User:Helgus/ Mathematical eventology

Mathematical eventology, is a mathematical language of eventology – a new direction the probability theory; is based on the principle of eventological duality of notion of a set of random events and a random set of events; studying eventological distributions of a set of random events and eventological structures of its dependencies.

Unlike probability theory, theory of random events focuses mainly on direct and regular studying of random events and their dependencies.

Allocation of the theory of random events into an independent direction of probability theory;
development of mathematical event language — the eventological device (eventological distribution, set of random events, random set of events, event-terrace, etc.);
fuzzy mathematical eventology that generalizes fuzzy set theory^[1]^[2]^[3]^[4]^[5], possibility theory^[6] and Dempster-Shafer theory of evidence^[7];
efficiency of theory of random events in many applied areas which is direct consequence of universality of mathematical event language

— can be considered as the basic results of mathematical eventology.

The basic achievements of mathematical eventology in actual fileds of application are:

eventological portfolio analysis (statement and the decision of inverse eventological Markowitz’s problems (Harry Markovitz^[8], the Nobel Prize on economy, 1990);
eventological models of supply and demand (eventological substantiation and expanded interpretation of classical market «Marchall’s cross»^[9] - «supply and demand cross»);
eventological interpretation of Herrnstein’s experiment with pigeons (1961) in psychology^[10] («The mind appears there and then, where and when there is an ability to make a probabilistic choice» - Vladimir Lefebvre, University of California at Irvin^[11], 2003);
eventological models of «Vickrey auctions»^[12] (William Vickrey, the Nobel Prize on economy, 1996);
eventological basis and expansion of prospect theory (Daniel Kahneman^[13] and Amos Tversky^[14]) (Daniel Kahneman, the Nobel Prize on economy, 2002);
eventological generalization of methods of experimental economics^[15] (Vernon Smith, the Nobel Prize on economy, 2002); and also in
statistical geometry (Dietrich Stoyan): the new notion of set-means for random sets (1975).^[16]

Following fields have been developed recently:

eventological theory of dependencies^[17] of random events including theory of eventological copula^[18];
eventological system analysis,
eventological decision theory and
eventological theory of set-preferences (eventological explanation for a long time known Blyth’s paradox^[19] in preference theory).

The basic eventological terms

Event, probability and value
Conditional event, conditional probability and conditional value
Set of events and random set of events
Event-terrace
Eventological duality
Eventological distribution
Eventological language
Eventological glossary
Additive set-functions and measures
Set-formulae of Mobius inversing events-terraces
Formulae of Mobius inversing eventological distributions
Set-means characteristics of a random set of events
Eventological Bayes's theorem
Frechet's covariances and correlations
The structure of dependencies of a set of events
Eventological copula

Applications of eventological theory

Eventological theory of fuzzy events
Eventological foundation of Kahneman and Tversky theory
Eventological portfolio analysis
Eventological system analysis
Eventology of making decision
Eventological theory of set-preferences
Eventological foundation of economics
Eventological scoring
Eventological direct and inverse Markowitz's problems
Eventological market "Marshall's Cross"
Eventological explaination of K.Blayh's paradox in theory of preferences

At bounds of eventology

Subjective events, subjective probability and subjective value
Gibbsean eventological model "probability of event — value of event"
The phantom eventological distributions

Eventological sections on International conferences

PhD theses on eventology (phys.-math. sciences, in Russian)

Vorob'ov Alexei (1998) Direct and inverse problems for models of spreading space risks. Krasnoyarsk: ICM of RAS
Goldenok Ellen (2002) Modeling dependence and interaction structures of random events in statistical systems. Krasnoyarsk: KGTEI
Kupriyanova Tatyana (2002) A problem of classification of subsets of random set and its application. Krasnoyarsk: Krasnoyarsk University
Semenova Daria (2002) Methods of constructing statistical dependencies of portfolio operations in market systems. Krasnoyarsk: ICM of RAS
Fomin Andrew (2002) Set-regressional analysis of dependencies of random events in statistical systems. Krasnoyarsk: ICM of RAS
Klotchkov Svyatoslav (2006) Eventological models of distributing and filling resources. Krasnoyarsk: KGTEI
Baranova Iren (2006) Methods of bipartitional sets of events in eventological analysis of complicated systems. Krasnoyarsk: Krasnoyarsk University
Tyaglova Hellena (2006) Game theory methods of analysis of random sets of events. Krasnoyarsk: ICM of RAS

Bibliography (in English)

Vorob'ov O.Yu. (1991) Set-summation. Soviet Math. Dokl. 1991, Vol.43,p.747-752
Vorob'ov O.Yu. (1993) The calculus of set-distribution. Rissian Acad. Sci., Dokl. Math., 1993, Vol.46, 301-306.
Vorob'ov O.Yu., A.O.Vorob'ov (1994) Summation of set-additive functions anf the Mobius inversion formula. Russian Acad. Sci. Dokl. Math., vol. 49, No. 2, 340-344.
Stoyan Dietrich, and Helga Stoyan (1994) Fractals, Random Shapes and Point Fields. Methods of Geometrical Statistics. John Wiley and Sons. Chichester, New York
(pp.107-116: Vorob'ov's set-means of a random set)
Vorob'ov O.Yu., A.O.Vorob'ov (1996) Inverse problems for generalized Richardson's model of spread. Computational Fluid Dynamics'96, John Wiley & Sons, 104-110.
Vorob'ov O.Yu. (1996) A random set analysis of fire spread. Fire Technology, NFPA (USA), v.32, N 2, 137--173.
Vorob'ov Oleg Yu., Arcady A. Novosyolov, Konstantin V. Simonov, and Andrew Yu. Fomin (2001) Portfolio Analysis of Financial Market Risks by Random Set Tools. Risks in Investment Accumulation Products of Financial Institutions. Simposium Proceedings held in January 1999, New York. Schaumburg, USA: The Society of Actuaries, 43--66.

