Logistic function

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

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

where

${\displaystyle x_{0}}$, the ${\displaystyle x}$ value of the sigmoid's midpoint;
${\displaystyle L}$, the curve's maximum value;
${\displaystyle k}$, the logistic growth rate or steepness of the curve.[1]

For values of ${\displaystyle x}$ in the domain of real numbers from ${\displaystyle -\infty }$ to ${\displaystyle +\infty }$, the S-curve shown on the right is obtained, with the graph of ${\displaystyle f}$ approaching ${\displaystyle L}$ as ${\displaystyle x}$ approaches ${\displaystyle +\infty }$ and approaching zero as ${\displaystyle x}$ approaches ${\displaystyle -\infty }$.

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. A generalization of the logistic function is the hyperbolastic function of type I.

The standard logistic function, where ${\displaystyle L=1,k=1,x_{0}=0}$, is sometimes simply called the sigmoid.[2] It is also sometimes called the expit, being the inverse of the logit.[3][4]