Sigmoid function

In mathematics, a unitary sigmoid function is a bounded sigmoid-type function normalized to the unit range, typically with lower and upper asymptotes at 0 and 1. The theory proposed by Grebenc^[1] distinguishes three kinds of unitary sigmoid functions according to their asymptotic behavior and the presence or absence of oscillation near the asymptotes.

A general form of a unitary sigmoid function is

${\displaystyle y=A\,S(f(x))+B,}$

where ${\displaystyle S}$ is an increasing sigmoid function, ${\displaystyle f(x)}$ is a transformation of the independent variable, and ${\displaystyle A}$ and ${\displaystyle B}$ are constants controlling scaling and translation.

Classification

1^st kind

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.

2^nd kind

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.

3^rd kind

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.

Taxonomy

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^st kind

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No	Type of sigmoid function	Unbounded interval ${\displaystyle x\in (-\infty ,+\infty )}$	Semi-bounded interval ${\displaystyle x\in [0,+\infty ),\ a>1,\ a\notin 2n,\ b>0}$	Unitary interval ${\displaystyle x\in (0,1),\ a,b>0}$
1	rational	${\displaystyle {\frac {1}{2}}\left(1+{\frac {x}{|a|+|x|}}\right)}$ (Elliot[2])	${\displaystyle \left(1+(b\,x)^{-a}\right)^{-c}}$	${\displaystyle {\frac {1}{2}}\left(1+{\frac {(b+{\frac {1}{0-x}}+{\frac {1}{1-x}})}{\left|a\right|+\left|b+{\frac {1}{0-x}}+{\frac {1}{1-x}}\right|}}\right)}$
2	irrational	${\displaystyle {\frac {1}{2}}\left(1+{\frac {x}{\sqrt {a^{2}+x^{2}}}}\right)}$ (algebraic) ${\displaystyle {\sqrt[{-b}]{1+e^{-b\,a\,x}}}}$ (Richards[3])	${\displaystyle {\frac {x^{a}}{\sqrt {b^{2}+x^{2a}}}}}$	${\displaystyle \left(1-\left(1-x^{a}\right)^{b}\right)^{c}}$
3	exponential	${\displaystyle \left(1+e^{-a\,x}\right)^{-1}}$ (Logistic/Verhulst[4]) ${\displaystyle {\frac {1}{2}}+{\frac {x}{2|x|}}\left(1-e^{-a|x|}\right)}$ (Laplace[5])	${\displaystyle 2^{-b^{-x^{a}}}}$; ${\displaystyle 1-e^{a\,x^{2}}}$ (Rayleigh[6])	${\displaystyle \left(1+e^{\frac {-a\,(2x^{b}-1)}{x^{b}(1-x^{b})}}\right)^{-1}}$
4	logarithmic	${\displaystyle \ln \!\left({\frac {1+e^{ax}e^{\frac {1}{2}}}{1+e^{ax}e^{-{\frac {1}{2}}}}}\right)}$	${\displaystyle \ln \!\left({\frac {1+x^{a}e^{\frac {1}{2}}}{1+x^{a}e^{-{\frac {1}{2}}}}}\right)}$	${\displaystyle \ln \!\left({\frac {1+e^{\frac {a(2x^{b}-1)}{x^{b}(1-x^{b})}}e^{\frac {1}{2}}}{1+e^{\frac {a(2x^{b}-1)}{x^{b}(1-x^{b})}}e^{-{\frac {1}{2}}}}}\right)}$
5	trigonometric	${\displaystyle \sin \!\left({\frac {\pi }{2+2\,e^{-a\,x}}}\right)}$	${\displaystyle \sin \!\left({\frac {\pi \,e^{a\ln(x)}}{2+2\,e^{a\ln(x)}}}\right)}$	${\displaystyle {\frac {1}{2}}\left(1-\cos(\pi \,x)\right)}$
6	inverse trigonometric	${\displaystyle {\frac {1}{2}}+{\frac {1}{\pi }}\arctan(a\,x)}$ (Lorenz[7])	${\displaystyle {\frac {1}{2}}+{\frac {1}{\pi }}\arctan(x^{a})}$	${\displaystyle {\frac {1}{2}}+{\frac {1}{\pi }}\arctan \!\left({\frac {a(2x^{b}-1)}{x^{b}(1-x^{b})}}\right)}$
7	hyperbolic	${\displaystyle {\frac {1}{2}}+{\frac {1}{2}}\tanh(a\,x)}$	${\displaystyle {\frac {1}{2}}+{\frac {1}{2}}\tanh(a\ln(x))}$ (Log-logistic[8])	${\displaystyle {\frac {1}{2}}+{\frac {1}{2}}\tanh \!\left({\frac {a(2x^{b}-1)}{x^{b}(1-x^{b})}}\right)}$
8	inverse hyperbolic	${\displaystyle {\frac {2}{\pi }}\arcsin \!\left({\frac {1}{1+e^{-a(x-c)}}}\right),\ e^{c}={\sqrt {2}}-1}$	${\displaystyle {\frac {2}{\pi }}\arcsin \!\left({\frac {x^{a}}{1+x^{a}}}\right)}$	${\displaystyle {\frac {2}{\pi }}\arcsin \!\left({\frac {1}{1+e^{a\cot(\pi x)}}}\right)}$
9	special	${\displaystyle {\frac {1}{2}}+{\frac {1}{2}}\,\operatorname {erf} \!\left({\frac {a}{\sqrt {2}}}\,x\right)}$ (Normal[9][10])	${\displaystyle {\frac {1}{2}}+{\frac {1}{2}}\,\operatorname {erf} \!\left({\frac {a}{\sqrt {2}}}\ln(x)\right)}$ (Log-normal[11])	${\displaystyle {\frac {1}{2}}+{\frac {1}{2}}\,\operatorname {erf} \!\left({\frac {a(2x^{b}-1)}{x^{b}(1-x^{b})}}\right)}$
10	stochastic	—	${\displaystyle e^{\left(a-{\frac {b^{2}}{2}}\right)x+b\,w_{i}}\cdot \left(y_{0}^{-1}+a\int _{0}^{x}e^{\left(a-{\frac {b^{2}}{2}}\right)s+b\,w_{s}}ds\right)^{-1}}$	—
11	chaotic	${\displaystyle {\frac {dx}{dt}}=-x+yz,\quad {\frac {dy}{dt}}=-ay+az,\quad {\frac {dz}{dt}}=-z+by-(b-1)xy}$ (Grebenc[12])	—	—

