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Neighborhood radius estimation for variable-neighborhood random fields

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  • Löcherbach, Eva
  • Orlandi, Enza

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 with the one-dimensional setting we call these neighborhood symbols the context associated to the region at hand. This framework is a natural extension, to d-dimensional fields, of the notion of variable length Markov chains introduced by Rissanen [24] 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.

Suggested Citation

  • Löcherbach, Eva & Orlandi, Enza, 2011. "Neighborhood radius estimation for variable-neighborhood random fields," Stochastic Processes and their Applications, Elsevier, vol. 121(9), pages 2151-2185, September.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:9:p:2151-2185
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    References listed on IDEAS

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    1. Fiorenzo Ferrari & Abraham Wyner, 2003. "Estimation of General Stationary Processes by Variable Length Markov Chains," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 30(3), pages 459-480, September.
    2. Peter D. Grünwald, 2007. "The Minimum Description Length Principle," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262072815, April.
    3. Dereudre, D. & Lavancier, F., 2011. "Practical simulation and estimation for Gibbs Delaunay-Voronoi tessellations with geometric hardcore interaction," Computational Statistics & Data Analysis, Elsevier, vol. 55(1), pages 498-519, January.
    4. Dzhaparidze, K. & van Zanten, J. H., 2001. "On Bernstein-type inequalities for martingales," Stochastic Processes and their Applications, Elsevier, vol. 93(1), pages 109-117, May.
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