How to generate random correlation matrix that has approximately normally distributed off-diagonal entries with given standard deviation?

I would like to generate a random correlation matrix such that the distribution of its off-diagonal elements looks approximately like normal. How can I do it?

The motivation is this. For a set of $n$ time series data, the correlation distribution often looks quite close to normal. I would like to generate many "normal" correlation matrices to represent the general situation and use them to calculate risk number.

I know one method, but the resulting standard deviation (of the distribution of the off-diagonal elements) is too small for my purpose: generate $n$ uniform or normal random rows of a matrix $\mathbf X$, standardize the rows (subtract the mean, divide by standard deviation), then the sample correlation matrix $\frac{1}{n-1}\mathbf X \mathbf X^\top$ has normally distributed off-diagonal entries [Update after comments: standard deviation will be $\sim n^{-1/2}$].

Can anyone suggest a better method with which I can control the standard deviation?

• @Richard, thanks for your question. Unfortunately, the method you describe above will not produce entries that are normally distributed. The diagonals are 1 with probability one and the off-diagonals are bounded between $-1$ and $+1$. Now, the rescaled entries will converge asymptotically to a normal distribution centered around zero. Can you give us more information about the problem you're actually trying to solve? And, why do you want "normally distributed" off diagonals? – cardinal Apr 28 '11 at 22:47
• @Richard, what I mean was, suppose $X = (X_1,X_2,\ldots,X_n)$ and $Y = (Y_1,Y_2,\ldots,Y_n)$ are two independent vectors such that the entries of each are i.i.d. standard normal. Compute $\hat{\rho}_n = s_{xy} / (s_x s_y)$; that is, the sample correlation between $X$ and $Y$. Then $n^{1/2} \hat{\rho}_n$ converges in distribution to a standard normal random variable. By "rescaled", I meant the multiplication by $n^{1/2}$ which is what is required to obtain a non degenerate limiting distribution. – cardinal Apr 28 '11 at 23:11
• @Richard, the essence of the "problem" is that by making two restrictions (a) that the norms of each row are 1 and (b) that the entries are generated from a random sample, you necessarily are forcing the correlations to be quite small (on the order of $n^{-1/2}$. The reason is that you can't have arbitrarily large correlations between rows and still get the norms of each row to be 1 in the presence of so much independence. – cardinal Apr 28 '11 at 23:18
• ...now, you can get larger correlations in magnitude by first correlating the rows among themselves before renormalizing. But, you essentially only have one parameter to play with, so both the asymptotic mean and variance will be tied to that parameter. So, that probably won't give you the flexibility you seem to want, either. – cardinal Apr 28 '11 at 23:22
• Sure, let's take a simple case. Call the generating matrix $X$, which we'll assume to be $m \times n$ without loss of generality. Now, generate the columns of $X$ as i.i.d. vectors such that the elements of each vector are standard normal random variables that are equicorrelated with correlation $\rho$. Now, use the procedure you have been. Let $\hat{\rho}_{ij}$ denote the sample correlation between the $i$th and $j$th *row* of $X$. Then for fixed $m$, letting $n \to \infty$, $n^{1/2} (\hat{\rho}_{ij} - \rho)$ converges in distribution to a $\mathcal{N}(0,(1-\rho^2)^2)$ random variable. – cardinal Apr 28 '11 at 23:38

I have first provided what I now believe is a sub-optimal answer; therefore I edited my answer to start with a better suggestion.

Using vine method

In this thread: How to efficiently generate random positive-semidefinite correlation matrices? -- I described and provided the code for two efficient algorithms of generating random correlation matrices. Both come from a paper by Lewandowski, Kurowicka, and Joe (2009).

Please see my answer there for a lot of figures and matlab code. Here I would only like to say that the vine method allows to generate random correlation matrices with any distribution of partial correlations (note the word "partial") and can be used to generate correlation matrices with large off-diagonal values. Here is the relevant figure from that thread:

The only thing that changes between subplots, is one parameter that controls how much the distribution of partial correlations is concentrated around $\pm 1$. As OP was asking for an approximately normal distribution off-diagonal, here is the plot with histograms of the off-diagonal elements (for the same matrices as above):

I think this distributions are reasonably "normal", and one can see how the standard deviation gradually increases. I should add that the algorithm is very fast. See linked thread for the details.

