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\mode<presentation> 

\title{Latent Model Rules\footnote{See last slide for copyright information.}}
\subtitle{STA2053 Fall 2022}
\date{} % To suppress date

\begin{document}

\begin{frame}
  \titlepage
\end{frame}

\begin{frame}
\frametitle{The two-stage model: $cov(\mathbf{d}_i)=\boldsymbol{\Sigma}$}
\framesubtitle{All variables are centered} 
{\LARGE
\begin{eqnarray*}
    \mathbf{y}_i &=& \boldsymbol{\beta} \mathbf{y}_i + \boldsymbol{\Gamma} \mathbf{x}_i + \boldsymbol{\epsilon}_i \\
\mathbf{F}_i &=& \left( \begin{array}{c}
         \mathbf{x}_i  \\ \mathbf{y}_i 
             \end{array} \right)  \\
\mathbf{d}_i &=& \boldsymbol{\Lambda}\mathbf{F}_i + \mathbf{e}_i 
\end{eqnarray*}
} % End size
\begin{itemize}
     \item $\mathbf{x}_i$ is $p \times 1$, $\mathbf{y}_i$ is $q \times 1$, $\mathbf{d}_i$ is $k \times 1$. 
     \item  $cov(\mathbf{x}_i)=\boldsymbol{\Phi}_x$, 
            $cov(\boldsymbol{\epsilon}_i)=\boldsymbol{\Psi}$
     \item $cov(\mathbf{F}_i) = cov\left( \begin{array}{c}
         \mathbf{x}_i  \\ \mathbf{y}_i 
             \end{array} \right)
     =\boldsymbol{\Phi} 
= \left( \begin{array}{c c}
         \boldsymbol{\Phi}_{11}  & \boldsymbol{\Phi}_{12} \\
         \boldsymbol{\Phi}_{12}^\top  & \boldsymbol{\Phi}_{22} \\
             \end{array} \right)$
     \item $cov(\mathbf{e}_i)=\boldsymbol{\Omega}$
\end{itemize}
\end{frame}


\begin{frame}
\frametitle{Identify parameter matrices in two steps}
\framesubtitle{It does not really matter which one you do first.}   
%{\footnotesize
    \begin{itemize}
        \item $\mathbf{y}_i = \boldsymbol{\beta} \mathbf{y}_i + \boldsymbol{\Gamma} \mathbf{x}_i + \boldsymbol{\epsilon}_i$
        \begin{itemize}
            \item[] $cov(\mathbf{x}_i)=\boldsymbol{\Phi}_x$, 
                    $cov(\boldsymbol{\epsilon}_i)=\boldsymbol{\Psi}$
        \end{itemize}  
        \item $\mathbf{d}_i = \boldsymbol{\Lambda}\mathbf{F}_i + \mathbf{e}_i $
        \begin{itemize}
            \item[] $cov(\mathbf{F}_i)=\boldsymbol{\Phi}$, 
                    $cov(\boldsymbol{e}_i)=\boldsymbol{\Omega}$
        \end{itemize}
    \end{itemize} 
%} % End size

\vspace{3mm}
\hrule 
\vspace{3mm}

%{\Large
\begin{enumerate}
     \item \emph{Latent model}: Show $\boldsymbol{\beta}$, $\boldsymbol{\Gamma}$, $\boldsymbol{\Phi}_x$ and $\boldsymbol{\Psi}$ can be recovered from 
     $\boldsymbol{\Phi} = cov\left( \begin{array}{c} \mathbf{x}_i \\ \mathbf{y}_i \end{array} \right)$. 
     \item \emph{Measurement model}: Show $\boldsymbol{\Phi}$ and $\boldsymbol{\Omega}$ can be recovered from $\boldsymbol{\Sigma}=cov(\mathbf{d}_i)$. 
\end{enumerate}
This means all the parameters can be recovered from  $\boldsymbol{\Sigma}$.
%} % End size

