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A Generalization of Rand and Jaccard Indices with Its Fuzzy Extension

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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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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Correspondence to Miin-Shen Yang.

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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

Keywords

  • Similarity measure
  • Rand index
  • Jaccard index
  • Fuzzy set
  • Compromised weight