A logistic function or logistic curve is a common S-shaped curve (sigmoid curve) with the equation
f ( x ) = L 1 + e − k ( x − x 0 ) {\displaystyle f(x)={\frac {L}{1+e^{-k(x-x_{0})}}}}
where
L {\displaystyle L} is the carrying capacity, the supremum of the values of the function;
k {\displaystyle k} is the logistic growth rate, the steepness of the curve; and
x 0 {\displaystyle x_{0}} is the x {\displaystyle x} value of the function's midpoint. The logistic function has domain the real numbers, the limit as x → − ∞ {\displaystyle x\to -\infty } is 0, and the limit as x → + ∞ {\displaystyle x\to +\infty } is L {\displaystyle L} .
The exponential function with negated argument ( e − x {\displaystyle e^{-x}} ) is used to define the standard logistic function where L = 1 , k = 1 , x 0 = 0 {\displaystyle L=1,k=1,x_{0}=0} , which has the equation
f ( x ) = 1 1 + e − x {\displaystyle f(x)={\frac {1}{1+e^{-x}}}}
and is sometimes simply called the sigmoid function. It is also sometimes called the expit, being the inverse function of the logit. 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.
History
The logistic function was introduced in a series of three papers by Pierre François Verhulst between 1838 and 1847, who devised it as a model of population growth by adjusting the exponential growth model, under the guidance of Adolphe Quetelet. Verhulst first devised the function in the mid 1830s, publishing a brief note in 1838, then presented an expanded analysis and named the function in 1844 (published 1845); the third paper adjusted the correction term in his model of Belgian population growth. The initial stage of growth is approximately exponential (geometric); then, as saturation begins, the growth slows to linear (arithmetic), and at maturity, growth approaches the limit with an exponentially decaying gap, like the initial stage in reverse. Verhulst did not explain the choice of the term "logistic" (French: logistique), but it is presumably in contrast to the logarithmic curve, and by analogy with arithmetic and geometric. His growth model is preceded by a discussion of arithmetic growth and geometric growth (whose curve he calls a logarithmic curve, instead of the modern term exponential curve), and thus "logistic growth" is presumably named by analogy, logistic being from Ancient Greek: λογιστικός, romanized: logistikós, a traditional division of Greek mathematics. As a word derived from ancient Greek mathematical terms, the name of this function is unrelated to the military and management term logistics, which is instead from French: logis "lodgings", though some believe the Greek term also influenced logistics; see Logistics § Origin for details.
Mathematical properties The standard logistic function is the logistic function with parameters k = 1 {\displaystyle k=1} , x 0 = 0 {\displaystyle x_{0}=0} , L = 1 {\displaystyle L=1} , which yields
f ( x ) = 1 1 + e − x = e x e x + 1 = e x / 2 e x / 2 + e − x / 2 . {\displaystyle f(x)={\frac {1}{1+e^{-x}}}={\frac {e^{x}}{e^{x}+1}}={\frac {e^{x/2}}{e^{x/2}+e^{-x/2}}}.}
… excerpt ends here. Continue reading the full article.
