Exact sampling and counting for fixed-margin matrices

TL;DR

This paper demonstrates that exact sampling and counting of fixed-margin matrices are feasible by exploiting symmetries, enabling precise null model testing in real-world ecological and contingency table data.

Contribution

It introduces a method to perform exact sampling and counting for fixed-margin matrices using symmetry exploitation, overcoming previous computational challenges.

Findings

Exact sampling is possible in many real-world cases.

The method applies to ecological co-occurrence matrices.

It improves null model testing accuracy.

Abstract

The uniform distribution on matrices with specified row and column sums is often a natural choice of null model when testing for structure in two-way tables (binary or nonnegative integer). Due to the difficulty of sampling from this distribution, many approximate methods have been developed. We will show that by exploiting certain symmetries, exact sampling and counting is in fact possible in many nontrivial real-world cases. We illustrate with real datasets including ecological co-occurrence matrices and contingency tables.