2013
10
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Cover Special Issue vol. 10, no. 2, April 2013
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RANDOM FUZZY SETS: A MATHEMATICAL TOOL TO
DEVELOP STATISTICAL FUZZY DATA ANALYSIS
RANDOM FUZZY SETS: A MATHEMATICAL TOOL TO
DEVELOP STATISTICAL FUZZY DATA ANALYSIS
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2
Data obtained in association with many reallife random experiments from different fields cannot be perfectly/exactly quantified.hspace{.1cm}Often the underlying imprecision can be suitably described in terms of fuzzy numbers/\values. For these random experiments, the scale of fuzzy numbers/values enables to capture more variability and subjectivity than that of categorical data, and more accuracy and expressiveness than that of numerical/vectorial data. On the other hand, random fuzzy numbers/sets model the random mechanisms generating experimental fuzzy data, and they are soundly formalized within the probabilistic setting.This paper aims to review a significant part of the recent literature concerning the statistical data analysis with fuzzy data and being developed around the concept of random fuzzy numbers/sets.
1
Data obtained in association with many reallife random experiments from different fields cannot be perfectly/exactly quantified.hspace{.1cm}Often the underlying imprecision can be suitably described in terms of fuzzy numbers/\values. For these random experiments, the scale of fuzzy numbers/values enables to capture more variability and subjectivity than that of categorical data, and more accuracy and expressiveness than that of numerical/vectorial data. On the other hand, random fuzzy numbers/sets model the random mechanisms generating experimental fuzzy data, and they are soundly formalized within the probabilistic setting.This paper aims to review a significant part of the recent literature concerning the statistical data analysis with fuzzy data and being developed around the concept of random fuzzy numbers/sets.
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28
A.
BlancoFernandez
A.
BlancoFernandez
Departamento de Estadstica e I.O. y D.M., Universidad de
Oviedo, Spain
Departamento de Estadstica e I.O. y D.M.,
Spain
blancoangela@uniovi.es
M. R.
Casals
M. R.
Casals
Departamento de Estadstica e I.O. y D.M., Universidad de Oviedo,
Spain
Departamento de Estadstica e I.O. y D.M.,
Spain
rmcasals@uniovi.es
A.
Colubi
A.
Colubi
Departamento de Estadstica e I.O. y D.M., Universidad de Oviedo, Spain
Departamento de Estadstica e I.O. y D.M.,
Spain
colubi@uniovi.es
N.
Corral
N.
Corral
Departamento de Estadstica e I.O. y D.M., Universidad de Oviedo, Spain
Departamento de Estadstica e I.O. y D.M.,
Spain
norbert@uniovi.es
M.
GarcaBarzana
M.
GarcaBarzana
Departamento de Estadstica e I.O. y D.M., Universidad de Oviedo,
Spain
Departamento de Estadstica e I.O. y D.M.,
Spain
martagb5@gmail.com
M. A.
Gil
M. A.
Gil
Departamento de Estadstica e I.O. y D.M., Universidad de Oviedo, Spain
Departamento de Estadstica e I.O. y D.M.,
Spain
magil@uniovi.es
G.
GonzalezRodrguez
G.
GonzalezRodrguez
Departamento de Estadstica e I.O. y D.M., Universidad de
Oviedo, Spain
Departamento de Estadstica e I.O. y D.M.,
Spain
gil@uniovi.es
M.T.
Lopez
M.T.
Lopez
Departamento de Estadstica e I.O. y D.M., Universidad de Oviedo, Spain
Departamento de Estadstica e I.O. y D.M.,
Spain
mtlopez@uniovi.es
M.
Montenegro
M.
Montenegro
Departamento de Estadstica e I.O. y D.M., Universidad de Oviedo,
Spain
Departamento de Estadstica e I.O. y D.M.,
Spain
mmontenegro@uniovi.es
M. A.
Lubiano
M. A.
Lubiano
Departamento de Estadstica e I.O. y D.M., Universidad de Oviedo,
Spain
Departamento de Estadstica e I.O. y D.M.,
Spain
lubiano@uniovi.es
A. B.
RamosGuajardo
A. B.
RamosGuajardo
Departamento de Estadstica e I.O. y D.M., Universidad de
Oviedo, Spain
Departamento de Estadstica e I.O. y D.M.,
Spain
ramosana@uniovi.es
S.
de la Rosa de Saa
S.
de la Rosa de Saa
Departamento de Estadstica e I.O. y D.M., Universidad de
Oviedo, Spain
Departamento de Estadstica e I.O. y D.M.,
Spain
delarosasara@uniovi.es
B.
Sinova
B.
