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
${\displaystyle \operatorname {smht} x={\frac {e^{ax}-e^{-bx}}{e^{cx}+e^{-dx}}}.}$

History

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

The function was used in economics for modelling consumption and investment, ^[4] to approximate current-voltage characteristics of field-effect transistors and light-emitting diodes, ^[5] and analyze plasma temperatures and densities in the divertor region of fusion reactors. ^[6]

Sensitivity to parameters

Derivative of the function is defined by the formula:

$${\displaystyle \operatorname {smht} '(x)\doteq {\frac {ae^{ax}+be^{-bx}}{e^{cx}+e^{-dx}}}-\operatorname {smht} (x){\frac {ce^{cx}-de^{-dx}}{e^{cx}+e^{-dx}}}}$$

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.^{[ citation needed ]} 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
${\displaystyle {\frac {e^{ax}-e^{-bx}}{e^{ax}+e^{-bx}}}={\frac {e^{bx}-e^{-ax}}{e^{bx}+e^{-ax}}}}$

The function is sensitive to ratio of the denominator coefficients and often is used without coefficients in the numerator:

Equation	Basic chart	Scaled function
${\displaystyle \operatorname {ssht} x={\frac {e^{x}-e^{-x}}{e^{\alpha x}+e^{-\beta x}}}.}$ Extremum estimates: ${\displaystyle x_{\min }={\frac {1}{2}}\ln {\frac {\beta -1}{\beta +1}};}$${\displaystyle x_{\max }={\frac {1}{2}}\ln {\frac {\alpha +1}{\alpha -1}}.}$

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

• 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

1. Soboleva proposed the name "modified hyperbolic tangent" (mtanh, mth), but since other authors used this name also for other functions, some authors have started to refer to this function as "Soboleva modified hyperbolic tangent".

References

1. Soboleva, Elena Vladimirovna; Beskorovainyi, Vladimir Valentinovich (2008). The utility function in problems of structural optimization of distributed objectsФункция для оценки полезности альтернатив в задачах структурной оптимизации территориально распределенных объектов. Четверта наукова конференція Харківського університету Повітряних Сил імені Івана 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.
2. 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.
3. 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 SbornikБИОНИКА 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)
4. 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. 125: 83–98. doi:10.1016/j.matcom.2016.01.001. ISSN   0378-4754.
5. 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)
6. Rubino, Giulio (2018-01-15) [2018-01-14]. Power Exhaust Data Analysis and Modeling Of Advanced Divertor Configuration (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 from the original on 2020-06-19. Retrieved 2020-06-19. (2+viii+3*iii+102 pages)

Further reading

• Iliev, Anton; Kyurkchiev, Nikolay; Markov, Svetoslav (2017). "A Note on the New Activation Function of Gompertz Type". Biomath Communications. 4 (2). 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)