Abstract
The paper presents the estimators of three relations: equivalence, tolerance and preference in a finite set on the basis of multiple pairwise comparisons, disturbed by random errors; they have been developed by the author. The estimators can rest on: binary (qualitative), multivalent (quantitative) and combined comparisons. The estimates are obtained on the basis of discrete programming problems. They require weak assumptions about distributions of comparisons errors, especially allow non-zero expected values. The estimators have good statistical properties, in particular are consistent. The estimates can be verified using statistical tests. The paper summarizes briefly the results obtained lastly by the author.
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References
Abraham, A., Grosan, C. (eds.): Swarm Intelligence in Data Mining. Springer (2006)
Brunk, H.D.: Mathematical models for ranking from paired comparisons. JASA Vol 55, 503–520 (1960)
Bradley, R.A.: Science, statistics and paired comparisons. Biometrics vol 32, 213–232 (1976)
Bradley, R.A.: Paired comparisons: some basic procedures and examples. In: Krishnaiah, P.R., Se, P.K. (eds.) Handbook of Statistics, vol. 4, pp. 299–326. North-Holland, Amsterdam (1984)
David, H.A.: Order Statistics. J. Wiley (1970)
David, H.A.: The Method of Paired Comparisons, 2nd edn. Griffin, London (1988)
Falkenauer, E.: Genetics Algorithms and Grouping Data. J. Wiley (1998)
Flinger, Verducci (eds.): Probability Models and Statistical Analyses for Ranking Data. Springer, Heidelberg (1993)
Gordon, A.D.: Classification, 2nd edn. Chapman&Hall/CRC (1999)
Hand, D.J.: Discrimination and Classification. J. Wiley (1986)
Hansen, P., Jaumard, B., Sanlaville, E.: Partitioning Problems in Cluster Analysis: A Review of Mathematical Programming Approaches. In: Studies in Classification, Data Analysis, and Knowledge Organization, Springer (1994)
Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning. Data Mining, Inference and Prediction. Springer (2002)
Hoeffding, W.: Probability inequalities for sums of bounded random variables. JASA 58, 13–30 (1963)
Kaufman, L., Rousseeuw, P.J.: Findings Groups in Data: An Introduction to Cluster Analysis. J. Wiley (1990)
Klukowski, L.: Algorithm for classification of samples in the case of unknown number variables generating them, Przeglad Statystyczny XXXVII, pp. 167–177 (1990) (in Polish)
Klukowski, L.: Some probabilistic properties of the nearest adjoining order method and its extensions. Annals of Operational Research 51, 241–261 (1994)
Klukowski, L.: The nearest adjoining order method for pairwise comparisons in the form of difference of ranks. Annals of Operations Research vol 97, 357–378 (2000)
Klukowski, L.: Methods of Estimation of Relations of: Equivalence, Tolerance, and Preference in a Finite Set. IBS PAN, Warsaw. Systems Research, vol. 69 (2011)
Kohonen, T.: Self-Organizing Maps. Springer (1995)
Ripley, B.D.: Stochastic Simulation. J. Wiley (2006)
Slater, P.: Inconsistencies in a schedule of paired comparisons. Biometrika 48, 303–312 (1961)
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Klukowski, L. (2015). Estimators of the Relations of: Equivalence, Tolerance and Preference on the Basis of Pairwise Comparisons with Random Errors. In: Angelov, P., et al. Intelligent Systems'2014. Advances in Intelligent Systems and Computing, vol 322. Springer, Cham. https://doi.org/10.1007/978-3-319-11313-5_34
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DOI: https://doi.org/10.1007/978-3-319-11313-5_34
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-11312-8
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