On the Hausdorff Distance Between the Heaviside Function and Some Transmuted Activation Functions
Mathematical Modelling and Applications
Volume 1, Issue 1, October 2016, Pages: 8-12
Received: Aug. 18, 2016;
Accepted: Oct. 12, 2016;
Published: Oct. 14, 2016
Views 3307 Downloads 94
Nikolay Kyurkchiev, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria
Anton Iliev, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria; Faculty of Mathematics and Informatics, Paisii Hilendarski University of Plovdiv, Plovdiv, Bulgaria
In this paper we study the one-sided Hausdorff distance between the Heaviside function and some transmuted activation functions. Precise upper and lower bounds for the Hausdorff distance have been obtained. Numerical examples are presented throughout the paper using the computer algebra system MATHEMATICA. The results can be successfully used in the field of applied insurance mathematics.
On the Hausdorff Distance Between the Heaviside Function and Some Transmuted Activation Functions, Mathematical Modelling and Applications.
Vol. 1, No. 1,
2016, pp. 8-12.
N. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions, Willey, NY, Vol. 1, 1994.
N. Guliyev and V. Ismailov, A single hidden layer feedforward network with only one neuron in the hidden layer san approximate any univariate function, Neural Computation, Vol. 28, 2016, pp. 1289–1304.
W. Shaw and I. Buckley, The alchemy of probability distributions: beyond Gram-Charlier expansion and a skew-kurtotic-normal distribution from rank transmutation map, Research report, 2007.
R. C. Gupta, O. Akman, and S. Lvin, A study of log–logistic model in survival analysis, Biometrical Journal, Vol. 41 (4), 1999, pp. 431–443.
F. Hausdorff, Set Theory (2 ed.) (Chelsea Publ., New York, (1962 ) (Republished by AMS-Chelsea 2005), ISBN: 978–0–821–83835–8.
B. Sendov, Hausdorff Approximations (Kluwer, Boston, 1990), doi: 10.1007/978-94-009-0673-0.
R. Anguelov and S. Markov, Hausdorff Continuous Interval Functions and Approximations, In: M. Nehmeier et al. (Eds), Scientific Computing, Computer Arithmetic, and Validated Numerics, 16th International Symposium, SCAN 2014, LNCS 9553, pp. 3–13, 2016, Springer, doi: 10.1007/978-3-319-31769-4
N. Kyurkchiev and A. Andreev, Approximation and antenna and filter synthesis: Some moduli in programming environment Mathematica, LAP LAMBERT Academic Publishing, Saarbrucken, 2014, ISBN: 978-3-659-53322-8.
D. Costarelli and R. Spigler, Approximation results for neural network operators activated by sigmoidal functions, Neural Networks, Vol. 44, 2013, pp. 101–106.
D. Costarelli and 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.
D. Costarelli and 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.
D. Costarelli and G. Vinti, Convergence for a family of neural network operators in Orlicz spaces, Mathematische Nachrichten 2016, doi: 10.1002/mana.20160006.
J. Dombi and Z. Gera, The Approximation of Piecewise Linear Membership Functions and Lukasiewicz Operators, Fuzzy Sets and Systems, Vol. 154 (2), 2005, pp. 275–286.
N. Kyurkchiev, On the Approximation of the step function by some cumulative distribution functions, Compt. rend. Acad. bulg. Sci., Vol. 68 (12), 2015, pp. 1475–1482.
N. Kyurkchiev and S. Markov, Sigmoid functions: Some Approximation and Modelling Aspects, LAP LAMBERT Academic Publishing, Saarbrucken, 2015, ISBN: 978-3-659-76045-7.
N. Kyurkchiev and S. Markov, On the Hausdorff distance between the Heaviside step function and Verhulst logistic function. J. Math. Chem., Vol. 54 (1), 2016, pp. 109–119, doi: 10.1007/S10910-015-0552-0.
N. Kyurkchiev and S. Markov, Sigmoidal functions: some computational and modelling aspects. Biomath Communications, Vol. 1 (2), 2014, pp. 30–48, doi: 10.11145/j.bmc.2015.03.081.
N. Kyurkchiev and S. Markov, On the approximation of the generalized cut function of degree by smooth sigmoid functions, Serdica J. Computing, Vol. 9 (1), 2015, pp. 101–112.
N. Kyurkchiev, A note on the new geometric representation for the parameters in the fibril elongation process, Compt. rend. Acad. bulg. Sci., Vol. 69 (8), 2016, pp. 963–972.
N. Kyurkchiev, S. Markov, and A. Iliev, A note on the Schnute growth model, Int. J. of Engineering Research and Development, vol. 12 (6), 2016, 47-54, ISSN: 2278-067X.
V. Kyurkchiev and N. Kyurkchiev, On the Approximation of the Step function by Raised-Cosine and Laplace Cumulative Distribution Functions. European International Journal of Science and Technology, Vol. 4 (9), 2016, pp. 75–84.
A. Iliev, N. Kyurkchiev, and S. Markov, On the Approximation of the Cut and Step Functions by Logistic and Gompertz Functions, BIOMATH, Vol. 4 (2), 2015, 1510101, doi: 10.11145/j.biomath.2015.10.101.
A. Iliev, N. Kyurkchiev, and S. Markov, On the Approximation of the step function by some sigmoid functions, Mathematics and Computers in Simulation, 2015, doi: 10.1016/j.matcom.2015.11.005.
N. Kyurkchiev and A. Iliev, A note on some growth curves arising from Box-Cox transformation, Int. J. of Engineering Works, Vol. 3 (6), 2016, pp. 47–51, ISSN: 2409-2770.
N. Kyurkchiev and A. Iliev, On some growth curve modeling: approximation theory and applications, Int. J. of Trends in Research and Development, Vol. 3 (3), 2016, pp. 466–471.