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North American Fuzzy Information Processing Society Annual Conference

NAFIPS 2018: Fuzzy Information Processing pp 96–107Cite as

Relevance of Classes in a Fuzzy Partition. A Study from a Group of Aggregation Operators

Part of the Communications in Computer and Information Science book series (CCIS,volume 831)

Abstract

This paper presents a study of the relevance property in a fuzzy partition from a fuzzy classification system. This study allows establishing a stopping criterion for the inclusion of a class in a fuzzy partition based on relevance. Such a criterion is constructed from a stable relationship on the commutative group formed by two new mappings (and the aggregation operators conjunctive and disjunctive) of the fuzzy classification system. The criterion is illustrated through an example on image analysis by the fuzzy c-means algorithm.

Keywords

  • Relevance
  • Covering
  • Overlap
  • Classification
  • Fuzzy partition

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Fig. 1.
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Notes

  1. 1.

    Recursiveness is a property of a sequence of operators \( \left\{ {\phi_{n} } \right\}_{n > 2} \) allowing the aggregation of any number of items: \( \phi_{2} \) tells us how to aggregate two items, \( \phi_{3} \) tells how to aggregate three items and so on. A recursive rule \( \phi \) is a family of aggregation functions \( \{ \phi_{n} :\left[ {0,1} \right]^{n} \to \left[ {0,1} \right]\}_{n > 1} \) allowing a sequential reckoning by means of a successive application of binary operators, once data have been properly ordered: the ordering rule assures that new data do not introduce modifications in the relative position of items already ordered. For more details see [5, 18].

  2. 2.

    Here we refer to strict negations of the type \( N\left( x \right) = f^{ - 1} \left( {f\left( 1 \right) - f\left( x \right)} \right) \) with \( f:\left[ {0,1} \right] \to \left[ {0,1} \right] \) increasing, bijective, \( f\left( 0 \right) = 0 \), and \( 0 < f\left( 1 \right) \le 1 \). In particular, if \( N\left( x \right) = 1 - x \), then \( f\left( x \right) = x \).

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Acknowledgements

This research has been partially supported by the Government of Spain (grant TIN2015-66471-P), the Government of Madrid (grant S2013/ICE-2845), and Complutense University (UCM Research Group 910149).

Author information

Authors and Affiliations

  1. Faculty of Economic, Administrative and Accounting Sciences, Gran Colombia University, Bogotá, Colombia

    Fabián Castiblanco

  2. Department of Industrial Engineering, Andes University, Bogotá, Colombia

    Camilo Franco

  3. Department of Statistics, Complutense University, Madrid, Spain

    Javier Montero & J. Tinguaro Rodríguez

Authors
  1. Fabián Castiblanco
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  2. Camilo Franco
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  3. Javier Montero
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  4. J. Tinguaro Rodríguez
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Corresponding author

Correspondence to Fabián Castiblanco .

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Editors and Affiliations

  1. Department of Teleinformatics Engineering, Federal University of Ceará, Fortaleza, Ceará, Brazil

    Guilherme A. Barreto

  2. Department of Statistics & Applied Mathematics, Federal University of Ceará, Fortaleza, Ceará, Brazil

    Ricardo Coelho

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Castiblanco, F., Franco, C., Montero, J., Rodríguez, J.T. (2018). Relevance of Classes in a Fuzzy Partition. A Study from a Group of Aggregation Operators. In: Barreto, G., Coelho, R. (eds) Fuzzy Information Processing. NAFIPS 2018. Communications in Computer and Information Science, vol 831. Springer, Cham. https://doi.org/10.1007/978-3-319-95312-0_9

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