Overview on the development of fuzzy random variables

Fuzzy Sets and Systems

Volumn 157, Issue 19, 2006, Pages 2546-2557

  Gil, María Ángeles ^a   López Díaz, Miguel ^a   Ralescu, Dan A ^b  

^a UNIVERSITY OF OVIEDO   (Spain)

^b UNIVERSITY OF CINCINNATI   (United States)

Author keywords

(W, symbol ) metric; d metric; Fuzzy arithmetic; Fuzzy random variables; Fuzzy values; Hausdorff metric; Random sets

Indexed keywords

MATHEMATICAL MODELS; PROBABILISTIC LOGICS; PROBLEM SOLVING; RANDOM PROCESSES; STATISTICAL METHODS;

FUZZY ARITHMETIC; FUZZY RANDOM VARIABLES; FUZZY VALUES; HAUSDORFF METRIC;

FUZZY SETS;

EID: 33748062355     PISSN: 01650114     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.fss.2006.05.002     Document Type: Article

References (62)

1. Bertoluzza C., Corral N., and Salas A. On a new class of distances between fuzzy numbers. Mathware & Soft Comput. 2 (1995) 71-84
2. A. Colubi, Strong laws of large numbers for fuzzy random variables, Ph.D. Thesis, Universidad de Oviedo, 2000.
3. Colubi A., Domínguez-Menchero J.S., López-Díaz M., and Gil M.A. A generalized strong law of large numbers. Probab. Theory Rel. Fields 114 (1999) 401-417
4. Colubi A., Domínguez-Menchero J.S., López-Díaz M., and Ralescu D.A. On the formalization of fuzzy random variables. Inform. Sci. 133 (2001) 3-6
5. E [0, 1]-representation of random upper semicontinuous functions. Proc. Amer. Math. Soc. 130 (2002) 3237-3242
6. Colubi A., Fernández-García C., and Gil M.A. Simulation of random fuzzy variables: an empirical approach to statistical/probabilistic studies with fuzzy experimental data. IEEE Trans. Fuzzy Systems 10 (2002) 384-390
7. Diamond P. Least squares fitting of compact set-valued data. J. Math. Anal. Appl. 147 (1990) 351-362
8. Feng Y. Decomposition theorems for fuzzy supermartingales and submartingales. Fuzzy Sets and Systems 116 (2000) 225-235
9. Gil M.A., and López-Díaz M. Fundamentals and Bayesian analyses of decision problems with fuzzy-valued utilities. Intenat. J. Approx. Reasoning 15 (1996) 203-224
10. Gil M.A., López-Díaz M., and López-García H. The fuzzy hyperbolic inequality index associated with fuzzy random variables. European J. Oper. Res. 110 (1998) 377-391
11. M.A. Gil, M. Montenegro, G. González-Rodríguez, A. Colubi, M.R. Casals, Bootstrap approach to the multi-sample test of means with imprecise data, Comp. Statist. Data Anal. 2006, in press.
12. Gong Z., Wu C., and Li B. On the problem of characterizing derivatives for the fuzzy-valued functions. Fuzzy Sets and Systems 127 (2002) 315-322
13. G. González-Rodríguez, M. Montenegro, A. Colubi, M.A. Gil, Bootstrap techniques and fuzzy random variables: synergy in hypothesis testing with fuzzy data, Fuzzy Sets and Systems, (2006) (in this issue).
14. Hirota K. Probabilistic sets: probabilistic extensions of fuzzy sets. In: Yager R.R., and Zadeh L.A. (Eds). An Introduction to Fuzzy Logic Applications in Intelligent Systems, Kluwer International Series in Engineering and Computer Science vol. 165 (1992), Kluwer, Boston 335-354
15. Inoue H., and Taylor R.L. A SLLN for arrays of rowwise exchangeable fuzzy random sets. Stochastic Anal. Appl. 13 (1995) 461-470
16. Kim Y.-K. Measurability for fuzzy valued functions. Fuzzy Sets and Systems 129 (2002) 105-109
