Nearest-neighbour modelling of reciprocal chains

This paper focuses on the class of finite-state, discrete-index, reciprocal processes (reciprocal chains). Such a class of processes seems to be a suitable setup in many applications and, in particular, it appears well-suited for image-processing. While addressing this issue, the aim is 2-fold: theo...

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Veröffentlicht in:Stochastics (Abingdon, Eng. : 2005) Jg. 80; H. 6; S. 525 - 584
1. Verfasser: Carravetta, Francesco
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Taylor & Francis Group 01.12.2008
Schlagworte:
Markov chains
Markov fields
nearest-neighbour models
reciprocal processes
smoothing algorithms
stochastic realization
ISSN:1744-2508, 1744-2516
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Zusammenfassung:This paper focuses on the class of finite-state, discrete-index, reciprocal processes (reciprocal chains). Such a class of processes seems to be a suitable setup in many applications and, in particular, it appears well-suited for image-processing. While addressing this issue, the aim is 2-fold: theoretic and practical. As to the theoretic purpose, some new results are provided: first, a general stochastic realization result is provided for reciprocal chains endowed with a known, arbitrary, distribution. Such a model has the form of a fixed-degree, nearest-neighbour polynomial model. Next, the polynomial model is shown to be exactly linearizable, which means it is equivalent to a nearest-neighbour linear model in a different set of variables. The latter model turns out to be formally identical to the Levi-Frezza-Krener linear model of a Gaussian reciprocal process, although actually non-linear with respect to the chain's values. As far as the practical purpose is concerned, in order to yield an example of application an estimation issue is addressed: a suboptimal (polynomial-optimal) solution is derived for the smoothing problem of a reciprocal chain partially observed under non-Gaussian noise. To this purpose, two kinds of boundary conditions (Dirichlet and Cyclic), specifying the reciprocal chain on a finite interval, are considered, and in both cases the model is shown to be well-posed, in a 'wide-sense'. Under this view, some well-known representation results about Gaussian reciprocal processes extend, in a sense, to a 'non-Gaussian' case.
ISSN:1744-2508
1744-2516
DOI:10.1080/17442500802088517