# Generating a string of random standard normal variables that are correlated

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Natialol on 6 Dec 2011
Hi Everyone,
I'm a sort of newbie, I would like to know how and what the implications are of 'Generating a string of random standard normal variables that are correlated with each other'.
To get by this problem, I have been generating and correlating my desired sequence to a different random variable and then calculating the correlation between my sequence.
ps: Generate random standard normal's A, B, C, D so that have a correlation and standard deviation of corr and std.
Thanks again

Daniel Shub on 6 Dec 2011
I am not sure if this is homework or not ...
Start off with two independent random variables with zero mean and standard deviation sigma.
sigma = 10;
X = sigma*randn(1e7, 1);
Y = sigma*randn(1e7, 1);
Then make two new random variables from these with correlation rho.
rho = 0.2;
A = X;
B = sqrt(rho^2)*X+sqrt(1-rho^2)*Y;
[std(A), std(B)]
corrcoef(A, B)
When you add a third random variable C you need to specify what you want rho_AB, rho_AC, and rho_AB to be. The basic idea is the same: start with N independent random variables and add them together with appropriate weighting to get N new random variables.
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Daniel Shub on 6 Dec 2011
I am very lost. It is odd to talk about the correlation between scalars like A_i and B_j. I think what you are asking is to create 5 random processes which have a particular covariance and then you want 1 value from each process. My answer provides 2 random processes which each have 1e7 values. If you extend it to 5 terms (A,B,C,D,E) then you can do a = A(i), b = B(j), c = C(k) ... and I think what you want is [a,b,c,d,e]. Of course if you only want 1 value, you do not need to generate 1e7 values.
Alexander Knetsch on 10 Nov 2020
The equation is quite good already, it doesn't allow for a negative corellation. If you change the expression for B, you can allow for this:
B = (rho/abs(rho))*sqrt(rho^2)*X+sqrt(1-rho^2)*Y;

Oleg Komarov on 6 Dec 2011
Given a correlation matrix C = A*A', then A = P*sqrt(D), where:
[P,D] = eig(C); % spectral decomposition
To get the correlated normal random series Z, use W = (W1, ...,W2)' (the normal random series):
Z = A*W;
Note that if you have 4 variables, then C is 4 by 4, and W should be 4 by nobs.
Natialol on 6 Dec 2011
I'm trying to read up on the answer and will accept as soon as I understand that it does the above. Please can you also comment on the above question to Daniel?

Chet Sharma on 30 Jan 2018
I think you should refer Hull's option book. In the 8th edition, chapter 20, page 450, I believe you will find out answer. He describes something called a "Cholesky decomposition" that is needed to generate the "correct" correlation matrix......
Garrison Greenwood on 24 Feb 2021
Cholesky won't work if the C matrix is positive semi-definite.

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