17. Klement E.P., Puri M.L., and Ralescu D.A. Law of large numbers and central limit theorems for fuzzy random variables. In: Trappl R. (Ed). Cybernetics and Systems Research 2 (1984), Elsevier, North-Holland, Amsterdam 525-529
18. Klement E.P., Puri M.L., and Ralescu D.A. Limit theorems for fuzzy random variables. Proc. Roy. Soc. London A 407 (1986) 171-182
19. Körner R. On the variance of fuzzy random variables. Fuzzy Sets and Systems 92 (1997) 83-93
20. Körner R. An asymptotic α-test for the expectation of random fuzzy variables. J. Statist. Plann. Inference 83 (2000) 331-346
21. Körner R., and Näther W. Linear regression with random fuzzy variables: extended classical estimates, best linear estimates, least squares estimates. Inform. Sci. 109 (1998) 95-118
22. Körner R., and Näther W. On the variance of random fuzzy variables. In: Bertoluzza C., Gil M.A., and Ralescu D.A. (Eds). Statistical Modeling (2002), Analysis and Management of Fuzzy Data, Physica-Verlag, Heidelberg 22-39
23. Krätschmer V. Limit theorems for fuzzy-random variables. Fuzzy Sets and Systems 126 (2002) 253-263
24. V. Krätschmer, Integrals of random fuzzy sets, Abst. 25th Linz Seminar on Fuzzy Set Theory 2004, pp. 93-103.
25. R. Kruse, K.D. Meyer, Statistics with Vague Data, Reidel Publ. Co. Dordrecht, 1987.
26. Kwakernaak H. Fuzzy random variables. Part I: definitions and theorems. Inform. Sci. 15 (1978) 1-29
27. Kwakernaak H. Fuzzy random variables. Part II: algorithms and examples for the discrete case. Inform. Sci. 17 (1979) 253-278
28. ∞. Fuzzy Sets and Systems 135 (2003) 391-399
29. Li S., Ogura Y., and Kreinovich V. Limit Theorems and Applications of Set Valued and Fuzzy Valued Random Variables (2002), Kluwer Academic Publishers, Dordrecht
30. Li S., Ogura Y., and Nguyen H.T. Gaussian processes and martingales for fuzzy valued random variables with continuous parameter. Inform. Sci. 133 (2001) 7-21
31. Li S., Ogura Y., Proske F.N., and Puri M.L. Central limit theorems for generalized set-valued random variables. J. Math. Anal. Appl. 285 (2003) 250-263
32. López-Díaz M., and Gil M.A. Reversing the order of integration in iterated expectations of fuzzy random variables, and statistical applications. J. Statist. Plann. Inference 74 (1998) 11-29
33. M.A. Lubiano, Variation measures for imprecise random elements, Ph.D. Thesis, Universidad de Oviedo, 1999.
34. Lubiano M.A., Gil M.A., López-Díaz M., and López M.T. The over(λ, →)-mean squared dispersion associated with a fuzzy random variable. Fuzzy Sets and Systems 111 (2000) 307-317
35. Molchanov I. On strong laws of large numbers for random upper semicontinuous functions. J. Math. Anal. Appl. 235 (1999) 349-355
36. M. Montenegro, Estadística con Datos Imprecisos Basada en una Métrica Generalizada, Ph.D. Thesis, Universidad de Oviedo, 2003.
37. Montenegro M., Casals M.R., Lubiano M.A., and Gil M.A. Two-sample hypothesis tests of means of a fuzzy random variable. Inform. Sci. 133 (2001) 89-100
38. Montenegro M., Colubi A., Casals M.R., and Gil M.A. Asymptotic and bootstrap techniques for testing the expected value of a fuzzy random variable. Metrika 59 (2004) 31-49
39. Montenegro M., González-Rodríguez G., Gil M.A., Colubi A., and Casals M.R. Introduction to ANOVA with fuzzy random variables. In: López-Díaz M., Gil M.A., Grzegorzewski P., Hryniewicz O., and Lawry J. (Eds). Soft Methodology and Random Information Systems (2004), Springer, Berlin 487-494
