Logistic function

A logistic function or logistic curve is a common S-shaped curve (sigmoid curve) with the equation

${\displaystyle f(x)={\frac {L}{1+e^{-k(x-x_{0})}}}}$

where

${\displaystyle L}$ is the carrying capacity, the supremum of the values of the function;

${\displaystyle k}$ is the logistic growth rate, the steepness of the curve; and

${\displaystyle x_{0}}$ is the ${\displaystyle x}$ value of the function's midpoint.^[1]

The logistic function has domain the real numbers, the limit as ${\displaystyle x\to -\infty }$ is 0, and the limit as ${\displaystyle x\to +\infty }$ is ${\displaystyle L}$.

The standard logistic function, depicted at right, where ${\displaystyle L=1,k=1,x_{0}=0}$, has the equation

${\displaystyle f(x)={\frac {1}{1+e^{-x}}}}$

and is sometimes simply called the sigmoid.^[2] It is also sometimes called the expit, being the inverse function of the logit.^[3][4]

The logistic function finds applications in a range of fields, including biology (especially ecology), biomathematics, chemistry, demography, economics, geoscience, mathematical psychology, probability, sociology, political science, linguistics, statistics, and artificial neural networks. There are various generalizations, depending on the field.