How can I sample a multivariate log-normal distribution in Python?

Question

Using Python, how can I sample data from a multivariate log-normal distribution? For instance, for a multivariate normal, there are two options. Let's assume we have a 3 x 3 covariance matrix and a 3-dimensional mean vector mu.

# Method 1
sample = np.random.multivariate_normal(mu, covariance)

# Method 2
L = np.linalg.cholesky(covariance)
sample = L.dot(np.random.randn(3)) + mu

I found numpy's numpy.random.lognormal, but that only seems to work for univariate samples. I also noticed scipy's scipy.stats.lognorm. This does seem to have the potential for multivariate samples. However, I can't figure out how to do this.

The docs suggest r = lognorm.rvs(s, size=1000) returns a list of 1000 random variables. s is a floating point number (a seed, I suppose), Why won't that do what you want? — Paul Cornelius
@PaulCornelius: Check those docs again. s is not a seed; it is the shape parameter of the distribution. — Warren Weckesser
I believe what you're referencing would draw 1000 samples from a univariate distribution. I want to sample once from a multivariate distribution with a covariance matrix and a vector of means. — Mack

CT Zhu CT Zhu · Accepted Answer · 2015-08-19T20:46:59

A multivariate lognormal distributed random variable Rv should have this property: log(Rv) should follow a normal distribution. Therefore, the problem is really just to generation a random variable of multivariate normal distribution and np.exp it.

In [1]: import numpy.random as nr

In [2]: cov = np.array([[1.0, 0.2, 0.3,],
                        [0.2, 1.0, 0.3,],
                        [0.3, 0.3, 1.0]])

In [3]: mu  = np.log([0.3, 0.4, 0.5])

In [4]: mvn = nr.multivariate_normal(mu, cov, size=5)

In [5]: mvn   # This is multivariate normal
Out[5]:
array([[-1.36808854, -1.32562291, -1.9706876 ],
       [-2.13119289,  1.28146425,  0.66000019],
       [-2.82590272, -1.22500654, -0.32635701],
       [-0.4967589 , -0.34469589, -2.04084115],
       [-0.85590235, -1.27133544, -0.70959595]])

In [6]: mvln = np.exp(mvn)

In [7]: mvln   # This is multivariate log-normal
Out[7]:
array([[ 0.25459314,  0.26563744,  0.139361  ],
       [ 0.11869562,  3.60190996,  1.9347927 ],
       [ 0.05925514,  0.29375578,  0.72154754],
       [ 0.60849968,  0.70843576,  0.12991938],
       [ 0.42489961,  0.28045684,  0.49184289]])

How can I sample a multivariate log-normal distribution in Python?

1 Answers