Generating Correlated Random Numbers: Cholesky Decomposition

When it comes to generate random numbers most of the people see this exercise as a black box. In order to add some colour to this. There are several approaches to generate correlated random numbers, of which two are mainly used, namely Cholesky decomposition and spectral (eigenvalue) decomposition. These types of decomposition are used in copula estimates.

Cholesky decomposition takes the correlation (or covariance) matrix along with randomly generated numbers and correlates them. Cholesky decomposition has two, let’s call it, versions: lower and upper triangular. The Cholesky decomposition can be done in Python via Numpy and SciPy linear algebra (linalg) libraries:

(1) np.linalg.cholesky(A) # using numply linear algebra library and

(2) scipy.linalg.cholesky(A, lower=True) # using SciPy linear algebra library with lower=True indicating we want lower triangular, if we want upper triangular: lower=False. 

I prefer to use the lower triangular matrix. I also prefer working with array instead of matrix.  

Put into practice and using Python, the things look like this (we work with 3 variables):

import numpy as np

from scipy.linalg import cholesky

from scipy.stats import norm

A=np.array([[1,0.1558, 0.2227], [0.1558,1,0.5118], [0.2227,0.5118,1]]) # set the correlation matrix

B= norm.rvs(size=(3, 10000)) # 10,000 simulations to generate 3 random normally distributed variables

Cholesky = cholesky(A, lower=True) # Cholesky decomposition - lower triangular matrix

result= np.dot(Cholesky, B) # generates correlated variables

correlation=np.corrcoef(result) #check the correlation of the simulated variables

By the way, checking if the initial matrix is positive definite is also quite easy on Python:

from scipy.linalg import eigh

eigh(A) # the first array is eigenvalues and the second is eigenvectors

In our example, the correlation matrix is positive definite, meaning that all its eigenvalues are positive.  The eigenvalues of the above correlation matrix are: 0.4832, 0.8903, 1.6265.



Cholesky decomposition can also be used in the opposite case - to uncorrelate variables that are correlated.