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On the theoretical basis of metric analysis of poorly formalized problems of recognition and classification

  • Mathematical Method in Pattern Recognition
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Abstract

In many fields of modern science, there are problems adequate formalization of which is indispensable for obtaining practically and theoretically important results. In the terminology of the scientific school of academician Yu.I. Zhuravlev, a formalized problem is uniquely defined by the matrix of information and the information matrix. In the present paper, a whole class of issues related to the formalization of recognition/classification problems is considered, and a universal formalism is proposed for carrying out a metric analysis of poorly formalized problems. Thus, the formalization of a problem can be represented as a successive transition from the set of original descriptions to a particular topology, then to a lattice, and then to a certain metric space. It is shown that the property of Zhuravlev’s regularity is sufficient for the existence of bijective mappings between these mathematical constructs. The possibilities of application of the apparatus developed are illustrated by several issues important for the formalization of the problems: introduction of metrics on the sets of the features and metrics on the sets of objects and analysis of “interactions” between dissimilar feature descriptions.

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References

  1. Yu. I. Zhuravlev, “Set-theoretical methods in logic algebra,” Probl. Kibernet. 8 (1), 25–45 (1962).

    MathSciNet  Google Scholar 

  2. Yu. I. Zhuravlev, “Correct algebras at the set of incorrect (heuristic) algorithms. 1,” Kibernetika, No. 4, 5–17 (1977).

    Google Scholar 

  3. Yu. I. Zhuravlev, “Correct algebras at the set of incorrect (heuristic) algorithms. 2,” Kibernetika, No. 6, 21–27 (1977).

    Google Scholar 

  4. Yu. I. Zhuravlev, “Correct algebras at the set of incorrect (heuristic) algorithms. 3,” Kibernetika, No. 2, 35–43 (1978).

    Google Scholar 

  5. Yu. I. Zhuravlev, “On algebraic approach for solving recognition and classification problems,” in Cybernetic Problems (Nauka, Moscow, 1978), Issue 33, pp. 5–68 [in Russian].

    MATH  Google Scholar 

  6. Yu. I. Zhuravlev, K. V. Rudakov, and I. Yu. Torshin, “Algebraic criteria of local solvability and regularity as a research tool for researching the amino acid sequences morphology,” Trudy Mosk. Fiz.-Tekhn. Inst. 3 (4), 67–76 (2011).

    Google Scholar 

  7. K. V. Rudakov and I. Yu. Torshin, “Solvability of protein secondary structure recognition problem,” Inf. Ee Prim. 4 (2), 25–35 (2010).

    Google Scholar 

  8. K. V. Rudakov and I. Yu. Torshin, “The way to analyze motifs informativeness on the base of a solvability criteria in a problem on protein secondary structure recognition,” Inf. Ee Prim. 5 (4), 40–50 (2011).

    Google Scholar 

  9. K. V. Rudakov and I. Yu. Torshin, “The way to choose indexes informative meanings on the base of a solvability criteria in a problem on protein secondary structure recognition,” Dokl. Akad. Nauk 441 (1), 1–5 (2011).

    Google Scholar 

  10. I. Yu. Torshin, “On solvability, regularity, and locality of the problem of genome annotation,” Pattern Recogn. Image Anal. 20 (3), 386–395 (2010).

    Article  Google Scholar 

  11. I. Yu. Torshin, “The study of the solvability of the genome annotation problem on sets of elementary motifs,” Pattern Recogn. Image Anal. 21 (4), 652–662 (2011).

    Article  Google Scholar 

  12. I. Yu. Torshin, “Optimal dictionaries of the final information on the basis of the solvability criterion and their applications in bioinformatics,” Pattern Recogn. Image Anal. 23 (2), 319–327 (2013).

    Article  Google Scholar 

  13. Yu. I. Zhuravlev and K. V. Rudakov, “Algebraic correction for information processing procedures,” in Problems of Applied Mathematics and Informatics (Nauka, Moscow, 1987), pp. 187–198 [in Russian].

    Google Scholar 

  14. K. V. Rudakov, “Universe and local limitations in the problem on correcting the heuristic algorithms,” Kibernetika, No. 2, 30–35 (1987).

    MathSciNet  Google Scholar 

  15. K. V. Rudakov, “Universal limitations for researching the classification algorithms,” Kibernetika, No. 1, 1–5 (1988).

    MathSciNet  Google Scholar 

  16. I. Yu. Torshin and K. V. Rudakov, “On the application of the combinatorial theory of solvability to the analysis of chemographs. Part 1: Fundamentals of modern chemical bonding theory and the concept of the chemograph,” Pattern Recogn. Image Anal. 24 (1), 11–23 (2014).

    Article  Google Scholar 

  17. I. Yu. Torshin and K. V. Rudakov, “On the application of the combinatorial theory of solvability to the analysis of chemographs. Part 2. Local completeness of the chemographs’ invariants in view of the combinatorial theory of solvability,” Pattern Recogn. Image Anal. 24 (2), 196–208 (2014).

    Article  Google Scholar 

  18. O. A. Gromova, I. Yu. Torshin, A. G. Kalacheva, L. E. Fedotova, A. N. Gromov, and K. V. Rudakov, “Chemoinformational analysis for orotic acid molecule demonstrates anti-inflammatory, neuroprotective, and cardioprotective properties of magnesium ligand,” Farmateka, No. 13 (2013).

  19. I. Yu. Torshin and O. A. Gromova, Data Expert Analysis in Molecular Pharmacology (Moscow Center for Continuous Mathematical Education, Moscow, 2012) [in Russian].

