×

zbMATH — the first resource for mathematics

Fuzzy partitions: a way to integrate expert knowledge into distance calculations. (English) Zbl 1321.62072
Summary: This work proposes a new pseudo-metric based on fuzzy partitions (FPs). This pseudo-metric allows for the introduction of expert knowledge into distance computations performed on numerical data and can be used in various types of statistical clustering or other applications. The knowledge is formalized by a FP, in which each fuzzy set represents a linguistic concept. The pseudo-metric is designed to respect the FP semantics. The univariate case is first studied, and the pseudo-metric behavior is discussed using synthetic experiments. Then, a multivariate version is proposed as a Minkowski-like combination of univariate distances or semi-distances. The value of the proposal is illustrated with two real-world case studies in the fields of Biology and Precision Agriculture.

MSC:
62H30 Classification and discrimination; cluster analysis (statistical aspects)
62H86 Multivariate analysis and fuzziness
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bezdek, J. C., Pattern recognition with fuzzy objective functions algorithms, (1981), Plenum Press New York · Zbl 0503.68069
[2] Bloch, I., On fuzzy distances and their use in image processing under imprecision, Pattern Recognition, 32, 1873-1895, (1999)
[3] S. Boriah, V. Chandola, V. Kumar, Similarity measures for categorical data: A comparative evaluation, in: SIAM Data Mining Conference, Atlanta, GA, 2008, pp. 243-254.
[4] Chaudhuri, B. B.; Rosenfeld, A., On a metric distance between fuzzy sets, Pattern Recognition Letters, 17, 1157-1160, (1996)
[5] Coppola, C.; Pacelli, T., Approximate distances, pointless geometry and incomplete information, Fuzzy Sets and Systems, 157, 2371-2383, (2006) · Zbl 1103.68837
[6] Diamond, P.; Kloeden, P., Metric spaces of fuzzy sets, Fuzzy Sets and Systems, 35, 241-249, (1990) · Zbl 0704.54006
[7] W.J. Dixon, BMDP statistical software manual: to accompany the 1990 software release, BDMP (1990).
[8] Dubois, D.; Prade, H., Fuzzy sets and systems: theory and applications, (1980), Academic Press New York · Zbl 0444.94049
[9] Egghe, L.; Rousseau, R., Classical retrieval and overlap measures satisfy the requirements for rankings based on a Lorenz curve, Information Processing and Management, 42, 106-120, (2006) · Zbl 1081.68593
[10] Fan, J.; Xie, W., Distance measure and induced fuzzy entropy, Fuzzy Sets and Systems, 104, 305-314, (1999) · Zbl 0968.94025
[11] Guillaume, S.; Charnomordic, B., Generating an interpretable family of fuzzy partitions, IEEE Transactions on Fuzzy Systems, 12, 3, 324-335, (2004)
[12] Guillaume, S.; Charnomordic, B., Learning interpretable fuzzy inference systems with fispro, Information Sciences, 181, 4409-4427, (2011)
[13] S. Guillaume, B. Charnomordic, P. Loisel, A numerical distance based on fuzzy partitions, in: S. Galichet, J. Montero, G. Mauris (Eds.), Proceedings of the 7th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-2011) and LFA-2011, Annecy, France, 2011, pp. 1000-1006. · Zbl 1254.68206
[14] Hammah, R. E.; Curran, J. H., On distance measures for the fuzzy k-means algorithm for joint data, Rock Mechanics and Rock Engineering, 32, 1, 1-27, (1999)
[15] Hartigan, J. A., Clustering algorithms, (1975), Wiley · Zbl 0372.62040
[16] Jousselme, A-L.; Maupin, P., Distances in evidence theory: comprehensive survey and generalizations, International Journal of Approximate Reasoning, 53, 2, 118-145, (2012) · Zbl 1280.68258
[17] Kaufman, L.; Rousseeuw, P., Finding groups in data: an introduction to cluster analysis, (1990), Wiley Interscience New York · Zbl 1345.62009
[18] Liang, J.; Li, R.; Qian, Y., Distance: a more comprehensible perspective for measures in rough set theory, Knowledge-Based Systems, 27, 0, 126-136, (2012)
[19] Lowen, R.; Peeters, W., Distance between fuzzy sets representing grey level images, Fuzzy Sets and Systems, 99, 135-149, (1998) · Zbl 0944.68190
[20] Mansingh, G.; Osei-Bryson, K-M.; Reichgelt, H., Using ontologies to facilitate post-processing of association rules by domain experts, Information Sciences, 181, 3, 419-434, (2011)
[21] Pedroso, M.; Taylor, J.; Tisseyre, B.; Charnomordic, B.; Guillaume, S., A segmentation algorithm for the delineation of management zones, Computer and Electronics in Agriculture, 70, 199-208, (2010)
[22] Pedrycz, W., Why triangular membership functions?, Fuzzy sets and Systems, 64, 1, 21-30, (1994)
[23] R Development Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, ISBN 3-900051-07-0, 2008. <http://www.R-project.org>.
[24] Rousseeuw, P. J., Silhouettes: a graphical aid to the interpretation and validation of cluster analysis, Journal of Computational and Applied Mathematics, 20, 53-65, (1987) · Zbl 0636.62059
[25] Tisseyre, B.; McBratney, A. B., A technical opportunity index based on mathematical morphology for site-specific management: an application to viticulture, Precision Agriculture, 9, 101-113, (2008)
[26] Torra, V.; Narukawa, Y., On a comparison between Mahalanobis distance and Choquet integral: the Choquet Mahalanobis operator, Information Sciences, 190, 0, 56-63, (2012) · Zbl 1250.28015
[27] Trutschnig, W.; González-Rodríguez, G.; Colubi, A.; Gil, M. A., A new family of metrics for compact, convex (fuzzy) sets based on a generalized concept of mid and spread, Information Sciences, 179, 23, 3964-3972, (2009) · Zbl 1181.62016
[28] Xu, Z.; Chen, J., An overview of distance and similarity measures of intuitionistic fuzzy sets, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 16, 529-555, (2008) · Zbl 1154.03317
[29] Xu, Z.; Xia, M., Distance and similarity measures for hesitant fuzzy sets, Information Sciences, 181, 2128-2138, (2010) · Zbl 1219.03064
[30] Zadeh, L. A., The concept of linguistic variable and its application to approximate reasoning - parts i, ii, and iii, Information Sciences, 8-9, 199-249, (1975), 301-357, 43-80 · Zbl 0397.68071
[31] Zeng, W.; Guo, P., Normalized distance, similarity measure, inclusion measure and entropy of interval-valued fuzzy sets and their relationship, Information Sciences, 178, 1334-1342, (2008) · Zbl 1134.68059
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.