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Shadowed sets provide a meaningful description of information granules by abstracting the corresponding fuzzy sets into three categories: full acceptance, full rejection, and uncertain (represented by shadows). One of the main motivating points to derive shadowed sets from fuzzy sets is the determination and explanation of the separation thresholds based on a specific optimization mechanism. The available optimization objective functions are mainly discussed on semantic interpretations and their mathematical properties; constructive algorithms for optimal solutions have rarely been reported. In this paper, the continuous and convex properties of Pedrycz's optimization objective function to construct shadowed sets, as well as the existence and uniqueness of solution points, are analyzed in detail. It is demonstrated that different approximation region partitions would be generated even under the same optimization model, which requires further criteria to make the constructed shadowed sets well‐defined. To address this limitation, the notions of passive and active constrained shadowed sets are introduced. A fast algorithm to obtain the proposed constrained shadowed sets is also designed based on the analyzed mathematical properties. Its performance is then illustrated by some typical fuzzy sets and some real data from the UCI repository. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Highlights • A novel rough possibilistic C-means clustering approach is presented. • The partition threshold for each cluster is automatically optimized. • The uncertainty generated by a single fuzzifier value is captured. • The scale parameters are adaptively adjusted. • A framework for updating prototypes based on ensemble strategies is formed. Abstract The management of uncertain information in a data set is crucial for clustering models. In this study, we present a rough possibilistic C-means clustering approach based on multigranulation approximation regions and shadowed sets, which can handle the uncertainties implicated in data and generated by the model parameters simultaneously. In particular, all patterns are first partitioned into three approximation regions with respect to a fixed cluster according to their possibilistic membership degrees based on shadowed set theory, which can help capture the natural topology of the data, especially when dealing with outliers and noisy data. The multigranulation approximation regions of each cluster can then be formed under a series of fuzzifier values, where the uncertainty caused by a specific fuzzifier value can be detected based on variations in the approximation regions with different levels of granularity. We also introduce a framework for updating prototypes based on ensemble strategies to attenuate the distortions due to iteration during clustering procedures. Finally, an adaptive mechanism is developed for dynamically adjusting scale parameters based on the notion of the maximal compatible regions of clusters. By integrating various granular computing techniques, i.e., rough sets, fuzzy sets, shadowed sets, and the notion of multigranulation, the uncertainties implicated in data and produced by model parameters can be adequately addressed, and the possibilistic membership values involved make the method sufficiently robust to deal with noisy environments. The improved performance of the proposed approach was demonstrated in experiments based on the comparisons with other available fuzzy and possibilistic clustering methods. [ABSTRACT FROM AUTHOR]