Abstract
The possibility theory as a mathematical model of randomness and fuzziness phenomena is considered in a variant that enables the modeling of both probabilistic randomness, including that inherent in unpredictably evolving stochastic objects whose probabilistic models cannot be empirically reconstructed and nonprobabilistic randomness (fuzziness) inherent in real physical, technical, and economical objects, human–machine and expert systems, etc. Some principal distinctions between the considered variant and the known possibility theory variants, in particular, in mathematical formalism and its relationship with probability theory, substantive interpretation, and applications exemplified by solving the problems of identification and estimation optimization, empirical reconstruction of a fuzzy model for a studied object, measurement data analysis and interpretation, etc. (in the paper “Mathematical Modeling of Randomness and Fuzziness Phenomena in Scientific Studies. II. Applications”) are shown.
Similar content being viewed by others
References
G. Choquet, Ann. Inst. Fourier 5, 131 (1953–1954).
L. A. Zadeh, Inf. Control 8, 235 (1965).
A. P. Dempster, Ann. Math. Stat. 38, 325 (1967).
A. P. Dempster, J. R. Stat. Soc. B 30, 205 (1968).
L. J. Savage, The Foundations of Statistics (Dover, New York, 1972).
M. Sugeno, PhD Thesis (Tokyo Inst. of Technology, Tokyo, 1974).
G. Shafer, A Mathematical Theory of Evidence (Princeton Univ. Press, 1976).
L. A. Zadeh, Fuzzy Sets Syst., No. 1, 3 (1978).
S. A. Orlovskii, Problems of Decision-Making under Uncertain Original Information (Nauka, Moscow, 1981).
Fuzzy Sets and Possibility Theory. Recent Developments, Ed. by P. P. Yager (Pergamon Press, 1982).
Fuzzy Sets in Control and Artificial Intelligence Models, Ed. by D. A. Pospelov (Nauka, Moscow, 1986).
D. Dubois and H. Prade, Theorie des Possibilites (Masson, 1988).
Yu. P. Pyt’ev, Pattern Recognit. Image Anal. 4, 177 (1994).
Yu. P. Pyt’ev, Pattern Recognit. Image Anal. 5, 13 (1995).
G. de Cooman and E. E. Kerre, Fuzzy Sets Syst. 77, 207 (1996).
G. de Cooman, Int. J. Gen. Syst. 25, 291 (1997).
G. de Cooman, Int. J. Gen. Syst. 25, 325 (1997).
G. de Cooman, Int. J. Gen. Syst. 25, 353 (1997).
Yu. P. Pyt’ev, Pattern Recognit. Image Anal. 7, 338 (1997).
D. Dubois and H. Prade, in Quantified Representation of Uncertainty and Imprecision, Ed. by D. M. Gabbay and P. Smets (Springer, 1998), p. 169.
O. Wolkenhauer, Possibility Theory with Applications to Data Analysis (Research Studies Press, 1998).
Applied Fuzzy Systems, Ed. by T. Terano, K. Asai, and M. Sugeno (Elsevier, 1989).
Yu. P. Pyt’ev, Pattern Recognit. Image Anal. 8, 1 (1998).
Yu. P. Pyt’ev, Pattern Recognit. Image Anal. 9, 416 (1999).
D. Dubois, H. T. Nguyen, and H. Prade, in Fundamentals of Fuzzy Sets, Ed. by D. Dubois and H. Prade (Kluwer Academic Publishers, Boston, 2000), p. 343.
Yu. P. Pyt’ev, Pattern Recognit. Image Anal. 10, 43 (2000).
Yu. P. Pyt’ev, Pattern Recognit. Image Anal. 10, 447 (2000).
Yu. P. Pyt’ev, Possibility. Elements of Theory and Applications (Editorial URSS, Moscow, 2000).
D. Dubois, H. Prade, and S. Sandri, in Proc. 4th IFSA Congress, Brussels, Belgium, 1991 (Kluwer Academic Publishers, 1993), p. 103.
I. V. D’yakonova, T. V. Matveeva, and Yu. P. Pyt’ev, Pattern Recognit. Image Anal. 11, 711 (2001).
Yu. P. Pyt’ev, Intellekt. Sist. 6, 25 (2001).
Yu. P. Pyt’ev, Pattern Recognit. Image Anal. 12, 107 (2002).
Yu. P. Pyt’ev and O. V. Zhucko, Pattern Recognit. Image Anal. 12, 116 (2002).
T. V. Matveeva and Yu. P. Pyt’ev, Pattern Recognit. Image Anal. 12, 316 (2002).
Yu. P. Pyt’ev, Pattern Recognit. Image Anal. 12, 376 (2002).
Yu. P. Pyt’ev and I. V. Mazaeva, Moscow Univ. Phys. Bull. 57 (5), 27 (2002).
Yu. P. Pyt’ev and G. S. Zhivotnikov, Pattern Recognit. Image Anal. 14, 60 (2004).
Yu. P. Pyt’ev, Pattern Recognit. Image Anal. 14, 529 (2004).
Yu. P. Pyt’ev, Intellekt. Sist. 8, 147 (2004).
Yu. P. Pyt’ev, Pattern Recognit. Image Anal. 14, 541 (2004).
Yu. P. Pyt’ev, Pattern Recognit. Image Anal. 16 (3), 1 (2006).
