Mathematical function having a characteristic S-shaped curve or sigmoid curve
A sigmoid function is any
mathematical function whose
graph has 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:[1]
Other standard 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 an alias for the logistic function.
Special cases of the sigmoid function 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.
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[1][2] and exactly one
inflection point.
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.
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.[6]
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 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.[8]
Mitchell, Tom M. (1997). Machine Learning. WCB
McGraw–Hill.
ISBN978-0-07-042807-2.. (NB. In particular see "Chapter 4: Artificial Neural Networks" (in particular pp. 96–97) where Mitchell uses the word "logistic function" and the "sigmoid function" synonymously – this function he also calls the "squashing function" – and the sigmoid (aka logistic) function is used to compress the outputs of the "neurons" in multi-layer neural nets.)
Humphrys, Mark.
"Continuous output, the sigmoid function".
Archived from the original on 2022-07-14. Retrieved 2022-07-14. (NB. Properties of the sigmoid, including how it can shift along axes and how its domain may be transformed.)