    Google Scholar 

  20. O. A. Gromova, A. G. Kalacheva, I. Yu. Torshin, K. V. Rudakov, U. E. Grustlivaya, N. V. Yudina, E. Yu. Egorova, O. A. Limanova, L. E. Fedotova, O. N. Gracheva, N. V. Nikifororva, T. E. Satarina, I. V. Gogoleva, T. R. Grishina, D. B. Kuramshina, L. B. Novikova, E. Yu. Lisitsyna, N. V. Kerimkulova, I. S. Vladimirova, M. N. Chekmareva, et al., “Magnesium deficiency is an authentic risk factor for comorbide states. Results of large-scale screening of magnesium status in Russian regions,” Farmateka, No. 6 (259), 116–129 (2013).

    Google Scholar 

  21. N. V. Kerimkulova, N. V. Nikifororva, I. S. Vladimirova, I. Yu. Torshin, and O. A. Gromova, “Effect of connective tissue undifferentiated dysplasia onto pregnancy and delivery termination and outcome. Complex examination of pregnant women with connective tissue dysplasia by using intellectual data analysis,” Zemskii Vrach, No. 2 (19), 34–38 (2013).

    Google Scholar 

  22. T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd ed. (Springer-Verlag, 2009).

    Book  Google Scholar 

  23. Yu. I. Zhuravlev, V. V. Ryazanov, and O. V. Sen’ko, Recognition. Mathematical Methods. Program System. Practical Applications (Fazis, Moscow, 2006) [in Russian].

    Google Scholar 

  24. K. Bailey, “Numerical taxonomy and cluster analysis,” in Typologies and Taxonomies (New York, 1994), p. 34.

    Google Scholar 

  25. V. Estivill-Castro, “Why so many clustering algorithms,” ACM SIGKDD Explor. Newslett 4 (1), 65–75 (2002).

    Article  MathSciNet  Google Scholar 

  26. D. Defays, “An efficient algorithm for a complete link method,” J. Brit. Comp. Soc. 20 (4), 364–366 (1977).

    MathSciNet  MATH  Google Scholar 

  27. S. Lloyd, “Least squares quantization in PCM,” IEEE Trans. Inf. Theory 28 (2), 129–137 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  28. M. Ester, H. P. Kriegel, J. Sander, and X. Xu, “A density-based algorithm for discovering clusters in large spatial databases with noise,” in Proc. 2nd Int. Conf. on Knowledge Discovery and Data Mining (KDD-96) (AAAI Press, 1996), pp. 226–231.

    Google Scholar 

  29. A. N. Kolmogorov and S. V. Fomin, Elements of Function Theory and Functional Analysis, 4th ed. (Nauka, Moscow, 1976) [in Russian].

    Google Scholar 

  30. P. S. Aleksandrov, Introduction into Set Theory and General Topology (Nauka, Moscow, 1977) [in Russian].

    Google Scholar 

  31. J. G. Hocking and G. S. Young, Topology (Dover, New York, 1961), pp. 5–6.

    MATH  Google Scholar 

  32. V. I. Ponomarev, “Open-closed set,” in Encyclopaedia of Mathematics (M. Hazewinkel (Hrsg.), Springer-Verlag, Berlin, 2002).

    Google Scholar 

  33. M. H. Stone, “The theory of representations of Boolean algebras,” Trans. Am. Math. Soc. 40, 37–111 (1936).

    Google Scholar 

  34. G. Birkhoff, Lattice Theory (Am. Math. Soc. Colloq. Publ., New York, 1967).

    MATH  Google Scholar 

  35. O. Frink, “Topology in lattices,” Trans. Amer. Math. Soc. 51, 568–582 (1942).

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Moscow Institute of Physics and Technology (State University), Institutskii per. 9, Dolgoprudnyi, Moscow oblast, 141700, Russia

    I. Yu. Torshin & K. V. Rudakov

  2. Dorodnitsyn Computing Centre, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119333, Russia

    K. V. Rudakov

Authors
  1. I. Yu. Torshin
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  2. K. V. Rudakov
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Correspondence to I. Yu. Torshin.

Additional information

Ivan Yur’evich Torshin. Born 1972. Graduated from the Department of Chemistry, Moscow State University, in 1995. Received candidates degrees in chemistry in 1997 and in physics and mathematics in 2011. Currently is an associate professor at Moscow Institute of Physics and Technology, lecturer at the Faculty of Computational Mathematics and Cybernetics, Moscow State University, leading scientist at the Russian Branch of the Trace Elements Institute for UNESCO, and a member of the Center of Forecasting and Recognition. Author of 205 publications in per-reviewed journals in biology, chemistry, medicine, and informatics and of 3 monographs in the series “Bioinformatics in Post-genomic Era” (Nova Biomedical Publishers, NY, 2006-2009).

Konstantin Vladimirovich Rudakov. Born 1954. Russian mathematician, corresponding member of the Russian Academy of Sciences, head of the Department of Computational Methods of Forecasting at the Dortodnitsyn Computing Centre, Russian Academy of Sciences, and head of the Chair “Intelligent Systems” at the Moscow Institute of Physics and Technology.

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Torshin, I.Y., Rudakov, K.V. On the theoretical basis of metric analysis of poorly formalized problems of recognition and classification. Pattern Recognit. Image Anal. 25, 577–587 (2015). https://doi.org/10.1134/S1054661815040252

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  • DOI: https://doi.org/10.1134/S1054661815040252

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