Mathematical activation function in data analysis
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
History.
This function was originally proposed as "modified hyperbolic tangent" by Ukrainian scientist Elena V. Soboleva () as a utility function for multi-objective optimization and choice modelling in decision-making. 
Practical usage.
The function has since been introduced into neural network theory and practice.
It was also used in economics for modelling consumption and investment, to approximate current-voltage characteristics of field-effect transistors and light-emitting diodes, to design antenna feeders,[predatory publisher] 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:
formula_1
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:
The function is sensitive to ratio of the denominator coefficients and often is used without coefficients in the numerator:
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").
Notes.
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References.
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