drawing surface plots on the IR³ simplex

simplexAs a result of a corridor conversation in Warwick, I started looking at distributions on the IR³ simplex,

\{(x_1,x_2,x_3)\in\mathbb{R}_+^3;\ x_1+x_2+x_3=1\},

and wanted to plot the density in a nice way. As I could not find a proper package on CRAN, the closer being the BMAmevt (for Bayesian Model Averaging for Multivariate Extremes) R package developed by a former TSI Master student, Anne Sabourin, I ended up programming the thing myself. And producing the picture above. Here is the code, for all it is worth:

# setting the limits
par(mar=c(0,0,0,0),bg="black")
plot(c(0,1),col="white",axes=F,xlab="",ylab="",
xlim=c(-1,1)*1.1/sqrt(2),ylim=c(-.1,sqrt(3/2))*1.1)

# density on a grid with NAs outside, as in image()
gride=matrix(NA,ncol=520,nrow=520)
ww3=ww2=seq(.01,.99,le=520)
for (i in 1:520){
   cur=ww2[i];op=1-cur
   for (j in 1:(521-i))
      gride[i,j]=mydensity(c(cur,ww3[j],op-ww3[j]))
   }

# preparing the graph
subset=(1:length(gride))[!is.na(gride)]
logride=log(gride[subset])
grida=(logride-min(logride))/diff(range(logride))
grolor=terrain.colors(250)[1+trunc(as.vector(grida)*250)]
iis=(subset%%520)+520*(subset==520)
jis=(subset%/%520)+1

# plotting the value of the (log-)density
# at each point of the grid
points(x=(ww3[jis]-ww2[iis])/sqrt(2),
   y=(1-ww3[jis]-ww2[iis])/sqrt(2/3),
   pch=20,col=grolor,cex=.3)

One Response to “drawing surface plots on the IR³ simplex”

  1. Mike Betancourt has a little paper on the arXiv on the transformation y_i^2 = x_i. Under this transformation, the normalization constraint becomes the equation of a sphere; positivity restricts the manifold to the all-positive orthant. That’s kind of pretty, isn’t it?

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Follow

Get every new post delivered to your Inbox.

Join 619 other followers