Learning functional data on-line: An evolving functional fuzzy-neural system approach

Functional data analysis (FDA) has attracted great attention from the statistical community, but there are few fuzzy systems approaches. Therefore, this is a great opportunity for the fuzzy systems community. Focusing on the function-on-function (FOF) regression problem in FDA, the existing models are batch learning models with fixed model structure and parameters, which makes them unsuitable for solving the time-varying regression problem that requires the model to be updated dynamically and continuously with streams of functional data. Furthermore, as far as we are aware, there is no work on the online learning approach for FOF regression problems. To fill this gap, this paper proposes a nonlinear online FOF regression approach called evolving functional fuzzy-neural system (EFFNS) to learn functional data streams in real-time, with both input and output being functions. As a completely new type of evolving fuzzy system designed to learn from data represented as functions rather than vectors or matrices, EFFNS has a flexible model structure which could start from an empty rule base, expand and shrink the rule base, and tune the parameters dynamically depending on the knowledge learned from the rapidly coming functions. Various benchmark examples verify that EFFNS outperforms many of the state-of-the-art approaches. • An evolving functional fuzzy-neural system (EFFNS) is proposed. • EFFNS is the first online fuzzy model for function-on-function regression. • EFFNS solves the regression problem in infinite-dimensional spaces online. • EFFNS can dynamically update neurons and parameters with data.