= Package QltvRespModel =

Max-likelihood and bayesian estimation of qualitative response models.

== Weighted Boolean Regresions ==

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 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].

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}{\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 
bounds for all variables for which he/she has robust knowledge.[[BR]] [[BR]]
[[LatexEquation( \beta_k \sim N\left(\nu_k, \sigma_k \right) )]] [[BR]] [[BR]]
[[LatexEquation( l_k \le \beta_k \le u_k \wedge l_k < u_k)]] [[BR]] [[BR]]
When [[LatexEquation( \sigma_k )]] is infinite or unknown we will express a uniform
prior.
When [[LatexEquation( l_k = -\infty)]] or unknown we will express that variable 
has no lower bound.
When [[LatexEquation( u_k = +\infty)]] or unknown we will express that variable
has no upper bound.

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]]

=== Weighted Logit Regression ===
Class [source:/tolp/OfficialTolArchiveNetwork/QltvRespModel/WgtLogit.tol @WgtLogit] 
is an specialization of class 
[source:/tolp/OfficialTolArchiveNetwork/QltvRespModel/WgtBoolReg.tol @WgtBoolReg]
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 ===
Class [source:/tolp/OfficialTolArchiveNetwork/QltvRespModel/WgtProbit.tol @WgtProbit] 
is an specialization of class 
[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} )]]