Predictive tools in data mining and \(k\)-means clustering: universal inequalities. (English) Zbl 1285.60014
Summary: Grouping data into meaningful clusters is very important in data mining. \(k\)-means clustering is a fast method for finding clusters in data. The integral inequalities are a predictive tool in data mining and \(k\)-means clustering. Many papers have been published on speeding up \(k\)-means or nearest neighbor search using inequalities that are specific for Euclidean distance. An extended inequality related to Hölder type for universal integral is obtained in a rather general form. Previous results of H. Agahi et al. [Result. Math. 61, No. 1–2, 179–194 (2012; Zbl 1404.28025)] are generalized by relaxing some of their requirements, thus closing the series of papers on this topic. Chebyshev’s, Hölder’s, Minkowski’s, Stolarsky’s, Jensen’s and Lyapunov’s type inequalities for the universal integral are obtained.
MSC:
| 60E15 | Inequalities; stochastic orderings |
| 39B62 | Functional inequalities, including subadditivity, convexity, etc. |
Keywords:
monotone measure; universal integral; data mining; Chebyshev’s inequality; Stolarsky’s inequality; Minkowski’s inequality; Jensen’s inequality; Lyapunov’s inequality; Hölder’s inequalityCitations:
Zbl 1404.28025References:
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