Posts Tagged: tails


11
Apr 13

High Obesity levels found among fat-tailed distributions

In my never ending quest to find the perfect measure of tail fatness, I ran across this recent paper by Cooke, Nieboer, and Misiewicz. They created a measure called the “Obesity index.” Here’s how it works:

  • Step 1: Sample four times from a distribution. The sample points should be independent and identically distributed (did your mind just say “IID”?)
  • Step 2: Sort the points from lowest to highest (that’s right, order statistics)
  • Step 3: Test whether the sum of the smallest and greatest number is larger than the sum of the two middle.

The Obesity index is the probability that the sum of these end points is larger than the sum of the middle numbers. In mathy symbols:

Ob(X) = P (X_1 + X_4 > X_2 + X_3 | X_1 \leq X_2 \leq X_3 \leq X_4), X_i~IID



The graph at the top of this post shows how the Obesity index converges for different distributions. As always, I’ve included my R code at the end of this article, so you can run this simulation for yourself (though, as usual, I forgot to set a random seed so that you can run it exactly like I did).

The dots in the graph represent the mean results from 8, 16, 32, and so on, up to 4096 trials from each of the distributions I tested. Note that each trial involves taking 4 sample points. Confused? Think of it this way: each sample of 4 points gives us one Bernoulli trial from a single distribution, which returns a 0 or 1. Find the average result after doing 4096 of these trials, and you get one of the colored dots at the far right of the graph. For example, the red dots are averages from a Uniform distribution. The more trials you do, the closer results from the Uniform will cluster around 0.5, which is the “true” Obesity value for this distribution. The Uniform distribution is, not coincidentally, symmetric. For symmetric distributions like the Normal, we only consider positive values.

The graph gives a feel for how many trials would be needed to distinguish between different distributions based on their Obesity index. I’ve done it this way as part of my Grand Master Plan to map every possible distribution based on how it performs in a variety of tail indices. Apparently the Obesity index can be used to estimate quantiles; I haven’t done this yet.

My initial impressions of this measure (and these are very initial!) are mixed. With a large enough number of trials, it does a good job of ordering distributions in a way that seems intuitively correct. On the other hand, I’d like to see a greater distance between the Uniform and Beta(0.01, 0.01) distribution, as the latter is an extreme case of small tails.

Note that Obesity is invariant to scaling:

Ob(x) = Ob(k*X)



but not to translations:

Ob(X) \neq Ob(X+c)



This could be a bug or a feature, depending on what you want to use the index for.

Extra special karma points to the first person who comes up with a distribution whose Obesity index is between the Uniform and Normal, and that isn’t a variant of one I already tested.

Here’s the code:

# Code by Matt Asher for StatisticsBlog.com
# Feel free to redistribute, but please keep this notice
 
# Create random varaibles from the function named in the string
generateFromList = function(n, dist, ...) {
	match.fun(paste('r', dist, sep=''))(n, ...)
}
 
# Powers of 2 for testAt
testAt = 3:12
testAtSeq = 2^testAt
testsPerLevel = 30
 
 
distros = c()
 
distros[1] = 'generateFromList(4,"norm")'
distros[2] = 'generateFromList(4,"unif")'
distros[3] = 'generateFromList(4,"cauchy")'
distros[4] = 'generateFromList(4,"exp")'
distros[5] = 'generateFromList(4,"chisq",1)'
distros[6] = 'generateFromList(4,"beta",.01,.01)'
distros[7] = 'generateFromList(4,"lnorm")'
distros[8] = 'generateFromList(4,"weibull",1,1)'
 
# Gotta be a better way to do this.
dWords = c("Normal", "Uniform", "Cauchy", "Exponential", "Chisquare", "Beta", "Lognormal", "Weibull")
 
par(mar=c(4,5,1.5,.5))
plot(0,0,col="white",xlim=c(min(testAt),max(testAt)), ylim=c(-.5,1), xlab="Sample size, expressed in powers of 2", ylab="Obesity index measure", main="Test of tail fatness using Obesity index")
 
abline(h=0)
 
colorList = list()
colorList[[1]]=rgb(0,0,1,.2)
colorList[[2]]=rgb(1,0,0,.2)
colorList[[3]]=rgb(0,1,0,.2)
colorList[[4]]=rgb(1,1,0,.2)
colorList[[5]]=rgb(1,0,1,.2)
colorList[[6]]=rgb(0,1,1,.2)
colorList[[7]]=rgb(0,0,0,.2)
colorList[[8]]=rgb(.5,.5,0,.2)
 
