Statistics

Realization (probability)

Applied mathematics

Set (abstract data type)

Data mining

Computer science

Fuzzy set

Fuzzy number

Approximations of π

Mathematics

Fuzzy logic

Artificial intelligence

Programming language

Machine learning

Fuzzy Sets and Systems

Three-way approximations of fuzzy sets are an important scheme of granular computing, by abstracting a fuzzy set to its discrete three option-alternatives which adhere to human cognitive behaviors and reduce the computational burden. The key point of such three-way approximations of fuzzy sets is how to choose a suitable design leading to their realization. Undesired three-way approximations might be produced if the selected mechanism is unsuitable to data distribution. In this study, the principles for constructing three-way approximations of fuzzy sets are summarized. The following taxonomy of these principles is provided, namely (i) uncertainty balance-based principle, (ii) prototype-based principle, and (iii) model-based invoking the tradeoff between classification error and the number of data that have to be classified. A number of detailed optimization models are discussed in detail. To evaluate the performance of different construction principles, a general unsupervised learning framework based on three-way approximations of fuzzy sets is exhibited. Some synthetic data sets along with sixteen data sets from UCI repository are involved for experiments. Friedman testing followed by Holm-Bonferroni testing are exploited to test the performance significance of the proposed criteria, which can provide insights and deliver guidance when choosing a principle for constructing three-way approximations of fuzzy sets in the real-world scenarios. The research methods in this paper can also be extended to supervised and semi-supervised learning areas.

Principles for constructing three-way approximations of fuzzy sets: A comparative evaluation based on unsupervised learning