A sigmoid function is a function whose graph follows the logistic function. It is defined by the formula:
\sigma(x)=
1 | |
1+e-x |
=
ex | |
1+ex |
=1-\sigma(-x).
In many fields, especially in the context of artificial neural networks, the term "sigmoid function" is correctly recognized as a synonym for the logistic function. While other S-shaped curves, such as the Gompertz curve or the ogee curve, may resemble sigmoid functions, they are distinct mathematical functions with different properties and applications.
Sigmoid functions, particularly the logistic function, have a domain of all real numbers and typically produce output values in the range from 0 to 1, although some variations, like the hyperbolic tangent, produce output values between −1 and 1. These functions are commonly used as activation functions in artificial neurons and as cumulative distribution functions in statistics. The logistic sigmoid is also invertible, with its inverse being the logit function.
A sigmoid function is a bounded, differentiable, real function that is defined for all real input values and has a non-negative derivative at each point and exactly one inflection 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
x → \pminfty
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.
, & |x| \le 1 \\\\\sgn(x) & |x| \ge 1 \\\end \quad N \in \mathbb \ge 1
\alpha<1
\beta<1
using the hyperbolic tangent mentioned above. Here,
m
x=0
\sqrt{3}
x\leq-1
x\geq1
Cinfty
x=\pm1
Many natural processes, such as those of complex system learning curves, exhibit a progression from small beginnings that accelerates and approaches a climax over time. When a specific mathematical model is lacking, a sigmoid function is often used.
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, sometimes non-smooth functions are used instead 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.
In biochemistry and pharmacology, the Hill and Hill–Langmuir equations are sigmoid functions.
In computer graphics and real-time rendering, some of the sigmoid functions are used to blend colors or geometry between two values, smoothly and 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.