Sinova
Departamento de Estadstica e I.O. y D.M., Universidad de Oviedo, Spain
Departamento de Estadstica e I.O. y D.M.,
Spain
sinovabeatriz@uniovi.es
Distances between fuzzy numbers/values
Fuzzy numbers/values
Fuzzy arithmetic
Random fuzzy numbers/sets
Statistical methodology
[[1] M. C. Alonso, T. Brezmes, M. A. Lubiano and C. Bertoluzza, A generalized realvalued##measure of the inequality associated with a fuzzy random variable, Int. J. Approx. Reas., 26##(2001), 47–66.##[2] C. Bertoluzza, N. Corral and A. Salas, On a new class of distances between fuzzy numbers,##Mathware & Soft Computing, 2 (1995), 71–84.##[3] A. Colubi, Statistical inference about the means of fuzzy random variables: applications to##the analysis of fuzzy and realvalued data, Fuzzy Sets and Systems, 160 (2009), 344–356.##[4] A. Colubi, J. S. Dom´ınguezMenchero, M. L´opezD´ıaz and D. A. Ralescu, On the formalization##of fuzzy random variables, Information Sciences, 133 (2001), 3–6.##[5] A. Colubi, G. Gonz´alezRodr´ıguez, M. A. Gil and W. Trutschnig, Nonparametric criteria for##supervised classification of fuzzy data, Int. J. Approx. Reas., 52 (2011), 1272–1282.##[6] A. Colubi, M. L´opezD´ıaz, J. S. Dom´ınguezMenchero and M. A. Gil, A generalized strong##law of large numbers, Prob. Theor. Rel. Fields, 114 (1999), 401–417.##[7] P. Diamond and P. Kloeden, Metric spaces of fuzzy sets, Fuzzy Sets and Systems, 100 (1999),##63–71.##[8] M. B. Ferraro, R. Coppi, G. Gonz´alezRodr´ıguez and A. Colubi, A linear regression model##for imprecise response, Int. J. Approx. Reas., 51 (2010), 759–770.##[9] D. Garc´ıa, M. A. Lubiano and M. C. Alonso, Estimating the expected value of fuzzy random##variables in the stratified random sampling from finite populations, Information Sciences, 138##(2001), 165–184.##[10] M. A. Gil, M. L´opezD´ıaz and H. L´opezGarc´ıa, The fuzzy hyperbolic inequality index associated##with fuzzy random variables, Eur. J. Oper. Res., 110 (1998), 377–391.##[11] M. A. Gil, M. A. Lubiano, M. Montenegro and M. T. L´opez, Least squares fitting of an affine##function and strength of association for intervalvalued data, Metrika, 56 (2002), 97–111.##[12] M. A. Gil, M. Montenegro, G. Gonz´alezRodr´ıguez, A. Colubi and M. R. Casals, Bootstrap##approach to the multisample test of means with imprecise data, Comp. Stat. Data Anal., 51##(2006), 148–162.##[13] E. Gin´e and J. Zinn, Bootstrapping general empirical measures, Ann. Probab., 18 (1990),##851–869.##[14] G. Gonz´alezRodr´ıguez, A. Blanco, A. Colubi and M. A. Lubiano, Estimation of a simple##linear regression model for fuzzy random variables, Fuzzy Sets and Systems, 160 (2009),##[15] G. Gonz´alezRodr´ıguez, A. Colubi and M. A. Gil, A fuzzy representation of random variables:##an operational tool in exploratory analysis and hypothesis testing, Comp. Stat. Data Anal.,##51 (2006), 163–176.##[16] G. Gonz´alezRodr´ıguez, A. Colubi and M. A. Gil, Fuzzy data treated as functional data. A##oneway ANOVA test approach, Comp. Stat. Data Anal., 56 (2012), 943955.##[17] G. Gonz´alezRodr´ıguez, A. Colubi, M. A. Gil and P. D’Urso, An asymptotic two dependent##samples test of equality of means of fuzzy random variables, In: Proc. COMPSTAT’2006,##(2006), http://www.stat.unipg.it/iasc/Proceedings/2006/COMPSTAT/CD/145.pdf.##[18] G. Gonz´alezRodr´ıguez, M. Montenegro, A. Colubi and M. A. Gil, Bootstrap techniques##and fuzzy random variables: Synergy in hypothesis testing with fuzzy data, Fuzzy Sets and##Systems, 157 (2006), 2608–2613.##[19] G. Gonz´alezRodr´ıguez, W. Trutschnig and A. Colubi., Confidence regions for##the mean of a fuzzy random variable, In: Abstracts of IFSAEUSFLAT 2009,##http://www.eusflat.org/publications/proceedings/IFSAEUSFLAT 2009/pdf/tema 1433.pdf.##[20] T. Hesketh, R. Pryor and B. Hesketh, An application of a computerized fuzzy graphic rating##scale to the psychological measurement of individual differences, Int. J. ManMachine Studies,##29 (1988), 21–35. ##[21] B. Hesketh, T. Hesketh, J. I. Hansen and D. Goranson, Use of fuzzy variables in developing##new scales from the strong interest inventory, J. Counseling Psychology, 42 (1995), 85–99.##[22] M. Hukuhara, Int´egration des applications measurables dont la valeur est un compact convexe,##Funkcial. Ekvac., 10 (1967), 205223.##[23] E. P. Klement, M. L. Puri and D. A. Ralescu, Limit theorems for fuzzy random variables,##Proc. R. Soc. Lond. A, 407 (1986), 171–182.##[24] R. K¨orner, An asymptotic test for the expectation of random fuzzy variables, J. Stat. Plann.##Infer., 83 (2000), 331–346.##[25] R. K¨orner and W. N¨ather, On the variance of random fuzzy variables, In: C. Bertoluzza,##M. A. Gil and D. A. Ralescu, eds., Statistical Modeling, Analysis and Management of Fuzzy##Data, PhysicaVerlag, Heidelberg, (2002), 22–39.