from _cvxhhull and _cvxhcnt to create new variables _cvxh<G>mindex and _cvxh<G>maxdex, where <G> represents the group number from _cvxhgrp. The counts are returned as the 1st, 2nd etc elements of each variable, so no longer necessarily correspond to the group variable. There are, however, no formulae for the expected values of these counts but they may be found for any distribution by runningsimulate.Unless the nograph option is used, the command will draw a scatterplot with the hulls joined. Plots may be drawn separately from the call using the related command

cvxplot. The general graph commands are, of course, available but the data may need to be sorted or the sort option used (see examp > les). Whether plots of multiple groups on one scatterplot are useful may depend on the degree of overlap, and needs to be decided by trial and error.The value of complex hulls in describing bivariate datasets is discussed in Barnett ("Understanding Multivariate Data" 1981, chapter by P J Green). A useful survey of algorithms is provided by Hempel on his website www.pixelish.com (Convex Hulls - a tutorial Sept 2002). An earlier Stata routine (gr16 by Grey & McGuire) had the disadvantage that it restructured the working data. In cvxhull I use a variant of the Jarvis March algorithm, which I propose should be named the Hull Football algorithm, on the grounds that:

a) the first code was written on the day England won the Rugby Union World Cup b) a convex hull is a funny-shaped ball c) it's a game of two halves, Brian.

ParametersThere must be precisely two variables, forming the y and x of the plot. Observations that are missing on either variable will not be used.

The

ifandinqualifiers act as usual to prescribe a subset of observations to be considered. When qualifiers are in force, the plot will show only observations from this subset. See thegroup()option for an alternative.

OptionsMain options

hulls(numlist)either one or two integers. The first defines the maximum level of hulls to be computed for each set of points. Defaults to 1. When the second integer has a value n, only the first and every n'th hull are plotted or retained.

mdplot(integer)maximum depth of hulls to be drawn. Defaults to 8 hulls for each group of points. Further hulls as specified by the data and the hulls option will be calculated and retained, but not drawn.group(varname)separate hulls are defined for each set of points identified with a different value ofvarname. The variable may be numeric or string. The group membership is returned as a new integer variable, with missing for observations excluded byiforinqualifiers. A table is output showing the equivalence of the integers with the original values.

select(numlist)only groups indexed in the list will have hulls constructed, but other groups will be plotted as points. Different plots may therefore be obtained by the commands cvxhull y x if group==1 and cvxhull y x, group(group) select(1)Group values in the numlist do not have to be listed in sorted order.

meansGroup means of y-x will be added to the graph as solid triangles.

scatopt(string)provides a string of options that will be applied to datapoints on the scatterplot. Any scatterplot options can be applied, obvious examples being scatopt(ms(i)) to make the points invisible leaving only the hull outlines, scatopt(mlab(var)) to mark each point with a value. Some spurious comments relating to pen styles may be output for hull levels > when large numbers of hulls are defined.As the scatopt string is the last option added when building the plotting command, it is possible to insert a whole extra plotting command. Eg: cvxhull y x, scat("||line x x") adds a diagonal reference line.

prefix(string)provides a prefix for the names of variables created within the routine. Variables as generated by cvxhull using this prefix will be dropped when the program is called. Defaults to _cvxh.

noretainmakes all results internal to the program, so new variables are not added to the dataset. Existing variables of the same names will still be dropped.

noreportsuppresses progress messages output while the program is running.

nographsuppresses drawing of the graph within the program.

saving(graph_filename[,replace])saves the graph in a .gph file.

Examples.

cvxhull yvar xvardraws a default scatterplot with one convex hull containing all the observations..

cvxhull yvar xvar if year==99draws a scatterplot showing only the subset of observations and the hull round that subset.

cvxhull yvar xvar, group(year) hulls(3) ///scat(mlab(year) mlabpos(c) msym(i) ysc(r(0,1000)))draws a scatterplot with defined y range and each point marked with its group value and no symbol. Up to three layers of hull will be computed and plotted for each set of observations sharing a group value..

cvxhull yvar xvar, group(year) select(3) scat(mlab(year))draws a scatterplot of all groups with each point labelled with its group value. A hull is drawn round group 3, where the equivalence of 3 and a particular year value is identified in the output table..

cvxhull yvar xvar, hulls(15) group(group)will plot the maximum 8 hulls for each group. However, up to 15 hulls (subject to there being data points) will be calculated, and variables _cvxh1l-_cvxh15r etc will be added to the dataset..

cvxhull yvar xvar, hulls(10 3)will plot hulls 1, 4, 7 and 10.See cvxplot for a command to draw hulls using variables saved by cvxhull. If you insist on going it alone, the following example gives an idea of what is required. It may be convenient to divide this into several lines and write the command as a

.dofile. local hs "_cvxh1l-_cvxh2r" scatter y x, ms(i) mlab(id,pos(c)) /// || line `hs' x if group==1, sort(x y) /// clc(black black black black black black) legend(off) /// || line `hs' x if group==2, sort(x y) /// clc(blue blue blue blue blue blue) legend(off) /// || line `hs' x if group==3, sort(x y) /// clc(black black blue blue green green) legend(off) /// || line `hs' x if group==4, sort(x y) /// clc(black black black black black black) legend(off) /// ... etc

AuthorR Allan Reese, University of Hull, UK r.a.reese@hull.ac.uk

Also seeOn-line: help for

cvxplotcvxindexgraphscatterManual: [R] graph, [R] connectlines