Logistic function

The lo­gis­tic func­tion is a sig­moid func­tion that maps the real num­bers to the unit in­ter­val \((0, 1)\) us­ing the for­mula \(\displaystyle f(x) = \frac{1}{1 + e^{-x}}\).

More gen­er­ally, there ex­ists a fam­ily of lo­gis­tic func­tions that can be writ­ten as \(\displaystyle f(x) = \frac{M}{1 + \alpha^{c(x_0 - x)}}\), where:

  • \(M\) is the up­per bound of the func­tion (in which case the func­tion maps to the in­ter­val \((0, M)\) in­stead). When \(M = 1\), the lo­gis­tic func­tion is usu­ally be­ing used to calcu­late some prob­a­bil­ity or pro­por­tion of a to­tal.

  • \(x_0\) is the in­flec­tion point of the curve, or the value of \(x\) where the func­tion’s growth stops speed­ing up and starts slow­ing down.

  • \(\alpha\) is a vari­able con­trol­ling the steep­ness of the curve.

  • \(c\) is a scal­ing fac­tor for the dis­tance.

Applications

  • The lo­gis­tic func­tion is used to model growth rates of pop­u­la­tions in an ecosys­tem with a limited car­ry­ing ca­pac­ity.

  • The in­verse lo­gis­tic func­tion (with \(\alpha = 2\)) is used to con­vert a prob­a­bil­ity to log-odds form for use in Bayes’ rule.

  • The lo­gis­tic func­tion (with \(\alpha = 10\) and \(c = 1/400\)) is used to calcu­late the ex­pected prob­a­bil­ity of a player win­ning given a spe­cific differ­ence in rat­ing in the Elo rat­ing sys­tem.

Parents:

  • Mathematics

    Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.