Sigmoid function

The logistic curve Plot of the error function

A sigmoid function is any mathematical function whose graph has a characteristic S-shaped or sigmoid curve.

A common example of a sigmoid function is the logistic function.

Other sigmoid functions are given in the Examples section. In some fields, most notably in the context of artificial neural networks, the term "sigmoid function" is used as a synonym for "logistic function".

Special cases of sigmoid functions include the Gompertz curve (used in modeling systems that saturate at large values of x) and the ogee curve (used in the spillway of some dams). Sigmoid functions have domain of all real numbers, with return (response) value commonly monotonically increasing but could be decreasing. Sigmoid functions most often show a return value (y axis) in the range 0 to 1. Another commonly used range is from −1 to 1.

There is also the Heaviside step function, which instantaneously transitions between 0 and 1.

A wide variety of sigmoid functions including the logistic and hyperbolic tangent functions have been used as the activation function of artificial neurons. Sigmoid curves are also common in statistics as cumulative distribution functions (which go from 0 to 1), such as the integrals of the logistic density, the normal density, and Student's t probability density functions. The logistic sigmoid function is invertible, and its inverse is the logit function.

S is an increasing sigmoid function, B are constants controlling scaling and translation.

A unitary sigmoid function of the first kind is a bounded increasing function that approaches its lower and upper asymptotes monotonically, without oscillation. This class includes many of the standard sigmoid functions used in statistics, biomathematics, and engineering, such as the logistic function and related generalizations.

A unitary sigmoid function of the second kind is a bounded increasing function that oscillates near the upper asymptote while preserving an overall sigmoid transition.

A unitary sigmoid function of the third kind is a bounded increasing function that oscillates near both the lower and upper asymptotes. These functions retain the global shape of a sigmoid curve but exhibit oscillatory behavior in the vicinity of both limiting states.

Fig. 1. Graphical representation of the three kinds of sigmoid functions on the infinite domain: the 1 kind (red solid line), the 2 kind (violet dashed line), the 3 kind (blue dotted line).

The tables below show the taxonomy of unitary sigmoid functions of all three kinds.

Table 1. Taxonomy matrix with examples of sigmoid functions of the 1 kind

No	Type of sigmoid function	Unbounded interval x∈(0,1), a,b>0
1	rational	12(1+(b+10−x+11−x)|a|+|b+10−x+11−x|)
2	irrational	xab2+x2a	12+x2|x|(1−e−a|x|) (Laplace)	(1+e−a(2xb−1)xb(1−xb))−1
4	logarithmic	ln⁡(1+ea(2xb−1)xb(1−xb)e121+ea(2xb−1)xb(1−xb)e−12)
5	trigonometric	12(1−cos⁡(πx))
6	inverse trigonometric	12+1πarctan⁡(a(2xb−1)xb(1−xb))
7	hyperbolic	12+12tanh⁡(a(2xb−1)xb(1−xb))
8	inverse hyperbolic	2πarcsin⁡(11+eacot⁡(πx))
9	special	12+12erf⁡(a(2xb−1)xb(1−xb))
10	stochastic	—	x∈(−∞,+∞), A<1, b>0, A+B=1	Explanation
1	All functions from Table 1	s(ax) of the 3 column of the Table 1, where As(ax)+B8(1+2C(ax))2+B8(1+2S(ax))2	1−sin⁡(beax)beax
5	special	x∈(−∞,+∞), A+B=1, B≠0	Explanation
1	special	As(ax) of the 3 column of the Table 1, where B(12+eCi⁡(ax)2) to the functions As(ax)+B(12+S(ax))	C(ax) is the Fresnel C integral

The same theory presents a list of 30 methods for constructing sigmoid functions.. These include algebraic transformations, integration and convolution methods, constructions from bell-shaped functions, solutions of ordinary and partial differential equations, recursive schemes, stochastic differential equations, feedback systems, and chaotic systems.

• M0: Construction method for sigmoid functions not evident or intuitive
• M1: Inverse of singularity functions
• M2: Sigmoid functions of embedded positive functions
• M3: Rising a sigmoid function to the power
• M4: Exponentiating a sigmoid function
• M5: Symmetric sigmoid functions derived from asymmetric ones
• M6: Sigmoid functions of the reciprocal independent variable
• M7: Embedding a sigmoid function into other function
• M8: Sum of sigmoid functions
• M9: Multiplication of sigmoid functions
• M10: Integral of the product of an increasing and a decreasing function
• M11: Derivation from lambda (bell-shaped) functions
• M12: Integration of lambda (bell-shaped) function
• M13: Integration of the sum of lambda (bell-shaped) functions
• M14: Integration of the product of two lambda (bell-shaped) functions
• M15: Integration of the difference of two shifted sigmoid functions
• M16: Integration of the product of two shifted sigmoid functions
• M17: Convolution of sigmoid functions
• M18: Integration of the product of lambda and sigmoid function
• M19: Solutions of ordinary differential equations
• M20: Solutions of partial differential equation (PDE)
• M21: Solutions of functional differential equation (FDE)
• M22: Sum of a sigmoid function and some derivatives
• M23: Combination of sigmoid functions, its derivative and integral
• M24: Filtering sigmoid functions
• M25: Special cases of Gauss hypergeometric functions
• M26: Feedback closed-loop systems
• M27: Recursive functions
• M28: Recursive time-delayed feed-forward loops
• M29: Solutions of stochastic differential equation
• M30: Chaotic sigmoid functions

Consult reference for more details.

A sigmoid function is a bounded, differentiable, real function that is defined for all real input values and has a positive derivative at each point.

In general, a sigmoid function is monotonic, and has a first derivative which is bell shaped. Conversely, the integral of any continuous, non-negative, bell-shaped function (with one local maximum and no local minimum, unless degenerate) will be sigmoidal. Thus the cumulative distribution functions for many common probability distributions are sigmoidal. One such example is the error function, which is related to the cumulative distribution function of a normal distribution; another is the arctan function, which is related to the cumulative distribution function of a Cauchy distribution.

A sigmoid function is constrained by a pair of horizontal asymptotes as f(x)=tanh⁡x=ex−e−xex+e−x

Arctangent function f(x)=erf⁡(x)=2π∫0xe−t2dt

Generalised logistic function 0 "}}'>f(x)=(1+e−x)−α,α>0

Smoothstep function f(x)=x(1+|x|k)1/k

Up to shifts and scaling, many sigmoids are special cases of α<1 and