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Table 2. Taxonomy matrix with examples of sigmoid functions of the 2^nd kind on the unbounded interval

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No	Type of sigmoid function	Unbounded interval ${\displaystyle x\in (-\infty ,+\infty ),\ A<1,\ b>0,\ A+B=1}$	Explanation
1	All functions from Table 1	${\displaystyle s(a\,x)+A\cdot \operatorname {Ai} (-b\,x)}$	Adding ${\displaystyle A\cdot \operatorname {Ai} (-bx)}$ to the functions ${\displaystyle s(ax)}$ of the 3rd column of the Table 1, where ${\displaystyle \operatorname {Ai} }$ is the Airy Ai function
2	special	${\displaystyle a\int \operatorname {Ai} (-a\,x)\,dx+{\tfrac {1}{3}}}$
3	special	${\displaystyle A\,s(a\,x)+{\frac {B}{8}}\left(1+2\,C(a\,x)\right)^{2}+{\frac {B}{8}}\left(1+2\,S(a\,x)\right)^{2}}$	${\displaystyle C}$, ${\displaystyle S}$ are Fresnel integrals
4	special	${\displaystyle 1-{\frac {\sin(b\,e^{a\,x})}{b\,e^{a\,x}}}}$
5	special	${\displaystyle 1-J_{0}\!\left(b\,e^{a\,x}\right)}$	${\displaystyle J_{0}}$ is a Bessel J function

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Table 3. Taxonomy matrix with examples of sigmoid functions of the 3^rd kind

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No	Type	Unbounded interval ${\displaystyle x\in (-\infty ,+\infty ),\ A+B=1,\ B\neq 0}$	Explanation
1	special	${\displaystyle A\,s(a\,x)+B\!\left({\tfrac {1}{2}}+{\frac {\operatorname {Si} (ax)}{\pi }}\right)}$	Adding ${\displaystyle B\!\left({\frac {1}{2}}+{\frac {\operatorname {Si} (ax)}{\pi }}\right)}$ to the functions ${\displaystyle A\,s(ax)}$ of the 3rd column of the Table 1, where ${\displaystyle \operatorname {Si} }$ is the sine integral function
2	special	${\displaystyle A\,s(a\,x)+B\!\left({\tfrac {1}{2}}+{\tfrac {1}{2}}e^{\operatorname {Ci} (ax)}\right)}$	Adding ${\displaystyle B\!\left({\frac {1}{2}}+{\frac {e^{\operatorname {Ci} (ax)}}{2}}\right)}$ to the functions ${\displaystyle A\,s(ax)}$ of the 3rd column of Table 1, where ${\displaystyle \operatorname {Ci} }$ is the cosine integral function
3	special	${\displaystyle A\,s(a\,x)+B\!\left({\tfrac {1}{2}}+S(a\,x)\right)}$	${\displaystyle S(ax)}$ is the Fresnel S integral
4	special	${\displaystyle A\,s(a\,x)+B\!\left({\tfrac {1}{2}}+C(a\,x)\right)}$	${\displaystyle C(ax)}$ is the Fresnel C integral

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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.^[13][14]

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 ${\displaystyle x\rightarrow \pm \infty }$.

A sigmoid function is convex for values less than a particular point, and it is concave for values greater than that point: in many of the examples here, that point is 0.

