Supervised Learning of Fuzzy Sets for Fuzzy Markov Chains

In a recent article, we mathematically extended conventional discrete-time finite Markov chains, characterized by an $N \times N$ transition probability matrix, to discrete-time finite fuzzy Markov chains capable of modeling fuzzy states and fuzzy events, which frequently arise in fields such as biomedicine. This advancement is built upon the theory of stochastic fuzzy discrete event systems (SFDESs) and the supervised learning algorithm for FDESs, previously published by the authors. The fuzzy Markov chain is represented by a single-event SFDES comprising $N^{2}$ FDES, each with its own occurrence probability and an associated $N \times N$ event transition matrix, which is automatically learned using the aforementioned learning algorithm. Additionally, each FDES is associated with a set of fuzzy sets that fuzzify the random variable values and are required to satisfy specific constraints. Manually designing these fuzzy sets can be challenging, especially for modelers with little or no prior knowledge of fuzzy set theory. To overcome this challenge, we develop stochastic gradient descent-based algorithms that simultaneously learn constrained Gaussian fuzzy sets and the event transition matrices. To reduce the complexity of parameter learning, the Gaussian fuzzy sets are designed such that their means are computed directly from the terminal points of the subintervals that divide the ranges of the random variables, rather than being learned. Furthermore, dependencies between the Gaussian fuzzy sets for each random variable are introduced, reducing the number of standard deviations to be learned to $N$ for all Gaussian fuzzy sets in an FDES, while the remaining standard deviations are computed based on these dependencies. In addition, we establish that these new algorithms are fully applicable to continuous-time finite fuzzy Markov chains, extending their utility to a broader range of applications. An illustrative example is provided to demonstrate the effectiveness of the learning algorithms. These new algorithms make fuzzy Markov chains more accessible to modelers, regardless of their familiarity with fuzzy sets, while enhancing the overall practicality of the approach.