Discrete-Time Finite Fuzzy Markov Chains Realized Through Supervised Learning Stochastic Fuzzy Discrete Event Systems

The binary nature of the states and transitions in discrete-time finite Markov chains makes this modeling methodology unsuitable for many practical systems, such as those found in biomedicine. To address this fundamental limitation, we have extended in this article Markov chains to fuzzy Markov chai...

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Bibliographic Details
Published in:IEEE transactions on fuzzy systems Vol. 32; no. 11; pp. 6088 - 6100
Main Authors: Ying, Hao, Lin, Feng
Format: Journal Article
Language:English
Published: IEEE 01.11.2024
Subjects:
Accuracy
Automata
Discrete-event systems
Fuzzy automaton
fuzzy Markov chains (FMC)
Fuzzy systems
Markov chains (MC)
Markov processes
stochastic fuzzy discrete event systems (SFDES)
stochastic modeling
Stochastic processes
Supervised learning
ISSN:1063-6706, 1941-0034
Online Access:Get full text
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Summary:The binary nature of the states and transitions in discrete-time finite Markov chains makes this modeling methodology unsuitable for many practical systems, such as those found in biomedicine. To address this fundamental limitation, we have extended in this article Markov chains to fuzzy Markov chains capable of handling fuzzy states and fuzzy events. This innovative and significant advancement is founded on the theory of stochastic fuzzy discrete event systems (SFDES) and the supervised learning algorithm for fuzzy discrete event systems (FDES), recently published by the authors. We mathematically generalize a traditional Markov chain with <inline-formula><tex-math notation="LaTeX">N</tex-math></inline-formula> states to a fuzzy Markov chain with <inline-formula><tex-math notation="LaTeX">N</tex-math></inline-formula> fuzzy states, which is represented by an SFDES consisting of <inline-formula><tex-math notation="LaTeX">N^{2}</tex-math></inline-formula> FDES. Each FDES has its own <inline-formula><tex-math notation="LaTeX">N \times N</tex-math></inline-formula> event transition matrix that is automatically learned by the aforementioned learning algorithm. Crucially, the fuzzy Markov chain fully preserves the stochastic characteristics defined by the transition probability matrix of the binary Markov chain, ensuring identical stochastic behaviors. A defuzzifier is used to yield crisp model output. The structurally more complex fuzzy Markov chain encompasses its binary counterpart as a special case and degenerates into it when fuzzy states degenerate into binary states. A simulation example is provided to illustrate the systematic design procedure and demonstrate the higher prediction accuracy of the fuzzy Markov chain over its binary counterpart. Due to their advantages, fuzzy Markov chains have the potential to address real-world stochastic problems beyond the reach of conventional Markov chains, especially in biomedicine.
ISSN:1063-6706
1941-0034
DOI:10.1109/TFUZZ.2024.3440184