D. Dubois, Comput. Stat. Data Anal. 51, 47 (2006).
G. De Campos and J. F. Huete, Int. J. Gen. Syst. 30, 309 (2001).
M. Masson and T. Denoeux, Fuzzy Sets Syst. 157, 319 (2006).
Yu. P. Pyt’ev, Intellekt. Sist. 11, 277 (2007).
D. Dubois and H. Prade, in Decision Making Process: Concepts and Methods, Ed. by D. Bouyssou, D. Dubois, M. Pirlot, and H. Prade (Wiley-ISTE, London, 2009), p. 85.
S. S. Papilin and Yu. P. Pyt’ev, Math. Models Comput. Simul. 3, 528 (2011).
S. S. Papilin and Yu. P. Pyt’ev, Math. Models Comput. Simul. 3, 528 (2011).
R. R. Yager, IEEE Trans. Fuzzy Syst. 20, 46 (2012).
R. R. Yager, IEEE Trans. Fuzzy Syst. 20, 526 (2012).
Yu. P. Pyt’ev, Possibility as an Alternative to Probability, 2nd ed. (Fizmatlit, Moscow, 2016).
Yu. P. Pyt’ev, Math. Models Comput. Simul. 5, 538 (2013).
Yu. P. Pyt’ev, Math. Models Comput. Simul. 5, 538 (2013). doi 10.1134/S2070048213060094
R. L. Kashyap and A. R. Rao, Dynamic Stochastic Models from Empirical Data (Academic Press, 1983).
Yu. A. Rozanov, Probability Theory, Stochastic Processes, and Mathematical Statistics (Fizmatlit, Moscow, 1985).
Yu. P. Pyt’ev, Autom. Remote Control 71, 486 (2010).
G. J. Klir, Uncertainty and Information: Foundations of Generalized Information Theory (Wiley-IEEE Press, 2005).
W. Hoeffding, J. Am. Stat. Assoc. 58, 213 (1963).
Yu. P. Pyt’ev, Mathematical Modeling of Measuring and Computing Systems, 3rd ed. (Fizmatlit, Moscow, 2011).
P. J. Huber, Robust Statistics (Wiley-Interscience, 1981).
A. I. Orlov, Applied Statistics (Ekzamen, Moscow, 2004).
http://studopedia.ru.
Yu. P. Pyt’ev, Math. USSR-Sbornik 46, 17 (1983).
V. F. Dem’yanov and V. N. Malozemov, Introduction to Minimax Theory (Nauka, Moscow, 1972).
Yu. P. Pyt’ev, Moscow Univ. Phys. Bull. 52 (3), 1 (1997). http://vmu.phys.msu.ru/file/1997/3/en-97- 52-3-001.pdf.
Yu. P. Pyt’ev, Vestn. Mosk. Univ., Ser. 3: Fiz. Astron. No. 4, 3 (1997).
Yu. P. Pyt’ev, Vestn. Mosk. Univ., Ser. 3: Fiz. Astron. No. 6, 3 (1997).
Yu. P. Pyt’ev, Moscow Univ. Phys. Bull. 53 (1), 1 (1998). http://vmu.phys.msu.ru/file/1998/1/en-98- 53-1-001.pdf.
Yu. P. Pyt’ev, Moscow Univ. Phys. Bull. 53 (2), 1 (1998).
Yu. P. Pyt’ev, Moscow Univ. Phys. Bull. 54 (5), 1 (1999). http://vmu.phys.msu.ru/file/1999/5/en-99- 54-5-001.pdf.
Yu. P. Pyt’ev, Moscow Univ. Phys. Bull. 54 (6), 1 (1999). http://vmu.phys.msu.ru/file/1999/6/en-99- 54-6-001.pdf.
Yu. P. Pyt’ev, Moscow Univ. Phys. Bull. 53 (6), 1 (1998). http://vmu.phys.msu.ru/file/1998/6/en-98- 53-6-001.pdf.
Yu. P. Pyt’ev, Moscow Univ. Phys. Bull. 54 (1), 1 (1999). http://vmu.phys.msu.ru/file/1999/1/en-99- 54-1-001.pdf.
G. De Cooman and D. Aeyels, IEEE Trans. Syst., Man, Cybern. 30, 124 (2000).
H. T. Nguyen and B. Bouchon-Meunier, Soft Comput. 8, 61 (2003).
Yu. P. Pyt’ev and I. V. Mazaeva, Moscow Univ. Phys. Bull. 57 (5), 27 (2002). http://vmu.phys.msu.ru/file/2002/5/en-02-57-5- 027.pdf.
Yu. P. Pyt’ev, Moscow Univ. Phys. Bull. (in press).
Yu. P. Pyt’ev, Autom. Remote Control 71, 486 (2010).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © Yu.P. Pyt’ev, 2017, published in Vestnik Moskovskogo Universiteta, Seriya 3: Fizika, Astronomiya, 2017, No. 1, pp. 3–16.
An object whose model is a certain probabilistic space at any moment is called stochastic.
About this article
Cite this article
Pyt’ev, Y.P. Mathematical modeling of randomness and fuzziness phenomena in scientific studies: I. Mathematical and empirical foundations. Moscow Univ. Phys. 72, 1–15 (2017). https://doi.org/10.3103/S002713491701012X
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S002713491701012X