Abstract
Measures of rank correlation are commonly used in statistics to capture the degree of concordance between two orderings of the same set of items. Standard measures like Kendall’s tau and Spearman’s rho coefficient put equal emphasis on each position of a ranking. Yet, motivated by applications in which some of the positions (typically those on the top) are more important than others, a few weighted variants of these measures have been proposed. Most of these generalizations fail to meet desirable formal properties, however. Besides, they are often quite inflexible in the sense of committing to a fixed weighing scheme. In this paper, we propose a weighted rank correlation measure on the basis of fuzzy order relations. Our measure, called scaled gamma, is related to Goodman and Kruskal’s gamma rank correlation. It is parametrized by a fuzzy equivalence relation on the rank positions, which in turn is specified conveniently by a so-called scaling function. This approach combines soundness with flexibility: it has a sound formal foundation and allows for weighing rank positions in a flexible way. The usefulness of our class of weighted rank correlation measures is shown by means of experimental studies using both synthetic and real-world ranking data.
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Balasubramaniyan, R., Hüllermeier, E., Weskamp, N., Kämper, J.: Clustering of gene expression data using a local shape-based similarity measure. Bioinformatics 21(7), 1069–1077 (2005)
Bodenhofer, U.: A similarity-based generalization of fuzzy orderings preserving the classical axioms. Int. J. Uncertainty, Fuzziness and Knowledge-Based Systems 8(5), 593–610 (2000)
Bodenhofer, U.: Representations and constructions of similarity-based fuzzy orderings. Fuzzy Sets and Systems 137, 113–136 (2003)
Bodenhofer, U., Demirci, M.: Strict fuzzy orderings with a given context of similarity. Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems 16(2), 147–178 (2008)
Bodenhofer, U., Klawonn, F.: Robust rank correlation coefficients on the basis of fuzzy orderings: Initial steps. Mathware & Soft Computing 15, 5–20 (2008)
Pinto da Costa, J., Soares, C.: A weighted rank measure of correlation. Australian & New Zealand Journal of Statistics 47(4), 515–529 (2005)
Fagin, R., Kumar, R., Sivakumar, D.: Comparing top k lists. SIAM Journal on Discrete Mathematics 17(1), 134–160 (2003)
Goodman, L.A., Kruskal, W.H.: Measures of Association for Cross Classifications. Springer-Verlag, New York (1979)
Henzgen, S., Hüllermeier, E.: Weighted rank correlation measures based on fuzzy order relations. In: Hoffmann, F., Hüllermeier, E. (eds.) Proceedings 23. Workshop Computational Intelligence, pp. 227–236. KIT Scientific Publishing, Dortmund, Germany (2013)
Kaye, D.: A weighted rank correlation coefficient for the comparison of relevance judgements. Journal of Documentation 29(4), 380–389 (1973)
Kendall, M.G.: Rank correlation methods. Charles Griffin, London (1955)
Klawonn, F.: Fuzzy sets and vague environment. Fuzzy Sets and Systems 66, 207–221 (1994)
Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers (2002)
Kumar, R., Vassilvitskii, S.: Generalized distances between rankings. In: Proc. WWW, 19. International Conference on World Wide Web, pp. 571–580 (2010)
Marden, J.I.: Analyzing and Modeling Rank Data. CRC Press (1996)
Maturi, T.A., Abdelfattah, E.H.: A new weighted rank correlation. Journal of Mathematics and Statistics 4(4), 226 (2008)
Dolorez Ruiz, M., Hüllermeier, E.: A formal and empirical analysis of the fuzzy gamma rank correlation coefficient. Information Sciences 206, 1–17 (2012)
Schölkopf, B., Smola, A., Müller, K.R.: Kernel principal component analysis. In: Advances in Kernel Methods: Support Vector Learning, pp. 327–352. MIT Press (1999)
Shieh, G.S.: A weighted Kendall’s tau statistic. Statistics & Probability Letters 39(1), 17–24 (1998)
Spearman, C.: The proof and measurement for association between two things. Amer. Journal of Psychology 15, 72–101 (1904)
Yilmaz, E., Aslam, J.A., Robertson, S.: A new rank correlation coefficient for information retrieval. In: Proc. 31st Annual International ACM SIGIR Conference on Research and Development in Information Retrieval, pp. 587–594. ACM (2008)
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Henzgen, S., Hüllermeier, E. (2015). Weighted Rank Correlation: A Flexible Approach Based on Fuzzy Order Relations. In: Appice, A., Rodrigues, P., Santos Costa, V., Gama, J., Jorge, A., Soares, C. (eds) Machine Learning and Knowledge Discovery in Databases. ECML PKDD 2015. Lecture Notes in Computer Science(), vol 9285. Springer, Cham. https://doi.org/10.1007/978-3-319-23525-7_26
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DOI: https://doi.org/10.1007/978-3-319-23525-7_26
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