321 lines
8.8 KiB
TeX
321 lines
8.8 KiB
TeX
\chapter{Linear Model}
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\section{Simple Linear Regression}
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\[
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Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i
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\]
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\[
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\Y = \X \beta + \varepsilon.
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\]
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\[
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\begin{pmatrix}
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Y_1 \\
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Y_2 \\
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\vdots \\
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Y_n
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\end{pmatrix}
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=
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\begin{pmatrix}
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1 & X_1 \\
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1 & X_2 \\
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\vdots & \vdots \\
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1 & X_n
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\end{pmatrix}
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\begin{pmatrix}
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\beta_0 \\
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\beta_1
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\end{pmatrix}
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+
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\begin{pmatrix}
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\varepsilon_1 \\
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\varepsilon_2 \\
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\vdots
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\varepsilon_n
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\end{pmatrix}
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\]
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\paragraph*{Assumptions}
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\begin{enumerate}[label={\color{primary}{($A_\arabic*$)}}]
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\item $\varepsilon_i$ are independent;
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\item $\varepsilon_i$ are identically distributed;
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\item $\varepsilon_i$ are i.i.d $\sim \Norm(0, \sigma^2)$ (homoscedasticity).
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\end{enumerate}
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\section{Generalized Linear Model}
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\[
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g(\EE(Y)) = X \beta
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\]
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with $g$ being
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\begin{itemize}
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\item Logistic regression: $g(v) = \log \left(\frac{v}{1-v}\right)$, for instance for boolean values,
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\item Poisson regression: $g(v) = \log(v)$, for instance for discrete variables.
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\end{itemize}
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\subsection{Penalized Regression}
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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.
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In order to estimate the parameters, we can use penalties (additional terms).
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Lasso regression, Elastic Net, etc.
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\subsection{Statistical Analysis Workflow}
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\begin{enumerate}[label={\bfseries\color{primary}Step \arabic*.}]
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\item Graphical representation;
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\item ...
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\end{enumerate}
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\[
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Y = X \beta + \varepsilon,
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\]
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is noted equivalently as
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\[
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\begin{pmatrix}
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y_1 \\
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y_2 \\
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y_3 \\
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y_4
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\end{pmatrix}
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= \begin{pmatrix}
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1 & x_{11} & x_{12} \\
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1 & x_{21} & x_{22} \\
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1 & x_{31} & x_{32} \\
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1 & x_{41} & x_{42}
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\end{pmatrix}
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\begin{pmatrix}
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\beta_0 \\
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\beta_1 \\
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\beta_2
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\end{pmatrix} +
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\begin{pmatrix}
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\varepsilon_1 \\
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\varepsilon_2 \\
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\varepsilon_3 \\
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\varepsilon_4
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\end{pmatrix}.
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\]
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\section{Parameter Estimation}
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\subsection{Simple Linear Regression}
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\subsection{General Case}
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If $\X^T\X$ is invertible, the OLS estimator is:
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\begin{equation}
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\hat{\beta} = (\X^T\X)^{-1} \X^T \Y
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\end{equation}
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\subsection{Ordinary Least Square Algorithm}
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We want to minimize the distance between $\X\beta$ and $\Y$:
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\[
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\min \norm{\Y - \X\beta}^2
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\]
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(See \autoref{ch:elements-of-linear-algebra}).
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\begin{align*}
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\Rightarrow& \X \beta = proj^{(1, \X)} \Y\\
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\Rightarrow& \forall v \in w,\, vy = v proj^w(y)\\
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\Rightarrow& \forall i: \\
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& \X_i \Y = \X_i X\hat{\beta} \qquad \text{where $\hat{\beta}$ is the estimator of $\beta$} \\
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\Rightarrow& \X^T \Y = \X^T \X \hat{\beta} \\
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\Rightarrow& {\color{gray}(\X^T \X)^{-1}} \X^T \Y = {\color{gray}(\X^T \X)^{-1}} (\X^T\X) \hat{\beta} \\
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\Rightarrow& \hat{\beta} = (\X^T\X)^{-1} \X^T \Y
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\end{align*}
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This formula comes from the orthogonal projection of $\Y$ on the vector subspace defined by the explanatory variables $\X$
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$\X \hat{\beta}$ is the closest point to $\Y$ in the subspace generated by $\X$.
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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 $\X\hat{\beta}$.
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\section{Sum of squares}
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$\Y - \X \hat{\beta} \perp \X \hat{\beta} - \Y \One$ if $\One \in V$, so
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\[
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\underbrace{\norm{\Y - \bar{\Y}\One}}_{\text{Total SS}} = \underbrace{\norm{\Y - \X \hat{\beta}}^2}_{\text{Residual SS}} + \underbrace{\norm{\X \hat{\beta} - \bar{\Y} \One}^2}_{\text{Explicated SS}}
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\]
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\section{Coefficient of Determination: \texorpdfstring{$R^2$}{R\textsuperscript{2}}}
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\begin{definition}[$R^2$]
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\[
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0 \leq R^2 = \frac{\norm{\X\hat{\beta} - \bar{\Y}\One}^2}{\norm{\Y - \bar{\Y}\One}^2} = 1 - \frac{\norm{\Y - \X\hat{\beta}}^2}{\norm{\Y - \bar{\Y}\One}^2} \leq 1
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\] proportion of variation of $\Y$ explained by the model.
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\end{definition}
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\begin{figure}
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\centering
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\includestandalone{figures/schemes/orthogonal_projection}
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\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 $\hat{\varepsilon} = \norm{\Y - \hat{\beta}\X}^2$} and $\color{blue}c$ corresponds to $\norm{Y - \bar{Y}}^2$.
