Neighborhood radius estimation in Variable-neighborhood Random Fields

TL;DR

This paper introduces an algorithm to estimate the neighborhood radius in variable-neighborhood random fields, extending variable-length Markov chains to higher dimensions, with proven consistency and explicit error bounds.

Contribution

It proposes a new estimator for the context radius in multi-dimensional random fields, generalizing variable-length Markov chains and providing theoretical guarantees.

Findings

Estimator is consistent for the context radius.

Explicit bounds on the probability of incorrect estimation.

Extension of variable-length Markov chains to higher dimensions.

Abstract

We consider random fields defined by finite-region conditional probabilities depending on a neighborhood of the region which changes with the boundary conditions. To predict the symbols within any finite region it is necessary to inspect a random number of neighborhood symbols which might change according to the value of them. In analogy to the one dimensional setting we call these neighborhood symbols the context of the region. This framework is a natural extension, to d-dimensional fields, of the notion of variable-length Markov chains introduced by Rissanen (1983) in his classical paper. We define an algorithm to estimate the radius of the smallest ball containing the context based on a realization of the field. We prove the consistency of this estimator. Our proofs are constructive and yield explicit upper bounds for the probability of wrong estimation of the radius of the context.