Revision 1073 trunk/docs/manuals/admb-re/admbre.tex

admbre.tex (revision 1073)
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\newcommand{\scGLM}{\textsc{glm}}
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\newcommand{\scGLMM}{\textsc{glmm}}
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\newcommand{\scLIDAR}{\textsc{lidar}}
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\newcommand\admbversion{11.1}
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\makeindex
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\title{%
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    \largetitlepart{Random Effects in\\ \ADM}
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    \smalltitlepart{ADMB-RE User Guide}
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    \vspace{4.5ex}\textsf{\textit{Version 11.1~~(2013-05-01)}}\vspace{3ex}
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    \vspace{4.5ex}\textsf{\textit{Version \admbversion~~(2013-05-01)\\[3pt]
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      Revised manual~~(2013-06-24)}}\vspace{3ex}
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}
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% Author definition.
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\author{\textsf{\textit{Hans Skaug \& David Fournier}}}
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~\vfill
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\noindent ADMB Foundation, Honolulu.\\\\
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\noindent This is the manual for AD Model Builder with Random Effects (ADMB-RE)
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version 11.1.\\\\
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\noindent Copyright \copyright\ 2004, 2006, 2008, 2009, 2011 Hans Skaug \& David
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version \admbversion.\\\\
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\noindent Copyright \copyright\ 2004, 2006, 2008, 2009, 2011, 2013 Hans Skaug \& David
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Fournier\\\\
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\noindent The latest edition of the manual is available at:\\
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\url{http://admb-project.org/documentation/manuals/admb-user-manuals}
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\url{http://admb-project.org/documentation/manuals}
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\tableofcontents
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  \item Hyper-parameters (variance components, etc.) estimated by maximum
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  likelihood.
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  \item Marginal likelihood evaluated by the Laplace approximation, (adaptive) importance
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	sampling or Gauss-Hermite integration.
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  \item Marginal likelihood evaluated by the Laplace approximation, (adaptive)
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  importance sampling or Gauss-Hermite integration.
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  \item Exact derivatives calculated using Automatic Differentiation.
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term borrowed from Bayesian statistics.
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A central concept that originates from generalized linear models is that of a
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``linear predictor.'' Let $x_{1},\ldots ,x_{p}$ denote observed covariates
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(explanatory variables), and let $\beta _{1},\ldots ,\beta _{p}$ be the
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``linear predictor.'' Let $x_{1},\ldots,x_{p}$ denote observed covariates
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(explanatory variables), and let $\beta _{1},\ldots,\beta _{p}$ be the
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corresponding regression parameters to be estimated. Many of the examples in
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this manual involve a linear predictor $\eta_{i}=\beta_{1}x_{1,i}+\cdots
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+\beta_{p}x_{p,i}$, which we also will write in vector form as
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exemplify the use of random effects. The statistical model underlying this
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example is the simple linear regression
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\[
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Y_i=ax_i+b+\varepsilon_i,\qquad i=1,\ldots ,n,
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Y_i=ax_i+b+\varepsilon_i,\qquad i=1,\ldots,n,
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\]
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where $Y_i$ and $x_i$ are the data, $a$ and $b$ are the unknown parameters to be
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estimated, and $\varepsilon_i\sim N(0,\sigma ^{2})$ is an error term.
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know the value of $\sigma_e$, so we shall pretend that~$\sigma_e=0.5$.
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Because $x_i$ is not observed, we model it as a random effect with $x_i\sim
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N(\mu ,\sigma_{x}^{2})$. In \scAR, you are allowed to make such definitions
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N(\mu,\sigma_{x}^{2})$. In \scAR, you are allowed to make such definitions
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through the new parameter type called \texttt{random\_effects\_vector}.
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\index{random effects!random effects vector} (There is also a
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\texttt{random\_effects\_matrix}, which allows you to define a matrix of random
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  a better term.
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  \item The unknown parameters in our model are: $a$, $b$, $\mu$, $\sigma$,
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  $\sigma_{x}$, and $x_{1},\ldots ,x_{n}$. We have agreed to call
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  $\sigma_{x}$, and $x_{1},\ldots,x_{n}$. We have agreed to call
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  $x_{1},\ldots,x_{n}$ ``random effects.'' The rest of the parameters are called
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  ``hyper-parameters.'' Note that we place no prior distribution on the
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  hyper-parameters.
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\section{The flexibility of \scAR\label{lognormal}}
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Say that you doubt the distributional assumption $x_i\sim N(\mu ,\sigma
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Say that you doubt the distributional assumption $x_i\sim N(\mu,\sigma
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_{x}^{2})$ made in \texttt{simple.tpl}, and that you want to check if a skewed
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distribution gives a better fit. You could, for instance,~take
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\[
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\section{The random effects distribution (prior)}
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In \texttt{simple.tpl}, we declared $x_{1},\ldots ,x_{n}$ to be of type
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In \texttt{simple.tpl}, we declared $x_{1},\ldots,x_{n}$ to be of type
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