Functional Fuzzy System: A Nonlinear Regression Model and Its Learning Algorithm for Function-on-Function Regression

Functional data analysis (FDA) in which each data sample is a function rather than a vector or a matrix has attracted a lot of attention in the statistics community in recent years. However, most of the existing functional data regression works focus on linear models; and there is a lack of research on general nonlinear functional regression models that can fit nonlinear relationships between functional data, especially when the input and output of the models are both functions. Furthermore, in the fuzzy system research domain, there is no fuzzy system approach to FDA so far as we are aware of. To fill in these dual gaps, this article develops the first fuzzy system approach to FDA by proposing a functional fuzzy regression model known as functional fuzzy system (FFS) and its learning method from data. FFS is a general nonlinear functional regression model, which has inputs and outputs are functions in infinite dimensional spaces. Furthermore, constructed with a collection of the “If-Then” fuzzy rules, on the one hand, FFS has a flexible structure, and can model the functional data without model structure assumptions; On the other hand, the fuzzy rule base enables FFS to be an interpretable model. An identification method for FFS is proposed for minimizing the mean squared prediction errors. The proposed FFS is compared with many existing state-of-the-art functional regression models upon benchmark examples using both artificial and real datasets, and shows that FFS is an effective model and can obtain preferable results.