Fitting Distribution X to Data From Distribution Y
I had someone ask me about fitting a beta distribution to data drawn from a gamma distribution and how well the distribution would fit. I’m not a “closed form” kinda guy. I’m more of a “numerical simulation” type of fellow. So I whipped up a little R code to illustrate the process then we changed the parameters of the gamma distribution to see how it impacted fit. An exercise like this is what I call building a “toy model” and I think this is invaluable as a method for building intuition and a visceral understanding of data.
Here’s some example code which we played with:
set.seed(3) x <- rgamma(1e5, 2, .2) plot(density(x)) # normalize the gamma so it's between 0 & 1 # .0001 added because having exactly 1 causes fail xt <- x / ( max( x ) + .0001 ) # fit a beta distribution to xt library( MASS ) fit.beta <- fitdistr( xt, "beta", start = list( shape1=2, shape2=5 ) ) x.beta <- rbeta(1e5,fit.beta$estimate[],fit.beta$estimate[]) ## plot the pdfs on top of each other plot(density(xt)) lines(density(x.beta), col="red" ) ## plot the qqplots qqplot(xt, x.beta)
It’s not illustrated above, but it’s probably useful to transform the simulated data (x.beta) back into pre normalized space by multiplying by max( x ) + .0001 . (I swore I’d never say this but I lied) I’ll leave that as an exercise for the reader.
Another very useful tool in building a mental road map of distributions is the graphical chart of distribution relationships that John Cook introduced me to.