Alpha-cut based compositional representation of fuzzy sets and exploration of associated fuzzy set regression

The compositional representation of data and associated statistical approaches is a powerful framework for modelling and reasoning about quantities which reflect proportions of a whole. Recently, an increasing body of work has started exploring the adoption of a compositional representation for modelling interval-valued data reflecting uncertainty or vagueness, for example interval-valued questionnaire responses. Results have flagged the intriguing potential of this approach, such as the elegant handling of traditional inference challenges, including implicitly ensuring coherence in linear regression for interval data, i.e. ensuring the estimated left bound of intervals is smaller than the right one. Building on these insights, extending the compositional representation via alpha-cut decomposition to fuzzy sets is an intuitive next step. In this paper, we discuss this compositional representation of fuzzy sets, building on prior interval work. We proceed to explore the adoption of compositional regression approaches to conduct linear regression on fuzzy set valued data sets. We demonstrate the approach, discuss results and in particular flag shortcomings and the challenges for next steps.