\end{frame}


\begin{frame}
\frametitle{Latent Model Rules}
  \begin{itemize}
    \item $\mathbf{y}_i = \boldsymbol{\beta} \mathbf{y}_i + \boldsymbol{\Gamma} \mathbf{x}_i + \boldsymbol{\epsilon}_i$ \pause 
    \item  Here, identifiability means that the parameters $\boldsymbol{\beta}$, $\boldsymbol{\Gamma}$, $\boldsymbol{\Phi}_x$ and $\boldsymbol{\Psi}$ are functions of $cov(\mathbf{F}_i)=\boldsymbol{\Phi}$.
  \end{itemize}
\end{frame}

\begin{frame}
\frametitle{Regression Rule}
\framesubtitle{Someimes called the Null Beta Rule}  
Suppose 
  \begin{itemize}
    \item No endogenous variables influence other endogenous variables. \pause 
    \item[]
    \item $\mathbf{y}_i = \boldsymbol{\Gamma} \mathbf{x}_i + \boldsymbol{\epsilon}_i$ \pause 
    \item Of course $cov(\mathbf{x}_i, \boldsymbol{\epsilon}_i) = \mathbf{0}$, always. \pause 
    \item $\boldsymbol{\Psi} = cov(\boldsymbol{\epsilon}_i)$ need not be diagonal. \pause 
  \end{itemize}
\vspace{5mm}
Then $\boldsymbol{\Gamma}$ and $\boldsymbol{\Psi}$ are identifiable.
\end{frame}

\begin{frame}
\frametitle{Acyclic Rule}
\framesubtitle{Acyclic models are frequently called ``recursive."}  \pause 
Parameters of the Latent Variable Model are identifiable if the model is acyclic  (no feedback loops through straight arrows) and the following conditions hold.  \pause 
  \begin{itemize}
    \item Organize the variables that are not error terms into sets. Set 0 consists of all the exogenous variables.  
    \item For $j=1,\ldots ,m$, each endogenous variable in set $j$ is influenced by at least one variable in set $j-1$, and also possibly by variables in earlier sets. 
    \item Error terms may be correlated within sets, but not between sets. \pause 
  \end{itemize}

\vspace{5mm}
Proof: Repeated application of the Regression Rule.
\end{frame}

\begin{frame}
\frametitle{An Acyclic model}
%\framesubtitle{} 
\begin{center}
\includegraphics[width=4.5in]{AcyclicHand}
\end{center}
\end{frame}


\begin{frame}
\frametitle{Brand awareness model}
%\framesubtitle{} 
\begin{center}
\includegraphics[width=3.6in]{Doughnut5}
\end{center}
\end{frame}

\begin{frame}
\frametitle{Parameters of this model are just identifiable}
\framesubtitle{Example from Ch. 5 of Duncan's \emph{Introduction to Structural Equation Models}} 
\begin{center}
\includegraphics[width=4in]{Duncan-Just-Ident}
\end{center}
Shows that the acyclic rule is sufficient but not necessary.
\end{frame}

\begin{frame}
\frametitle{The Pinwheel Model}
\framesubtitle{Parameters are identifiable} 
\begin{center}
\includegraphics[width=2.5in]{Pinwheel}
\end{center}
\end{frame}

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\begin{frame}
\frametitle{Covariance matrix for the pinwheel model}
%\framesubtitle{} 

