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 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 $N$ states to a fuzzy Markov chain with $N$ fuzzy states, which is represented by an SFDES consisting of $N^{2}$ FDES. Each FDES has its own $N \times N$ 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.