ANALYSIS AND DATA PROCESSING SYSTEMS

The basic idea of constructing locally-adaptive regression models (LAR models) consists in the use of regressors defined on the local subregions of factor values. The belonging of factor values to a particular local subdomain is set by indicator functions. Indicator functions by their nature are close to the well-known concepts of membership functions from the theory of fuzzy systems (Fuzzy Systems). As a rule, to provide the required smoothness of the required dependence of the response on the acting factors such local subdomains are defined with overlapping - in the form of the so-called fuzzy partitions. Type or type of indicator functions may be very different: triangular, trapezoidal, and non-linear. Specifying one or another type of indicator function determines the scheme of weighing local models. Each indicator function must be defined for the entire range of the corresponding factor. Triangular-type functions are used as indicator functions in this work. Linear factor models are considered as local models. It is noted that in their original form the proposed LAR models are not identifiable. The issue of identification of such models in the case of joint estimation of all parameters is considered. The procedure of model reduction is introduced. The resulting model is written out in the space of functions that allow estimation. In the case of dividing the domain of factor determination into two, three or four fuzzy partitions we propose the basis of functions allowing evaluation. The results of computational experiment on regression dependence reconstruction by ordinary polynomials of different degrees and by LAR models are given. The efficiency of LAR models in comparison with polynomials of degree 3 and 4 is noted.

1. Vapnik V. Statistical Learning Theory. New York, John Wiley, 1998. 736 p.

2. Takagi T., Sugeno M. Fuzzy identification of systems and its applications to modeling and control. IEEE Transactions on Systems, Man and Cybernetics, 1985, vol. 15, no. 1, pp. 116–132.

3. Dickerson J.A., Kosko B. Fuzzy function approximation with ellipsoidal rules. IEEE Transaction on Fuzzy Systems, 1996, vol. 26, no. 4, pp. 542–560.

4. Kosko B. Fuzzy systems as universal approximators. Proceedings First IEEE International Conference on Fuzzy Systems, San Diego, CA, USA, 1992, pp. 1153–1162. DOI: 10.1109/FUZZY.1992.258720.

5. Butkiewicz B., Rutkowski L., Kacprzyk J. Simple modification of Takagi-Sugeno model. Neural Networks and Soft Computing, 2003, no. 11, pp. 504–509.

6. Babus?ka R. Fuzzy modeling for control. London, Boston, Kluwer Academic Publ., 1998. 257 p.

7. Lilly J.H. Fuzzy control and identification. Hoboken, NJ, Wiley, 2010. 231 p.

8. Piegat A. Fuzzy modeling and control. Heidelberg, Physica-Verlag, 2001 (Russ. ed.: Pegat A. Nechetkoe modelirovanie i upravlenie. 2nd ed. Moscow, Binom Publ, 2013. 798 p.).

9. Popov A.A. Regressionnoe modelirovanie na osnove nechetkikh pravil [Regression modeling based on fuzzy rules]. Sbornik nauchnykh trudov Novosibirskogo gosudarstvennogo tekhnicheskogo universiteta = Transaction of scientific papers of the Novosibirsk state technical university, 2000, no. 2 (19), pp. 49–57.

10. Hodashinsky I.A., Sarin K.S. Metodika postroeniya kompaktnykh i tochnykh nechetkikh sistem tipa Takagi–Sugeno [Technique for designing accurate and compact Takagi–Sugeno fuzzy systems]. Doklady TUSUR = Proceedings of TUSUR University, 2014, vol. 19, no. 1, pp. 50–56.

11. Kotyukov V.I. Mnogofaktornye kusochno-lineinye modeli [Multifactor piecewise linear models]. Moscow, Finansy i statistika Publ., 1984. 216 p.

12. Yen K.K., Ghoshray S., Roig G. A linear regression model using triangular fuzzy number coefficients. Fuzzy Sets and Systems, 1999, vol. 106, iss. 2, pp. 167–177.

13. Popov A.A. [Identification of locally adaptive regression models]. Obrabotka informatsii i matematicheskoe modelirovanie: materialy konferentsii [Information processing and mathematical modeling]. Conference proceedings, Novosibirsk, 23–24 April, 2020, pp. 155–160. (In Russian). Available at: https://sibsutis.ru/workgroups/w/group/46/files/Материалы%20конференций/РНТК-2020 (accessed 29.05.2023).

14. Popov A.A. Konstruirovanie diskretnykh i nepreryvno-diskretnykh modelei regressionnogo tipa [Construction of discrete and continuous-discrete models such as regression]. Sbornik nauchnykh trudov Novosibirskogo gosudarstvennogo tekhnicheskogo universiteta = Transaction of scientific papers of the Novosibirsk state technical university, 1996, no. 1 (3), pp. 21–30.

15. Popov A.A. Postroenie derev'ev reshenii dlya prognozirovaniya kolichestvennogo priznaka na klasse logicheskikh funktsii ot lingvisticheskikh peremennykh [Construction of decision trees to predict the quantitative trait in the class of logic functions of linguistic variables]. Nauchnyi vestnik Novosibirskogo gosudarstvennogo tekhnicheskogo universiteta = Science bulletin of the Novosibirsk state technical university, 2009, no. 3 (36), pp. 77–86.

16. Popov A.A., Karmanov V.S. Postroenie stoikostnoi modeli sverleniya s ispol'zovaniem lokal'no adaptivnykh regressionnykh modelei [Building a wear resistance model of drilling operation using locally adaptive regression models]. Dinamika sistem, mekhanizmov i mashin = Dynamics of Systems, Mechanisms and Machines, 2020, vol. 8, no. 1, pp. 141–146. DOI: 10.25206/2310-9793-8-1-141-146.