Probabilistic cellular automata with general alphabets possessing a Markov chain as an invariant distribution

2016 ◽  
Vol 48 (2) ◽  
pp. 369-391 ◽  
Author(s):  
Jérôme Casse
Keyword(s):  
Integral Equation ◽  
Markov Chain ◽  
Cellular Automata ◽  
Transition Matrix ◽  
Invariant Distribution ◽  
Finite Alphabet ◽  
Transition Kernel ◽  
Probabilistic Cellular Automata ◽  
Existence Of A Solution ◽  
Positive Rate

Abstract This paper is devoted to probabilistic cellular automata (PCAs) on N,Z or Z / nZ, depending on two neighbors with a general alphabet E (finite or infinite, discrete or not). We study the following question: under which conditions does a PCA possess a Markov chain as an invariant distribution? Previous results in the literature give some conditions on the transition matrix (for positive rate PCAs) when the alphabet E is finite. Here we obtain conditions on the transition kernel of a PCA with a general alphabet E. In particular, we show that the existence of an invariant Markov chain is equivalent to the existence of a solution to a cubic integral equation. One of the difficulties in passing from a finite alphabet to a general alphabet comes from the problem of measurability, and a large part of this work is devoted to clarifying these issues.

Truncation approximations of invariant measures for Markov chains

1998 ◽  
Vol 35 (03) ◽  
pp. 517-536 ◽  
Author(s):  
R. L. Tweedie
Keyword(s):  
Markov Chain ◽  
Probability Distribution ◽  
Markov Chains ◽  
Transition Matrix ◽  
Invariant Measures ◽  
Invariant Distribution ◽  
Special Cases ◽  
Recurrent Markov Chain ◽  
Positive Recurrent

Let P be the transition matrix of a positive recurrent Markov chain on the integers, with invariant distribution π. If (n) P denotes the n x n ‘northwest truncation’ of P, it is known that approximations to π(j)/π(0) can be constructed from (n) P, but these are known to converge to the probability distribution itself in special cases only. We show that such convergence always occurs for three further general classes of chains, geometrically ergodic chains, stochastically monotone chains, and those dominated by stochastically monotone chains. We show that all ‘finite’ perturbations of stochastically monotone chains can be considered to be dominated by such chains, and thus the results hold for a much wider class than is first apparent. In the cases of uniformly ergodic chains, and chains dominated by irreducible stochastically monotone chains, we find practical bounds on the accuracy of the approximations.

Truncation approximations of invariant measures for Markov chains

1998 ◽  
Vol 35 (3) ◽  
pp. 517-536 ◽  
Author(s):  
R. L. Tweedie
Keyword(s):  
Markov Chain ◽  
Probability Distribution ◽  
Markov Chains ◽  
Transition Matrix ◽  
Invariant Measures ◽  
Invariant Distribution ◽  
Special Cases ◽  
Recurrent Markov Chain ◽  
Positive Recurrent

Let P be the transition matrix of a positive recurrent Markov chain on the integers, with invariant distribution π. If (n)P denotes the n x n ‘northwest truncation’ of P, it is known that approximations to π(j)/π(0) can be constructed from (n)P, but these are known to converge to the probability distribution itself in special cases only. We show that such convergence always occurs for three further general classes of chains, geometrically ergodic chains, stochastically monotone chains, and those dominated by stochastically monotone chains. We show that all ‘finite’ perturbations of stochastically monotone chains can be considered to be dominated by such chains, and thus the results hold for a much wider class than is first apparent. In the cases of uniformly ergodic chains, and chains dominated by irreducible stochastically monotone chains, we find practical bounds on the accuracy of the approximations.

scholarly journals Landscape pattern and economic factors’ effect on prediction accuracy of cellular automata-Markov chain model on county scale

2020 ◽  
Vol 12 (1) ◽  
pp. 626-636
Author(s):  
Wang Song ◽  
Zhao Yunlin ◽  
Xu Zhenggang ◽  
Yang Guiyan ◽  
Huang Tian ◽  
...  
Keyword(s):  
Land Use ◽  
Markov Chain ◽  
Cellular Automata ◽  
Markov Model ◽  
Prediction Accuracy ◽  
Driving Mechanism ◽  
Economic Factors ◽  
Chain Model ◽  
The Impact ◽  
County Scale

AbstractUnderstanding and modeling of land use change is of great significance to environmental protection and land use planning. The cellular automata-Markov chain (CA-Markov) model is a powerful tool to predict the change of land use, and the prediction accuracy is limited by many factors. To explore the impact of land use and socio-economic factors on the prediction of CA-Markov model on county scale, this paper uses the CA-Markov model to simulate the land use of Anren County in 2016, based on the land use of 1996 and 2006. Then, the correlation between the land use, socio-economic data and the prediction accuracy was analyzed. The results show that Shannon’s evenness index and population density having an important impact on the accuracy of model predictions, negatively correlate with kappa coefficient. The research not only provides a reference for correct use of the model but also helps us to understand the driving mechanism of landscape changes.

scholarly journals Around probabilistic cellular automata

2014 ◽  
Vol 559 ◽  
pp. 42-72 ◽  
Author(s):  
Jean Mairesse ◽  
Irène Marcovici
Keyword(s):  
Cellular Automata ◽  
Probabilistic Cellular Automata

scholarly journals Percolation games, probabilistic cellular automata, and the hard-core model

2018 ◽  
Vol 174 (3-4) ◽  
pp. 1187-1217 ◽  
Author(s):  
Alexander E. Holroyd ◽  
Irène Marcovici ◽  
James B. Martin
Keyword(s):  
Cellular Automata ◽  
Hard Core ◽  
Core Model ◽  
Probabilistic Cellular Automata ◽  
Hard Core Model
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