# Create the legend
for(d in 1:length(distros)) {
	x = abs(rnorm(20,min(testAt),.1))
	y = rep(-d/16,20)
	points(x, y, col=colorList[[d]], pch=20)
	text(min(testAt)+.25, y[1], dWords[d], cex=.7, pos=4)
 
}
 
 
dCounter = 1
for(d in 1:length(distros)) {
	for(l in testAtSeq) {
		for(i in 1:testsPerLevel) {
			count = 0
			for(m in 1:l) {
 
				# Get the estimate at that level, plot it testsPerLevel times	
				x = sort(abs(eval(parse( text=distros[dCounter] ))))
				if ( (x[4]+x[1])>(x[2]+x[3]) ) {
					count = count + 1
				}
			}
 
			# Tiny bit of scatter added
			ratio = count/l
			points(log(l, base=2), ( ratio+rnorm(1,0,ratio/100)), col=colorList[[dCounter]], pch=20)
		}
	}
 
	dCounter = dCounter + 1
}

25
Oct 12

How fat are your tails?

Lately I’ve been thinking about how to measure the fatness of the tails of a distribution. After some searching, I came across the Pareto Tail Index method. This seems to be used mostly in economics. It works by finding the decay rate of the tail. It’s complicated, both in formula and in it’s R implementation (I couldn’t get “awstindex” to run, which supposedly can be used to calculate it). The Index also has the disadvantage of being a “curve fitting” approach, where you start by assuming a particular distribution, then see which parameter gives the best fit. If this doesn’t seem morally abhorrent to you, perhaps you have a future as a high-paid econometrician.

In the past I’ve looked at how to visualize the impact of the tails on expectation, but what I really wanted was a single number to measure fatness. Poking around the interwebs, I found a more promising approach. The Mean Absolute Deviation (or MAD, not to be confused with the Median Absolute Distribution, or MAD) measures the average absolute distance between a random variable and it’s mean. Unlike the Standard Deviation (SD), the MAD contains no squared terms, which makes it less volatile to outliers.

As a result, we can use the MAD/SD ratio as a gauge of fat-tailedness. The closer the number is to zero, the fatter the tails. The closer the number is to 1 (it can never exceed 1!), the thinner the tails. For example, the normal distribution has a MAD/SD ratio of 0.7970, which happens to be the square root of 2 over pi (not a coincidence, try proving this if you rock at solving integrals).

The graph at the beginning of this post shows a Monte Carlo estimation of the MAD/SD ratio for the Student T distribution as it goes from very high Degrees of Freedom (1024) to very low (1/4). You may know that the T distro converges to the Normal at high degrees of freedom (hence the result of nearly .8 for high DF), but did you know that the T distro on 1 Degree of Freedom is the same as the infamously fat-tailed Cauchy? And why stop at 1? We can keep going into fractional DFs. I’ve plotted the ratio all the way down to 1/4. As always, code in R is at the end of the post.

One more thing: there is at least one continuous distribution for which the MAD/SD ratio reaches it’s maximum possible value of one. First person to guess this maximally thin-tailed distribution gets a free copy of the comic I worked on.

# Start with a Normal, move to a Cauchy
dfs = 2^(10:-2)
results = c()
for(i in dfs) {
	x = rt(1000000,i)
	results = c(results, mean(mean(abs(x))/sd(x)))
}
 
# Note the wonky x-axis limit and order
plot(rev(-2:10), results, col="blue", pch=20, xlim=rev(range(-2:10)), xlab="Degrees of Freedom (binary log scale)", ylab="MAD/SD ratio")