Abstract
The paper considers the problem of classification error in pattern recognition. This model of classification is primarily based on the Bayes rule and secondarily on the notion of intuitionistic or interval-valued fuzzy sets. A probability of misclassifications is derived for a classifier under the assumption that the features are class-conditionally statistically independent, and we have intuitionistic or interval-valued fuzzy information on object features instead of exact information. A probability of the intuitionistic or interval-valued fuzzy event is represented by the real number. Additionally, the received results are compared with the bound on the probability of error based on information energy. Numerical example concludes the work.
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References
Zadeh LA (1968) Probability measures of fuzzy events. J Math Anal Appl 23:421–427
Goguen J (1967) L-fuzzy sets. J Math Anal Appl 18(1):145–174
Pawlak Z (1985) Rough sets and fuzzy sets. Fuzzy Set Syst 17:99–102
Atanassov K (1986) Intuitionistic fuzzy sets. Fuzzy Set Syst 20:87–96
Atanassov K, Georgeiv C (1993) Intuitionistic fuzzy prolog. Fuzzy Set Syst 53:121–128
Szmidt E, Kacprzyk J (2002) Using intuitionistic fuzzy sets in group decision making. Control Cybern 31(4):1037–1053
Szmidt E, Kacprzyk J (2003) A consensus-reaching process under intuitionistic fuzzy preference relations. Int J Intell Syst 18(7):837–852
Gerstenkorn T, Mańko J (1988) Bifuzzy probability of intuitionistic sets. Note Intuition Fuzzy Set 4:8–14
Gerstenkorn, T, Mańko J (1990) Probability of fuzzy intuitionistic sets. BUSEFAL 45:128–136
Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning, part I. Inf Sci 8:199–249
Dengfeng L, Chuntian C (2002) New similarity measure of intuitionistic fuzzy sets and application to pattern recognitions. Pattern Recognit Lett 23:221–225
Szmidt E, Kacprzyk J (2001) Intuitionistic fuzzy sets in iteligent data analysis for medical diagnostics. LNCS 2074:263–271
Szmidt E, Kacprzyk J (2004) A similarity measure of intuitionistic fuzzy sets and its application in supporting medical diagnostics reasoning. LNAI 3070:388–393
Vlachos IK, Sergiadis, GD (2007) Intuitionistic fuzzy information—aplication to pattern recognition. Pattern Recognit Lett 28:197–206
De SK, Biswas R, Roy AR (2001) An application of intitionistic fuzzy sets in medical diagnosis. Fuzzy Set Syst 117:209–213
Mitchell HB (2005) Pattern recognition using type-II fuzzy sets. Inf Sci 170:409–418
Zeng J, Liu Y-Q (2008) Type-2 fuzzy markov random fields and their application to handwritten chinese character recognition. IEEE Trans Fuzzy Syst 16(3):747–760
Bhatt RB, Gopal M (2008) FRCT: fuzzy-rough classification trees. Pattern Anal Appl 11(1):73–88
Mitchell HB (2003) On the Dengfang-Chuntain similarity measure and its application to pattern cecognition. Pattern Recognit Lett 24:3101–3104
Stańczyk U (2010) Rough set-based analysis of characteristic features for ANN classiifier. Lecture notes in artifical intelligence, vol 6076. Springer, Berlin, pp 565–572
Okuda T, Tanaka H, Asai K (1978) A formulation of fuzzy decision problems with fuzzy information using probability measures of fuzzy events. Inf Control 38:135–147
Antos A, Devroye LL, Gyorfi L (1999) Lower bounds for Bayes error estimation. IEEE Trans Pattern Anal Mach Intell 21:643–645
Avi-Itzhak H, Diep T (1996) Arbitrarily tight upper and lower bounds on the bayesian probability of error. IEEE Trans Pattern Anal Mach Intell 18:89–91
Hashlamoun WA, Varshney PK, Samarasooriya VNS (1994) A tight upper bound on the Bayesian probability of error. IEEE Trans Pattern Anal Mach Intell 16(2):220–224
Kittler, J (1998) Combining classifiers: a theoretical framework. Pattern Anal Appl 1:18–27
Woźniak M (2008) Experiments on linear combiners. Advances in soft computing, vol 47. Springer, Berlin, pp 445–452
Kulkarni A (1978) On the mean accuracy of hierarchical classifiers. IEEE Trans Comput 27:771–776
Kurzyński M (1988) On the multistage Bayes classifier. Pattern Recognit 21:355–365
Burduk R (2010) Classification error in Bayes multistage recognition task with fuzzy observations. Pattern Anal Appl 13(1):85–91
Pardo L, Menendez ML (1991) Some bounds on probability of error in fuzzy discrimination problems. Eur J Oper Res 53:362–370
Pardo JA, Taneja IJ (1992) On the probability of error in fuzzy discrimination Problems. Kybernetes 21(6):43–52
Kuncheva LI (2004) Combining pattern classifier: methods and algorithms. Wiley, New York
Tizhoosh HR (2008) Interval-valued versus intuitionistics fuzzy sets: isomorphism versus semantics. Pattern Recognit 41:1812–1813
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This work is supported by The Polish Ministry of Science and Higher Education under the grant which is being realized in years 2010–2013.
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Burduk, R. Imprecise information in Bayes classifier. Pattern Anal Applic 15, 147–153 (2012). https://doi.org/10.1007/s10044-011-0201-6
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DOI: https://doi.org/10.1007/s10044-011-0201-6