Logistic function

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

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

where

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

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

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 $L=1,k=1,x_{0}=0$ , is sometimes simply called the sigmoid. It is also sometimes called the expit, being the inverse of the logit.