Soboleva modified hyperbolic tangent
The Soboleva modified hyperbolic tangent, also known as (parametric) Soboleva modified hyperbolic tangent activation function ([P]SMHTAF),[nb 1] is a special S-shaped function based on the hyperbolic tangent, given by
Equation | Left tail control | Right tail control |
---|---|---|
History[edit]
This function was originally proposed as "modified hyperbolic tangent"[nb 1] by Ukrainian scientist Elena V. Soboleva (Елена В. Соболева) as a utility function for multi-objective optimization and choice modelling in decision-making.[1][2][3]
Practical usage[edit]
The function has since been introduced into neural network theory and practice.[4]
It was also used in economics for modelling consumption and investment,[5] to approximate current-voltage characteristics of field-effect transistors and light-emitting diodes,[6] to design antenna feeders,[7] and analyze plasma temperatures and densities in the divertor region of fusion reactors.[8]
Sensitivity to parameters[edit]
Derivative of the function is defined by the formula:
The following conditions are keeping the function limited on y-axes: a ≤ c, b ≤ d.
A family of recurrence-generated parametric Soboleva modified hyperbolic tangent activation functions (NPSMHTAF, FPSMHTAF) was studied with parameters a = c and b = d.[9] It is worth noting that in this case, the function is not sensitive to flipping the left and right-sides parameters:
Equation | Left prevalence | Right prevalence |
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The function is sensitive to ratio of the denominator coefficients and often is used without coefficients in the numerator:
Equation | Basic chart | Scaled function |
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Extremum estimates: |
With parameters a = b = c = d = 1 the modified hyperbolic tangent function reduces to the conventional tanh(x) function, whereas for a = b = 1 and c = d = 0, the term becomes equal to sinh(x).
See also[edit]
- Activation function
- e (mathematical constant)
- Equal incircles theorem, based on sinh
- Hausdorff distance
- Inverse hyperbolic functions
- List of integrals of hyperbolic functions
- Poinsot's spirals
- Sigmoid function
Notes[edit]
References[edit]
- ^ Soboleva, Elena Vladimirovna; Beskorovainyi, Vladimir Valentinovich (2008). The utility function in problems of structural optimization of distributed objects Функция для оценки полезности альтернатив в задачах структурной оптимизации территориально распределенных объектов. Четверта наукова конференція Харківського університету Повітряних Сил імені ��вана Кожедуба, 16–17 квітня 2008 (The fourth scientific conference of the Ivan Kozhedub Kharkiv University of Air Forces, 16–17 April 2008) (in Russian). Kharkiv, Ukraine: Kharkiv University of Air Force (HUPS/ХУПС). p. 121.
- ^ Soboleva, Elena Vladimirovna (2009). S-образная функция полезности част-ных критериев для многофакторной оценки проектных решений [The S-shaped utility function of individual criteria for multi-objective decision-making in design]. Материалы XIII Международного молодежного форума «Радиоэлектро-ника и молодежь в XXI веке» (Materials of the 13th international youth forum "Radioelectronics and youth in the 21st century") (in Russian). Kharkiv, Ukraine: Kharkiv National University of Radioelectronics (KNURE/ХНУРЕ). p. 247.
- ^ Beskorovainyi, Vladimir Valentinovich; Soboleva, Elena Vladimirovna (2010). ИДЕНТИФИКАЦИЯ ЧАСТНОй ПОлЕЗНОСТИ МНОГОФАКТОРНЫХ АлЬТЕРНАТИВ С ПОМОЩЬЮ S-ОБРАЗНЫХ ФУНКЦИй [Identification of utility functions in multi-objective choice modelling by using S-shaped functions] (PDF). Problemy Bioniki: Respublikanskij Mežvedomstvennyj Naučno-Techničeskij Sbornik БИОНИКА ИНТЕЛЛЕКТА [Bionics of Intelligence] (in Russian). Vol. 72, no. 1. Kharkiv National University of Radioelectronics (KNURE/ХНУРЕ). pp. 50–54. ISSN 0555-2656. UDK 519.688: 004.896. Archived (PDF) from the original on 2022-06-21. Retrieved 2020-06-19. (5 pages) [1]
- ^ Malinova, Anna; Golev, Angel; Iliev, Anton; Kyurkchiev, Nikolay (August 2017). "A Family Of Recurrence Generating Activation Functions Based On Gudermann Function" (PDF). International Journal of Engineering Researches and Management Studies. 4 (8). Faculty of Mathematics and Informatics, University of Plovdiv "Paisii Hilendarski", Plovdiv, Bulgaria: 38–48. ISSN 2394-7659. Archived (PDF) from the original on 2022-07-14. Retrieved 2020-06-19. (11 pages) [2]
- ^ Orlando, Giuseppe (2016-07-01). "A discrete mathematical model for chaotic dynamics in economics: Kaldor's model on business cycle". Mathematics and Computers in Simulation. 8th Workshop STRUCTURAL DYNAMICAL SYSTEMS: Computational Aspects; Edited by Nicoletta Del Buono, Roberto Garrappa and Giulia Spaletta and Nonstandard Applications of Computer Algebra (ACA’2013); Edited by Francisco Botana, Antonio Hernando, Eugenio Roanes-Lozano and Michael J. Wester. 125: 83–98. doi:10.1016/j.matcom.2016.01.001. ISSN 0378-4754.
