Generate multivariate symmetric distribution

Question

I have a (bivariate) spherically symmetric distribution, in the sense that I can generate iid values distributed according to it. But there is a detail - I can't get the generated values as a whole, but only their single component, i.e. x coordinate and not y. So I don't know the underlying distribution - it's an empirical one basically, but I need to generate values from it for different purposes: e.g. estimating distribution of some function f(x, y).

Of course, y has the same distribution as x (which we can generate) - so one would think that we can just generate two iid values and use them for x and y, but this doesn't work: for the underlying distribution to be spherically symmetric x and y have to have some dependence (in general case). So, distribution generated this was will not be symmetric. Simple modifications of this method, like rotating the resulting vector by a random angle, don't seem to help. So, any ideas, suggestions? I couldn't find anything on this topic.

For the sake of example you can think of the underlying distribution as being a multivariate t distribution. But NOT multivariate normal of course, as it is the same as two independent univariate ones.

Answer 1

I don't entirely follow your question, but maybe I can still help, or at least get you part-way there.

In order for a bivariate distribution to be circularly symmetric, its $Y$-distribution must be the same as its $X$-distribution. Thus, if you know the distribution for $X$, you can use that to generate realized $Y$-values. On the other hand, if you can draw existing realized $X$-values from an existing sample, you could use independent draws from that sample to approximate draws of the sample's $Y$-values. Here is a simple example, coded in R:

set.seed(2614)    # this makes the simulation exactly reproducible
N = 500           # we'll generate 500 data
x = rt(N, df=5)   # x & y will be distributed as t w/ 5 df
y = rt(N, df=5)

range(c(x,y))     # [1] -8.293021  9.571094
SD    = 5/3       # 1.666667
theta = seq(0, 2*pi, length=200)
windows()
  plot(x, y, xlim=c(-10,10), ylim=c(-10,10), col=rgb(.5,.5,.5,.75))
  lines(x=SD*sin(theta),   y=SD*cos(theta),   col="red3")       # plotting circles
  lines(x=2*SD*sin(theta), y=2*SD*cos(theta), col="orangered")

This looks reasonably circularly symmetrical.

Now let's try to figure out what the distribution is from the $X$-variable (to which you have access):

library(fitdistrplus)                   # we'll use this package
xn = fitdist(x, "norm")                 # we'll check the normal distribution
xt = fitdist(x, "t", start=list(df=2))  #  & the t-distribution starting w/ 2 df
summary(xn)
# Fitting of the distribution ' norm ' by maximum likelihood 
# Parameters : 
#         estimate Std. Error
# mean -0.05376445 0.06156805
# sd    1.37670350 0.04353508
# Loglikelihood:  -869.3152   AIC:  1742.63   BIC:  1751.06 
summary(xt)
# Fitting of the distribution ' t ' by maximum likelihood 
# Parameters : 
#    estimate Std. Error
# df 4.512022  0.6678295
# Loglikelihood:  -832.3588   AIC:  1666.718   BIC:  1670.932 
windows()
  plot(xn)

windows()
  plot(xt)

From this, our best guess is that $X$ is distributed as $t$ with $4.5$ degrees of freedom. Let's see what we would have guessed for $Y$, if we had had access to it:

yn = fitdist(y, "norm")
yt = fitdist(y, "t", start=list(df=2))
summary(yn)
# Fitting of the distribution ' norm ' by maximum likelihood 
# Parameters : 
#         estimate Std. Error
# mean -0.03244677 0.05624948
# sd    1.25777662 0.03977428
# Loglikelihood:  -824.1421   AIC:  1652.284   BIC:  1660.713 
summary(yt)
# Fitting of the distribution ' t ' by maximum likelihood 
# Parameters : 
#    estimate Std. Error
# df 5.504327  0.9605105
# Loglikelihood:  -807.3761   AIC:  1616.752   BIC:  1620.967 
# windows()  # plots omitted for brevity
#   plot(yn)
# windows()
#   plot(yt)

It seems we would have guessed $Y$ was distributed as $t$ with $5.5$ degrees of freedom. These two guesses are not far from each other, or from the true data generating process (a $t$-distribution with $5.0$ degrees of freedom).

What if you want to sample from $X$ directly? That works reasonably well, too:

x.gen = sample(x, size=N, replace=TRUE)
y.gen = sample(x, size=N, replace=TRUE)

range(c(x.gen, y.gen))            # [1] -8.293021  9.571094
SD    = sd(c(x.gen, y.gen));  SD  # [1] 1.329146
theta = seq(0, 2*pi, length=200)
windows()
  plot(x.gen, y.gen, xlim=c(-10,10), ylim=c(-10,10), col=rgb(.5,.5,.5,.75))
  lines(x=SD*sin(theta),   y=SD*cos(theta),   col="red3")
  lines(x=2*SD*sin(theta), y=2*SD*cos(theta), col="orangered")

The plot / result doesn't look that bad:

All of these results assume you don't have access to $Y$, but can (correctly) assume that the bivariate distribution is circularly symmetric. We also tend to get good results because I simulated a moderately large dataset—if you had only had, say, $N = 25$ data, the approximations probably wouldn't be very good.