How do I generate correlated random numbers in Python?

Question

How do I create a set of n vectors of dimensionality d such that elements have correlation c (ie, if a vector has one large element, the other elements are likely to be large)?

For demonstration, let's say n=5, d=3, and c=0.7.

Is there some way to set up conv here: https://numpy.org/doc/stable/reference/random/generated/numpy.random.multivariate_normal.html

This may be too much to ask, but what if I want the numbers drawn from a normal distribution?

Thanks!

Edit: Basically I'm trying to create a synthetic population whose individuals differ in some latent variable, and ideally this latent variable would follow a normal distribution. For instance, the psychometric g factor summarizes performance on multiple tests, and explains a certain amount of variance between people on a given test. So I'd like to create n vectors (population size) of dimensionality d (number of tasks), but maybe c needs to be a vector of d numbers? And I might need to specify a vector of d numbers for the latent variable scores (eg, g), or maybe that emerges from how the vectors for the individuals are created?

Answer 1

Might this be what you are looking for?

import numpy as np


def gen_random(n: int, d: int, covar: float) -> np.ndarray:
    """
    Paramters
    ---------
    n : int
        number of samples generated
    d : int
        dimensionality of samples
    covar : float
        uniform covariance for samples
    
    Returns
    -------
    samples : np.ndarray
        samples in as (n, d)-matrix
    """
    cov_mat = np.ones((d, d)) * covar; np.fill_diagonal(cov_mat, 1)
    offset = np.zeros(d)

    return np.random.multivariate_normal(offset, cov_mat, size=n)


v = gen_random(n=10_000, d=3, covar=0.7)
print(v)
# [[ 0.03031736  0.18227023 -0.1302022 ]
#  [-0.17770689  0.70979971 -0.74631702]
#  [-0.78485455 -0.73942846 -0.04819704]
#  ...
#  [ 2.5928135   2.43727782  1.59459156]
#  [ 0.33443158 -0.74126937 -0.7542286 ]
#  [ 0.11238505 -0.1940429   0.7397402 ]]

# sanity check
print(np.corrcoef(v, rowvar=False))
# [[1.         0.6985444  0.69802535]
#  [0.6985444  1.         0.70168241]
#  [0.69802535 0.70168241 1.        ]]