Using Expander Graphs to test whether samples are i.i.d

08/03/2020

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by   Stefan Steinerberger, et al.

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The purpose of this note is to point out that the theory of expander graphs leads to an interesting test whether n real numbers x_1, …, x_n could be n independent samples of a random variable. To any distinct, real numbers x_1, …, x_n, we associate a 4-regular graph G as follows: using π to denote the permutation ordering the elements, x_π(1) < x_π(2) < … < x_π(n), we build a graph on {1, …, n} by connecting i and i+1 (cyclically) and π(i) and π(i+1) (cyclically). If the numbers are i.i.d. samples, then a result of Friedman implies that G is close to Ramanujan. This suggests a test for whether these numbers are i.i.d: compute the second largest (in absolute value) eigenvalue of the adjacency matrix. The larger λ - 2√(3), the less likely it is for the numbers to be i.i.d. We explain why this is a reasonable test and give many examples.