Spherical platykurtic random cloud

Question

I'd like to generate a multivariate continuous data which is globular cloud, like multivariate standard normal data is, but which is more platykurtic than normal data.

There are many ways to get platykurtic data (e.g. from beta distribution) but the more platykurtic they are the more they are rectangular shape in multivariate space. But I want globular, hyperspherical random data. Unimodal data. Preferably from not bounded distribution, if possible. And with an option to vary the amount of flat kurtosis.

Can you suggest a distribution or a trick to generate?

P.S. Saying "hyperspherical" I mean "any dimensionality" (not "high dimensionality"). That is, I imply 2D case to be just particular case; I'm interested in it as well.

Answer 1

Although this is obviously related to my previous answer, I think it's different enough to be considered separately. Given a point in an $n$-dimensional cloud of iid standard normals, shrink it radially a chosen proportion of the way to the corresponding fractile of a uniform distribution in the unit-radius $n$-ball.

For iid standard normals in $n$ dimensions, the distance from the origin has a $\chi_n$ distribution. Let $F_n(d)$ denote its cdf. For a uniform distribution in the unit-radius $n$-ball, the cdf of the distance from the origin is $d^n$, and the inverse cdf is $p^{1/n}$. So multiply each point's coordinate vector by $1 + \alpha(F_n(d)^{1/n}/d - 1)$, where $\alpha$ is the chosen proportion and $d$ is the observed distance of the point from the origin.

EDIT -- A minor improvement, that makes the results easier to interpret: Scale the ball so that the mean square distance of the points from the origin is the same as the normal, $n$. The corresponding multiplier on the vector of normals is $1 + \alpha\,(F_n(d)^{1/n}\sqrt{n+2}\,/d - 1)$.

I have no proof that the marginal distribution is unimodal, but I have looked at histograms of marginal distributions with $\alpha = 0, .05, \ldots, 1$ for $n = 1, \ldots, 10$, and they all look as they should, varying smoothly from $\mathrm{N}(0,1)$ to a shifted and scaled $\mathrm{Beta}(\frac{n+1}{2},\frac{n+1}{2})$. For $n = 2$ I have also looked at scatter plots with the same set of $\alpha$-values, and they too look as they should, with no "bald spot" in the middle.

Here is Mathematica code whose results are organized to facilitate exploring the effect of $\alpha$.

{m, n} = { sample size, # of dimensions };
z = RandomReal[NormalDistribution[], {m, n}];
y = z * With[{dd = Total[z^2, {2}]},
    CDF[ChiSquareDistribution[n], dd]^(1/n) * Sqrt[(n + 2)/dd]];

Then x = alpha*y + (1-alpha)*z will be a matrix that varies smoothly with alpha between iid normal (alpha = 0; pure z) and a uniform n-ball (alpha = 1; pure y). Here are scatter plots of 5000 points in 2 dimensions for alpha = {0, .25, .50, .75, 1}.

Answer 2

For a cloud of i.i.d. standard normals, try shrinking the points toward the origin logarithmically. That is, multiply the coordinates of each point by $\log(1+\alpha\, d)/(\alpha\, d)$, where $d$ is the distance of the point from the origin, and $\alpha$ controls the amount of shrinkage, with larger values shrinking more.

Answer 3

You might look into the generalized normal distribution, a generalization of the normal distribution that introduces a parameter that controls the kurtosis. It includes as special cases the laplace, normal, and uniform distributions. There appears to be a citation on the wikipedia page that discusses the multivariate generalized normal distribution. Just a note of caution -- there is at least one other distribution called the "generalized normal" distribution; the alternate one I am thinking of involves parametric control of skewness, not kurtosis, so make sure you are reading about the "correct" generalized normal distribution!

Answer 4

The Johnson $S_U$ distribution (see citation below) has infinite tails and can be platykurtotic based on your parameter choices. It is easily generated from normally distributed data as follows:

$$ Z = \gamma + \delta \; \text{sinh}^{-1} \left(\frac{X-\xi}{\lambda} \right) $$

where $X \sim N(0,1)$. You probably just want to set $\gamma=\xi=0$ and $\lambda=1$. The parameter $\delta$ controls kurtosis, with higher values resulting in smaller kurtosis.

Here is a contour plot (with R code) showing the bivariate density of independent $S_U$ variables:

library(SuppDists)

parms = list(gamma=0,delta=2,xi=0,lambda=2,type='SU')
coords = seq(-3,3,0.01)
zz = outer(
    X=coords,Y=coords,
    FUN=function(x,y) c(dJohnson(x,parms)*dJohnson(y,parms))
)
contour(coords,coords,zz)

The shape of the distribution is a bit more clear if we look at univariate densities:

curve(dJohnson(x,parms),from=-3,to=3)

Note that $\xi$ is the location parameter, meaning that it shifts the distribution left and right leaving variance, skewness, and kurtosis unchanged. $\lambda$ is the scale parameter, meaning that it changes the variance but leaves mean, skewness, and kurtosis unchanged. $\gamma$ changes skewness and $\delta$ changes kurtosis, but they affect lower moments as well.

For example, we can calculate theoretical variance and kurtosis in R when $\gamma=0,\xi=0,\delta=1,\lambda=1$:

lam1 = sJohnson(list(gamma=0,delta=1,xi=0,lambda=1,type='SU'))
lam1$Variance
##[1] 3.194528
lam1$Kurtosis
##[1] 33.18813

Note that if we change $\lambda$ to 2 and hold the other parameters constant variance changes but kurtosis remains the same:

lam2 = sJohnson(list(gamma=0,delta=1,xi=0,lambda=2,type='SU'))
lam2$Variance
##[1] 12.77811
lam2$Kurtosis
##[1] 33.18813

Note that if we then change $\delta$ to 2 and hold the other parameters constant (keep $\lambda=2$), both variance and kurtosis change:

del2lam2 = sJohnson(list(gamma=0,delta=2,xi=0,lambda=2,type='SU'))
del2lam2$Variance
##[1] 1.297443
del2lam2$Kurtosis
##[1] 1.507862

Citation: Johnson NL (1949). Systems of Frequency Curves Generated by Methods of Translation. Biometrika 36(1/2), 149-176. See in particular equation 21.