ON THE APPROXIMATION OF THE GENERALIZED CUT FUNCTION OF DEGREE p+1 BY SMOOTH HYPER–LOG–LOGISTIC FUNCTION

TitleON THE APPROXIMATION OF THE GENERALIZED CUT FUNCTION OF DEGREE p+1 BY SMOOTH HYPER–LOG–LOGISTIC FUNCTION
Publication TypeJournal Article
Year of Publication208
AuthorsMARKOV SVETOSLAV, KYURKCHIEV NIKOLAY, ILIEV ANTON, RAHNEV ASEN
JournalDynamic Systems and Applications
Volume27
Issue4
Start Page715
Pagination14
Date Published09/2018
ISSN1056-2176
AMS Subject Classification41A46
Abstract

We introduce a modification of the familiar cut function by replacing the linear part in its definition by a polynomial of degree p + 1 obtaining thus a sigmoid function called generalized cut function of degree p+1 (GCFP).We then study the uniform approximation of the (GCFP) by smooth sigmoid functions such as the hyper–log–logistic and the shifted hyper–log–logistic functions. The limiting case of the interval-valued Heaviside step function is also discussed which imposes the use of Hausdorff metric. Numerical examples are presented using CAS MATHEMATICA.

PDFhttps://acadsol.eu/dsa/articles/27/4/2.pdf
DOI10.12732/dsa.v27i4.2
Refereed DesignationRefereed
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REFERENCES
[1] N. Kyurkchiev, S. Markov, On the approximation of the generalized cut function of degree p + 1 by smooth sigmoid functions, Serdica J. Computing, 9 No. 1 (2015), 101–112.
[2] R. Anguelov, S. Markov, Hausdorff Continuous Interval Functions and Approximations, SCAN 2014 Revised Selected Papers of the 16th International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics, 9553, 3–13 .
[3] R. Anguelov, S. Markov, B. Sendov, The Set of Hausdorff Continuous Functions - the Largest Linear Space of Interval Functions, Reliable Computing, 12 (2006), 337–363.
[4] J. Dombi, Z. Gera, The Approximation of Piecewise Linear Membership Functions and Lukasiewicz Operators, Fuzzy Sets and Systems, 154 No. 2 (2005), 275–286.
[5] A. Grauel, L. Ludwig, Construction of differentiable membership functions, Fuzzy Sets and Systems, 101 (1999), 219–225.
[6] A. Iliev, N. Kyurkchiev, S. Markov, On the approximation of the cut and step functions by logistic and Gompertz functions, Biomath, 4 No. 2 (2015), 2–13.
[7] W. Kramer, J. Wolff v. Gudenberg, Scientific Computing, Validated Numerics, Interval Methods, Proceedings of the conference Scan-2000/Interval-2000, Kluwer/Plenum (2001).
[8] N. Kyurkchiev, S. Markov, On the Hausdorff distance between the Heaviside step function and Verhulst logistic function, J. Math. Chem., 54 No. 1 (2016), 109–119.
[9] N. Kyurkchiev, S. Markov, Sigmoid functions: Some Approximation and Modelling Aspects, LAP LAMBERT Academic Publishing, Saarbrucken (2015).
[10] B. Sendov, Hausdorff Approximations, Kluwer, Boston (1990).
[11] J. H. Van derWalt, The Linear Space of Hausdorff Continuous Interval Functions, Biomath, 2 (2013).
[12] P.-F. Verhulst, Notice sur la loi que la population poursuit dans son accroissement, Correspondance mathematique et physique, 10 (1838), 113–121.
[13] N. Kyurkchiev, S. Markov, Sigmoidal Functions: some Computational and Modelling Aspects, Biomath Communications, 1 No. 2 (2014), 19 pp.
[14] A. Iliev, N. Kyurkchiev, S.Markov, On the Approximation of the step function by some sigmoid functions, Mathematics and Computers in Simulation, 133 (2017), 223–234.
[15] N. Kyurkchiev, A. Iliev, On some growth curve modeling: approximation theory and applications, Int. J. of Trends in Research and Development, 3 No. 3 (2016), 466–471.
[16] S. Markov, N. Kyurkchiev, A. Iliev, A. Rahnev, A Note on the Loglogistic and Transmuted Loglogistic Models. Some Applications, Dynamic Systems and Applications, 27 No. 3 (2018), 593–607.
[17] N. Kyurkchiev, A. Iliev, A note on some growth curves arising from Box-Cox transformation, Int. J. of Engineering Works, 3 No. 6 (2016), 47–51.
[18] N. Kyurkchiev, S. Markov, A. Iliev, A note on the Schnute growth model, Int. J. of Engineering Research and Development, 12 No. 6 (2016), 47–54. 
[19] A. Iliev, N. Kyurkchiev, S. Markov, On the Hausdorff distance between the shifted Heaviside step function and the transmuted Stannard growth function, Biomath, 5 No. 2, 2016.
[20] N. Kyurkchiev, On the Approximation of the step function by some cumulative distribution functions, Compt. rend. Acad. bulg. Sci., 68 No. 12 (2015), 1475–1482.
