|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| [[Image:Generalized logistic function A0 K1 B1.5 Q0.5 ν0.5 M0.5.png|thumb|right|A=0, K=1, B=3, Q=ν=0.5, M=0]]
| | Hello, dear friend! My name is Scott. I smile that I could join to the whole world. I live in Netherlands, in the south region. I dream to see the different nations, to look for acquainted with interesting individuals.<br><br>Here is my page :: [http://www.ariston-oliveoil.gr/index.asp?Keyword=0000001824 σαπουνι με ελαιολαδο] |
| | |
| The '''generalised logistic curve''' or '''function''', also known as '''Richards' curve''' is a widely used and flexible [[sigmoid function]] for growth modelling, extending the well-known [[logistic function]].
| |
| | |
| :<math>Y(t) = A + { K-A \over (1 + Q e^{-B(t - M)}) ^ {1 / \nu} }</math>
| |
| | |
| where ''Y'' = weight, height, size etc., and ''t'' = time.
| |
| | |
| It has six parameters:
| |
| *''A'': the lower asymptote;
| |
| *''K'': the upper asymptote. If ''A''=0 then ''K'' is called the carrying capacity;
| |
| *''B'': the growth rate;
| |
| *ν>0 : affects near which asymptote maximum growth occurs.
| |
| *''Q'': depends on the value ''Y''(0)
| |
| *''M'': the time of maximum growth if ''Q''=ν
| |
| | |
| == Generalised logistic differential equation ==
| |
| A particular case of Richard's function is:
| |
| | |
| :<math>Y(t) = { K \over (1 + Q e^{- \alpha \nu (t - t_0)}) ^ {1 / \nu} }</math>
| |
| | |
| which is the solution of the so-called Richard'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 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 <math>\nu \rightarrow 0^+</math> provided that:
| |
| | |
| :<math>\alpha = O\left(\frac{1}{\nu}\right)</math>
| |
| | |
| In fact, for small ν 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 suits to model many growth phenomena, including the growth of tumours. Concerning its applications in oncology, its main biological features are similar to those of [[Logistic curve]] model.
| |
| | |
| ==Gradient==
| |
| When estimating parameters from data, it is often necessary to compute the partial derivatives of the parameters at a given data point ''t'' (see <ref name=fekedulegn1999parameter>{{cite journal|last=Fekedulegn|first=Desta|coauthors=Mairitin P. Mac Siurtain, Jim J. Colbert|title=Parameter Estimation of Nonlinear Growth Models in Forestry|journal=Silva Fennica|year=1999|volume=33|issue=4|pages=327–336|url=http://www.metla.fi/silvafennica/full/sf33/sf334327.pdf|accessdate=2011-05-31}}</ref>):
| |
| :<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)Be^{-B(t-M)}}{\nu(1 + Qe^{-B(t-M)})^{\frac{1}{\nu}+1}}
| |
| \end{align}
| |
| </math>
| |
| | |
| ==See also==
| |
| *[[Logistic function]]
| |
| *[[Gompertz curve]]
| |
| *[[Ludwig von Bertalanffy]]
| |
| | |
| ==Citations==
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| * Richards, F.J. 1959 ''A flexible growth function for empirical use''. J. Exp. Bot. 10: 290-300.
| |
| * Pella JS and PK Tomlinson. 1969. ''A generalised stock-production model''. Bull. IATTC 13: 421-496.
| |
| * Lei, Y.C. and Zhang, S.Y. 2004. ''Features and Partial Derivatives of Bertalanffy-Richards Growth Model in Forestry''. Nonlinear Analysis: Modelling and Control, Vol 9, No. 1:65-73
| |
| | |
| | |
| | |
| | |
| [[Category:Curves]]
| |
Hello, dear friend! My name is Scott. I smile that I could join to the whole world. I live in Netherlands, in the south region. I dream to see the different nations, to look for acquainted with interesting individuals.
Here is my page :: σαπουνι με ελαιολαδο