Editing Generalised logistic function

{{Short description|Mathematical function}}
[[Image:Generalized logistic function A0 K1 B1.5 Q0.5 ν0.5 M0.5.png|thumb|right|A=M=0, K=C=1, B=3, ν=0.5, Q=0.5]]
[[File:GeneralizedLogisticA.svg|thumb|right|Effect of varying parameter A.  All other parameters are 1.]]
[[File:GeneralizedLogisticB.svg|thumb|right|Effect of varying parameter B.  A = 0, all other parameters are 1.]]
[[File:GeneralizedLogisticC.svg|thumb|right|Effect of varying parameter C.  A = 0, all other parameters are 1.]]
[[File:GeneralizedLogisticK.svg|thumb|right|Effect of varying parameter K.  A = 0, all other parameters are 1.]]
[[File:GeneralizedLogisticQ.svg|thumb|right|Effect of varying parameter Q.  A = 0, all other parameters are 1.]]
[[File:GeneralizedLogisticNu.svg|thumb|right|Effect of varying parameter <math>\nu</math>.  A = 0, all other parameters are 1.]]
The '''generalized logistic function''' or '''curve''' is an extension of the [[logistic function|logistic]] or [[sigmoid function|sigmoid]] functions. Originally developed for growth modelling, it allows for more flexible S-shaped curves. The function is sometimes named '''Richards's curve''' after [[Francis John Richards|F.{{nbsp}}J.{{nbsp}}Richards]], who proposed the general form for the family of models in 1959.

==Definition==
Richards's curve has the following form:
:<math>Y(t) = A + { K-A \over (C + Q e^{-B t}) ^ {1 / \nu} }</math>
where <math>Y</math> = weight, height, size etc., and <math>t</math> = time. It has six parameters:
*<math>A</math>: the left horizontal asymptote;
*<math>K</math>: the right horizontal asymptote when <math>C=1</math>. If <math>A=0</math> and <math>C=1</math> then <math>K</math> is called the [[carrying capacity]];
*<math>B</math>: the growth rate;
*<math>\nu > 0</math> : affects near which asymptote maximum growth occurs.
*<math>Q</math>: is related to the value <math>Y(0)</math>
*<math>C</math>: typically takes a value of 1. Otherwise, the upper asymptote is <math>A + {K - A \over C^{\, 1 / \nu}}</math>

The equation can also be written:

:<math>Y(t) = A + { K-A \over (C + e^{-B(t - M)}) ^ {1 / \nu} }</math>

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

:<math>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>Y</math> at that time.

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

==Generalised logistic differential equation==
A particular case of the generalised logistic function is:

:<math>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>Y^{\prime}(t) = \alpha  \left(1 - \left(\frac{Y}{K} \right)^{\nu} \right)Y </math>

with initial condition

:<math>Y(t_0) = Y_0 </math>

where

:<math>Q = -1 + \left(\frac {K}{Y_0} \right)^{\nu}</math>

provided that <math>\nu > 0</math> and <math>\alpha > 0</math>

The classical logistic differential equation is a particular case of the above equation, with <math>\nu =1</math>, whereas the [[Gompertz curve]] can be recovered in the limit <math>\nu \rightarrow 0^+</math> provided that:

:<math>\alpha = O\left(\frac{1}{\nu}\right)</math>

In fact, for small <math>\nu</math> it is

:<math>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>t</math> (see<ref name=fekedulegn1999parameter>{{cite journal|last=Fekedulegn|first=Desta|author2=Mairitin P. Mac Siurtain|author3=Jim J. Colbert|title=Parameter Estimation of Nonlinear Growth Models in Forestry|journal=Silva Fennica|year=1999|volume=33|issue=4|pages=327–336|doi=10.14214/sf.653|url=http://www.metla.fi/silvafennica/full/sf33/sf334327.pdf|access-date=2011-05-31|archive-url=https://web.archive.org/web/20110929005929/http://www.metla.fi/silvafennica/full/sf33/sf334327.pdf|archive-date=2011-09-29|url-status=dead}}</ref>).  For the case where <math>C = 1</math>, 
:<math>
\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><!-- DY/dt missing .... -->


==Special cases==
The following functions are specific cases of Richards's curves:
* [[Logistic function]]
* [[Gompertz curve]]
* [[Von Bertalanffy function]]
* Monomolecular curve

==Footnotes==
{{reflist}}

==References==
*{{cite journal |last=Richards |first=F. J. |year=1959 |title=A Flexible Growth Function for Empirical Use |journal=[[Journal of Experimental Botany]] |volume=10 |issue=2 |pages=290–300 |doi=10.1093/jxb/10.2.290 }}
*{{cite journal |last1=Pella |first1=J. S. |first2=P. K. |last2=Tomlinson |year=1969 |title=A Generalised Stock-Production Model |journal=Bull. Inter-Am. Trop. Tuna Comm |volume=13 |pages=421–496 }}
*{{cite journal |last1=Lei |first1=Y. C. |last2=Zhang |first2=S. Y. |year=2004 |title=Features and Partial Derivatives of Bertalanffy–Richards Growth Model in Forestry |journal=Nonlinear Analysis: Modelling and Control |volume=9 |issue=1 |pages=65–73 |doi=10.15388/NA.2004.9.1.15171 }}
[[Category:Growth curves]]
[[Category:Mathematical modeling]]