##[26] R. Kruse and K. D. Meyer, Statistics with vague data, D. Reidel Publishing Company, Dordrecht,##[27] H. Kwakernaak, Fuzzy random variablesI. definitions and theorems, Information Sciences,##15 (1978), 1–29.##[28] H. Kwakernaak, Fuzzy random variablesII. algorithms and examples for the discrete case,##Information Sciences, 17 (1979), 253–278.##[29] H. L´opezGarc´ıa, M. A. Gil, N. Corral and M. T. L´opez, Estimating the fuzzy inequality##associated with a fuzzy random variable in random samplings from finite populations, Kybernetika,##34 (1998), 149–161.##[30] M. A. Lubiano, M. C. Alonso and M. A. Gil, Statistical inferences on the Smean squared##dispersion of a fuzzy random variable, In: B. de Baets, J. Fodor and L. T. Koczy, eds.,##Proceedings of EUROFUSESIC99, University of Veterinary Science, Budapest, (1999), 532–##[31] M. A. Lubiano and M. A. Gil, Estimating the expected value of fuzzy random variables in##random samplings from finite populations, Stat. Pap., 40(1999), 277–295.##[32] M. A. Lubiano, M. A. Gil and M. L´opezD´ıaz, On the RaoBlackwell theorem for fuzzy##random variables, Kybernetika, 35 (1999), 167–175.##[33] M. A. Lubiano, M. A. Gil, M. L´opezD´ıaz and M. T. L´opez, The ! mean squared dispersion##associated with a fuzzy random variable, Fuzzy Sets and Systems, 111 (2000), 307–317.##[34] M. Montenegro, M. R. Casals, M. A. Lubiano and M. A. Gil, Twosample hypothesis tests of##means of a fuzzy random variable, Information Sciences, 133 (2001), 89–100.##[35] M. Montenegro, A. Colubi, M. R. Casals and M. A. Gil, Asymptotic and Bootstrap techniques##for testing the expected value of a fuzzy random variable, Metrika, 59 (2004), 31–49.##[36] M. Montenegro, M. T. L´opezGarc´ıa, M. A. Lubiano and G. Gonz´alezRodr´ıguez, A dependent##multisample test for fuzzy means, In: Abst. 2nd Workshop ERCIM WG Comput. & Statist,##(2009), 102.##[37] T. Nakama, A. Colubi and M. A. Lubiano, Factorial analysis of variance for fuzzy data, In:##Abst. CFE’10 & ERCIM’10, (2010), 88.##[38] H. T. Nguyen, A note on the extension principle for fuzzy sets, J. Math. Anal. Appl., 64##(1978), 369–380.##[39] M. L. Puri and D. A. Ralescu, Differentials of fuzzy functions, J. Math. Anal. Appl., 91##(1983), 552–558.##[40] M. L. Puri and D. A. Ralescu, The concept of normality for fuzzy random variables, Ann.##Probab., 11 (1985), 1373–1379.##[41] M. L. Puri and D. A. Ralescu, Fuzzy random variables, J. Math. Anal. Appl., 114 (1986),##409–422.##[42] S. Ramezanzadeh, M. Memariani and S. Saati, Data envelopment analysis with fuzzy random##inputs and outputs: a chanceconstrained programming approach, Iranian Journal of Fuzzy##Systems, 2 (2005), 21–29.##[43] A. B. RamosGuajardo, A. Colubi, G. Gonz´alezRodr´ıguez and M. A. Gil, One sample tests##for a generalized Fr´echet variance of a fuzzy random variable, Metrika, 71 (2010), 185–202. ##[44] A. B. RamosGuajardo and M. A. Lubiano, Ksample tests for equality of variances of##random fuzzy sets, Comp. Stat. Data Anal., 56 (2012), 956–966.##[45] B. Sinova, M. A. Gil, A. Colubi and S. Van Aelst, The median of a random fuzzy number.##The 1norm distance approach, Fuzzy Sets and Systems, 200 (2011), 99115.##[46] B. Sinova, S. de la Rosa de S´aa and M. A. Gil, A generalized L1type metric between fuzzy##numbers for an approach to central tendency of fuzzy data, Information Sciences, under 2nd##[47] S. M. Taheri and M. Kelkinnama, Fuzzy linear regression based on least absolutes deviations,##Iranian Journal of Fuzzy Systems, 9(1) (2012), 121140.##[48] P. Ter´an, A strong law of large numbers for random upper semicontinuous functions under##exchangeability conditions, Statist. Prob. Lett., 65 (2003), 251–258.##[49] W. Trutschnig, G. Gonz´alezRodr´ıguez, A. Colubi and M. A. Gil, A new family of metrics for##compact, convex (fuzzy) sets based on a generalized concept of mid and spread, Information##Sciences, 179 (2009), 3964–3972.##[50] W. Trutschnig and M. A. Lubiano, SAFD: statistical analysis of fuzzy data, http://cran.rproject.##org/web/packages/SAFD/index.html.##[51] R. Viertl and D. Hareter, Fuzzy information and stochastics, Iranian Journal of Fuzzy Systems,##1 (2004), 43–56.##[52] L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning,##Part 1, Information Sciences, 8 (1975), 199–249; Part 2, Information Sciences, 8 (1975), 301–##353; Part 3, Information Sciences, 9 (1975), 43–80.##[53] L. A. Zadeh, Discussion: probability theory and fuzzy logic are complementary rather than##competitive, Technometrics, 37 (1995), 271–276.##]
AGE REPLACEMENT POLICY IN UNCERTAIN
ENVIRONMENT
AGE REPLACEMENT POLICY IN UNCERTAIN
ENVIRONMENT
2
2
Age replacement policy is concerned with finding an optional time tominimize the cost, at which time the unit is replaced even if itdoes not fail. So far, age replacement policy involving random agehas been proposed. This paper will assume the age of the unit is anuncertain variable, and find the optimal time to replace the unit.
