Abstract
In this chapter, we will create and use generalized combined comparison measures from t-norms (T) and t-conorms (S) for comparison of data. Norms are combined by the use of generalized mean, where t-norms give minimum and t-conorms give maximum compensation. From this intuitively thinking follows that when these norms are aggregated together, these new comparison measures should be able to find the best possible classification result in between minimum and maximum. We will use classification as our test bench for the suitability of these new comparison measures created. In these classification tasks, we have tested five different types of combined comparison measures (CCM), with t-norms and t-conorms. That were Dombi family, Frank family, Schweizer-Sklar family, Yager family, and Yu family. In classification, we used the following datasets: ionosphere, iris, and wine. We will compare the results achieved with CCM to the ones achieved with pseudo equivalences and show that these new measures tend to give better results.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Tversky, A., Krantz, D.H.: The dimensional representation and the metric structure of similarity data. J. Math. Psychol. 7(3), 572–596 (1970)
Santini, S., Jain, R.: Similarity measures. IEEE Trans. Pattern Anal. Mach. Intell. 21(9), 871–883 (1999)
France, R.K.: Weights and Measures: An Axiomatic Model for Similarity Computations. Internal Report, Virginia Tech (1994)
De Cock, M., Kerre, E.: Why fuzzy T-equivalence relations do not resolve the Poincar paradox, and related issues. Fuzzy Sets Syst. 133(2), 181–192 (2003)
Tversky, A.: Features of similarity. Psychol. Rev. 84(4), 327–352 (1977)
Zadeh, L.A., Fuzzy sets and their application to pattern classification and clustering analysis. In: Van Ryzin, J. (Ed.) Classification and Clustering. Academic Press, pp. 251–299 (1977)
Pal, S.K., Dutta-Majumder, D.K.: Fuzzy Mathematical Approach to Pattern Recognition. Wiley (Halsted), N. Y. (1986)
Saastamoinen, K.: Classification of data with similarity classifier. In: International Work-Conference on Time Series, Keynote Speech, ITISE2016 Proceedings
Lowen, R.R.: Fuzzy Set Theory: Basic Concepts, Techniques, and Bibliography. Kluwer Academic Publishers, Dordrecht (1996)
Bellman, R., Giertz, M.: On the analytic formalism of the theory of fuzzy sets. Inf. Sci. 5, 149–165 (1973)
Kang, T., Chen, G.: Modifications of Bellman-Giertz’s theorem. Fuzzy Sets Syst. 94(3), 349–353 (1998)
Saastamoinen, K., Ketola, J.: Using generalized combination measure from Dombi and Yager type of T-norms and T-conorms in classification. In: Proceedings of the ECTI-CON 2005 Conference (2005)
Dyckhoff, H., Pedrycz, W.: Generalized means as model of compensative connectives. Fuzzy Sets Syst. 14, 143–154 (1984)
Zimmermann, H.-J., Zysno, P.: Latent connectives in human decision making. Fuzzy Sets Syst. 4, 37–51 (1980)
Bilgiç, T., Türkşen, I.B.: Measurement-theoretic justification of connectives in fuzzy set theory. Fuzzy Sets Syst. 76(3), 289–308 (1995)
Dombi, J.: Basic concepts for a theory of evaluation: the aggregative operator. Eur. J. Oper. Res. 10, 282–293 (1982)
Frank, M.J.: On the simultaneous associativity of \(F(x, y)\) and \(x+y-F(x, y)\). Aeqationes Math. 19, 194–226 (1979)
Schweizer, B., Sklar, A.: Associative functions and abstract semigroups. Publ. Math. Debr. 10, 69–81 (1963)
Yager, R.R.: On a general class of fuzzy connectives. Fuzzy Sets Syst. 4, 235–242 (1980)
Yu, Y.: Triangular norms and TNF-sigma algebras. Fuzzy Sets Syst. 16, 251–264 (1985)
Sklar, A.: Fonctions de r\(\acute{e}\)partition \(\acute{a}\) n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8, 229–231 (1959)
Fisher, N.I.: Copulas. In: Kotz, S., Read, C.B., Banks, D.L. (eds.) Encyclopedia of Statistical Sciences, vol. 1, pp. 159–163. Wiley, New York (1997)
Hastie, T., Tibshirani, R.: The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer Series in Statistics, Springer, New York (2001)
Goldberg, D.E.: Real-coded genetic algorithms, virtual alphabets, and blocking. Technical Report 9001, University of Illinois at Urbana-Champain (1990)
Mantel, B., Periaux, J., Sefrioui, M.: Gradient and genetic optimizers for aerodynamic desing. In: ICIAM 95 Conference, Hamburgh (1995)
Michalewics, Z.: Genetic Algorithms + Data Structures = Evolution Programs Artificial Intelligence. Springer, New York (1992)
Grefenstette, J.J.: Optimization of control parameters for genetic algorithms. IEEE Trans. Syst. Man Cybern. 16(1), 122–128 (1986)
Price, K.V., Storn, R.M., Lampinen, J.A.: Differential Evolution—A Practical Approach to Global Optimization. Springer, Natural Computing Series (2005)
UC Irvine Machine Learning Repository. http://archive.ics.uci.edu/ml/. Accessed 15 Jan 2017
Saastamoinen, K.: Many valued algebraic structures as measures of comparison, Acta Universitatis Lappeenrantaensis. Ph.D. Thesis (2008)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this paper
Cite this paper
Saastamoinen, K. (2018). Logical Comparison Measures in Classification of Data—Nonmetric Measures. In: Rojas, I., Pomares, H., Valenzuela, O. (eds) Time Series Analysis and Forecasting. ITISE 2017. Contributions to Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-96944-2_12
Download citation
DOI: https://doi.org/10.1007/978-3-319-96944-2_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-96943-5
Online ISBN: 978-3-319-96944-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)