Neighborhood radius estimation in Variable-neighborhood Random Fields

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.

References in corpus (3)

• Consistent estimation of the basic neighborhood of Markov random fields
• Stochastic chains with memory of variable length
• Existence of Gibbsian point processes with geometry-dependent interactions

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• Markov and almost Markov properties in one, two or more directions
• Consistent model selection for estimating functional interactions among stochastic neurons with variable-length memory