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= Package GrzLinModel =

Max-likelihood and bayesian estimation of 
[http://en.wikipedia.org/wiki/Generalized_linear_model generalized linear models].

== Weighted Generalized Regresions ==

Abstract class 
[source:/tolp/OfficialTolArchiveNetwork/GrzLinModel/WgtReg.tol @WgtReg] 
is the base to inherit weighted generalized linear regressions as poisson, 
binomial, normal or any other, given just the scalar link 
function [[LatexEquation( g )]] and the density function [[LatexEquation( f )]]. 

For boolean and qualitative response outputs like logit or probit there is an 
specialization on package [wiki:OfficialTolArchiveNetworkQltvRespModel QltvRespModel]

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.

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 
 * [[LatexEquation( \beta\in\mathbb{R}^{n} )]] the regression coefficients 
 * [[LatexEquation( \eta=X\beta\in\mathbb{R}^{n} )]] the linear prediction
 * [[LatexEquation( g )]] the link function
 * [[LatexEquation( f)]] the density function of a distribution of the 
   [http://en.wikipedia.org/wiki/Exponential_family exponential family]

Then we purpose that the average of the output is the inverse of the link function 
applyied to the linear predictor

  [[LatexEquation( E\left[y\right]=\mu=g^{-1}\left(X\beta\right) )]]

The density function becomes as a real valuated function of at least two parameters 

  [[LatexEquation( f\left(y_{k};\mu_{k}\right) )]]

the output [[LatexEquation( y_k )]] and the average 

  [[LatexEquation( \mu_{k}=g^{-1}\left(\eta_{k}\right)=g^{-1}\left(x_{k}\beta\right) )]] 

for each row [[LatexEquation( k=1 \dots  n)]]:

If there are more unknown parameters about the density we will supose 
  

=== Weighted Poisson Regresion ===