1
Age replacement policy is concerned with finding an optional time tominimize the cost, at which time the unit is replaced even if itdoes not fail. So far, age replacement policy involving random agehas been proposed. This paper will assume the age of the unit is anuncertain variable, and find the optimal time to replace the unit.
29
39
Kai
Yao
Kai
Yao
Department of Mathematical Sciences, Tsinghua University, Beijing 100084,
China
Department of Mathematical Sciences, Tsinghua
China
yaok09@mails.tsinghua.edu.cn
Dan A.
Ralescu
Dan A.
Ralescu
Department of Mathematical Sciences, University of Cincinnati,
Cincinnati, OH 452210025, USA
Department of Mathematical Sciences, University
United States
ralescd@ucmail.uc.edu
Uncertainty theory
Renewal process
Age replacement
Maintenance
[[1] R. E. Barlow and F. Proschan, Mathematical theory of reliability, Wiley and Sons, New York,##[2] P. J. Boland and F. Proschan, Periodic replacement with increasing minimal repair costs at##failure, Operations Research, 30(6) (1982), 1183{1189.##[3] X. Chen, American option pricing formula for uncertain financial market, International##Journal of Operations Research, 8(2) (2011), 32{37.##[4] X. Chen and W. Dai, Maximum entropy principle for uncertain variables, International##Journal of Fuzzy Systems, 13(3) (2011), 232{236.##[5] R. Cleroux, S. Dubuc and C. Tilquin, The age replacement problem with minimal repair and##random repair costs, Operations Research, 27(6) (1979), 1158{1167.##[6] W. Dai and X. Chen, Entropy of function of uncertain variables, Mathematical and Computer##Modelling, 55(34) (2012), 754{760.##[7] B. Fox, Age replacement with discounting, Operations Research, 14(3) (1966), 533{537.##[8] X. Gao, Some properties of continuous uncertain measure, International Journal of Uncer##tainty, Fuzziness and KnowledgeBased Systems, 17(3) (2009), 419{426.##[9] J. Gao, Q. Zhang and P. Shen, Coalitional game with fuzzy payoffs and credibilistic Shapley##value, Iranian Journal of Fuzzy Systems, 8(4) (2011), 107{117.##[10] J. Gao, Uncertain bimatrix game with applications, Fuzzy Optimization and Decision Making,##12(1) (2013), 6578.##[11] D. Kahneman and A. Tversky, Prospect theory: An analysis of decisions under risk, Econo##metrica, 47(2) (1979), 263{291.##[12] B. Liu, Uncertainty Theory, 2nd ed., SpringerVerlag, Berlin, 2007.##[13] B. Liu, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems,##2(1) (2008), 3{16.##[14] B. Liu, Theory and Practice of Uncertain Programming, 2nd ed., SpringerVerlag, Berlin,##[15] B. Liu, Uncertain set theory and uncertain inference rule with application to uncertain control##, Journal of Uncertain Systems, 4(2) (2010), 83{98.##[16] B. Liu, Uncertain risk analysis and uncertain reliability analysis, Journal of Uncertain Sys##tems, 4(3) (2010), 163{170.##[17] B. Liu, Uncertainty theory: a branch of mathematics for modeling human uncertainty,##SpringerVerlag, Berlin, 2011.##[18] B. Liu, Uncertain logic for modeling human language, Journal of Uncertain Systems, 5(1)##(2011), 3{20.##[19] B. Liu, Why is there a need for uncertainty theory?, Journal of Uncertain Systems, 6(1)##(2012), 3{10. ##[20] B. Liu, Extreme value theorems of uncertain process with application to insurance risk model,##Soft Computing, accepted.##[21] Y.H. Liu and M. Ha, Expected value of function of uncertain variables, Journal of Uncertain##Systems, 4(3) (2010), 181{186.##[22] V. P. Marathe and K. P. K. Nair, Multistage planned replacement strategies, Operations##Research, 14(5) (1966), 874{887.##[23] T. Nakagawa, Maintenance theory of reliability, SpringerVerlag, London, 2005.##[24] J. Peng and K. Yao, A new option pricing model for stocks in uncertainty markets, Interna##tional Journal of Operations Research, 8(2) (2011), 18{26.##[25] Z. Peng and K. Iwamura, A sufficient and necessary condition of uncertainty distribution,##Journal of Interdisciplinary Mathematics, 13(3) (2010), 277{285.##[26] C. Tilquin and R. Cleroux, Block replacement policies with general cost structures, Techno##metrics, 17(3) (1975), 291{298.##[27] C. Tilquin and R. Cleroux, Periodic replacement with minimal repair at failure and adjustment##costs, Naval Research Logistics Quarterly, 22(2) (1975), 243{254.##[28] X. Wang, Z. Gao and H. Guo, Uncertain hypothesis testing for two experts’ empirical data,##Mathematical and Computer Modelling, 55 (2012), 1478{1482.##[29] K. Yao, Uncertain calculus with renewal process, Fuzzy Optimization and Decision Making,##11(3) (2012), 285{297.##[30] K. Yao and X. Li, Uncertain alternating renewal process and its application, IEEE Transac##tions on Fuzzy Systems, 20(6) (2012), 11541160.##[31] K. Yao, Noarbitrage determinant theorems on meanreverting stock model in uncertain market##, KnowledgeBased Systems, 35 (2012), 259263.##[32] K. Yao, Block repalcement policy in uncertain environment, http://orsc.edu.cn/online/##110612.pdf.##[33] K. Yao, Some properties of uncertain renewal process, http://orsc.edu.cn/online/110602.pdf.##[34] C. You, Some convergence theorems of uncertain sequences, Mathematical and Computer##Modelling, 49(34) (2009), 482{487.##[35] Y. Zhu, Uncertain optimal control with application to a portfolio selection model, Cybernetics##and Systems, 41(7) (2010), 535{547.##]
