Exact goodness-of-fit testing for the Ising model

TL;DR

This paper introduces an exact goodness-of-fit test for the finite-lattice Ising model that avoids complex computations, enabling more efficient analysis of spatial data in various scientific fields.

Contribution

It develops a Monte Carlo method for exact goodness-of-fit testing of the Ising model that bypasses Markov basis computation, improving efficiency and convergence.

Findings

The proposed method effectively tests the Ising model fit.

Application to cell membrane receptor data demonstrates practical utility.

Method achieves faster convergence than traditional approaches.

Abstract

The Ising model is one of the simplest and most famous models of interacting systems. It was originally proposed to model ferromagnetic interactions in statistical physics and is now widely used to model spatial processes in many areas such as ecology, sociology, and genetics, usually without testing its goodness of fit. Here, we propose various test statistics and an exact goodness-of-fit test for the finite-lattice Ising model. The theory of Markov bases has been developed in algebraic statistics for exact goodness-of-fit testing using a Monte Carlo approach. However, finding a Markov basis is often computationally intractable. Thus, we develop a Monte Carlo method for exact goodness-of-fit testing for the Ising model which avoids computing a Markov basis and also leads to a better connectivity of the Markov chain and hence to a faster convergence. We show how this method can be applied to analyze the spatial organization of receptors on the cell membrane.