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.

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.

Definition

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Richards's curve has the following form:

 

where   = weight, height, size etc., and   = time. It has six parameters:

  •  : the left horizontal asymptote;
  •  : the right horizontal asymptote when  . If   and   then   is called the carrying capacity;
  •  : the growth rate;
  •   : affects near which asymptote maximum growth occurs.
  •  : is related to the value  
  •  : typically takes a value of 1. Otherwise, the upper asymptote is  

The equation can also be written:

 

where   can be thought of as a starting time, at which  . Including both   and   can be convenient:

 

this representation simplifies the setting of both a starting time and the value of   at that time.

The logistic function, with maximum growth rate at time  , is the case where  .

Generalised logistic differential equation

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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   and  

The classical logistic differential equation is a particular case of the above equation, with  , whereas the Gompertz curve can be recovered in the limit   provided that:

 

In fact, for small   it is

 

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

Gradient of generalized logistic function

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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   (see[1]). For the case where  ,

 


Special cases

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

Footnotes

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  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.

References

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  • Richards, F. J. (1959). "A Flexible Growth Function for Empirical Use". Journal of Experimental Botany. 10 (2): 290–300. doi:10.1093/jxb/10.2.290.
  • Pella, J. S.; Tomlinson, P. K. (1969). "A Generalised Stock-Production Model". Bull. Inter-Am. Trop. Tuna Comm. 13: 421–496.
  • Lei, Y. C.; Zhang, S. Y. (2004). "Features and Partial Derivatives of Bertalanffy–Richards Growth Model in Forestry". Nonlinear Analysis: Modelling and Control. 9 (1): 65–73. doi:10.15388/NA.2004.9.1.15171.