'Aspect ratio' as a measure of the 'tall-and-skinny' property of unimodal distributions

Question

Kurtosis has not entirely lived up to being a measure of the 'tall-and-skinny' property. If you can access it, see Westfall 2014 for a valuable discussion of the pitfalls of using kurtosis as 'peakedness'.

Intuitively an aspect ratio of the height of a distribution divided by its width would convey a sort of 'tall-and-skinny' property. But since probability density functions (PDFs) have infinite width, such a value would be zero. It might be tempting to take the denominator to be the width at the average value of the PDF, but in general can lead to a factor of $\frac{1}{\infty - (- \infty)}$.

I remembered back to when I did chromatography that we instead used "width-at-half-height" to quantify the tall-and-skinny property of analyte peaks. Taking a look around online I found the notion of full width at half maximum (FWHM) that seems to be an equivalent term.

In correspondance with the diagram above, we could consider the ratio $\frac{f_{\max}}{|x_2 - x_1|}$ where $f$ is the probability density function for random variable $X$ with instances $x_1$ and $x_2$ corresponding to the width endpoints at half of the maximum of $f$.

At least, that is what I have thought on the matter so far. Any tradeoffs in choosing this as an estimate of 'tall-and-skinniness' for unimodal distributions?

Answer 1

You already got my +1 for drawing attention to Westfall (2014). A few thoughts:

Unimodality is of course important, and also quite a restriction. If you have multiple peaks, then all kinds of strange things can happen. And if you don't have a peak at all (e.g., a gamma distribution with shape $k<1$), then your $f_{\text{max}}$ is not even defined.

The next problem is when your density is noncontinuous. There may not be a point where it takes a value of $\frac{f_{\text{max}}}{2}$. But of course you can get around this by using suprema and infima as appropriate.

Here are a few well-behaved (that is, unimodal and symmetric) examples, all with your KPI value of $2$ (axes aligned for comparability):

I don't quite know whether I would call all of these "equally tall-and-skinny". Equally tall, yes, but not really equally skinny. Essentially, the two flanks rotate around their halfway point (indicated by red dots) between the four panels, and this constant halfway point is at the constant $x_{1,2}$. But of course all this is subjective.

These distributions all have the following form:

$$ f(x) = \begin{cases} 0, & x<-b \\ f_{\text{max}}\cdot\frac{b+x}{b-a}, & -b\leq x < -a \\ f_{\text{max}}, & -a\leq x<a \\ f_{\text{max}}\cdot\frac{b-x}{b-a}, & a\leq x <b \\ 0, & b\leq x \end{cases} $$ for appropriate parameters $0\leq a<b$ and $f_{\text{max}}$. To ensure these integrate to $1$, we set for given $a$ and $b$

$$ f_{\text{max}}:= \frac{1}{a+b}. $$

To ensure that they have a prespecified value of your KPI

$$ \text{KPI} = \frac{f_{\text{max}}}{a+b}=\frac{1}{(a+b)^2} $$

(because $x_{1,2}=\pm\frac{a+b}{2}$), we use $a$ as a free parameter and set

$$ b := \frac{1}{\sqrt{\text{KPI}}} - a. $$

The plots have $a\in\big\{0,\frac{1}{10},\frac{2}{10},\frac{3}{10}\big\}$.

R code:

KPI <- 2
opar <- par(mfrow=c(2,2),las=1,mai=c(.5,.5,.1,.1))
    y_max <- sqrt(KPI)

    aa <- c(0,0.1,0.2,0.3)
    bb <- 1/sqrt(KPI)-aa
    f_max <- 1/(aa+bb)
    xx <- seq(-1.1/sqrt(KPI),1.1/sqrt(KPI),by=0.01)

    ff <- Vectorize(FUN=function(xx,aa,bb,f_max) {
        if ( xx < -bb ) return(0)
        if ( -bb <= xx & xx < -aa ) return(f_max*(bb+xx)/(bb-aa))
        if ( -aa <= xx & xx < aa ) return(f_max)
        if ( aa <= xx & xx < bb ) return(f_max*(bb-xx)/(bb-aa))
        if ( bb <= xx ) return(0)
        },vectorize.args="xx")

    for ( ii in seq_along(aa) ) {
        plot(xx,ff(xx,aa[ii],bb[ii],f_max[ii]),ylim=c(0,y_max),type="l",xlab="",ylab="")
        halfway <- c(-1,1)*(aa[ii]+bb[ii])/2
        points(halfway,ff(halfway,aa[ii],bb[ii],f_max[ii]),pch=19,cex=1.5,col="red")
    }
par(opar)

Answer 2

Before you get to the other drawbacks of your measure, the first thing to note is that you appear to be comparing apples and oranges. The idea of using kurtosis as a measure of "peakedness" of a distribution (as flawed as that is) is that it is a measure that adjusts for variance, so the "peakedness" looks at the shape of the distribution rather than its scale. Your measure does not do this --- it is proportionate to the inverse of the variance.$^\dagger$

So, before you get to anything else, you are going to need to decide whether you want your measure of what is "tall-and-skinny" to be effectively just another measure of (inverse) variance, or whether you want it to be a measure that is determined by the shape of the density rather than its scale. If the former then your measure is not really a measure of "peakedness" in the sense that kurtosis is sometimes (erroneously) used. If the latter, one thing you could do is to multiply your measure by the variance of the distribution, so that it is now "scale free".

Even with this adjustment, two other obvious drawbacks are: (1) there might not be a maximum density value (i.e., the density might be unbounded); and (2) the choice of using half the density maximum is arbitrary. You could generalise your analysis in a way that alleviates this problem by looking at the "intensity function" for the distribution. Consider a continuous random variable $X$ scaled to have unit variance and suppose it has a unimodal density $f$. The "intensity" of the distribution can be defined by:

$$H(a) \equiv \mathbb{P}(f(X) \geqslant a) \quad \quad \quad \text{for all } 0 \leqslant a \leqslant \infty.$$

This is a non-increasing function with $H(0) = 1$ and $H(\infty) = 0$ (a simple consequence of the properties of probability densities). In general, a density that is "tall-and-skinny" is going to have an intensity that decreases down to zero slowly as $a$ increases, whereas a density that is "short-and-fat" is going to have an intensity that decreases down to zero rapidly as $a$ increases. Thus, you could get an idea of how "tall-and-skinny" the distribution is by looking at how rapidly the "intensity" of the distribution decreases.

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$^\dagger$ To see this, note that if you were to double the random variable then this would halve the maximum density value, your numerator, and double the width in the denominator.