Soboleva modified hyperbolic tangent

The Soboleva modified hyperbolic tangent, also known as (parametric) Soboleva modified hyperbolic tangent activation function ([P]SMHTAF), is a special S-shaped function based on the hyperbolic tangent, given by

Equation	Left tail control	Right tail control
$\mathrm{smht}x=\frac{e^{ax}-e^{-bx}}{e^{cx}+e^{-dx}}.$

History

This function was originally proposed as "modified hyperbolic tangent" by Ukrainian scientist Elena V. Soboleva (Елена В. Соболева) as a utility function for multi-objective optimization and choice modelling in decision-making.

Practical usage

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

Sensitivity to parameters

Derivative of the function is defined by the formula:

$${\mathrm{smht}}^{\prime }(x)\doteq \frac{a{e}^{ax}+b{e}^{-bx}}{e^{cx}+e^{-dx}}-\mathrm{smht}(x)\frac{c{e}^{cx}-d{e}^{-dx}}{e^{cx}+e^{-dx}}$$

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. 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
$\frac{e^{ax}-e^{-bx}}{e^{ax}+e^{-bx}}=\frac{e^{bx}-e^{-ax}}{e^{bx}+e^{-ax}}$

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

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

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

Further reading

• Iliev, Anton; Kyurkchiev, Nikolay; Markov, Svetoslav (2017). "A Note on the New Activation Function of Gompertz Type". Biomath Communications. Biomath Forum (BF). 4 (2). 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) [3]

This article "Soboleva modified hyperbolic tangent" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Soboleva modified hyperbolic tangent. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

📰 Article(s) of the same category(ies)

• Simone Muratelli
• Government Polytechnic Muzaffarpur
• Lymm (MCR)
• Harry Jarvis
• Crocker Island
• Eachy
• Michael Patrick Coyle

---