# Uniform distribution of correlation coefficients in correlation matrix

This question is purely academic in nature; I have no application for the knowledge scholastic or otherwise other than curiousity.

Suppose you have a KxN matrix of data, with each row representing a variable, and each column representing an observation of each variable, called M.

Suppose you now generate a correlation matrix using the data in M. Is there a simple distribution for the variables in the K rows such that the correlation coefficients of the non-diagonal elements of the matrix are distributed approximately Uniform(0,1)? Or alternatively, for a covariance matrix rather than a correlation matrix?

It's not only possible, it's easy to create any distribution $F$ whatsoever supported on the interval $[-1/(N-2), 1]$, provided only that $K \le N-2$. Here's one way. It creates datasets in which all the variables have the same correlation with each other.

Let $\rho$ be a random variable with distribution $F$. Define $U \ge 1/(N-1)$ as the unique solution to

$$\rho = \frac{1 + 2 U - (N-1)U^2}{2 - 2(N-2)U + (N-1)(N-2)U^2}.$$

Set $V = (N-2)U-1$ and construct the $K$ vectors, each of length $N$, given by

\left\{\eqalign{ X_1 &= (1, V, -U, -U, \ldots, -U) \\ X_2 &= (1, -U, V, -U, -U, \ldots, -U) \\ &\ldots \\ X_K &= (1, -U, -U, \ldots, -U, V, -U, \ldots, -U). }\right.

Each has a $1$ in the first place, $V$ in the $K+1^\text{st}$ place, and $-U$ everywhere else.

A computation (which is simple because all the $X_i$ have zero means and the same variance) shows that $\rho$ is the correlation coefficient between each $X_i$ and $X_j$. Therefore all the correlation coefficients of these $K$ random vectors of length $N$ equal $\rho$, QED.

### Appendix: Illustration via simulation

This R code simulates from a given distribution $F$. It displays histograms of the correlation coefficients and tests them for uniformity. The comments explain the details.

#
# Specify the situation.
#
N <- 20       # Dataset size
K <- 4        # Number of variables
n.sim <- 1e4  # Simulation size
#
# Predefine some objects.
#
f <- function(rho, n) { # Maps rho to U
(1 + (n-2)*rho + sqrt(n * (1-rho)*(1+(n-2)*rho))) / ((n-1) * (1+(n-2)*rho))
}
pattern <- cbind(diag(rep(1, K)), matrix(0, K, N-K))
#
# Conduct the simulation.
#
# rF <- runif      # The random number generator
# qF <- qunif      # The quantile function
# dF <- dunif      # The density function
rF <- function(n) rbeta(n, 1, 3)
qF <- function(q) qbeta(q, 1, 3)
dF <- function(x) dbeta(x, 1, 3)
rho <- rF(n.sim)   # Draw values of rho
#
# Construct the data and compute their correlation coefficients.
# Each row of sim will record one particular correlation coefficient.
# Its columns are the iterations.
#
U <- f(rho, N)
sim <- sapply(U, function(u) {
v <- (N-1)*u - 1
x <- matrix(rep(c(rep(-u, N-1), 1), K), nrow=K, byrow=TRUE) + v*pattern
})
#
# Display the distributions of the correlation coefficients.
#
n.plots <- choose(K,2)
n.rows <- floor(sqrt(n.plots))
n.cols <- ceiling(n.plots/n.rows)
par(mfrow=c(n.rows, n.cols))
breaks <- qF(seq(0, 1, by=1/20))
invisible(apply(sim, 1, function(x) {
H <<- hist(x, main="Marginal Histogram", freq=FALSE, breaks=breaks)
p <- chisq.test(H$counts)$p.value