\begin{columns} % Use Beamer's columns to use more of the margins
\column{1.15\textwidth}
{\tiny
$\left(\begin{array}{cccc}
\phi & -\frac{\gamma \phi}{\beta_{1} \beta_{2} \beta_{3} - 1} &
-\frac{\beta_{2} \gamma \phi}{\beta_{1} \beta_{2} \beta_{3} - 1} &
-\frac{\beta_{2} \beta_{3} \gamma \phi}{\beta_{1} \beta_{2} \beta_{3} -
1} \\
 &
\frac{\beta_{1}^{2} \beta_{3}^{2} \psi_{2} + \gamma^{2} \phi +
\beta_{1}^{2} \psi_{3} + \psi_{1}}{{\left(\beta_{1} \beta_{2} \beta_{3}
- 1\right)}^{2}} & \frac{\beta_{2} \gamma^{2} \phi + \beta_{1}^{2}
\beta_{2} \psi_{3} + \beta_{1} \beta_{3} \psi_{2} + \beta_{2}
\psi_{1}}{{\left(\beta_{1} \beta_{2} \beta_{3} - 1\right)}^{2}} &
\frac{\beta_{2} \beta_{3} \gamma^{2} \phi + \beta_{1} \beta_{3}^{2}
\psi_{2} + \beta_{2} \beta_{3} \psi_{1} + \beta_{1}
\psi_{3}}{{\left(\beta_{1} \beta_{2} \beta_{3} - 1\right)}^{2}} \\
 &
 & \frac{\beta_{2}^{2}
\gamma^{2} \phi + \beta_{1}^{2} \beta_{2}^{2} \psi_{3} + \beta_{2}^{2}
\psi_{1} + \psi_{2}}{{\left(\beta_{1} \beta_{2} \beta_{3} -
1\right)}^{2}} & \frac{\beta_{2}^{2} \beta_{3} \gamma^{2} \phi +
\beta_{2}^{2} \beta_{3} \psi_{1} + \beta_{1} \beta_{2} \psi_{3} +
\beta_{3} \psi_{2}}{{\left(\beta_{1} \beta_{2} \beta_{3} -
1\right)}^{2}} \\
 &  &  &
\frac{\beta_{2}^{2} \beta_{3}^{2} \gamma^{2} \phi + \beta_{2}^{2}
\beta_{3}^{2} \psi_{1} + \beta_{3}^{2} \psi_{2} +
\psi_{3}}{{\left(\beta_{1} \beta_{2} \beta_{3} - 1\right)}^{2}}
\end{array}\right)$
} % End size
\end{columns}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Copyright Information}

This slide show was prepared by  \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Statistical Sciences, University of Toronto. It is licensed under a 
\href{http://creativecommons.org/licenses/by-sa/3.0/deed.en_US}
     {Creative Commons Attribution - ShareAlike 3.0 Unported License}. Use any part of it as you like and share the result freely. The \LaTeX~source code is available from the course website:
\href{http://www.utstat.toronto.edu/~brunner/oldclass/2053f22} {\small\texttt{http://www.utstat.toronto.edu/brunner/oldclass/2053f22}}

\end{frame}


\end{document}

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\begin{frame}
\frametitle{}
%\framesubtitle{} 
  \begin{itemize}
    \item 
    \item 
    \item 
  \end{itemize}
\end{frame}


{\LARGE
\begin{displaymath}
    
\end{displaymath} }




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# Pinwheel model with sagemath: Call the sage notebook pinwheel3

# load sem package, set up  Beta, Gamma, Phi, Psi
sem = 'http://www.utstat.toronto.edu/~brunner/openSEM/sage/sem.sage'
load('~/sem.sage') # Local version
BETA = ZeroMatrix(3,3)
BETA[0,2] = var('beta1'); BETA[1,0] = var('beta2')
BETA[2,1] = var('beta3'); show(BETA)
GAMMA = ZeroMatrix(3,1); GAMMA[0,0] = var('gamma'); show(GAMMA)
PHI = ZeroMatrix(1,1); PHI[0,0] = var('phi'); show(PHI)
PSI = DiagonalMatrix(3,'psi'); show(PSI)

# Check the determinant of (I-beta). It can't be zero.
det(IdentityMatrix(3)-BETA)

# Calculate the covariance matrix.
Sigma = NoGammaCov(BETA,PSI)
show(Sigma)


# Covariance structure equations
eqns = SetupEqns(Sigma)
for item in eqns: show(item)



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