H.А. Kravtsov, Cand. Sc. (Eng.), V.I. Koshel, Post-graduate, A. V. Dolgorukov, Post-graduate
Pukhov Institute for Problems in Electrical Engineering, 15 General Naumov St, Kyiv, 03164, Ukraine, e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.   This email address is being protected from spambots. You need JavaScript enabled to view it. This email address is being protected from spambots. You need JavaScript enabled to view it.
V.V. Tsurkan, Sc. (Eng.),
National Technical Institute of Ukraine “Ihor Sikorsky Kyiv Polytechnic Institute” (22, Bldg, 37 Pobeda St, Kyiv, 03056, Ukraine, e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

Èlektron. model. 2018, 40(3):63-76


The authors investigate the classical concept of measure in accordance with symmetry conditions, reflexivity and triangle inequality. The requirements to the measure have been formulated for its further use in the theory of the calculus over classification. Some features of the distance function, correlation coefficient, cosine measure of similarity are signification restrictions for applying them for the theory. All measures used in practice have been studied. The results of research have shown that new measure should be introduced that has been proposed. The authors have given formal definition of the trainable model of calculus over classification.


similarity measure, difference measure, distance function, calculation model, model trainability, continuum of equivalent measures


1. Kravtsov, H.A. (2016), “Measure of difference between classifications”, Elektronnoe modelirovanie, Vol. 38, no. 1, pp. 73-87.
2. Diuran, B. and Odell, P. (1977), Klasternyi analiz [Cluster analysis], Translated by E.Z. Demidenko, Statistika, Moscow, USSR.
3. Semkin, B.I. and Gorshkov, M.V. (2008), “The axiomatic introduction of similarity measures, differences measures, compatibility and dependencies for components of the biological variety”, Vestnik Tikhookeanskogo gosudarstvennogo ekonomicheskogo universiteta, no. 4, pp. 31-46.
4. Kim, J.-O., Miuller, Ch.U., Klekka, U.R., et al. (1989), Faktornyi diskriminantnyi i klasternyi analiz [Factorial, discriminant and cluster analysis], Translated from English, Finansy i statistika, Moscow, USSR.
5. Jaccard, P. (1901), Distribution de la flore alpine dans le Bassin des Dranses et dans quelques regions voisines, Bulletin de la Societe Vaudoise des Sciences Naturelles, Vol. 37 (140), pp. 241-272, DOI : 10.5169/seals-266440.
6. Levandowsky, M. and Winter, D. (1971), Distance between Sets, Nature, Vol. 234, pp. 34-35, DOI : 10.1038/234034a0.
7. Sörensen, T. (1948), A method of establishing groups of equal amplitude in plant sociology based on similarity of species content, Biologiske Skrifter, Vol. 5, no. 4, pp. 1-34.
8. Kulczynski, S. (1927), Zespoly róslin w Pieninach (Die Pflanzenassociationen der Pienenen), Bulletin International de L’Acad´emie Polonaise des Sciences et des Letters, Classe des Sciences Mathematiques et Naturelles, Serie B, Suppl´ement II, 2, pp. 57-203.
9. Sokal, R.R. and Sneath, P.H.A. (1963), Principles of numerical taxonomy, W.H. Freeman&Co., New York, USA.
10. Szymkiewicz, D. (1934), Une contribution statistique a la gographie floristique, Acta Soc. Bot. Polon, Vol. 34, no. 3, pp. 249-265.
11. Simpson, G.G. (1947), Holarctic mammalian faunas and continental relationship during the Cenozoic, Bull. Geol. Sci. America, Vol. 58, no. 2, pp. 613-688.[613:HMFACR]2.0.CO;2
12. Braun-Blanquet, J. (1951), Pflanzensoziologie Grundzüge der Vegetationskunde, Springer-Verlag Wien, Berlin, Germany, DOI : 10.1007/978-3-7091-4078-9.
13. Ochiai, A. (1957), Zoogeographical studies on the soleoid fishes found Japan and its neighboring regions-II, Bull. Jap. Soc. sci. Fish, Vol. 22, no. 9, pp. 526-530, DOI : 10.2331/suisan.22.526.
14. Semkin, B.I. (1979), “The equality of similarity measures and hierarchical classification of multidimensional data”, Hierarchical structures built over classifications in the geographical ecology and systematics, pp. 97-112.

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