Bibliography (in Russian)

Vorob'ov Oleg (1978) Probabilistic set modeling. — Novosibirsk: Nauka. — 131 p.
Vorob'ov Oleg (1984) Mean measure modeling. — Moscow: Nauka. — 133 p.
Vorob'ov Oleg (1993) Set-summation. — Novosibirsk: Nauka. — 137 p.
Kovyazin S.A. (1999) Mean measure set. — Probability and Mathematical Statistics. Encyclopaedia. Ed. Yu.V.Prokhorov. Moscow: BRE.
(Mean measure set: Vorob’ov’s set-means of a random set. – p.644.)
Vorob'ov Oleg (2006) Introduction to eventology. — Krasnoyarsk: ICM of RAS, Krasnoyarsk University. — 466 p.
Vorob'ov Oleg (2006) Eventology. — Krasnoyarsk: ICM of RAS, Krasnoyarsk University. — 347 p.
Vorob'ov Oleg (2007, to appear) Eventology and its applications. — Krasnoyarsk: ICM of RAS, Krasnoyarsk University. — 512 p.

References

^ Blyth C.R. (1972) On Simpson's Paradox and the Sure --- Thing Principle. - Journal of the American Statistical Association, June, 67, P.367-381.
^ Dubois D., H.Prade (1988) Possibility theory. - New York: Plenum Press.
Feynman R.P. (1982) Simulating physics with computers. - International Journal of Theoretical Physics, Vol. 21, nos. 6/7, 467-488.
^ Fr'echet M. (1935) G'en'eralisations du th'eor'eme des probabilit'es totales - Fundamenta Mathematica. - 25.
Hajek, Alan (2003) Interpretations of Probability. - The Stanford Encyclopedia of Philosophy (Summer 2003 Edition), Edward N.Zalta (ed.)
^ Herrnstein R.J. (1961) Relative and Absolute strength of Response as a Function of Frequency of Reinforcement. - Journal of the Experimental Analysis of Behavior, 4, 267-272.
^ Kahneman D., Tversky A. (1979) Prospect theory: An analysis of decisios under risk. - Econometrica, 47, 313-327.
^ Lefebvre V.A. (2001) Algebra of conscience. - Kluwer Academic Publishers. Dordrecht, Boston, London.
^ Markowitz Harry (1952) Portfolio Selection. - The Journal of Finance. Vol. VII, No. 1, March, 77-91.
^ Marshall Alfred A collection of Marshall's published works
^ Nelsen R.B. (1999) An Introduction to Copulas. - Lecture Notes in Statistics, Springer-Verlag, New York, v.139.
^ Russell Bertrand (1945) A History of Western Philosophy and Its Connection with Political and Social Circumstances from the Earliest Times to the Present Day, New York: Simon and Schuster.
^ Russell Bertrand (1948) Human Knowledge: Its Scope and Limits, London: George Allen & Unwin.
Schrodinger Erwin (1959) Mind and Matter. - Cambridge, at the University Press.
^ Shafer G. (1976). A Mathematical Theory of Evidence. – Princeton University Press.
^ Smith Vernon (2002) Nobel Lecture.
^ Stoyan D., and H. Stoyan (1994) Fractals, Random Shapes and Point Fields. - Chichester: John Wiley & Sons.
^ Tversky A., Kahneman D. (1992) Advances in prospect theory: cumulative representation of uncertainty. - Journal of Risk and Uncertainty, 5, 297-323.
^ Vickrey William Paper on the history of Vickrey auctions in stamp collecting
^ Zadeh L.A. (1965) Fuzzy Sets. - Information and Control. - Vol.8. - P.338-353.
^ Zadeh L.A. (1968) Probability Measures of Fuzzy Events. - Journal of Mathematical Analysis and Applications. - Vol.10. - P.421-427.
^ Zadeh L.A. (1978). Fuzzy Sets as a Basis for a Theory of Possibility. – Fuzzy Sets and Systems. - Vol.1. - P.3-28.
^ Zadeh L.A. (2005). Toward a Generalized Theory of Uncertainty (GTU) - An Outline. - Information sciences (to appear).
^ Zadeh L.A. (2005). Toward a computational theory of precisiation of meaning based on fuzzy logic - the concept of cointensive precisiation. - Proceedings of IFSA-2005 World Congress.} - Beijing: Tsinghua University Press, Springer.

External links

Eventology and Mathematical eventology at the site of the author of theory.