Changes between Version 10 and Version 11 of OfficialTolArchiveNetworkQltvRespModel

Timestamp:

Dec 21, 2010, 12:17:10 PM (15 years ago)

Author:

Víctor de Buen Remiro

Comment:

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OfficialTolArchiveNetworkQltvRespModel

@@ -74 +74 @@
   [[LatexEquation( f'\left(z\right) = - f\left(z\right) F\left(z\right) {\left(1-e^{-z}\right)} )]] [[BR]] [[BR]]
 
-  [[LatexEquation( L\left(\beta\right)=\underset{i}{\sum}w_{i}\left(y_{i}x_{i}^{t}\beta-\ln\left(1+e^{x_{i}^{t}\beta}\right)\right) )]] 
+  [[LatexEquation( L\left(\beta\right)=\underset{i}{\sum}w_{i}\left(y_{i}x_{i}^{t}\beta-\ln\left(1+e^{-x_{i}^{t}\beta}\right)\right) )]] 
 
-  [[LatexEquation( \frac{\partial L\left(\beta\right)}{\partial\beta_{j}}=\underset{i}{\sum}w_{i}\left(y_{i}x_{ij}-x_{ij}\left(\frac{e^{x_{i}^{t}\beta}}{1+e^{x_{i}^{t}\beta}}\right)\right)=\underset{i}{\sum}w_{i}x_{ij}\left(y_{i}-\left(\frac{e^{x_{i}^{t}\beta}}{1+e^{x_{i}^{t}\beta}}\right)\right) )]] 
+  [[LatexEquation( \frac{\partial L\left(\beta\right)}{\partial\beta_{j}}=\underset{i}{\sum}w_{i}\left(y_{i}x_{ij}-x_{ij}\left(\frac{e^{-x_{i}^{t}\beta}}{1+e^{-x_{i}^{t}\beta}}\right)\right)=\underset{i}{\sum}w_{i}x_{ij}\left(y_{i}-\left(\frac{e^{-x_{i}^{t}\beta}}{1+e^{-x_{i}^{t}\beta}}\right)\right) )]] 
 
-  [[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) )]] 
+  [[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
+
+  [[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
+  
+  [[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]]
 
 
@@ -100 +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} )]] 
+
+