A study on fuzzy hidden markov chain and long term behavior of cyclic non homogeneous fuzzy markov chain

Abstract

Uncertainty comes in many forms and in real world problems it is usually impossible to avoid uncertainties. Fuzzy theory is basically a theory of graded concepts and the mathematical apparatus of the theory of fuzzy sets provides a natural basis for the theory of possibility. This significance of fuzzy sets motivates us to propose three types of fuzzy-based hidden Markov models namely, Fuzzy Hidden Markov chain, Hierarchical Fuzzy Hidden Markov Chain, Non Homogeneous Fuzzy Hidden Markov Chain on possibility space where hidden Markov chain can be seen as an extension of Markov chain to the case that the observation is a possibilistic function of the state. Three problems are calculating the likelihood of the given observation sequence; finding the most likelihood state sequence for the given observation sequence; re-estimating the parameters of proposed model must be solved for fuzzy-based hidden Markov models. We have applied our proposed models FHMC, HFHMC, NHFHMC to our institution s website and the three problems viewed for the website are: The first problem reveals that for a given sequence of web pages, the extent of the possibility value of website viewed by the web users; The second problem exposes the most likelihood path which is the best to explain the given sequence of web pages; and The third problem reveals the improved possibility value of given sequence of web pages. We have defined fuzzy hidden Markov chain on possibility space and introduced algorithms, namely forward system, modified Viterbi algorithm, and backward system to solve the three problems. We have defined Hierarchical Fuzzy Hidden Markov Chain on possibility space and presented the algorithms, namely forward system, generalized Viterbi algorithm, and backward system. Then we have derived the Chapman Kolmogorov equation for NHFMC. We have defined Non Homogeneous Fuzzy Hidden Markov Chain and solved the three problems using the algorithms forward system, Viterbi algorithm for NHFHMC, and backward system. newline newline newline