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Credal-based fuzzy number data clustering

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Abstract

It remains challenging in characterizing uncertain and imprecise information when clustering fuzzy number data. To solve such a problem, this paper investigates a new credal-based fuzzy number data clustering (CFNDC) method based on the theory of belief functions (TBF). The CFNDC method attempts to learn a general and flexible credal partition to gain deeper insights into fuzzy number data. Specifically, the CFNDC can provide the behavior (support degrees) for each fuzzy number in different clusters to characterize uncertainty. Additionally, it can reasonably assign the fuzzy numbers in overlapping regions to related meta-clusters, defined as the disjunction of several close specific clusters, to characterize imprecision. Doing so allows prudent decision-making and reduces the risk of errors. We evaluate the performance of different methods using various evaluation indexes and introduce a new measure called evidential purity (EPU). The proposed CFNDC method achieves an average EPU of 0.8330, which demonstrates that CFNDC is a promising tool for effectively handling fuzzy number data.

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Data availability

All data information is included in this paper.

Notes

  1. Here uncertainty refers to the lack of certainty caused by insufficient information. For example, I am not sure that the student got full marks in both subjects. In comparison, imprecision refers to the lack of precision caused by the fuzziness of knowledge. For example, the student got full marks in at least two subjects.

  2. Some works regard fuzzy numbers as fuzzy data to represent imprecise or fuzzy information, see (Dubois 1980), He et al. (2018). In contrast, some works also define fuzzy data as a set of fuzzy membership functions, see (Han et al. 2013), Yang et al. (2016). From our point of view, fuzzy number is essentially a generalization of real number, which means only knowing around a certain and precise value. Fuzzy number data can be considered as a set of fuzzy numbers. Therefore, to avoid ambiguity, we use fuzzy number data to represent this case in this paper.

  3. LR fuzzy number (data) can also be transformed to triangular fuzzy number, interval number and crisp one in some specific cases (D’Urso and De Giovanni 2014).

  4. So far, there is no work on credal partition of fuzzy number data, so we take object data as an example to explain credal partition.

  5. In this paper, in order to avoid ambiguity, we use “center” to represent the center of fuzzy number data and “prototype” to represent the center of cluster.

  6. The weighted distance in Coppi et al. (2012) uses an appropriate weighting system to consider the center and spread distance of fuzzy numbers, respectively.

  7. Here, we only show the optimization process of \(\mathcal {V}^{Z_1}\), and the optimization process of \(\mathcal {V}^{Z_2}\), \(\mathcal {V}^{L}\) and \(\mathcal {V}^{R}\) is similar to that of \(\mathcal {V}^{Z_1}\).

  8. For credal partition, we can also employ pignistic probabilities (Smets and Kennes 1994) to adapt the clustering results to these indexes.

  9. For visualization, each fuzzy number \(\tilde{\textbf{x}}_{i}\) is represented by a rectangle with the rectangle box representing the 0-level of the membership function (Ramos-Guajardo and Ferraro 2020).

  10. Basic information about the dataset can be found at: https://archive.ics.uci.edu.

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  1. School of Computer Sciences, Universiti Sains Malaysia, 11800, Gelugor, Penang, Malaysia

    Zhe Liu

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  1. Zhe Liu
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ZL: Conceptualization, Methodology, Writing—review & editing.

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Liu, Z. Credal-based fuzzy number data clustering. Granul. Comput. 8, 1907–1924 (2023). https://doi.org/10.1007/s41066-023-00410-0

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