How to generate correlated binary variables

Question

I need to generate a series of N random binary variables with a given correlation function. Let x = {x_i} be a series of binary variables (taking the value 0 or 1, i running from 1 to N). The marginal probability is given Pr(x_i = 1) = p, and the variables should be correlated in the following way:

Corr[ x_ix_j ] = const × |i−j|^−α (for i!=j)

where α is a positive number.

If it is easier, consider the correlation function:

Corr[ x_ix_j ] = (|i−j|+1)^−α

The essential part is that I want to investigate the behavior when the correlation function goes like a power law. (not α^|i−j| )

Is it possible to generate a series like this, preferably in Python?

Answer 1

Thanks for all your inputs. I found an answer to my question in the cute little article by Chul Gyu Park et al., so in case anyone run into the same problem, look up:

"A simple method for Generating Correlated Binary Variates" (jstor.org.stable/2684925)

for a simple algorithm. The algorithm works if all the elements in the correlation matrix are positive, and for a general marginal distribution Pr(x_i)=p_i.

j

Answer 2

You're describing a random process, and it looks like a tough one to me... if you eliminated the binary (0,1) requirement, and instead specified the expected value and variance, it would be possible to describe this as a white noise generator feeding through a 1-pole low-pass filter, which I think would give you the α^|i-j| characteristic.

This actually might meet the bar for mathoverflow.net, depending on how it is phrased. Let me try asking....

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update: I did ask on mathoverflow.net for the α^|i-j| case. But perhaps there are some ideas there that can be adapted to your case.