Generalised logistic 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 [math]\displaystyle{ \nu }[/math]. 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:

[math]\displaystyle{ Y(t) = A + { K-A \over (C + Q e^{-B t}) ^ {1 / \nu} } }[/math]

where [math]\displaystyle{ Y }[/math] = weight, height, size etc., and [math]\displaystyle{ t }[/math] = time. It has six parameters:

* [math]\displaystyle{ A }[/math]: the left horizontal asymptote;
* [math]\displaystyle{ K }[/math]: the right horizontal asymptote when [math]\displaystyle{ C=1 }[/math]. If [math]\displaystyle{ A=0 }[/math] and [math]\displaystyle{ C=1 }[/math] then [math]\displaystyle{ K }[/math] is called the carrying capacity;
* [math]\displaystyle{ B }[/math]: the growth rate;
* [math]\displaystyle{ \nu \gt 0 }[/math] : affects near which asymptote maximum growth occurs.
* [math]\displaystyle{ Q }[/math]: is related to the value [math]\displaystyle{ Y(0) }[/math]
* [math]\displaystyle{ C }[/math]: typically takes a value of 1. Otherwise, the upper asymptote is [math]\displaystyle{ A + {K - A \over C^{\, 1 / \nu}} }[/math]

The equation can also be written:

[math]\displaystyle{ Y(t) = A + { K-A \over (C + e^{-B(t - M)}) ^ {1 / \nu} } }[/math]

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

[math]\displaystyle{ Y(t) = A + { K-A \over (C + Q e^{-B(t - M)}) ^ {1 / \nu} } }[/math]

this representation simplifies the setting of both a starting time and the value of [math]\displaystyle{ Y }[/math] at that time.

The logistic function, with maximum growth rate at time [math]\displaystyle{ M }[/math], is the case where [math]\displaystyle{ Q = \nu = 1 }[/math].

## Generalised logistic differential equation

A particular case of the generalised logistic function is:

[math]\displaystyle{ Y(t) = { K \over (1 + Q e^{- \alpha \nu (t - t_0)}) ^ {1 / \nu} } }[/math]

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

[math]\displaystyle{ Y^{\prime}(t) = \alpha \left(1 - \left(\frac{Y}{K} \right)^{\nu} \right)Y }[/math]

with initial condition

[math]\displaystyle{ Y(t_0) = Y_0 }[/math]

where

[math]\displaystyle{ Q = -1 + \left(\frac {K}{Y_0} \right)^{\nu} }[/math]

provided that [math]\displaystyle{ v \gt 0 }[/math] and [math]\displaystyle{ \alpha \gt 0 }[/math]

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 [math]\displaystyle{ \nu \rightarrow 0^+ }[/math] provided that:

[math]\displaystyle{ \alpha = O\left(\frac{1}{\nu}\right) }[/math]

In fact, for small ν it is

[math]\displaystyle{ Y^{\prime}(t) = Y r \frac{1-\exp\left(\nu \ln\left(\frac{Y}{K}\right) \right)}{\nu} \approx r Y \ln\left(\frac{Y}{K}\right) }[/math]

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 [math]\displaystyle{ t }[/math] (see[1]). For the case where [math]\displaystyle{ C = 1 }[/math],

[math]\displaystyle{ \begin{align} \\\ \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{align} }[/math]

## 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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