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.

= weight, height, size etc., and

can be thought of as a starting time, at which

can be convenient: this representation simplifies the setting of both a starting time and the value of

The logistic function, with maximum growth rate at time

A particular case of the generalised logistic function is: which is the solution of the Richards's differential equation (RDE): with initial condition where provided that

, whereas the Gompertz curve can be recovered in the limit

it is The RDE models many growth phenomena, arising in fields such as oncology and epidemiology.

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

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

A=M=0, K=C=1, B=3, ν=0.5, Q=0.5
Effect of varying parameter A. All other parameters are 1.
Effect of varying parameter B. A = 0, all other parameters are 1.
Effect of varying parameter C. A = 0, all other parameters are 1.
Effect of varying parameter K. A = 0, all other parameters are 1.
Effect of varying parameter Q. A = 0, all other parameters are 1.
Effect of varying parameter . A = 0, all other parameters are 1.