- ^ Tuev, Vasily I.; Uzhanin, Maxim V. (2009). ПРИМЕНЕНИЕ МОДИФИЦИРОВАННОЙ ФУНКЦИИ ГИПЕРБОЛИЧЕСКОГО ТАНГЕНСА ДЛЯ АППРОКСИМАЦИИ ВОЛЬТАМПЕРНЫХ ХАРАКТЕРИСТИК ПОЛЕВЫХ ТРАНЗИСТОРОВ [Using modified hyperbolic tangent function to approximate the current-voltage characteristics of field-effect transistors] (in Russian). Tomsk, Russia: Tomsk Politehnic University (TPU/ТПУ). pp. 135–138. No. 4/314. Archived from the original on 2017-08-15. Retrieved 2015-11-05. (4 pages) [3]
- ^ Golev, Angel; Djamiykov, Todor; Kyurkchiev, Nikolay (2017-11-23) [2017-10-09, 2017-08-19]. "Sigmoidal Functions In Antenna-Feeder Technique" (PDF). International Journal of Pure and Applied Mathematics. 116 (4). Faculty of Mathematics and Informatics, University of Plovdiv "Paisii Hilendarski", Plovdiv, Bulgaria / Technical University of Sofia, Sofia, Bulgaria: Academic Publications, Ltd.: 1081–1092. doi:10.12732/ijpam.v116i4.23 (inactive 2024-01-31). ISSN 1311-8080. Archived (PDF) from the original on 2020-06-19. Retrieved 2020-06-19.
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: CS1 maint: DOI inactive as of January 2024 (link) (12 pages) - ^ Rubino, Giulio (2018-01-15) [2018-01-14]. Power Exhaust Data Analysis and Modeling Of Advanced Divertor Configuration (PDF) (Thesis). Joint Research Doctorate In Fusion Science And Engineering Cycle XXX (in English, Italian, and Portuguese). Padova, Italy: Centro Ricerche Fusione (CRF), Università degli Studi di Padova / Università degli Studi di Napoli Federico II / Instituto Superior Técnico (IST), Universidade de Lisboa. p. 84. ID 10811. Archived (PDF) from the original on 2020-06-19. Retrieved 2020-06-19. [4] (2+viii+3*iii+102 pages)
- ^ Golev, Angel; Iliev, Anton; Kyurkchiev, Nikolay (June 2017). "A Note on the Soboleva' Modified Hyperbolic Tangent Activation Function" (PDF). International Journal of Innovative Science, Engineering & Technology (JISET). 4 (6). Faculty of Mathematics and Informatics, University of Plovdiv "Paisii Hilendarski", Plovdiv, Bulgaria: 177–182. ISSN 2348-7968. Archived (PDF) from the original on 2020-06-19. Retrieved 2020-06-19. (6 pages) [5]
Further reading[edit]
- Iliev, Anton; Kyurkchiev, Nikolay; Markov, Svetoslav (2017). "A Note on the New Activation Function of Gompertz Type". Biomath Communications. 4 (2). Faculty of Mathematics and Informatics, University of Plovdiv "Paisii Hilendarski", Plovdiv, Bulgaria / Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria: Biomath Forum (BF). doi:10.11145/10.11145/bmc.2017.10.201. ISSN 2367-5233. Archived from the original on 2020-06-20. Retrieved 2020-06-19. (20 pages) [6]