A straight-forward modification of your method might do the trick (depending on how close you want the distribution to be to normal). This answer was inspired by @cardinal's comments above and by @psarka's answer to my own question How to generate a large full-rank random correlation matrix with some strong correlations present?

The trick is to make samples of your $\mathbf X$ correlated (not features, but samples). Here is an example: I generate random matrix $\mathbf X$ of $1000 \times 100$ size (all elements from standard normal), and then add a random number from $[-a/2, a/2]$ to each row, for $a=0,1,2,5$. For $a=0$ the correlation matrix $\mathbf X^\top \mathbf X$ (after standardizing the features) will have off-diagonal elements approximately normally distributed with standard deviation $1/\sqrt{1000}$. For $a>0$, I compute correlation matrix without centering the variables (this preserves the inserted correlations), and the standard deviation of the off-diagonal elements grow with $a$ as shown on this figure (rows correspond to $a=0,1,2,5$):

All these matrices are of course positive definite. Here is the matlab code:

offsets = [0 1 2 5];
n = 1000;
p = 100;

rng(42) %// random seed

figure
for offset = 1:length(offsets)
X = randn(n,p);
for i=1:p
X(:,i) = X(:,i) + (rand-0.5) * offsets(offset);
end
C = 1/(n-1)*transpose(X)*X; %// covariance matrix (non-centred!)

%// convert to correlation
d = diag(C);
C = diag(1./sqrt(d))*C*diag(1./sqrt(d));

%// displaying C
subplot(length(offsets),3,(offset-1)*3+1)
imagesc(C, [-1 1])

%// histogram of the off-diagonal elements
subplot(length(offsets),3,(offset-1)*3+2)
offd = C(logical(ones(size(C))-eye(size(C))));
hist(offd)
xlim([-1 1])

%// QQ-plot to check the normality
subplot(length(offsets),3,(offset-1)*3+3)
qqplot(offd)

%// eigenvalues
eigv = eig(C);
display([num2str(min(eigv),2) ' ... ' num2str(max(eigv),2)])
end


The output of this code (minimum and maximum eigenvalues) is:

0.51 ... 1.7
0.44 ... 8.6
0.32 ... 22
0.1 ... 48

• can you plot the value of the smallest eigenvalues you obtain using this method alongside you plots? – user603 Nov 19 '14 at 13:35
• Without changing the figure, I can simply write here that the smallest eigenvalues are 0.5, 0.4, 0.3, and 0.1 respectively (for each row of my figure). The largest ones grow from 1.7 to 48. – amoeba Nov 19 '14 at 13:39
• but are these the eigenvalues of the correlation matrix or those of X'X?. – user603 Nov 19 '14 at 16:29
• These are the eigenvalues of my $C$ matrix, which is normalized to have ones on the diagonal, -- so of the correlation matrix. I updated my answer so that you can see it in the code. May I ask what makes you doubt that this is possible? Is there any reason to think that large correlation matrices should have very small off-diagonal elements? – amoeba Nov 19 '14 at 16:40
• I don't think its impossible, I just couldn't see it from the code (having not used matlab for years at this point) – user603 Nov 19 '14 at 16:53

You might be interested in some of the code at the following link:

Correlation and Co-integration

If you are trying to generate random correlation matrices, consider sampling from the Wishart distribution. This following question provides information the Wishart distribution as well as advice on how to sample: How to efficiently generate random positive-semidefinite correlation matrices?

• But can one control the standard deviation of the resulting off-diagonal elements with parameters of the Wishart distribution? If so, how? – amoeba Nov 19 '14 at 13:43

This is not a very sophisticated answer, but I can't help but think it's still a good answer...

If your motivation is that correlation parameters produced by time series data tend to look normal, why not just simulate time series data, calculate the correlation parameters and use those?

You may have a good reason for not doing this, but it's not clear to me from your question.