REGION MERGING STRATEGY FOR BRAIN MRI
SEGMENTATION USING DEMPSTERSHAFER THEORY
REGION MERGING STRATEGY FOR BRAIN MRI
SEGMENTATION USING DEMPSTERSHAFER THEORY
2
2
Detection of brain tissues using magnetic resonance imaging (MRI) is an active and challenging research area in computational neuroscience. Brain MRI artifacts lead to an uncertainty in pixel values. Therefore, brain MRI segmentation is a complicated concern which is tackled by a novel data fusion approach. The proposed algorithm has two main steps. In the first step the brain MRI is divided to some main and ancillary cluster which is done using Fuzzy cmean (FCM). In the second step, the considering ancillary clusters are merged with main clusters employing DempsterShafer Theory. The proposed method was validated on simulated brain images from the commonly used BrainWeb dataset. The results of the proposed method are evaluated by using Dice and Tanimoto coefficients which demonstrate well performance and robustness of this algorithm.
1
Detection of brain tissues using magnetic resonance imaging (MRI) is an active and challenging research area in computational neuroscience. Brain MRI artifacts lead to an uncertainty in pixel values. Therefore, brain MRI segmentation is a complicated concern which is tackled by a novel data fusion approach. The proposed algorithm has two main steps. In the first step the brain MRI is divided to some main and ancillary cluster which is done using Fuzzy cmean (FCM). In the second step, the considering ancillary clusters are merged with main clusters employing DempsterShafer Theory. The proposed method was validated on simulated brain images from the commonly used BrainWeb dataset. The results of the proposed method are evaluated by using Dice and Tanimoto coefficients which demonstrate well performance and robustness of this algorithm.
49
56
Jamal
Ghasemi
Jamal
Ghasemi
Faculty of Engineering and Technology, University of Mazan
daran, Babolsar, Iran
Faculty of Engineering and Technology, University
Iran
j.ghasemi@umz.ac.ir
Mohamad Reza
Karami Mollaei
Mohamad Reza
Karami Mollaei
Faculty of Electrical and Computer Engeniering,
Babol University of Technology, P.O.Box 484, Babol, Iran
Faculty of Electrical and Computer Engeniering,
Ba
Iran
mkarami@nit.ac.ir
Reza
Ghaderi
Reza
Ghaderi
Shahid Beheshti University, Tehran, Iran
Shahid Beheshti University, Tehran, Iran
Iran
r_ghaderi@sbu.ac.ir
Ali
Hojjatoleslami
Ali Hojjatoleslami
Hojjatoleslami
School of computing, University of Kent, Canterbury,CT2 7PT
UK
School of computing, University of Kent,
United Kingdom
s.a.hojjatoleslami@kent.ac.uk
MRI
Fuzzy cmean
Brain MRI Segmentation
DempsterShafer Theory
[[1] W. AbdAlmageed, A. ElOsery and C. Smith, A fuzzystatistical contour model for MRI##segmentation and target tracking, presented at the SPIE, Orlando, FL, USA, (2004), 25{33.##[2] M. N. Ahmed, S. M. Yamany, N. Mohamed, A. A. Farag and T. Moriarty, A modied fuzzy c##means algorithm for bias eld estimation and segmentation of MRI data, IEEE transactions##on medical imaging, 21(3) (2002), 193{199.##[3] S. P. Awate, H. Zhang, T. J. Simon and J. C. Gee, Multivariate segmentation of brain tissues##by fusion of MRI and DTI data, presented at the Proceedings of the 2008 IEEE International##Symposium on Biomedical Imaging: From Nano to Macro, Paris, France, (2008).##[4] M. Balafar, A. Ramli, M. Saripan and S. Mashohor, Review of brain MRI image segmentation##methods, Articial Intelligence Review, 33(3) (2010), 261{274.##[5] M. Beynon, D. Cosker and D. Marshall, An expert system for multicriteria decision making##using Dempster Shafer theory, Expert Systems with Applications, 20(4) (2001), 357{367.##[6] E. Binaghi and P. Madella, Fuzzy DempsterShafer reasoning for rulebased classiers, Inter##national Journal of Intelligent Systems, 14(6) (1999), 559583.##[7] I. Bloch, Some aspects of DempsterShafer evidence theory for classication of multimodality##medical images taking partial volume eect into account, Pattern Recognition Letters, 17(8)##(1996), 905{919.