40. Nguyen H.T. A note on the extension principle for fuzzy sets. J. Math. Anal. Appl. 64 (1978) 369-380
41. Nguyen H.T., Kreinovich V., and Wu B. Fuzzy/probability ∼ fractal/smooth. Internat. J. Uncert. Fuzzy Knowledge-Based Systems 7 (1999) 363-370
42. Ogura Y., and Li S. Separability for graph convergence of sequences of fuzzy-valued random variables. Fuzzy Sets and Systems 123 (2001) 19-27
43. Ogura Y., and Li S. Limit theorems for random fuzzy sets including large deviation principles. In: López M., Gil M.A., Grzegorzewski P., Hryniewicz O., and Lawry J. (Eds). Soft Methodology and Random Information Systems (2004), Springer, Heidelberg 32-44
44. Okuda T., Tanaka H., and Asai K. A formulation of fuzzy decision problems with fuzzy information using probability measures of fuzzy events. Inform. and Control 38 (1978) 135-147
45. Proske F.N., and Puri M.L. Strong law of large numbers for Banach space valued fuzzy random variables. J. Theor. Probab. 15 (2002) 543-551
46. Puri M.L., and Ralescu D.A. Différentielle d'une fonction floue. C.R. Acad. Sci. Paris, Sér. A 293 (1981) 237-239
47. Puri M.L., and Ralescu D.A. Differentials of fuzzy functions. J. Math. Anal. Appl. 91 (1983) 552-558
48. Puri M.L., and Ralescu D.A. The concept of normality for fuzzy random variables. Ann. Probab. 11 (1985) 1373-1379
49. Puri M.L., and Ralescu D.A. Fuzzy random variables. J. Math. Anal. Appl. 114 (1986) 409-422
50. Puri M.L., and Ralescu D.A. Convergence theorem for fuzzy martingales. J. Math. Anal. Appl. 160 (1991) 107-122
51. D.A. Ralescu, Fuzzy probabilities and their applications to statistical inference, in: B. Bouchon-Meunier, R.R. Yager, L.A. Zadeh (Eds.), Advances in Intelligent Computing-IPMU'94, Lecture Notes in Computer Science, vol. 945, Springer, Paris, 1995, pp. 217-222.
52. Rodríguez-Muñiz L., López-Díaz M., and Gil M.A. The s-differentiability of a fuzzy-valued mapping. Inform. Sci. 151 (2003) 283-299
53. Rodríguez-Muñiz L., López-Díaz M., and Gil M.A. Differentiating random upper semicontinuous functions under the integral sign. Test 12 (2003) 241-258
54. Rodríguez-Muñiz L., López-Díaz M., and Gil M.A. Solving influence diagrams with fuzzy chance and value nodes. European J. Oper. Res. 167 (2005) 444-460
55. Román-Flores H., and Rojas-Medar M.A. Differentiability of fuzzy-valued mapping. Rev. Mat. Estatist 16 (1998) 223-239
56. Stojaković M. Fuzzy random variables, expectation, and martingales. J. Math. Anal. Appl. 184 (1994) 594-606
57. Terán P. Cones and decomposition of sub- and supermartingales. Fuzzy Sets and Systems 147 (2004) 465-474
58. Terán P. An embedding theorem for convex fuzzy sets. Fuzzy Sets and Systems 152 (2005) 191-208
59. P. Terán, A large deviation principle for random upper semicontinuous functions, Proc. Amer. Math. Soc. 134 (2006) 571-580.
60. P. Terán, On Borel measurability and large deviations for fuzzy random variables, Fuzzy Sets and Systems, (2006) (in this issue).
61. L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, Part 1. Inform. Sci. 8 (1975) 199-249; Part 2. Inform. Sci. 8 (1975) 301-353; Part 3. Inform. Sci. 9 (1975) 43-80.
62. Zadeh L.A. Is probability theory sufficient for dealing with uncertainty in AI: A negative view. Mach. Intell. Pattern Recognition 4 (1986) 103-116