Abstract
The Jaccard and Rand indices are the best-known and used similarity measures. In general, the Jaccard index is relatively conservative, but the Rand index is relatively optimistic. In the paper, we make a generalization of Rand and Jaccard indices with its fuzzy extension. We first define a compromised weight to improve the Rand and Jaccard indices and provide the weight parameter selection. We then further advance this into fuzzy extension so that it can be used to measure similarities between fuzzy partitions and crisp reference partitions and those between fuzzy partitions and fuzzy reference partitions. Therefore, the proposed method is more flexible and reasonable to provide a useful way that can be applied in practical studies according to actual demands. Finally, we use simulation and make comparisons to complete more explanations and further discussions.
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Jaccard, P.: Novelles recherches sur la distribution florale. Bull. Soc. Vaud. Sci. Nat. 44, 223–270 (1908)
Rand, W.M.: Objective criteria for the evaluation of clustering methods. J. Am. Stat. Assoc. 66, 846–850 (1971)
Fowlkes, E.B., Mallows, C.L.: A method for comparing two hierarchical clusterings. J. Am. Stat. Assoc. 78, 553–569 (1983)
Hubert, L., Arabie, P.: Comparing partitions. J. Classif 2, 193–218 (1985)
Carpineto, C., Romano, G.: Consensus clustering based on a new probabilistic rand index with application to subtopic retrieval. IEEE Trans. Pattern Anal. Mach. Intell. 34, 2315–2326 (2012)
Yin, Y., Yasuda, K.: Similarity coefficient methods applied to the cell formation problem: a taxonomy and review. Int. J. Prod. Econ. 101, 329–352 (2006)
MacQueen, J.: Some methods for classification and analysis of multivariate observations. In: Proceedings of 5th Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, pp. 281–297. University of California Press, Berkley (1967)
Pollard, D.: Quantization and the method of k-means. IEEE Trans. Inf. Theory 28, 199–205 (1982)
Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–356 (1965)
Bezdek, J.C.: Pattern Recognition with Fuzzy Objective Function Algorithms. Plenum Press, New York (1981)
Yang, M.S.: A survey of fuzzy clustering. Math. Comput. Model. 18, 1–16 (1993)
Baraldi, A., Blonda, P.: A survey of fuzzy clustering algorithms for pattern recognition-part I and II. IEEE Trans. Syst. Man Cybern. B Cybern. 29, 778–801 (1999)
Campello, R.J.G.B.: A fuzzy extension of the rand index and other related indexes for clustering and classification assessment. Pattern Recognit. Lett. 28, 833–841 (2007)
Brouwer, R.K.: Extending the rand, adjusted rand, and Jaccard indices to fuzzy partitions. J. Intell. Inf. Syst. 32, 213–235 (2009)
Anderson, D.T., Bezdek, J.C., Popescu, M., Keller, J.M.: Comparing fuzzy, probabilistic, and possibilistic partitions. IEEE Trans. Fuzzy Syst. 18, 906–918 (2010)
Hullermeier, E., Rifqi, M., Henzgen, S., Senge, R.: Comparing fuzzy partitions: a generalization of the Rand Index and related measures. IEEE Trans. Fuzzy Syst. 20, 546–556 (2012)
Halkidi, M., Batistakis, Y., Vazirgiannis, M.: On clustering validation techniques. J. Intell. Inform. Syst. 17, 107–145 (2001)
Epstein, S., Meier, P.: Constructive thinking: a board coping variable with specific components. J. Pers. Soc. Psychol. 57, 332–350 (1989)
Tennen, H., Affleck, G.: The costs and benefits of optimistic explanations and dispositional optimism. J. Pers. 55, 377–393 (1987)
Peterson, C.: Explanatory style as a risk factor for illness. Cogn. Ther. Res. 12, 119–132 (1988)
Cantor, N., Norem, J.K.: Defensive pessimism and stress and coping”. Soc. Cognit. 7, 92–112 (1989)
Quere, R., Le Capitaine, H., Fraisseix, N., Frelicot, C.: On normalizing fuzzy coincidence matrices to compare fuzzy and/or possibilistic partitions with the rand index. In Proceedings of IEEE International Conference on Data Mining, pp. 977–982 (2010)
Krishnapuram, R., Keller, J.M.: A possibilistic approach to clustering. IEEE Trans. Fuzzy Syst. 1, 98–110 (1993)
Yang, M.S., Lai, C.Y.: A robust automatic merging possibilistic clustering method. IEEE Trans. Fuzzy Syst. 19, 26–41 (2011)
Acknowledgments
The authors would like to thank the anonymous referees for their helpful comments in improving the presentation of this paper. This work was supported in part by the Ministry of Science and Technology (MOST) of Taiwan under Grant MOST-103-2118-M-033-001-MY2.
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Yeh, CC., Yang, MS. A Generalization of Rand and Jaccard Indices with Its Fuzzy Extension. Int. J. Fuzzy Syst. 18, 1008–1018 (2016). https://doi.org/10.1007/s40815-016-0263-0
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DOI: https://doi.org/10.1007/s40815-016-0263-0