##[8] M. Bomans, K. H. Hohne, U. Tiede and M. Riemer, 3D segmentation of MR images of the##head for 3D display, IEEE transactions on medical imaging, 9(2) (1990), 177{183.##[9] C. Brechbhler, G. Gerig and G. Szkely, Compensation of spatial inhomogeneity in MRI based##on a multivalued image model and a parametric bias estimate, In Visualization in Biomedical##Computing, (1996), 141{146.##[10] K. S. Chuang, H. L. Tzeng, S. Chen, J. Wu and T. J. Chen, Fuzzy cmeans clustering with##spatial information for image segmentation, Computerized Medical Imaging and Graphics :##the Ocial Journal of the Computerized Medical Imaging Society, 30(1) (2006), 9151.##[11] A. Demirhan and I. Gler, Combining stationary wavelet transform and selforganizing maps##for brain MR image segmentation, Engineering Applications of Articial Intelligence, 24(2)##(2011), 358{367.##[12] J. Ghasemi, R. Ghaderi, M. R. Karami Mollaei and A. Hojjatoleslami, Separation of brain tis##sues in MRI based on multidimensional FCM and spatial information, Eighth International##Conference on in Fuzzy Systems and Knowledge Discovery (FSKD), (2011), 247{251.##[13] J. Ghasemi, M. R. Karami Mollaei, R. Ghaderi and A. Hojjatoleslami, Brain tissue segmen##tation based on spatial information fusion by DempsterShafer theory, Journal of Zhejiang##University  Science C, 13(7) (2012), 520{533.##[14] J. D. Gispert, S. Reig, J. Pascau, J. J. Vaquero, P. GarciaBarreno and M. Desco, Method for##bias eld correction of brain T1weighted magnetic resonance images minimizing segmenta##tion error, Human brain mapping, 22(2) (2004), 133{144.##[15] M. Hasanzadeh and S. Kasaei, Multispectral Brain MRI Segmentation based on Fuzzy Clas##siers and Evidence Theory, presented at the 15th Iranian Conference on Electrical Engi##neering, ICEE, Tehran, Iran, 2007.##[16] T. Heinonen, P. Dastidar, H. Eskola, H. Frey, P. Ryymin and E. Laasonen, Applicability of##semiautomatic segmentation for volumetric analysis of brain lesions, Journal of Medical##Engineering And Technology, 22(4) (1998), 173{178.##[17] S. K. Jha and R. D. S. Yadava, Denoising by singular value decomposition and its application##to electronic nose data processing, IEEE Sensors Journal, 11(1) (2011), 35{44.##[18] L. Ji and H. Yan, An attractable snakes based on the greedy algorithm for contour extraction,##Pattern Recognition, 35(4) (2002), 791{806. ##[19] Z. X. Ji, Q. S. Sun and D. S. 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An Empirical Comparison between Grade of Membership and Principal Component Analysis
An Empirical Comparison between Grade of Membership and Principal Component Analysis
2
2
t is the purpose of this paper to contribute to the discussion initiated byWachter about the parallelism between principal component (PC) and atypological grade of membership (GoM) analysis. The author testedempirically the close relationship between both analysis in a lowdimensional framework comprising up to nine dichotomous variables and twotypologies. Our contribution to the subject is also empirical. It relies ona dataset from a survey which was especially designed to study the reward ofskills in the banking sector in Portugal. The statistical data comprisethirty polythomous variables and were decomposed in four typologies using anoptimality criterion. The empirical evidence shows a high correlationbetween the first PC scores and individual GoM scores. No correlation withthe remaining PCs was found, however. In addtion to that, the first PC alsoproved effective to rank individuals by skill following the particularity ofdata distribution meanwhile unveiled in GoM analysis.
1
t is the purpose of this paper to contribute to the discussion initiated byWachter about the parallelism between principal component (PC) and atypological grade of membership (GoM) analysis. The author testedempirically the close relationship between both analysis in a lowdimensional framework comprising up to nine dichotomous variables and twotypologies. Our contribution to the subject is also empirical. It relies ona dataset from a survey which was especially designed to study the reward ofskills in the banking sector in Portugal. The statistical data comprisethirty polythomous variables and were decomposed in four typologies using anoptimality criterion. The empirical evidence shows a high correlationbetween the first PC scores and individual GoM scores. No correlation withthe remaining PCs was found, however. In addtion to that, the first PC alsoproved effective to rank individuals by skill following the particularity ofdata distribution meanwhile unveiled in GoM analysis.
