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\title{Chapter One of  \emph{Regression Analysis}: Overview\footnote{See last slide for copyright information.}}
\subtitle{STA302 Fall 2017}
\date{} % To suppress date

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% \section{Section One}

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\frametitle{Simple regression model} \pause
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{ \Large
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    y_i = \beta_0 + \beta_1 x_i + \epsilon_i \pause ,  
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where
\begin{itemize}
     \item[] $x_1, \ldots, x_n$ are observed, known constants. \pause
     \item[] $\epsilon_1, \ldots, \epsilon_n$ are random variables satisfying the 
             \emph{Gauss-Markov conditions}. \pause
        \begin{itemize}
            \item[] $E(\epsilon_i)=0$ \pause
            \item[] $Var(\epsilon_i)=\sigma^2$ \pause
            \item[] $Cov(\epsilon_i,\epsilon_j)=0$ for $i \neq j$. \pause
        \end{itemize}
     \item[] $\beta_0$, $\beta_1$ and $\sigma^2$ are unknown constants with $\sigma^2>0$. 
\end{itemize}  % 24
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\begin{frame}
\frametitle{Least Squares}
\framesubtitle{Background that's not in the text} \pause
  \begin{itemize}
    \item The random variable $y$ has a distribution that depends on the parameter $\theta$. \pause
    \item How can we estimate $\theta$ from data $y_1, \ldots, y_n$? \pause
    \item The expected value $E(y)$ is a function of $\theta$. \pause
    \item Write it $E_\theta(y)$. \pause
    \item Estimate $\theta$ by the value that gets the observed data values as close as possible to their expected values. \pause
    \item Minimize
\begin{displaymath}
\mathcal{S} =  \sum_{i=1}^n\left(y_i-E_\theta(y_i)\right)^2  
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over all $\theta$. \pause
    \item The value of $\theta$ that minimizes $\mathcal{S}$ is the \emph{least squares estimate}.
  \end{itemize}
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\frametitle{Simplest example of least squares}
\framesubtitle{Again, not in the text}  \pause
\begin{itemize}
     \item $y_1, \ldots, y_n$ all have $E(y_i) = \mu$. \pause
     \item The least squares estimate of $\mu$ is the value that makes the observed $y_i$ values as close as possible to what you would expect. \pause
     \item Minimize $\mathcal{S} = \sum_{i=1}^n\left(y_i-\mu\right)^2$ \pause
\end{itemize}

\begin{eqnarray*}
    \frac{d \mathcal{S}}{d\mu} & = & \frac{d}{d\mu} \sum_{i=1}^n\left(y_i-\mu\right)^2 \\
     & = & \sum_{i=1}^n \frac{d}{d\mu} \left(y_i-\mu\right)^2 \pause \\
     & = & 2 \sum_{i=1}^n \left(y_i-\mu\right) \, (-1) \pause \\
     & \stackrel{set}{=} & 0
\end{eqnarray*}

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\begin{frame}
\frametitle{Continuing the calculation: $-2 \sum_{i=1}^n \left(y_i-\mu\right) = 0$ \pause }
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\begin{eqnarray*}
     & \Rightarrow & \sum_{i=1}^n \left(y_i-\mu\right) = 0 \pause \\
     & \Rightarrow & \sum_{i=1}^n y_i - \sum_{i=1}^n\mu = 0 \pause \\
     & \Rightarrow & \sum_{i=1}^n y_i - n\mu = 0 \pause \\
     & \Rightarrow & \sum_{i=1}^n y_i = n\mu  \pause \\
     & \Rightarrow & \mu = \frac{1}{n}\sum_{i=1}^n y_i \pause = \overline{y} \pause\\
\end{eqnarray*} 
So the least-squares estimate of $\mu$ is $\overline{y}$.
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\frametitle{Least squares regression}
\framesubtitle{Minimize $\mathcal{S} =  \sum_{i=1}^n\left(y_i-E_\theta(y_i)\right)^2$}  \pause
  \begin{itemize}
    \item Model equation is $y_i = \beta_0 + \beta_1 x_i + \epsilon_i$ \pause
    \item $E(y_i) = \beta_0 + \beta_1 x_i$ \pause
    \item Minimize $\mathcal{S} =  \sum_{i=1}^n\left(y_i - \beta_0 - \beta_1 x_i \right)^2$ \pause over 
          $\theta = (\beta_0,\beta_1)$. \pause
    \item Take partial derivatives, set to zero, solve two equations in two unknowns.  \pause
    \item Least squares estimate of $\beta_0$ is $b_0$.  \pause Least squares estimate of $\beta_1$ is $b_1$.
  \end{itemize}
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\begin{frame}
\frametitle{Vocabulary and concepts in Chapter One}
\framesubtitle{A preview of almost the entire course}  \pause 
$y_i = \beta_0 + \beta_1 x_i + \epsilon_i$  \pause 
  \begin{itemize}
    \item Linear regresson means linear in the $\beta$ parameters.  \pause 
    \item Polynomial regression $y_i = \beta_0 + \beta_1 x_i + \beta_2 x_i^2 + \epsilon_i$, etc.  \pause 
    \item Centered model  $y_i = (\beta_0 + \beta_1\overline{x}) + \beta_1 (x_i-\overline{x}) + \epsilon_i$  \pause 
    \item Predicted value $\widehat{y}_i = b_0 + b_1 x_i$  \pause 
    \item Residual $e_i = y_i-\widehat{y}_i$  \pause 
    \item Plotting residuals (p.5) to diagnose problems with the model.  \pause 
    \item Gauss-Markov conditions.  \pause 
    \item Measure of model fit $R^2$  \pause 
    \item Mean and variance of $b_0$ and $b_1$.  \pause 
    \item Confidence intervals and tests.  \pause 
    \item Predicting future observations.
  \end{itemize}
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\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/302f17} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/302f17}}

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