Changes between Version 16 and Version 17 of OfficialTolArchiveNetworkGrzLinModel

Timestamp:

Mar 30, 2011, 4:06:02 PM (14 years ago)

Author:

Víctor de Buen Remiro

Comment:

--

OfficialTolArchiveNetworkGrzLinModel

@@ -38 +38 @@
   [[LatexEquation( f\left(y;\mu\right) )]]
 
-For each row [[LatexEquation( k=1 \dots  n)]] we will know the output [[LatexEquation( y_k )]] 
+For each row [[LatexEquation( i=1 \dots  n)]] we will know the output [[LatexEquation( y_i )]] 
 and the average 
 
-  [[LatexEquation( \mu_{k}=g^{-1}\left(\eta_{k}\right)=g^{-1}\left(x_{k}\beta\right) )]] 
+  [[LatexEquation( \mu_i=g^{-1}\left(\eta_i\right)=g^{-1}\left(x_i\beta\right) )]] 
 
 Each particular distribution may have its own additional parameters which will be treated 
@@ -53 +53 @@
  * the first and second partial derivatives of log-density function respect to the linear prediction [[BR]] [[BR]] 
    [[LatexEquation( \frac{\partial\ln f}{\partial\eta},\frac{\partial^{2}\ln f}{\partial\eta^{2}} )]] 
+
+The likelihood function of the weigthed regression is then 
+
+  [[LatexEquation( lk\left(\beta\right)=\overset{m}{\underset{i}{\prod}}f_{i}^{w_{i}}\:\wedge f_{i}=f\left(y_{i};\mu_{i}\right)\:\forall i=1\ldots m )]] 
+
+and its logarithm
+
+  [[LatexEquation( L\left(\beta\right)=\ln\left(lk\left(\beta\right)\right)=\overset{m}{\underset{i}{\sum}}w_{i}f_{i} )]] 
+
+The gradient of the logarithm of the likelihood function will be
+
+  [[LatexEquation( \frac{\partial L\left(\beta\right)}{\partial\beta_{j}}=\frac{\partial L\left(\beta\right)}{\partial\eta}\frac{\partial\eta}{\partial\beta_{j}}=\overset{m}{\underset{i}{\sum}}w_{i}\frac{\partial\ln f_{k}}{\partial\eta}x_{ij} )]] 
+
+and the hessian is
+
+  [[LatexEquation( \frac{\partial L\left(\beta\right)}{\partial\beta_{i}\partial\beta_{j}}=\underset{i}{\sum}w_{i}\frac{\partial^{2}\ln f_{i}}{\partial\eta^{2}}x_{ik}x_{jk} )]] 
 
 This class also implements these common features