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[[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]]
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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;
*&nu;>0 : affects near which asymptote maximum growth occurs.  
*''Q'': depends on the value ''Y''(0)
*''M'': the time of maximum growth if ''Q''=&nu;
 
== 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 &nu; &gt; 0 and &alpha; &gt; 0.
 
 
The classical logistic differential equation is a particular case of the above equation, with &nu; =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 &nu; 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]]

Revision as of 07:53, 3 March 2014

Today I am sharing a dark experience with we because I am from this darkness today. Whoa! Google Chrome has crashed was the disaster that struck on me early this morning. My blood flow seemed to stop for a few minutes. Then I regained my senses plus decided to fix Chrome crash.

Another answer is to provide the computer program with a new msvcr71 file. Often, once the file has been corrupted or damaged, it usually no longer be able to function like it did before thus it's just natural to substitute the file. Just download another msvcr71.dll file from the web. Often, the file will come in a zip structure. Extract the files within the zip folder plus region them accordingly inside this location: C:\Windows\System32. Afterward, re-register the file. Click Start and then choose Run. When the Run window appears, sort "cmd". Press Enter plus then kind "regsvr32 -u msvcr71.dll" followed by "regsvr32 msvcr71.dll". Press Enter again and the file ought to be registered accordingly.

So, this advanced double scan is not only among the better, however, it really is moreover freeware. And as of all this that numerous regard CCleaner 1 of the better registry products inside the marketplace today. I would add that I personally choose Regcure for the simple reason which it has a greater interface plus I learn for a fact it is ad-ware without charge.

Chrome allows customizing itself by applying range of themes accessible online. If you had recently used a theme that no longer functions correctly, it results in Chrome crash on Windows 7. It is suggested to set the authentic theme.

Whenever it comes to software, this might be the vital piece because it is the 1 running a system and alternative programs needed inside a functions. Always keep the cleanliness of the program from obsolete information by getting a advantageous tuneup utilities 2014. Protect it from a virus on the net by providing a workable virus security system. You should moreover have a monthly clean up by running the defragmenter system. This way it might enhance the performance of your computer plus for we to avoid any mistakes. If you think something is incorrect with all the software, plus we don't know how to fix it then refer to a technician.

If you think that there are issues with the d3d9.dll file, then you need to substitute it with a new functioning file. This could be done by conducting a series of procedures plus you are able to begin by downloading "d3d9.zip" within the host. Next you need to unzip the "d3d9.dll" file on the difficult drive of the computer. Proceed by locating "C:\Windows\System32" and then finding the existing "d3d9.dll" on the PC. Once found, rename the file "d3d9.dll to d3d9BACKUP.dll" plus then copy-paste this brand-new file to "C:\Windows\System32". After which, hit "Start" followed by "Run" or look "Run" on Windows Vista & 7. As soon because a box shows up, kind "cmd". A black screen may then appear and you need to type "regsvr32d3d9.dll" and then click "Enter". This task will help we to replace the old file with all the fresh copy.

By restoring the state of the program to an earlier date, error 1721 could not appear in Windows 7, Vista plus XP. There is a tool called System Restore that we have to utilize inside this procedure.

So in summary, when comparing registry cleaning, make sure the 1 you choose offers you the following.A backup and restore center, fast surgery, automatic deletion center, start-up administration, an easy technique of contact and a income back guarantee.

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