# Generalised logistic function

The generalized logistic function or curve is an extension of the logistic or sigmoid functions. Originally developed for growth modelling, it allows for more flexible S-shaped curves. The function is sometimes named Richards's curve after F. J. Richards, who proposed the general form for the family of models in 1959.

## Definition

Richards's curve has the following form:

$Y(t)=A+{K-A \over (C+Qe^{-Bt})^{1/\nu }}$ where $Y$ = weight, height, size etc., and $t$ = time. It has six parameters:

• $A$ : the lower (left) asymptote;
• $K$ : the upper (right) asymptote when $C=1$ . If $A=0$ and $C=1$ then $K$ is called the carrying capacity;
• $B$ : the growth rate;
• $\nu >0$ : affects near which asymptote maximum growth occurs.
• $Q$ : is related to the value $Y(0)$ • $C$ : typically takes a value of 1. Otherwise, the upper asymptote is $A+{K-A \over C^{\,1/\nu }}$ The equation can also be written:

$Y(t)=A+{K-A \over (C+e^{-B(t-M)})^{1/\nu }}$ where $M$ can be thought of as a starting time, at which $Y(M)=A+{K-A \over (C+1)^{1/\nu }}$ . Including both $Q$ and $M$ can be convenient:

$Y(t)=A+{K-A \over (C+Qe^{-B(t-M)})^{1/\nu }}$ this representation simplifies the setting of both a starting time and the value of $Y$ at that time.

The logistic function, with maximum growth rate at time $M$ , is the case where $Q=\nu =1$ .

## Generalised logistic differential equation

A particular case of the generalised logistic function is:

$Y(t)={K \over (1+Qe^{-\alpha \nu (t-t_{0})})^{1/\nu }}$ which is the solution of the Richards's differential equation (RDE):

$Y^{\prime }(t)=\alpha \left(1-\left({\frac {Y}{K}}\right)^{\nu }\right)Y$ with initial condition

$Y(t_{0})=Y_{0}$ where

$Q=-1+\left({\frac {K}{Y_{0}}}\right)^{\nu }$ provided that ν > 0 and α > 0.

The classical logistic differential equation is a particular case of the above equation, with ν =1, whereas the Gompertz curve can be recovered in the limit $\nu \rightarrow 0^{+}$ provided that:

$\alpha =O\left({\frac {1}{\nu }}\right)$ In fact, for small ν it is

$Y^{\prime }(t)=Yr{\frac {1-\exp \left(\nu \ln \left({\frac {Y}{K}}\right)\right)}{\nu }}\approx rY\ln \left({\frac {Y}{K}}\right)$ The RDE models many growth phenomena, arising in fields such as oncology and epidemiology.

## Gradient of generalized logistic function

When estimating parameters from data, it is often necessary to compute the partial derivatives of the logistic function with respect to parameters at a given data point $t$ (see). For the case where $C=1$ ,

{\begin{aligned}\\{\frac {\partial Y}{\partial A}}&=1-(1+Qe^{-B(t-M)})^{-1/\nu }\\\\{\frac {\partial Y}{\partial K}}&=(1+Qe^{-B(t-M)})^{-1/\nu }\\\\{\frac {\partial Y}{\partial B}}&={\frac {(K-A)(t-M)Qe^{-B(t-M)}}{\nu (1+Qe^{-B(t-M)})^{{\frac {1}{\nu }}+1}}}\\\\{\frac {\partial Y}{\partial \nu }}&={\frac {(K-A)\ln(1+Qe^{-B(t-M)})}{\nu ^{2}(1+Qe^{-B(t-M)})^{\frac {1}{\nu }}}}\\\\{\frac {\partial Y}{\partial Q}}&=-{\frac {(K-A)e^{-B(t-M)}}{\nu (1+Qe^{-B(t-M)})^{{\frac {1}{\nu }}+1}}}\\\\{\frac {\partial Y}{\partial M}}&=-{\frac {(K-A)QBe^{-B(t-M)}}{\nu (1+Qe^{-B(t-M)})^{{\frac {1}{\nu }}+1}}}\\\end{aligned}} ## Special cases

The following functions are specific cases of Richards's curves:

## Footnotes

1. ^ Fekedulegn, Desta; Mairitin P. Mac Siurtain; Jim J. Colbert (1999). "Parameter Estimation of Nonlinear Growth Models in Forestry" (PDF). Silva Fennica. 33 (4): 327–336. doi:10.14214/sf.653. Archived from the original (PDF) on 2011-09-29. Retrieved 2011-05-31.