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The Soboleva modified hyperbolic tangent, also known as (parametric) Soboleva modified hyperbolic tangent activation function ( MHTAF), is a special S-shaped
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
based on the
hyperbolic tangent In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the un ...
, given by :\operatornamex = \frac . This function was originally proposed as "modified hyperbolic tangent" by Elena V. Soboleva () as a utility function for
multi-objective optimization Multi-objective optimization (also known as multi-objective programming, vector optimization, multicriteria optimization, multiattribute optimization or Pareto optimization) is an area of multiple criteria decision making that is concerned with ...
and
choice modelling Choice modelling attempts to model the decision process of an individual or segment via revealed preferences or stated preferences made in a particular context or contexts. Typically, it attempts to use discrete choices (A over B; B over A, B & C) ...
in
decision-making In psychology, decision-making (also spelled decision making and decisionmaking) is regarded as the Cognition, cognitive process resulting in the selection of a belief or a course of action among several possible alternative options. It could be ...
. It has since been introduced into
neural network A neural network is a network or circuit of biological neurons, or, in a modern sense, an artificial neural network, composed of artificial neurons or nodes. Thus, a neural network is either a biological neural network, made up of biological ...
theory and practice. The function was also used to approximate current-voltage characteristics of
field-effect transistor The field-effect transistor (FET) is a type of transistor that uses an electric field to control the flow of current in a semiconductor. FETs (JFETs or MOSFETs) are devices with three terminals: ''source'', ''gate'', and ''drain''. FETs contro ...
s and
light-emitting diode A light-emitting diode (LED) is a semiconductor device that emits light when current flows through it. Electrons in the semiconductor recombine with electron holes, releasing energy in the form of photons. The color of the light (cor ...
s, to design
antenna feed A radio transmitter or receiver is connected to an antenna which emits or receives the radio waves. The antenna feed system or antenna feed is the cable or conductor, and other associated equipment, which connects the transmitter or receiver w ...
ers, and analyze
plasma Plasma or plasm may refer to: Science * Plasma (physics), one of the four fundamental states of matter * Plasma (mineral), a green translucent silica mineral * Quark–gluon plasma, a state of matter in quantum chromodynamics Biology * Blood pla ...
temperatures and densities in the
divertor In nuclear fusion power research, a divertor is a device within a tokamak or a stellarator that allows the online removal of waste material from the plasma while the reactor is still operating. This allows control over the buildup of fusion prod ...
region of
fusion reactor Fusion power is a proposed form of power generation that would generate electricity by using heat from nuclear fusion reactions. In a fusion process, two lighter atomic nuclei combine to form a heavier nucleus, while releasing energy. Devices des ...
s. A family of recurrence-generated parametric Soboleva modified hyperbolic tangent activation functions (NPSMHTAF, FPSMHTAF) was studied with parameters ''a'' = ''c'' and ''b'' = ''d''. With parameters ''a'' = ''b'' = ''c'' = ''d'' = 1 the modified hyperbolic tangent function reduces to the conventional
tanh In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the un ...
(''x'') function, whereas for ''a'' = ''b'' = 1 and ''c'' = ''d'' = 0, the term becomes equal to sinh(''x'').


See also

*
Activation function In artificial neural networks, the activation function of a node defines the output of that node given an input or set of inputs. A standard integrated circuit can be seen as a digital network of activation functions that can be "ON" (1) or "O ...
*
e (mathematical constant) The number , also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of the natural logarithms. It is the limit of as approaches infinity, an express ...
*
Equal incircles theorem In geometry, the equal incircles theorem derives from a Japanese Sangaku, and pertains to the following construction: a series of rays are drawn from a given point to a given line such that the inscribed circles of the triangles formed by adjacent ...
, based on sinh *
Hausdorff distance In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty compact subsets of a metric space into a metric ...
*
Inverse hyperbolic function In mathematics, the inverse hyperbolic functions are the inverse functions of the hyperbolic functions. For a given value of a hyperbolic function, the corresponding inverse hyperbolic function provides the corresponding hyperbolic angle. The s ...
s *
List of integrals of hyperbolic functions The following is a list of integrals (anti-derivative functions) of hyperbolic functions. For a complete list of integral functions, see list of integrals. In all formulas the constant ''a'' is assumed to be nonzero, and ''C'' denotes the constant ...
*
Poinsot's spirals In mathematics, Poinsot's spirals are two spirals represented by the polar equations : r = a\ \operatorname (n\theta) : r = a\ \operatorname (n\theta) where csch is the hyperbolic cosecant, and sech is the hyperbolic secant. They are named afte ...
*
Sigmoid function A sigmoid function is a mathematical function having a characteristic "S"-shaped curve or sigmoid curve. A common example of a sigmoid function is the logistic function shown in the first figure and defined by the formula: :S(x) = \frac = \f ...


Notes


References


Further reading

* (20 pages

{{DEFAULTSORT:Modified Hyperbolic Tangent Elementary special functions Exponentials Analytic functions Functions and mappings Artificial neural networks