Lift, leverage, and conviction are three of the best commonly known interest measures for crisp association rules. All of them are based on a comparison of observed support and the support that is expected if the antecedent and consequent part of the rule were stochastically independent. The aim of this paper is to provide a correct definition of lift, leverage, and conviction measures for fuzzy association rules and to study some of their interesting mathematical properties. 1. Introduction Searching for association rules is a broadly discussed, developed, and accepted data mining technique [1, 2]. An association rule is an expression , where antecedent and consequent are conditions, the former usually in the form of elementary conjunction and the latter being usually atomic. Such rules are usually interpreted as the following implicational statement: “if is satisfied then is true very often too.” Naturally, analysts are interested only in such rules that are somehow interesting, unusual, or exceptional. To assess rule interestingness objectively, there have been developed many measures of rule interest or intensity. Among the most essential, support and confidence are traditionally considered. An objective of association rules mining is to find rules with support and confidence above some user-defined thresholds. Searching for association rules fits particularly well on binary or categorical data and many have been written on that topic [1–4]. For association analysis on numeric data, a prior discretization is proposed, for example, by Srikant and Agrawal [5]. Another alternative is to take an advantage of fuzzy logic [6]. The use of fuzzy logic in connection with association rules has been motivated by many authors (see e.g., [7] for recent overview). Fuzzy association rules are appealing also because of the use of vague linguistic terms such as “small” and “very big” [8–11]. In this paper, we focus on three measures of rule intensity that are all based on comparison between the observed support and the support that is expected under the assumption of independence of the rule’s antecedent and consequent. These measures are lift, leverage, and conviction. All of them were initially developed for nonfuzzy (i.e., “crisp”) association rules. Lift was firstly described in [12] under its original name “interest.” It was well studied for association rules on binary data in [13, 14]. Lift is defined as a ratio of observed support to the support that is expected under the assumption of independence of and . On the other hand, leverage [15] measures the
References
[1]
M. Hahsler and K. Hornik, “New probabilistic interest measures for association rules,” Intelligent Data Analysis, vol. 11, no. 5, pp. 437–455, 2007.
[2]
R. Agrawal, T. Imielinski, and A. Swami, “Mining association rules between sets of items in large databases,” in Proceedings of the ACM SIGMOD International Conference on Management of Data, pp. 207–216, Washington, DC, USA, May 1993.
[3]
A. Berrado and G. C. Runger, “Using metarules to organize and group discovered association rules,” Data Mining and Knowledge Discovery, vol. 14, no. 3, pp. 409–431, 2007.
[4]
G. I. Webb, “Discovering significant patterns,” Machine Learning, vol. 68, no. 1, pp. 1–33, 2007.
[5]
R. Srikant and R. Agrawal, “Mining quantitative association rules in large relational tables,” SIGMOD Record, vol. 25, no. 2, pp. 1–12, 1996.
[6]
V. Novák, I. Perfilieva, and J. Mo？ko？, Mathematical Principles of Fuzzy Logic, Kluwer, Boston, Mass, USA, 1999.
[7]
H. Kalia, S. Dehuri, and A. Ghosh, “A survey on fuzzy association rule mining,” International Journal of Data Warehousing and Mining, vol. 9, no. 1, pp. 1–27, 2013.
[8]
K. C. C. Chen and W.-H. Au, Mining Fuzzy Association Rules, ACM, New York, NY, USA, 1997.
[9]
V. Novák, I. Perfilieva, A. Dvo？ák, G. Chen, Q. Wei, and P. Yan, “Mining pure linguistic associations from numerical data,” International Journal of Approximate Reasoning, vol. 48, no. 1, pp. 4–22, 2008.
[10]
M. Burda, “Fast evaluation of t-norms for fuzzy association rules mining,” in Proceedings of the 14th IEEE International Symposium on Computational Intelligence and Informatics (CINTI '13), pp. 465–470, Budapest, Hungary, November 2013.
[11]
M. Burda, V. Pavliska, and R. Valá？ek, “Parallel mining of fuzzy association rules on dense data sets,” in Proceedings of the IEEE International Conference on Fuzzy Systems (FUZZ-IEEE '14), pp. 2156–2162, IEEE, Beijing, China, 2014.
[12]
S. Brin, R. Motwani, J. D. Ullman, and S. Tsur, “Dynamic itemset counting and implication rules for market basket data,” in Proceedings ACM SIGMOD International Conference on Management of Data (SIGMOD '97), pp. 255–264, Tucson, Ariz, USA, 1997.
[13]
P. D. McNicholas, T. B. Murphy, and M. O'Regan, “Standardising the lift of an association rule,” Computational Statistics & Data Analysis, vol. 52, no. 10, pp. 4712–4721, 2008.
[14]
P. Hájek, I. Havel, and M. Chytil, “The GUHA method of automatic hypotheses determination,” Computing, vol. 1, no. 4, pp. 293–308, 1966.
[15]
G. Piatetsky-Shapiro, “Discovery, analysis, and presentation of strong rules,” in Knowledge Discovery in Databases, pp. 229–248, AAAI/MIT Press, 1991.
[16]
S. Lallich, O. Teytaud, and E. Prudhomme, “Association rule interestingness: measure and statistical validation,” in Quality Measures in Data Mining, vol. 43 of Studies in Computational Intelligence, pp. 251–275, Springer, Berlin, Germany, 2007.
[17]
A. L. Buczak and C. M. Gifford, “Fuzzy association rule mining for community crime pattern discovery,” in Proceedings of the ACM SIGKDD Workshop on Intelligence and Security Informatics (ISI-KDD '10), vol. 2, pp. 1–10, ACM, New York, NY, USA, 2010.
[18]
D. P. Pancho, J. M. Alonso, J. Alcalá-Fdez, and L. Magdalena, “Interpretability analysis of fuzzy association rules supported by fingrams,” in Proceedings of the 8th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT '13), pp. 469–474, September 2013.
[19]
R. Agrawal, “Fast discovery of association rules,” in Advances in Knowledge Discovery and Data Mining, pp. 307–328, AAAI Press/MIT Press, Cambridge, Mass, USA, 1996.
[20]
M. Burda, “Lift measure for fuzzy association rules,” in Proceedings of the 7th International Conference on Soft Methods in Probability and Statistics (SMPS '2014), Springer, Warsaw, Poland, September 2014.