57
72
Abdul
Suleman
Abdul
Suleman
Department of Quantitative Methods, Instituto Universitario de
Lisboa (ISCTE  IUL), BRUUNIDE, Av. Forcas Armadas, Lisbon, Portugal
Department of Quantitative Methods, Instituto
Portugal
abdul.suleman@iscte.pt
Grade of Membership
principal component analysis
Fuzzy partition
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HURST EXPONENTS FOR NONPRECISE DATA
HURST EXPONENTS FOR NONPRECISE DATA
2
2
We provide a framework for the study of statistical quantitiesrelated to the Hurst phenomenon when the data are nonprecise with boundedsupport.
1
We provide a framework for the study of statistical quantitiesrelated to the Hurst phenomenon when the data are nonprecise with boundedsupport.
73
81
Mayer
Alvo
Mayer
Alvo
Department of Mathematics & Statistics, University of Ottawa, 585
King Edward, Ottawa, ON (K1N 5N1), Canada
Department of Mathematics & Statistics, University
Canada
malvo@uottawa.ca
Francois
Theberge
Francois
Theberge
Department of Mathematics & Statistics, University of Ottawa,
585 King Edward, Ottawa, ON (K1N 5N1), Canada
Department of Mathematics & Statistics, University
Canada
ftheberg@uottawa.ca
Hurst phenomenon
Nonprecise data
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ADAPTIVE ORDERED WEIGHTED AVERAGING FOR
ANOMALY DETECTION IN CLUSTERBASED
MOBILE AD HOC NETWORKS
ADAPTIVE ORDERED WEIGHTED AVERAGING FOR
ANOMALY DETECTION IN CLUSTERBASED
MOBILE AD HOC NETWORKS
2
2
In this paper, an anomaly detection method in clusterbased mobile ad hoc networks with ad hoc on demand distance vector (AODV) routing protocol is proposed. In the method, the required features for describing the normal behavior of AODV are defined via step by step analysis of AODV and independent of any attack. In order to learn the normal behavior of AODV, a fuzzy averaging method is used for combining oneclass support vector machine (OCSVM), mixture of Gaussians (MoG), and selforganizing maps (SOM) oneclass classifiers and the combined model is utilized to partially detect the attacks in cluster members. The votes of cluster members are periodically transmitted to the cluster head and final decision on attack detection is carried out in the cluster head. In the proposed method, an adaptive ordered weighted averaging (OWA) operator is used for aggregating the votes of cluster members in the cluster head. Since the network topology, traffic, and environmental conditions of a MANET as well as the number of nodes in each cluster dynamically change, the mere use of a fixed quantifierbased weight generation approach for OWA operator is not efficient. We propose a conditionbased weight generation method for OWA operator in which the number of cluster members that participate in decision making may be varying in time and OWA weights are calculated periodically and dynamically based on the conditions of the network. Simulation results demonstrate the effectiveness of the proposed method in detecting rushing, RouteError fabrication, and wormhole attacks.
1
In this paper, an anomaly detection method in clusterbased mobile ad hoc networks with ad hoc on demand distance vector (AODV) routing protocol is proposed. In the method, the required features for describing the normal behavior of AODV are defined via step by step analysis of AODV and independent of any attack. In order to learn the normal behavior of AODV, a fuzzy averaging method is used for combining oneclass support vector machine (OCSVM), mixture of Gaussians (MoG), and selforganizing maps (SOM) oneclass classifiers and the combined model is utilized to partially detect the attacks in cluster members. The votes of cluster members are periodically transmitted to the cluster head and final decision on attack detection is carried out in the cluster head. In the proposed method, an adaptive ordered weighted averaging (OWA) operator is used for aggregating the votes of cluster members in the cluster head. Since the network topology, traffic, and environmental conditions of a MANET as well as the number of nodes in each cluster dynamically change, the mere use of a fixed quantifierbased weight generation approach for OWA operator is not efficient. We propose a conditionbased weight generation method for OWA operator in which the number of cluster members that participate in decision making may be varying in time and OWA weights are calculated periodically and dynamically based on the conditions of the network. Simulation results demonstrate the effectiveness of the proposed method in detecting rushing, RouteError fabrication, and wormhole attacks.