• Logistic function $${\displaystyle f(x)={\frac {1}{1+e^{-x}}}}$$
• Hyperbolic tangent (shifted and scaled version of the logistic function, above) $${\displaystyle f(x)=\tanh x={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}}$$
• Arctangent function $${\displaystyle f(x)=\arctan x}$$
• Gudermannian function $${\displaystyle f(x)=\operatorname {gd} (x)=\int _{0}^{x}{\frac {dt}{\cosh t}}=2\arctan \left(\tanh \left({\frac {x}{2}}\right)\right)}$$
• Error function $${\displaystyle f(x)=\operatorname {erf} (x)={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}\,dt}$$
• Generalised logistic function $${\displaystyle f(x)=\left(1+e^{-x}\right)^{-\alpha },\quad \alpha >0}$$
• Smoothstep function $${\displaystyle f(x)={\begin{cases}{\displaystyle \left(\int _{0}^{1}\left(1-u^{2}\right)^{N}du\right)^{-1}\int _{0}^{x}\left(1-u^{2}\right)^{N}\ du},&|x|\leq 1\\\\\operatorname {sgn}(x)&|x|\geq 1\\\end{cases}}\quad N\in \mathbb {Z} \geq 1}$$
• Some algebraic functions, for example $${\displaystyle f(x)={\frac {x}{\sqrt {1+x^{2}}}}}$$
• and in a more general form^[15] $${\displaystyle f(x)={\frac {x}{\left(1+|x|^{k}\right)^{1/k}}}}$$
• Up to shifts and scaling, many sigmoids are special cases of $${\displaystyle f(x)=\varphi (\varphi (x,\beta ),\alpha ),}$$ where $${\displaystyle \varphi (x,\lambda )={\begin{cases}(1-\lambda x)^{1/\lambda }&\lambda \neq 0\\e^{-x}&\lambda =0\\\end{cases}}}$$ is the inverse of the negative Box–Cox transformation, and ${\displaystyle \alpha <1}$ and ${\displaystyle \beta <1}$ are shape parameters.^[16]
• Smooth transition function^[17] normalized to (−1,1):

$${\displaystyle {\begin{aligned}f(x)&={\begin{cases}{\displaystyle {\frac {2}{1+e^{-2m{\frac {x}{1-x^{2}}}}}}-1},&|x|<1\\\\\operatorname {sgn}(x)&|x|\geq 1\\\end{cases}}\\&={\begin{cases}{\displaystyle \tanh \left(m{\frac {x}{1-x^{2}}}\right)},&|x|<1\\\\\operatorname {sgn}(x)&|x|\geq 1\\\end{cases}}\end{aligned}}}$$ using the hyperbolic tangent mentioned above. Here, ${\displaystyle m}$ is a free parameter encoding the slope at ${\displaystyle x=0}$, which must be greater than or equal to ${\displaystyle {\sqrt {3}}}$ because any smaller value will result in a function with multiple inflection points, which is therefore not a true sigmoid. This function is unusual because it actually attains the limiting values of −1 and 1 within a finite range, meaning that its value is constant at −1 for all ${\displaystyle x\leq -1}$ and at 1 for all ${\displaystyle x\geq 1}$. Nonetheless, it is smooth (infinitely differentiable, ${\displaystyle C^{\infty }}$) everywhere, including at ${\displaystyle x=\pm 1}$.

Many natural processes, such as those of complex systems learning curves, exhibit a progression from small beginnings that accelerates and approaches a climax over time.^[18] When a specific mathematical model is lacking, a sigmoid function is often used.^[19]

The van Genuchten–Gupta model is based on an inverted S-curve and applied to the response of crop yield to soil salinity.

Examples of the application of the logistic S-curve to the response of crop yield (wheat) to both the soil salinity and depth to water table in the soil are shown in modeling crop response in agriculture.

In artificial neural networks, non-smooth functions are sometimes used for efficiency; these are known as hard sigmoids.

In audio signal processing, sigmoid functions are used as waveshaper transfer functions to emulate the sound of analog circuitry clipping.^[20]

In Digital signal processing in general, sigmoid functions, due to their higher order of continuity, have much faster asymptotic rolloff in the frequency domain than a Heavyside step function, and therefore are useful to smooth discontinuities before sampling to reduce aliasing. This is, for example, used to generate square waves in many kinds of Digital synthesizer.

In biochemistry and pharmacology, the Hill and Hill–Langmuir equations are sigmoid functions.

In computer graphics and real-time rendering, sigmoid functions are used to blend colors or geometry between two values, producing smooth transitions without visible seams or discontinuities.

Titration curves between strong acids and strong bases have a sigmoid shape due to the logarithmic nature of the pH scale.

The logistic function can be calculated efficiently by utilizing type III Unums.^[21]

A hierarchy of sigmoid growth models with increasing complexity (number of parameters) was built^[22] with the primary goal to re-analyze kinetic data, the so-called N-t curves, from heterogeneous nucleation experiments,^[23] in electrochemistry. The hierarchy includes at present three models, with 1, 2, and 3 parameters, if not counting the maximal number of nuclei N_max, respectively—a tanh² based model called α₂₁^[24] originally devised to describe diffusion-limited crystal growth (not aggregation!) in 2D, the Johnson–Mehl–Avrami–Kolmogorov (JMAK) model,^[25] and the Richards model.^[26] It was shown that for the concrete purpose, even the simplest model works, and thus it was implied that the experiments revisited are an example of two-step nucleation with the first step being the growth of the metastable phase in which the nuclei of the stable phase form.^[22]