glm letex 笔记

这是我的letex学习笔记，由于时间有限，只能讲源码和结果贴出：

这里面是广义线性模型的推导过程：

\documentclass{article}
\usepackage{paralist}
\begin{document}
\title{Generate Linear Model Estimation Note}
\author{Xue Zoushi }
\date{April 28, 2016}
\maketitle
The general procedures:
\begin{compactenum}
\item General exponential family format
\begin{equation}
f(y|\theta) = exp \left ( \frac{y\theta + b(\theta)}{a(\phi)} + c(y,\phi) \right)
\end{equation}
\[
\ell(\theta|y) =log[f(y|\theta)]= \frac{y\theta + b(\theta)}{a(\phi)} + c(y,\phi)
\]
\item Some important attributes of log-likelihood
\[ E(Y) = \mu = \frac{\partial b(\theta)} {\partial \theta} \]
\[ E[S(\theta)]= 0 \]
\[ E[\frac{\partial S}{\partial \theta}] = -E[S(\theta)]^2 \]
\[ I(\theta)= Var(S(\theta))= E[S(\theta)]^2 - {E[S(\theta)]}^2 \]
\[ Var(Y) = a(\phi)[\frac{\partial^2 b(\theta)}{\partial \theta ^ 2}] \]
\item Newton-Raphson and Fisher-scoring
The scalar form of Taylor series
\[ \ell(\theta) \equiv \ell(\tilde{\theta}) + (\theta - \tilde{\theta})
\left. \frac{\partial \ell (\theta)}{\partial \theta} \right |_{\theta = \tilde{\theta}} +
\frac{1}{2} (\theta - \tilde{\theta})^2 \left. \frac{\partial \ell^2 (\theta)}{\partial \theta^2}
\right |_{\theta = \tilde{\theta}}\]

Set \( \partial \ell (\theta) / \partial \theta = 0 \) and rearranging terms yields:
\[ \theta \equiv \tilde{\theta} -
\left [ \left . \frac{\partial ^2 \ell (\theta)}{\partial \theta ^2} \right |_{\theta=\tilde{\theta}}\right ]^{-1}
\left [ \left . \frac{\partial \ell (\theta)}{\partial \theta} \right |_{\theta=\tilde{\theta}} \right ] \]

The basic matrix form of Newton-Raphson algorithm:
\begin{equation}
\theta \equiv \tilde{\theta} - [H(\tilde{\theta})]^{-1} S(\tilde{\theta})
\end{equation}

Replace hession matrix with the information matrix (i.e. \( E(H(\theta))= -Var[S(\theta)]= -I(\theta) \)),
we get Fisher scoring algorithm:
\begin{equation}
\theta \equiv \tilde{\theta} - [I(\tilde{\theta})]^{-1} S(\tilde{\theta})
\end{equation}

\item Estimate the coefficient \( \beta \).
Scalar form
\begin{equation}
\frac{\partial \ell (\beta)}{\partial \beta} =
\frac{\partial \ell (\theta) }{ \partial \theta} \frac{\partial \theta }{ \partial \mu }
\frac{\partial \mu }{ \partial \eta } \frac{\partial \eta }{ \partial \beta }
\end{equation}
Some results:
\begin{itemize}
\item
\[ \frac{\partial \ell (\theta)}{\partial \theta} = \frac{y-\mu}{a(\phi)} \]

\item
\[\frac{\partial\theta}{\partial\mu}=\left(\frac{\partial\mu}{\partial\theta}\right)^{-1}=\frac{1}{V(\mu)}\]
\item
\[ \frac{\partial \eta}{\partial \beta} = \frac{\partial X \beta}{\beta}\]
\item
\[ \frac{\partial \ell(\beta)}{\partial \beta} =
(y-\mu)\left( \frac{1}{V(y)} \right)\left(\frac{\partial \mu}{\partial \eta} \right) X \]
\end{itemize}

Matrix form
\begin{equation}
\frac{\partial\ell(\theta)}{\partial \beta} = X^{'} D^{-1} V^{-1}(y-\mu)
\end{equation}
where $y$ is the $n\times1$ vector of observations, $\ell(\theta)$ is the $n\times 1$ vector of log-likelihood
values associated with observations, $V = diag[Var(y_{i})]$ is the $n \times n$ variance matrix of the
observations, $D=diag[\partial \eta_{i} / \partial \mu_{i}]$ is the $n \times n$ matrix of derivatives, and $\mu$
is the $n \times 1$ mean vector.\\
Let $W=(DVD)^{-1}$, we can get:
\[ S(\beta) = \frac{\partial \ell (\theta)}{\partial \beta}
= X^{'} D^{-1}V^{-1}(D^{-1}D)(y-\mu) = X^{'}WD(y-\mu) \]
\[ Var[S(\beta)] =X^{'}WD[Var(y-\mu)]DWX =X^{'}WDVDWX=X^{'}WX \]


\item Pseudo-Likelihood for GLM \\
Using Fisher scoring equation yields
$\beta = \tilde{\beta} +(X^{'}\tilde{W}X)^{-1}X^{'}\tilde{W}\tilde{D}(y-\tilde{\mu})$,
where $\tilde{W},\tilde{D}$, and $\mu$ evaluated at $\tilde{\beta}$. So GLM estimating equations:
\begin{equation}
X^{'}\tilde{W}X\beta = X^{'}\tilde{W}y^{*}
\end{equation}
where $y^{*} = X\tilde{\beta} + \tilde{D}(y-\tilde{\mu}) = \tilde{eta} + \tilde{D}(y-\tilde{\mu})$, and
$y^{*}$ is called the pseudo-variable.

\[ E(y^{*}) =E[X\tilde{\beta} + \tilde{D}(y - \tilde{\mu})] = X\beta \]
\[ Var(y^{*}) = E[X\tilde{\beta} + \tilde{D}(y - \tilde{\mu})] = \tilde{D}\tilde{V}\tilde{D}=\tilde{W}^{-1} \]

\begin{equation}
X^{'}[Var(y^{*})]^{-1}X\beta = X^{'}[Var(y^{*})]^{-1} \Rightarrow X^{'}WX\beta = X^{'}Wy^{*}
\end{equation}

\end{compactenum}
\end{document}

使用emacs编辑，然后使用命令 pdfletex -glm_estimation.tex生成，生成文件在博客园的文件附件中。

下面是生成的pdf文件截图：