Changes between Version 11 and Version 12 of OfficialTolArchiveNetworkQltvRespModel

Timestamp:

Dec 21, 2010, 1:36:51 PM (15 years ago)

Author:

Víctor de Buen Remiro

Comment:

--

OfficialTolArchiveNetworkQltvRespModel

@@ -80 +80 @@
   [[LatexEquation( \frac{\partial L\left(\beta\right)}{\partial\beta_{i}\partial_{j}}=-\underset{k}{\sum}w_{k}\frac{\left(1+e^{-x_{k}^{t}\beta}\right)e^{-x_{k}^{t}\beta}x_{ki}x_{kj}-\left(e^{-x_{k}^{t}\beta}\right)^{2}x_{ki}x_{kj}}{\left(1+e^{-x_{k}^{t}\beta}\right)}=-\underset{k}{\sum}x_{ki}x_{kj}w_{k}\pi_{i}\left(1-\pi_{i}\right) )]] 
 
-Desde el punto de vista del cálculo numérico en aritmética discreta hay que tener en cuenta que
+From the standpoint of arithmetic discrete numerical calculation must take into account that[[BR]]
 
   [[LatexEquation( e^{710}=\infty )]] [[BR]] [[BR]]
   
-por lo que hay que tratar de forma cuidadosa con las expresiones que contienen la exponencial. 
-En este caso concreto se hará uso de las siguientes igualdades asintóticas
+For this reason we must carefully try to contain the exponential expressions.
+In this case it will use the following asymptotic equalities[[BR]] 
   
   [[LatexEquation( \ln\left(1+e^{-z}\right)\;\overset{z\rightarrow-\infty}{\longrightarrow}\;-z )]] [[BR]] [[BR]]
-  
   [[LatexEquation( \frac{e^{-z}}{1+e^{-z}}\;\overset{z\rightarrow-\infty}{\longrightarrow}\;1 )]] [[BR]] [[BR]]
 
@@ -112 +111 @@
   [[LatexEquation( \frac{\partial L\left(\beta\right)}{\partial\beta_{i}\partial_{j}}=-\underset{k}{\sum}w_{k}\phi\left(x_{k}\beta\right)\left(y_{k}\frac{z\Phi\left(x_{k}\beta\right)+\phi\left(x_{k}\beta\right)}{\Phi^{2}\left(x_{k}\beta\right)}+\left(1-y_{k}\right)\frac{-z \Phi\left(-x_{k}\beta\right)+\phi\left(x_{k}\beta\right)}{\Phi\left(-x_{k}\beta\right)^{2}}\right)x_{ik}x_{jk} )]] 
 
+To avoid numerical problems will use the following equality
   
+  [[LatexEquation( \ln\left(\Phi\left(z\right)\right)=\ln\left(1-erf\left(\frac{-z}{\sqrt{2}}\right)\right)-\ln2 )]] [[BR]] [[BR]]
+  
+The function logarithm of complemetary error function
+
+[[LatexEquation( \ln\left(1-erf\left(u\right)\right) )]] 
+
+is implemented as 
+[http://www.gnu.org/software/gsl/manual/html_node/Log-Complementary-Error-Function.html gsl_sf_log_erfc] 
+that is available in TOL.
+
+The Hazard function [[BR]] [[BR]]
+
+[[LatexEquation( h\left(z\right)=\frac{\phi\left(z\right)}{\Phi\left(z\right)} )]] 
+
+decreases rapidly as [[LatexEquation(z)]] approaches [[LatexEquation(-\infty)]] and asymptotes to 
+[[LatexEquation(h\left(z\right) \sim z )]] as [[LatexEquation(z)]] approaches [[LatexEquation(+\infty.)]]
+
+Hazard function is implemented as [http://www.gnu.org/software/gsl/manual/html_node/Probability-functions.html gsl_sf_hazard ] 
+that is also available in TOL.