\chapter{Linear Model}

\section{Simple Linear Regression}

\[
    Y_i = \beta_0 + \beta_1 X_i  + \varepsilon_i
\]
\[
    \Y = \X \beta + \varepsilon.
\]
\[
    \begin{pmatrix}
        Y_1 \\
        Y_2 \\
        \vdots \\
        Y_n
    \end{pmatrix}
    = 
    \begin{pmatrix}
        1 & X_1 \\
        1 & X_2 \\
        \vdots & \vdots \\
        1 & X_n 
    \end{pmatrix}
    \begin{pmatrix}
        \beta_0 \\
        \beta_1
    \end{pmatrix}
    +
    \begin{pmatrix}
        \varepsilon_1 \\
        \varepsilon_2 \\
        \vdots
        \varepsilon_n
    \end{pmatrix}
\]

\paragraph*{Assumptions}
\begin{enumerate}[label={\color{primary}{($A_\arabic*$)}}]
    \item $\varepsilon_i$ are independent;
    \item $\varepsilon_i$ are identically distributed;
    \item $\varepsilon_i$ are i.i.d $\sim \Norm(0, \sigma^2)$ (homoscedasticity).
\end{enumerate}

\section{Generalized Linear Model}

\[
    g(\EE(Y)) = X \beta
\]
with $g$ being
\begin{itemize}
    \item Logistic regression: $g(v) = \log \left(\frac{v}{1-v}\right)$, for instance for boolean values,
    \item Poisson regression: $g(v) = \log(v)$, for instance for discrete variables. 
\end{itemize}

\subsection{Penalized Regression}

When the number of variables is large, e.g, when the number of explanatory variable is above the number of observations, if $p >> n$ ($p$: the number of explanatory variable, $n$ is the number of observations), we cannot estimate the parameters.
In order to estimate the parameters, we can use penalties (additional terms).

Lasso regression, Elastic Net, etc.

\subsection{Statistical Analysis Workflow}

\begin{enumerate}[label={\bfseries\color{primary}Step \arabic*.}]
    \item Graphical representation;
    \item ...
\end{enumerate}
\[
    Y = X \beta + \varepsilon,
\]
is noted equivalently as
\[
        \begin{pmatrix}
            y_1 \\
            y_2 \\
            y_3 \\
            y_4
        \end{pmatrix}
        = \begin{pmatrix}
                1 & x_{11} & x_{12} \\
                1 & x_{21} & x_{22} \\
                1 & x_{31} & x_{32} \\
                1 & x_{41} & x_{42}
            \end{pmatrix}
        \begin{pmatrix}
            \beta_0 \\
            \beta_1 \\
            \beta_2
        \end{pmatrix} +
        \begin{pmatrix}
            \varepsilon_1 \\
            \varepsilon_2 \\
            \varepsilon_3 \\
            \varepsilon_4
        \end{pmatrix}.
\]
\section{Parameter Estimation}

\subsection{Simple Linear Regression}

\subsection{General Case}

If $\X^T\X$ is invertible, the OLS estimator is:
\begin{equation}
\hat{\beta} = (\X^T\X)^{-1} \X^T \Y
\end{equation}

\subsection{Ordinary Least Square Algorithm}

We want to minimize the distance between $\X\beta$ and $\Y$:
\[
    \min \norm{\Y - \X\beta}^2
\]
(See \autoref{ch:elements-of-linear-algebra}).
\begin{align*}
    \Rightarrow& \X \beta = proj^{(1, \X)} \Y\\
    \Rightarrow& \forall v \in w,\, vy = v proj^w(y)\\
    \Rightarrow& \forall i: \\
    & \X_i \Y = \X_i X\hat{\beta} \qquad \text{where $\hat{\beta}$ is the estimator of $\beta$} \\
    \Rightarrow& \X^T \Y = \X^T \X \hat{\beta} \\
    \Rightarrow& {\color{gray}(\X^T \X)^{-1}} \X^T \Y = {\color{gray}(\X^T \X)^{-1}} (\X^T\X) \hat{\beta} \\
    \Rightarrow& \hat{\beta} = (\X^T\X)^{-1} \X^T \Y
\end{align*}

This formula comes from the orthogonal projection of $\Y$ on the vector subspace defined by the explanatory variables $\X$

$\X \hat{\beta}$ is the closest point to $\Y$ in the subspace generated by $\X$.

If $H$ is the projection matrix of the subspace generated by $\X$, $X\Y$ is the projection on $\Y$ on this subspace, that corresponds to $\X\hat{\beta}$.

\section{Coefficient of Determination: \texorpdfstring{$R^2$}{R\textsuperscript{2}}}
\begin{definition}[$R^2$]
    \[
        0 \leq R^2 = \frac{\norm{\X\hat{\beta} - \bar{\Y}\One}^2}{\norm{\Y - \bar{\Y}\One}^2} = 1 - \frac{\norm{\Y - \X\hat{\beta}}^2}{\norm{\Y - \bar{\Y}\One}^2} \leq 1
    \] proportion of variation of $\Y$ explained by the model.
\end{definition}

\begin{figure}
    \centering
    \includestandalone{figures/schemes/orthogonal_projection}
    \caption{Orthogonal projection of $\Y$ on plan generated by the base described by $\X$. $\color{blue}a$ corresponds to $\norm{\X\hat{\beta} - \bar{\Y}}^2$ and $\color{blue}b$ corresponds to $\norm{\Y - \hat{\beta}\X}^2$}
    \label{fig:scheme-orthogonal-projection}
\end{figure}

\begin{figure}
    \centering
    \includestandalone{figures/schemes/ordinary_least_squares}
    \caption{Ordinary least squares and regression line with simulated data.}
    \label{fig:ordinary-least-squares}
\end{figure}