Generalised logistic function

Short description: Mathematical function

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 ν. A = 0, all other parameters are 1.

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.

## Contents

* 1 Definition
* 2 Generalised logistic differential equation
* 3 Gradient of generalized logistic function
* 4 Special cases
* 5 Footnotes
* 6 References

## Definition

Richards's curve has the following form:

Y(t)=A+K−A(C+Qe−Bt)1/ν

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

* A: the left horizontal asymptote;
* K: the right horizontal asymptote when C=1. If A=0 and C=1 then K is called the carrying capacity;
* B: the growth rate;
* ν>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−AC1/ν

The equation can also be written:

Y(t)=A+K−A(C+e−B(t−M))1/ν

where M can be thought of as a starting time, at which Y(M)=A+K−A(C+1)1/ν. Including both Q and M can be convenient:

Y(t)=A+K−A(C+Qe−B(t−M))1/ν

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=ν=1.

## Generalised logistic differential equation

A particular case of the generalised logistic function is:

Y(t)=K(1+Qe−αν(t−t0))1/ν

which is the solution of the Richards's differential equation (RDE):

Y′(t)=α(1−(YK)ν)Y

with initial condition

Y(t0)=Y0

where

Q=−1+(KY0)ν

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 ν->0+ provided that:

α=O(1ν)

In fact, for small ν it is

Y′(t)=Yr1−exp⁡(νln⁡(YK))ν≈rYln⁡(YK)

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[1]). For the case where C=1,

∂Y∂A=1−(1+Qe−B(t−M))−1/ν∂Y∂K=(1+Qe−B(t−M))−1/ν∂Y∂B=(K−A)(t−M)Qe−B(t−M)ν(1+Qe−B(t−M))1ν+1∂Y∂ν=(K−A)ln⁡(1+Qe−B(t−M))ν2(1+Qe−B(t−M))1ν∂Y∂Q=−(K−A)e−B(t−M)ν(1+Qe−B(t−M))1ν+1∂Y∂M=−(K−A)QBe−B(t−M)ν(1+Qe−B(t−M))1ν+1

## Special cases

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

* Logistic function
* Gompertz curve
* Von Bertalanffy function
* Monomolecular curve

## Footnotes

1. ↑ Fekedulegn, Desta; Mairitin P. Mac Siurtain; Jim J. Colbert (1999). "Parameter Estimation of Nonlinear Growth Models in Forestry". Silva Fennica 33 (4): 327–336. doi:10.14214/sf.653. http://www.metla.fi/silvafennica/full/sf33/sf334327.pdf. Retrieved 2011-05-31.

## References

* 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.

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