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.
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Notes
- 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.
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).
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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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