10-565: A sigmoid function is a function whose graph follows the logistic function . It is defined by the formula: 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
20-444: 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
30-668: 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 x ≤ − 1 {\displaystyle x\leq -1} and at 1 for all x ≥ 1 {\displaystyle x\geq 1} . Nonetheless, it is smooth (infinitely differentiable, C ∞ {\displaystyle C^{\infty }} ) everywhere , including at x = ± 1 {\displaystyle x=\pm 1} . Many natural processes, such as those of complex system learning curves , exhibit
40-1182: 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. f ( x ) = { 2 1 + e − 2 m x 1 − x 2 − 1 , | x | < 1 sgn ( x ) | x | ≥ 1 = { tanh ( m x 1 − x 2 ) , | x | < 1 sgn ( x ) | x | ≥ 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
50-550: 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 → ± ∞ {\displaystyle x\rightarrow \pm \infty } . A sigmoid function
60-489: 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
70-404: The hyperbolic tangent mentioned above. Here, m {\displaystyle m} is a free parameter encoding the slope at x = 0 {\displaystyle x=0} , which must be greater than or equal to 3 {\displaystyle {\sqrt {3}}} because any smaller value will result in a function with multiple inflection points, which is therefore not
80-423: 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
90-426: 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 . Graph of a function Too Many Requests If you report this error to
100-558: 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
#17982