Re: Generating a Matrix of Random values with a specified Correlation

On Thu, 25 Sep 2008 11:30:06 -0700, willgarrison wrote:
On Sep 25, 12:17 pm, Robert Israel... wrote:
willgarri...@xxxxxxxxx writes:
I'm trying to generate two columns of random values where each value
is between 1 and 5 where the matrix as a whole has a specified
correlation.

Ideally this is what I'm looking for:

Example algorithm:

1. Generate 20 random values between 1 and 5 in array A. 2. Generate
20 random values between 1 and 5 in array B. 3. Create a 3rd array
(array C), where array B is sorted so that array A and array C have a
correlation of X.
....
Start with your two arrays A and B.  Let a=Var(A), b=Var(B),
c=Cov(A,B), so the (Pearson) correlation of A and B is rho = c/sqrt(a
b).  Let's suppose the desired correlation x > rho (where of course x
<= 1). Consider C = tA + (1-t)B, where 0 <= t <= 1. We have Cov(A,C) =
ta + (1-t) c and
Var(C) = t^2 a + 2 t (1-t) c + (1-t)^2 b so the correlation of A and C
is (ta + (1-t)c)/sqrt(a(t^2 a + 2t(1-t)c + (1-t)^2 b)). Set this equal
to x and solve for t: there should be exactly one solution in the
interval 0 <= t <= 1.
On the other hand, if -1 <= x < rho,
try C = t R + (1-t) B where each R_j = 6 - A_j. --
Robert Israel
....
I think this is not only over my head but perhaps not the exact answer
I'm looking for. However, it being over my head, it's impossible for me
to know.

Let me phrase this as a simpler question with hopes of improving my
chances of understanding your response.
[snip restatement]

R.I. has given you a method of generating two vectors A and C
with corr(A,C) equal to a given value X.

In my earlier response, I misinterpreted your first post, and
thought you wanted to reorder the elements of a given matrix B to
form C to approximate a specified correlation, as opposed to
just generating a vector C with given correlation with A.

--
jiw
.

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