Supervised Learning of Fuzzy Sets for Fuzzy Markov Chains.

Area of Science:

• Stochastic Systems
• Fuzzy Logic
• Machine Learning

Background:

• Conventional discrete-time finite Markov chains model systems with discrete states and events.
• Fuzzy Markov chains extend this to model systems with fuzzy states and events, common in biomedicine.
• Previous work established the theory of stochastic fuzzy discrete event systems (SFDESs) and learning algorithms.

Purpose of the Study:

• To develop algorithms for simultaneously learning constrained Gaussian fuzzy sets and event transition matrices for fuzzy Markov chains.
• To overcome the challenge of manually designing fuzzy sets, especially for users unfamiliar with fuzzy set theory.
• To extend the applicability of fuzzy Markov chains to continuous-time models.

Main Methods:

• Developed stochastic gradient descent-based algorithms for simultaneous learning of fuzzy sets and transition matrices.
• Designed Gaussian fuzzy sets with means derived from variable ranges and introduced dependencies to reduce parameters.
• Extended algorithms for applicability to continuous-time finite fuzzy Markov chains.

Main Results:

• Successfully developed algorithms that automatically learn constrained Gaussian fuzzy sets and event transition matrices.
• Reduced the complexity of parameter learning by optimizing the design and dependencies of Gaussian fuzzy sets.
• Demonstrated the effectiveness of the learning algorithms through an illustrative example.

Conclusions:

• The new algorithms significantly enhance the practicality and accessibility of fuzzy Markov chains for modelers.
• These advancements broaden the utility of fuzzy Markov chains, including their application to continuous-time models.
• The approach empowers researchers and practitioners, regardless of their fuzzy set theory expertise.