Changes between Version 2 and Version 3 of OfficialTolArchiveNetworkQltvRespModel

Timestamp:

Dec 20, 2010, 9:33:46 PM (14 years ago)

Author:

Víctor de Buen Remiro

Comment:

--

OfficialTolArchiveNetworkQltvRespModel

@@ -7 +7 @@
 Abstract class 
 [source:/tolp/OfficialTolArchiveNetwork/QltvRespModel/WgtBoolReg.tol @WgtBoolReg] 
-is the base to inherit weighted boolean regressions as logit or probit or any other 
-given justthe scalar distribution function.
+is the base to inherit weighted boolean regressions as logit or probit or any other, 
+given just the scalar distribution function [[LatexEquation( F )]] and the 
+corresponding density function [[LatexEquation( f )]]. In a weighted regression 
+each row of input data has a distinct weight in the likelihood function. For
+example, it can be very usefull to handle with data extrated from an stratified 
+sample.
 
 This class implements max-likelihood by means of package 
-[wiki/OfficialTolArchiveNetworkNonLinGloOpt NonLinGloOpt] and bayesian estimation 
-using [wiki/OfficialTolArchiveNetworkBysSampler BysSampler].
+[wiki:OfficialTolArchiveNetworkNonLinGloOpt NonLinGloOpt] and bayesian estimation 
+using [wiki:OfficialTolArchiveNetworkBysSampler BysSampler].
+
+Let be 
+ * [[LatexEquation( X\in\mathbb{R}^{m\times n} )]] the regression input matrix 
+ * [[LatexEquation( w\in\mathbb{R}^{m} )]] the vector of weights of each register
+ * [[LatexEquation( y\in\mathbb{R}^{m} )]] the regression output matrix 
+ 
+The hypotesis is that [[LatexEquation( \forall i=1 \dots m )]] 
+
+  [[LatexEquation( y_{i}\sim Bernoulli\left(\pi_{i}\right) )]] 
+  [[LatexEquation( \pi_{i}=Pr\left[y_{i}=1\right] = F\left(X_{i}\beta\right) )]] 
+
+The likelihood function is then 
+
+  [[LatexEquation( lk\left(\beta\right)=\underset{i}{\underset{i}{\prod}\pi_{i}^{w_{i}y_{i}}\left(1-\pi_{i}\right)^{w_{i}\left(1-y_{i}\right)}} )]] 
+
+and its logarithm
+
+  [[LatexEquation( L\left(\beta\right)=\ln\left(lk\left(\beta\right)\right)=\underset{i}{\sum}w_{i}\left(y_{i}\ln\left(\pi_{i}\right)+\left(1-y_{i}\right)\ln\left(1-\pi_{i}\right)\right) )]] 
+
+The gradient of the logarithm of the likelihood function will be
+
+  [[LatexEquation( \frac{\partial L\left(\beta\right)}{\partial\beta_{j}}=\underset{i}{\sum}w_{i}\left(y_{i}\frac{f\left(x_{i}\beta\right)}{F\left(x_{i}\beta\right)}-\left(1-y_{i}\right)\frac{f\left(x_{i}\beta\right)}{1-F\left(x_{i}\beta\right)}\right)x_{ij} )]] 
+
+and the hessian is
+
+  [[LatexEquation( \frac{\partial L\left(\beta\right)}{\partial\beta_{i}\partial_{j}}=\underset{k}{\sum}w_{k}\left(y_{k}\frac{f'\left(x_{k}\beta\right)F\left(x_{k}\beta\right)-f^{2}\left(x_{k}\beta\right)}{F^{2}\left(x_{k}\beta\right)}-\left(1-y_{k}\right)\frac{f'\left(x_{k}\beta\right)\left(1-F\left(x_{k}\beta\right)\right)+f^{2}\left(x_{k}\beta\right)}{\left(1-F\left(x_{k}\beta\right)\right)^{2}}\right)x_{ik}x_{jk} )]] 
+
 
 User can and should define scalar truncated normal or uniform prior information and 
@@ -25 +56 @@
 has no upper bound.
 
-It's also allowed to give any set of constraining linear inequations [[BR]] [[BR]]
+It's also allowed to give any set of constraining linear inequations if they 
+are compatible with lower and upper bounds [[BR]] [[BR]]
 [[LatexEquation( A \beta \le a )]] [[BR]] [[BR]]
 
@@ -34 +66 @@
 that handles with weighted logit regressions.
 
+In this case we have that scalar distribution is the logistic one.
+
+  [[LatexEquation( F\left(z\right) = \frac{1}{1+e^{-z}} )]] [[BR]] [[BR]]
+
+  [[LatexEquation( f\left(z\right) = \frac{e^{-z}}{\left(1+e^{-z}\right)^2} ]] [[BR]] [[BR]]
+
+  [[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( \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) )]] 
+
 
 === Weighted Probit Regression ===
@@ -40 +86 @@
 [source:/tolp/OfficialTolArchiveNetwork/QltvRespModel/WgtBoolReg.tol @WgtBoolReg]
 that handles with weighted probit regressions.
+
+In this case we have that scalar distribution is the standard normal one.
+
+  [[LatexEquation( F\left(z\right) = F_{0,1}\left(z\right) )]] [[BR]] [[BR]]
+
+  [[LatexEquation( f\left(z\right) = f_{0,1}\left(z\right) )]] [[BR]] [[BR]]
+
+  [[LatexEquation( f'\left(z\right) = -z f_{0,1}\left(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( \frac{\partial L\left(\beta\right)}{\partial\beta_{j}}=\underset{i}{\sum}w_{i}\left(y_{i}\frac{f_{0,1}\left(x_{i}\beta\right)}{F_{0,1}\left(x_{i}\beta\right)}-\left(1-y_{i}\right)\frac{f_{0,1}\left(x_{i}\beta\right)}{1-F_{0,1}\left(x_{i}\beta\right)}\right)x_{ij} )]] 
+
+  [[LatexEquation( \frac{\partial L\left(\beta\right)}{\partial\beta_{i}\partial_{j}}=-\underset{k}{\sum}w_{k}f_{0,1}\left(x_{k}\beta\right)\left(y_{k}\frac{zF_{0,1}\left(x_{k}\beta\right)+f_{0,1}\left(x_{k}\beta\right)}{F_{0,1}^{2}\left(x_{k}\beta\right)}+\left(1-y_{k}\right)\frac{-z\left(1-F_{0,1}\left(x_{k}\beta\right)\right)+f_{0,1}\left(x_{k}\beta\right)}{\left(1-F_{0,1}\left(x_{k}\beta\right)\right)^{2}}\right)x_{ik}x_{jk} )]]