[21] V. Kyurkchiev, N. Kyurkchiev, On the Approximation of the Step function by Raised Cosine and Laplace Cumulative Distribution Functions, European International Journal of Science and Technology, 4 No. 9 (2016), 75–84.
[22] N. Kyurkchiev, S. Markov Approximation of the Cut Function by Some Generic Logistic Functions and Applications, Advances in Applied Sciences, 1 No. 2 (2016), 24–29.
[23] N. Kyurkchiev, A. Iliev, On the Hausdorff distance between the shifted Heaviside function and some generic growth functions, Int. J. of Engineering Works, 3 No. 10 (2016), 73–77.
[24] N. Kyurkchiev, S. Markov, On the numerical solution of the general kinetic K-angle reaction system, J. Math. Chem., 54 No. 3 (2016), 792–805.
[25] N. Kyurkchiev, S. Markov, Hausdorff approximation of the sign function by a class of parametric activation functions, Biomath Communications, 3 No. 2 (2016), 11 pp.
[26] A. Iliev, N. Kyurkchiev, S. Markov, A family of recurrence generated parametric activation functions with applications to neural networks, Int. J. Res. Inn. Eng. Sci. and Technology, 2 No. 1 (2017), 60–68.
[27] N. Kyurkchiev, A family of recurrence generated sigmoidal functions based on the Verhulst logistic function. Some approximation and modeling aspects, Biomath Communications, 3 No. 2 (2016), 18 pp.
[28] N. Kyurkchiev, A note on the new geometric representation for the parameters in the fibril elongation process, Compt. rend. Acad. bulg. Sci., 69 No. 8 (2016), 963–972.
[29] N. Kyurkchiev, A note on the Volmers activation function, Compt. rend. Acad. bulg. Sci., 70 No. 6 (2017), 769–776. 
[30] V. Kyurkchiev, N. Kyurkchiev, A family of recurrence generated functions based on the ”half-hyperbolic tangent activation function”, Biomedical Statistics and Informatics, 2 No. 3 (2017), 87–94.
[31] V. Kyurkchiev, A. Iliev, N. Kyurkchiev, On some families of recurrence generated activation functions, Int. J. of Sci. Eng. and Appl. Sci., 3 No. 3 (2017), 243–248.
[32] V. Kyurkchiev, A. Iliev, N. Kyurkchiev, Applications of some new transmuted cumulative distribution functions in population dynamics, Journal of Bioinformatics and Genomics, 1 No. 3 (2017).
[33] N. Kyurkchiev, A new transmuted cumulative distribution function based on Verhulst logistic function with application in population dynamics, Biomath Communications, 4 No. 1 (2017), 15 pp.
[34] D. Costarelli, R. Spigler, Approximation results for neural network operators activated by sigmoidal functions, Neural Networks, 44 (2013), 101–106. 
[35] D. Costarelli, G. Vinti, Pointwise and uniform approximation by multivariate neural network operators of the max-product type, Neural Networks, (2016), doi: 10.1016/j.neunet.2016.06.002.
[36] D. Costarelli, R. Spigler, Solving numerically nonlinear systems of balance laws by multivariate sigmoidal functions approximation, Computational and Applied Mathematics, (2016), doi: 10.1007/s40314-016-0334-8.
[37] D. Costarelli, G. Vinti, Convergence for a family of neural network operators in Orlicz spaces, Mathematische Nachrichten, (2016), doi: 10.1002/mana.20160006.
[38] N. Guliyev, V. Ismailov, A single hidden layer feedforward network with only one neuron in the hidden layer can approximate any univariate function, Neural Computation, 28 (2016), 1289–1304.
[39] N. Kyurkchiev, A. Iliev, S. Markov, Families of recurrence generated three and four parametric activation functions, Int. J. Sci. Res. and Development, 4 No. 12 (2017), 746–750.
[40] N. Kyurkchiev, A. Iliev, A Note On The New Fibonacci Hyperbolic Tangent Activation Function, International Journal of Innovative Science Engineering and Technology, 4 No. 5 (2017), 364–368.
[41] A. Golev, A. Iliev, N. Kyurkchiev, A Note on the Soboleva Modified Hyperbolic Tangent Activation Function, International Journal of Innovative Science Engineering and Technology, 4 No. 6 (2017), 177–182.
[42] A. Malinova, A. Golev, A. Iliev, N. Kyurkchiev, A Family of Recurrence Generating Activation Functions Based on Gudermann Function, International Journal of Engineering Researches and Management Studies, 4 No. 8 (2017), 38–48.
[43] N. Kyurkchiev, A. Iliev, S. Markov, Some Techniques for Recurrence Generating of Activation Functions: Some Modeling and Approximation Aspects, LAP Lambert Academic Publishing, 2017.
[44] N. Kyurkchiev, A Family of Recurrence Generated Parametric Functions Based on Volmer-Weber-Kaishew Activation Function, Pliska Studia Mathematica, 29 (2018), 69–80.