Changes between Version 18 and Version 19 of OfficialTolArchiveNetworkQltvRespModel

Timestamp:

Dec 22, 2010, 9:26:53 AM (14 years ago)

Author:

Víctor de Buen Remiro

Comment:

--

OfficialTolArchiveNetworkQltvRespModel

@@ -69 +69 @@
 In this case we have that scalar distribution is the logistic one.
 
+''Scalar cumulant'': [[BR]] 
   [[LatexEquation( F\left(z\right) = \frac{1}{1+e^{-z}} )]] [[BR]] [[BR]]
 
+''Scalar density'': [[BR]] 
   [[LatexEquation( f\left(z\right) = \frac{e^{-z}}{\left(1+e^{-z}\right)^2} )]] [[BR]] [[BR]]
 
+''Scalar density derivative'': [[BR]] 
   [[LatexEquation( f'\left(z\right) = - f\left(z\right) F\left(z\right) {\left(1-e^{-z}\right)} )]] [[BR]] [[BR]]
 
+''Logarithm of likelihood'': [[BR]]
   [[LatexEquation( L\left(\beta\right)=\underset{i}{-\sum}w_{i}\left(\ln\left(1+e^{-x_{i}^{t}\beta}\right)+\left(1-y_{i}\right)x_{i}^{t}\beta\right) )]] 
 
+''Gradient'': [[BR]] 
   [[LatexEquation( \frac{\partial L\left(\beta\right)}{\partial\beta_{j}}=\underset{i}{\sum}w_{i}\left(\frac{e^{-x_{i}^{t}\beta}}{1+e^{-x_{i}^{t}\beta}}-\left(1-y_{i}\right)\right)x_{ij} )]] 
 
-  [[LatexEquation( \frac{\partial L\left(\beta\right)}{\partial\beta_{i}\partial_{j}}=\underset{k}{\sum}w_{k}\frac{-e^{-x_{k}^{t}\beta}}{\left(1+e^{-x_{k}^{t}\beta}\right)^{2}}x_{ki}x_{kj}=-\underset{k}{\sum}x_{ki}x_{kj}w_{k}\pi_{i}\left(1-\pi_{i}\right) )]] 
+''Hessian'': [[BR]] 
+  [[LatexEquation( \frac{\partial L\left(\beta\right)}{\partial\beta_{i}\partial_{j}}=\underset{k}{\sum}w_{k}\frac{-e^{-x_{k}^{t}\beta}}{\left(1+e^{-x_{k}^{t}\beta}\right)^{2}}x_{ki}x_{kj}=-\underset{k}{\sum}x_{ki}x_{kj}w_{k}\pi_{k}\left(1-\pi_{k}\right) )]] 
 
 From the standpoint of arithmetic discrete numerical calculation must take into account that[[BR]]
@@ -100 +106 @@
 In this case we have that scalar distribution is the standard normal one.
 
+''Scalar cumulant'': [[BR]] 
   [[LatexEquation( F\left(z\right) = \Phi\left(z\right) )]] [[BR]] [[BR]]
 
+''Scalar density'': [[BR]] 
   [[LatexEquation( f\left(z\right) = \phi\left(z\right) )]] [[BR]] [[BR]]
 
+''Scalar density derivative'': [[BR]] 
   [[LatexEquation( f'\left(z\right) = -z \phi\left(z\right) )]] [[BR]] [[BR]]
 
+''Logarithm of likelihood'': [[BR]]
   [[LatexEquation( L\left(\beta\right)=\underset{i}{\sum}w_{i}\left(y_{i}\ln\left(\Phi\left(x_{i}\beta\right)\right)+\left(1-y_{i}\right)\ln\left(\Phi\left(-x_{i}\beta\right)\right)\right) )]] 
 
+''Gradient'': [[BR]] 
   [[LatexEquation( \frac{\partial L\left(\beta\right)}{\partial\beta_{j}}=\underset{i}{\sum}w_{i}\left(y_{i}\frac{\phi\left(x_{i}\beta\right)}{\Phi\left(x_{i}\beta\right)}-\left(1-y_{i}\right)\frac{\phi\left(x_{i}\beta\right)}{\Phi\left(-x_{i}\beta\right)}\right)x_{ij} )]] 
 
+''Hessian'': [[BR]] 
   [[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