Self-Learning Fuzzy Automaton With Input and Output Fuzzy Sets for System Modelling

Fuzzy Automaton plays a key role in modeling a system as a fuzzy discrete event system and determining the model's properties such as observability and predictability. We recently developed online supervised learning algorithms for the fuzzy automaton to learn its event transition matrix under three different conditions tied to availability of pre- and post-event states, which are the first and only studies in the current literature. In this paper, we first eliminate, in a theoretical manner, the restriction in our prior studies that the individual pre- and post-event states must be identical. We then tackle a more realistic and general condition-learning the transition matrix when neither pre-event nor post-event state is known, which covers the previous three conditions as special cases. The only information assumes to be available is values of the variables and they are only known to be vaguely related to the states. We link the variables to the states through (Gaussian) fuzzy sets owing to the conceptual relationship between a fuzzy set and a fuzzy state. The resulting innovative model is called the Fuzzy Automaton with Input and Output Fuzzy Sets. Stochastic-gradient-descent-based algorithms are derived for the model to iteratively learn online the transition matrix and all the parameters of the fuzzy sets simultaneously. Another significant extension is that unlike our previous models, the new model is capable of determining an individual post-event state even if it is associated with two or more variables, expanding model's utility. It is the first fuzzy discrete event system model that is capable of learning solely based on sensor data without relying on any subjective input from humans, removing a significant bottleneck for real-world applications. Computer simulation results are furnished to exhibit learning performance of this new model. This model will be valuable for solving practical problems in healthcare and various industries, and we discuss in detail its potential in modeling of disease treatment decision-making.