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Expanding the Possibilities of the Metric Approach Based on the Theory of Means and the Theory of Errors

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Abstract

It is proposed to expand the capabilities of the metric approach for solving special problems, for example, in scheduling theory, by weakening the requirements for the metric axioms or by introducing probability proximity measures. The author’s results are considered at the junction of the theory of means and the research area dealing with expert error indicators set axiomatically. The result obtained by Academician A.N. Kolmogorov when he considered a system of axioms for deriving an analytical formula for the associative mean is strengthened.

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Notes

  1. In earlier publications in Russian, this concept was referred to as “quasidistance” [5, p. 31]. There are publications where this concept is called semimetric [6].

  2. This theorem was first proved in the author’s dissertation and announced in its extended abstract [23].

  3. The order topology is the least system of subsets of a set \(Y \) closed with respect to union and finite intersection and containing all open half-lines [24].

REFERENCES

  1. Lazarev, A.A., Lemtyuzhnikova, D.V., and Werner, F., A metric approach for scheduling problems with minimizing the maximum penalty, Appl. Math. Model. Oxford, Elsevier, 2021, no. 89. 89, pp. 1163–1176.

  2. Rauschenbach, G.V., Proximity measures in the space of sets, in Algoritmy analiza dannykh sotsial’no-ekonomicheskikh issledovanii IE i OPP SO AN SSSR (Algorithms for Analysis of Data of Socio-Economic Research at the Institute of Economics and Industrial Organization of the Siberian Branch of the Academy of Sciences of the USSR), Novosibirsk, 1982, pp. 29–44.

  3. Uzdin, D.Z., Novye mery blizosti, funktsii sostoyanii i reshayushchie pravila v teorii raspoznavaniya sostoyanii (statisticheskoi klassifikatsii) (New Proximity Measures, State Functions and Decision Rules in the Theory of State Recognition (Statistical Classification)), Moscow: MAKS Press, 2016.

    Google Scholar 

  4. Kolmogorov, A.N. and Fomin, S.V., Elementy teorii funktsii i funktsional’nogo analiza (Elements of Function Theory and Functional Analysis), Moscow: Nauka, 1968.

    Google Scholar 

  5. Collatz, L., Funktionalanalysis und numerische Mathematik, Berlin: Springer-Verlag, 1964. Translated under the title: Funktsional’nyi analiz i vychislitel’naya matematika, Moscow: Mir, 1964.

    Book  Google Scholar 

  6. Iofina G.V., Выбор оптимальных метрик в задачах распознавания с порядковыми признаками, Cand. Sci. (Phys.-Math.) Dissertation, Moscow, 2010.

  7. Matematicheskaya entsiklopediya. T. 4 (Mathematical Encyclopedia. Vol. 4), Moscow: Sov. Entsikl., 1984.

  8. Zagoruiko, N.G., Prikladnye metody analiza dannykh i znanii (Applied Methods of Data and Knowledge Analysis), Novosibirsk: Izd. Inst. Mat. Sib. Otd. Ross. Akad. Nauk, 1999.

    Google Scholar 

  9. Mazurov, V.D., Metod komitetov v zadachakh optimizatsii i klassifikatsii (Committee Method in Optimization and Classification Problems), Moscow: Nauka, 1990.

    MATH  Google Scholar 

  10. Popper, K.R., Conjectures and Refutations. The Growth of Scientific Knowledge. Ch. 10, Sec. 3 , London/New York: Routledge & Kegan Paul/Basic Books, 1963.

    Google Scholar 

  11. Aidinyan, A.R. and Tsvetkova, O.L., The cluster algorithms for solving problems with asymmetric proximity measures, Numer. Anal. Appl., 2018, vol. 11, pp. 99–107.

    Article  MathSciNet  Google Scholar 

  12. Kryukov, K.V., Pankova, L.A., Pronina, V.A., and Shipilina, L.B., Semantic proximity measures in ontologies, Probl. Upr., 2010, no. 5, pp. 2–14.

  13. Bulskov, H., Knappe, R., and Andreasen, T., On measuring similarity for conceptual querying, in Flexible Query Answering Systems. Lecture Notes in Artificial Intelligence. Vol. 2522 , Andreasen, T., Motro, A., Christiansen, H., and Larsen, H.L., Eds., Berlin: Springer, 2002, pp. 100–111.

  14. Jun-Peng Bao, Jun-Yi Shen, Xiao-Dong Liu, and Hai-Yan Liu, Quick asymmetric text similarity measures, Proc. Second Int. Conf. Mach. Learn. Cybern. (Wan, November 2–5, 2003), vol. 1, pp. 374–379.

  15. Hubert, L.J., Min and max hierarchical clustering using asymmetric similarity measures, Psychometrika, 1973, no. 38, pp. 63–72.

  16. Faith, D.P., Asymmetric binary similarity measures, Oecologia, 1983, vol. 57, no. 3, pp. 287–290.

    Article  Google Scholar 

  17. Chen, H.H. and Giles, C.L., ASCOS++ an asymmetric similarity measure for weighted networks to address the problem of SimRank, ACM Trans. Knowl. Discovery Data (TKDD) (2015), vol. 10. no. 2, pp. 1–26.

  18. Sidel’nikov, Yu.V., Sistemnyi analiz tekhnologii ekspertnogo prognozirovaniya (System Analysis of Expert Forecasting Technology), Moscow: Izd. Mosk. Aviats. Inst.-PRINT, 2007.

    Google Scholar 

  19. Gini, C., Le medie. Translated under the title: Srednie velichiny, Moscow: Statistika, 1970.

  20. Kolmogorov, A.N., Sur la notion de moyenne, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 1930, no. (6)12, pp. 388–391.

  21. Kolmogorov, A.N., On determining the mean, in Matematika i mekhanika. Izbrann. tr. A.N. Kolmogorova. T. 1. (Mathematics and Mechanics. Sel. Works by A.N. Kolmogorov. Vol. 1), Nikol’skii, S.M. and Tikhomirov, V.M., Eds., Moscow: Nauka, 1985, pp. 136–138.

    Google Scholar 

  22. de Finetti, B., Sul concetto di media, G. Ist. Ital. Attuari, 1931, vol. 11, no. 3, pp. 365–396.

    MATH  Google Scholar 

  23. Sidel’nikov, Yu.V., Extended Abstract of Doctoral (Eng.) Dissertation “Expert Forecasting Technology,” Moscow: Inst. World Econ. Int. Relat. Russ. Acad. Sci., 2002.

  24. Pfanzagl, J., Theory of Measurement, Würzburg–Vienna: Physica-Verlag, 1971. Translated under the title: Teoriya izmerenii, Moscow: Mir, 1976.

    Book  Google Scholar 

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Authors and Affiliations

  1. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, 117997, Russia

    Yu. V. Sidel’nikov

  2. Moscow Aviation Institute (National Research University), Moscow, 125993, Russia

    Yu. V. Sidel’nikov

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  1. Yu. V. Sidel’nikov
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Translated by V. Potapchouck

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Sidel’nikov, Y.V. Expanding the Possibilities of the Metric Approach Based on the Theory of Means and the Theory of Errors. Autom Remote Control 82, 1912–1922 (2021). https://doi.org/10.1134/S0005117921110072

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