Generating Correlated Random Numbers

This article describes common methods that are used in generating correlated random numbers.

Contents

1 Generating two sequences of correlated random numbers

1.1 Before and after correlating

2 Generating multiple (more than two) sequences of correlated random variables

2.1 Example using a Cholesky decomposition
2.2 Example using a Eigenvector decomposition

3 Example Matlab code

Generating two sequences of correlated random numbers

Generating two sequences of random numbers with a given correlation is done in two simple steps:

Generate two sequences of uncorrelated normal distributed random numbers $X_1, X_2 \,$
Define a new sequence $Y_1 = \rho X_1 + \sqrt{1-\rho^2} X_2$

This new $Y_1 \,$ sequence will have a correlation of $\rho\,$ with the $X_1 \,$ sequence.

[edit] Before and after correlating

Generating multiple (more than two) sequences of correlated random variables

A general way to generate correlated (normal distributed) random numbers -with a given correlation matrix $C \,$ - is done by finding a matrix U such that

$U^T U = C \,$

Using this matrix, one can generate correlated random numbers $R_c\,$ from uncorrelated numbers $R\,$ by multiplying them with this matrix.

$R_c = R U \,$

There are multiple way to find such a matrix. The two most common methods are

A Cholesky decomposition of the correlation matrix
An Eigenvector decomposition of the correlation matrix (also known as a spectral decomposition)

Example using a Cholesky decomposition

We want to generate random numbers for three variables with the following correlations matrix:

$C=\begin{bmatrix} 1 & 0.6 & 0.3 \\ 0.6 & 1 & 0.5 \\ 0.3 & 0.5& 1 \end{bmatrix}\,$

Doing a Cholesky decomposition on the correlation matrix give the following matrix:

$U=\begin{bmatrix} 1 & 0.6 & 0.3 \\ 0 & 0.8 & 0.4 \\ 0 & 0 & 0.866 \end{bmatrix}\,$

Depending on the algorithm used for the Cholesky decomposition, you can also get the transposed of this matrix. This does doesn’t matter.

We can now transform three uncorrelated random numbers

$R=\begin{bmatrix} -0.3999 & -1.6041 & -1.0106 \end{bmatrix}\,$

into correlated numbers Rc by multiplying them with the U matrix.

$R_c = R U \,$

$= \begin{bmatrix} -0.3999 & -1.6041 & -1.0106 \end{bmatrix} \begin{bmatrix} 1 & 0.6 & 0.3 \\ 0 & 0.8 & 0.4 \\ 0 & 0 & 0.866 \end{bmatrix}$

$= \begin{bmatrix} -0.3999 & -1.5232 & -1.6368 \end{bmatrix}$

[edit] Example using a Eigenvector decomposition

The correlation matrix $C\,$ has the following eigenvectors

$E_1=\begin{bmatrix}0.5700\\ 0.6372\\ 0.5188\end{bmatrix}, E_2=\begin{bmatrix}0.6091\\ 0.0960\\ -0.7872\end{bmatrix}, E_3=\begin{bmatrix}0.5515\\ -0.7647\\ 0.3334\end{bmatrix}$

and eigenvalues

$\lambda_1=1.9439, \lambda_2=0.7069, \lambda_3=0.3493 \,$

If we define

$V= E_i Diag( \sqrt{ \lambda_i})$
$= \begin{bmatrix} 0.5700 & 0.6091 & 0.5515 \\ 0.6372 & 0.0960 & -0.7647 \\ 0.5188 & -0.7872 & 0.3334 \end{bmatrix} \begin{bmatrix} 1.3942 & 0 & 0 \\ 0 & 0.8402 & 0 \\ 0 & & 0.5911 \end{bmatrix}$
$= \begin{bmatrix} 0.7947 & 0.5121 & 0.3259 \\ 0.8884 & 0.0808 & -0.4520 \\ 0.7232 & -0.6619 & 0.1970 \end{bmatrix}$

we get a matrix $V\,$ that has the property

$V V^T = C \,$

which gives us the final result for the decomposition

$U^T U = C, U=V^T \,$

We can now transform three uncorrelated random numbers

$R=\begin{bmatrix} -0.3999 & -1.6041 & -1.0106 \end{bmatrix}\,$

into correlated numbers Rc by multiplying them with the U matrix.

$R_c = R U \,$

$= \begin{bmatrix} -0.3999 & -1.6041 & -1.0106 \end{bmatrix} \begin{bmatrix} 0.7947 & 0.8884 & 0.7232 \\ 0.5121 & 0.0808 & -0.6619 \\ 0.7232 & -0.4520 & 0.1970 \end{bmatrix}$
$= \begin{bmatrix} -1.4687 & -0.0280 & 0.5734 \end{bmatrix}$

[edit] Example Matlab code

The matlab code below decomposes the correlation matrix C into an upper matrix U using a Cholesky decomposition. Next 10 vectors with 3 random normal numbers are multiplied with the upper matrix to generate 10 correlated draws.

>> C= [ 1.0 0.6 0.3;   0.6 1.0 0.5;   0.3 0.5 1.0]

C =

    1.0000    0.6000    0.3000
    0.6000    1.0000    0.5000
    0.3000    0.5000    1.0000

>> U = chol(C)

U =

    1.0000    0.6000    0.3000
         0    0.8000    0.4000
         0         0    0.8660

>> randn(10,3)*U

ans =

   -0.4326   -0.4089    0.0505
   -1.6656   -0.4187   -1.3665
    0.1253   -0.3955    0.4209
    0.2877    1.9192    2.3656
   -1.1465   -0.7970   -0.9976
    1.1909    0.8057    1.1459
    1.1892    1.5669    1.8695
   -0.0376    0.0248   -1.3678
    0.3273    0.1199   -1.1880
    0.1746   -0.5611    0.2141

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