83
109
Mohammad
Rahmanimanesh
Mohammad
Rahmanimanesh
Department of Electrical and Computer Engineering,
Tarbiat Modares University, Tehran, Islamic Republic of Iran
Department of Electrical and Computer Engineering,
Iran
rahmanimanesh@modares.ac.ir
Saeed
Jalili
Saeed
Jalili
Department of Electrical and Computer Engineering, Tarbiat Modares
University, Tehran, Islamic Republic of Iran
Department of Electrical and Computer Engineering,
Iran
sjalili@modares.ac.ir
Ordered weighted averaging weight generation
Mobile ad hoc network
Anomaly detection
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Monitoring Fuzzy Capability Index $widetilde{C}_{pk}$ by Using
the EWMA Control Chart with Imprecise Data
Monitoring Fuzzy Capability Index $widetilde{C}_{pk}$ by Using
the EWMA Control Chart with Imprecise Data
2
2
A manufacturing process cannot be released to production until it has been proven to be stable. Also, we cannot begin to talk about process capability until we have demonstrated stability in our process. This means that the process variation is the result of random causes only and all assignable or special causes have been removed. In complicated manufacturing processes, such as drilling process, the natural instability of the process impedes the use of any control charts for the mean and standard deviation. However, a complicated manufacturing process can be capable in spite of this natural instability.In this paper we discuss the $widetilde{C}_{pk}$ process capability index. We find the membership function of $widetilde{C}_{pk}$ based on fuzzy data. Also, by using the definition of classical control charts and the method of V$ddot{a}$nnman and Castagliola, we propose new control charts that are constructed by the $alpha$cut sets of $widetilde{C}_{pk}$ for the natural instable manufacturing processes with fuzzy normal distributions. The results are concluded for $alpha=0.6$, that is chosen arbitrarily.
1
A manufacturing process cannot be released to production until it has been proven to be stable. Also, we cannot begin to talk about process capability until we have demonstrated stability in our process. This means that the process variation is the result of random causes only and all assignable or special causes have been removed. In complicated manufacturing processes, such as drilling process, the natural instability of the process impedes the use of any control charts for the mean and standard deviation. However, a complicated manufacturing process can be capable in spite of this natural instability.In this paper we discuss the $widetilde{C}_{pk}$ process capability index. We find the membership function of $widetilde{C}_{pk}$ based on fuzzy data. Also, by using the definition of classical control charts and the method of V$ddot{a}$nnman and Castagliola, we propose new control charts that are constructed by the $alpha$cut sets of $widetilde{C}_{pk}$ for the natural instable manufacturing processes with fuzzy normal distributions. The results are concluded for $alpha=0.6$, that is chosen arbitrarily.
111
132
Bahram
Sadeghpour Gildeh
Bahram
Sadeghpour Gildeh
Faculty of Mathematical Science, Department of Sta
tistics, University of Mazandaran, Babolsar, Iran and School of Mathematical Science,
Department of Statistics, Ferdowsi University of Mashhad, Postal Code : 9177948953,
Mashhad, Iran
Faculty of Mathematical Science, Department
Iran
sadeghpour@umz.ac.ir
Tala
Angoshtari
Tala
Angoshtari
Faculty of Mathematical Science, Department of Statistics, Uni
versity of Mazandaran, Babolsar, Iran
Faculty of Mathematical Science, Department
Iran
tala.angoshtari@gmail.com
Capability index
$D_{p
q}$distance
Fuzzy set
Membership Function
EWMA control chart
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ON INTERRELATIONSHIPS BETWEEN FUZZY
METRIC STRUCTURES
ON INTERRELATIONSHIPS BETWEEN FUZZY
METRIC STRUCTURES
2
2
Considering the increasing interest in fuzzy theory and possible applications,the concept of fuzzy metric space concept has been introduced by severalauthors from different perspectives. This paper interprets the theory in termsof metrics evaluated on fuzzy numbers and defines a strong Hausdorff topology.We study interrelationships between this theory and other fuzzy theories suchas intuitionistic fuzzy metric spaces, Kramosil and Michalek's spaces, Kalevaand Seikkala's spaces, probabilistic metric spaces, probabilisticmetric cospaces, Menger spaces and intuitionistic probabilistic metricspaces, determining their position in the framework of theses different theories.
1
Considering the increasing interest in fuzzy theory and possible applications,the concept of fuzzy metric space concept has been introduced by severalauthors from different perspectives. This paper interprets the theory in termsof metrics evaluated on fuzzy numbers and defines a strong Hausdorff topology.We study interrelationships between this theory and other fuzzy theories suchas intuitionistic fuzzy metric spaces, Kramosil and Michalek's spaces, Kalevaand Seikkala's spaces, probabilistic metric spaces, probabilisticmetric cospaces, Menger spaces and intuitionistic probabilistic metricspaces, determining their position in the framework of theses different theories.
133
150
Antonio
Roldan
Antonio
Roldan
Department of Statistics and Operations Research, University of
Jaen, Campus Las Lagunillas, s/n, E23071, Jaen, Spain
Department of Statistics and Operations Research,
Spain
afroldan@ujaen.es
Juan
MartnezMoreno
Juan
MartnezMoreno
Department of Mathematics, University of Jaen, Campus Las
Lagunillas, s/n, E23071, Jaen, Spain
Department of Mathematics, University of
Spain
jmmoreno@ujaen.es
Concepcion
Roldan
Concepcion
Roldan
Department of Statistics and Operations Research, University
of Granada, Campus Fuentenueva s/n, E18071, Granada, Spain
Department of Statistics and Operations Research,
Spain
iroldan@ugr.es
Fuzzy metric
Fuzzy metric space
Fuzzy number
Fuzzy topology
Links between dierent models
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