How to Generate Multivariate Random Numbers with Different Marginal Distributions

How to generate multivariate random numbers with different marginal distributions?

If you have Statistics Toolbox you can generate random numbers from copulas using the function copularnd. There are several examples in the documentation. To convert between using Kendall's tau and Pearson's rho, take a look at copulaparam and copulastat.

Generate copula-correlated samples with specified marginals in Python

The following code implements the Clayton and AMH copulas. Page 4 of this paper shows how other kinds of copulas can be implemented.

import random
import math
import scipy.stats as st
def clayton(theta, n):
    v=random.gammavariate(1/theta,1)
    uf=[random.expovariate(1)/v for _ in range(n)]
    return [(k+1)**(-1.0/theta) for k in uf]

def amh(theta, n):
    # NOTE: Use SciPy RNG for convenience here
    v=st.geom(1-theta).rvs()
    uf=[random.expovariate(1)/v for _ in range(n)]
    return [(1-theta)/(math.exp(k)-theta) for k in uf]

scipy - generate random variables with correlations

If you just want correlation through a Gaussian Copula (*), then it can be calculated in a few steps with numpy and scipy.

• create multivariate random variables with desired covariance, numpy.random.multivariate_normal, and creating a (nobs by k_variables) array

• apply scipy.stats.norm.cdf to transform normal to uniform random variables, for each column/variable to get uniform marginal distributions

• apply dist.ppf to transform uniform margin to the desired distribution, where dist can be one of the distributions in scipy.stats

(*) Gaussian copula is only one choice and it is not the best when we are interested in tail behavior, but it is the easiest to work with
for example http://archive.wired.com/techbiz/it/magazine/17-03/wp_quant?currentPage=all

two references

https://stats.stackexchange.com/questions/37424/how-to-simulate-from-a-gaussian-copula

http://www.mathworks.com/products/demos/statistics/copulademo.html

(I might have done this a while ago in python, but don't have any scripts or function right now.)

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