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\label{fig:scheme-orthogonal-projection}
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\end{figure}
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\begin{figure}
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\centering
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\includestandalone{figures/schemes/ordinary_least_squares}
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\caption{Ordinary least squares and regression line with simulated data.}
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\label{fig:ordinary-least-squares}
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\end{figure}
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\begin{definition}[Model dimension]
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Let $\M$ be a model.
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The dimension of $\M$ is the dimension of the subspace generated by $\X$, that is the number of parameters in the $\beta$ vector.
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\textit{Nb.} The dimension of the model is not the number of parameter, as $\sigma^2$ is one of the model parameters.
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\end{definition}
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\section{Gaussian vectors}
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\begin{definition}[Normal distribution]
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\end{definition}
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\begin{definition}[Gaussian vector]
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A random vector $\Y \in \RR[n]$ is a gaussian vector if every linear combination of its component is ...
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\end{definition}
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\begin{property}
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$m = \EE(Y) = (m_1, \ldots, m_n)^T$, where $m_i = \EE(Y_i)$
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...
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\[
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\Y \sim \Norm_n(m, \Sigma)
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\]
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where $\Sigma$ is the variance-covariance matrix!
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\[
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\Sigma = \E\left[(\Y -m)(\Y - m)^T\right].
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\]
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\end{property}
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\begin{remark}
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\[
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\Cov(Y_i, Y_i) = \Var(Y_i)
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\]
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\end{remark}
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\begin{definition}[Covariance]
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\[
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\Cov(Y_i, Y_j) = \EE\left((Y_i-\EE(Y_j))(Y_j-\EE(Y_j))\right)
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\]
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\end{definition}
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When two variable are linked, the covariance is large.
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If two variables $X, Y$ are independent, $\Cov(X, Y) = 0$.
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\begin{definition}[Correlation coefficient]
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\[
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\Cor(Y_i, Y_j) = \frac{\EE\left((Y_i-\EE(Y_j))(Y_j-\EE(Y_j))\right)}{\sqrt{\EE(Y_i - \EE(Y_i)) \cdot \EE(Y_j - \EE(Y_j))}}
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\]
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\end{definition}
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Covariance is really sensitive to scale of variables. For instance, if we measure distance in millimeters, the covariance would be larger than in the case of a measure expressed in metters. Thus the correlation coefficient, which is a sort of normalized covariance is useful, to be able to compare the values.
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\begin{remark}
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\begin{align*}
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\Cov(Y_i, Y_i) &= \EE((Y_i - \EE(Y_i)) (Y_i - \EE(Y_i))) \\
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&= \EE((Y_i - \EE(Y_i))^2) \\
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&= \Var(Y_i)
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\end{align*}
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\end{remark}
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\begin{equation}
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\Sigma = \begin{pNiceMatrix}
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\VVar(Y_1) & & & &\\
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& \Ddots & & & \\
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& \Cov(Y_i, Y_j) & \VVar(Y_i) & & \\
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& & & \Ddots & \\
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& & & & \VVar(Y_n)
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\end{pNiceMatrix}
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\end{equation}
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\begin{definition}[Identity matrix]
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\[
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\mathcal{I}_n = \begin{pNiceMatrix}
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1 & 0 & 0 \\
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0 & \Ddots & 0\\
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0 & 0 & 1
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\end{pNiceMatrix}
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\]
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\end{definition}
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\begin{theorem}[Cochran Theorem (Consequence)]
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Let $\mathbf{Z}$ be a gaussian vector: $\mathbf{Z} \sim \Norm_n(0_n, I_n)$.
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\begin{itemize}
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\item If $V_1, V_n$ are orthogonal subspaces of $\RR[n]$ with dimensions $n_1, n_2$ such that
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\[
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\RR[n] = V_1 \overset{\perp}{\oplus} V_2.
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\]
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\item If $Z_1, Z_2$ are orthogonal of $\mathbf{Z}$ on $V_1$ and $V_2$ i.e. $Z_1 = \Pi_{V_1}(\mathbf{Z}) = \Pi_1 \Y$ and $Z_2 = \Pi_{V_2} (\mathbf{Z}) = \Pi_2 \Y$...
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(\textcolor{red}{look to the slides})
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\end{itemize}
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\end{theorem}
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\begin{definition}[Chi 2 distribution]
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If $X_1, \ldots, X_n$ i.i.d. $\sim \Norm(0, 1)$, then;,
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\[
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X_1^2 + \ldots X_n^2 \sim \chi_n^2
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\]
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\end{definition}
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\subsection{Estimator's properties}
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\[
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\Pi_V = \X(\X^T\X)^{-1} \X^T
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\]
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\begin{align*}
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\hat{m} &= \X \hat{\beta} = \X(\X^T\X)^{-1} \X^T \Y \\
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\text{so} \\
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&= \Pi_V \Y
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\end{align*}
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According to Cochran theorem, we can deduce that the estimator of the predicted value $\hat{m}$ is independent $\hat{\sigma}^2$
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All the sum of squares follows a $\chi^2$ distribution:
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\[
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...
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\]
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\begin{property}
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\end{property}
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\subsection{Estimators consistency}
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If $q < n$,
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\begin{itemize}
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\item $\hat{\sigma}^2 \overunderset{\PP}{n\to\infty} \sigma^{*2}$.
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\item If $(\X^T\X)^{-1}$...
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\item ...
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\end{itemize}
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We can derive statistical test from these properties.
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\section{Statistical tests}
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\subsection{Student $t$-test}
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\[
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\frac{\hat{\theta}-\theta}{\sqrt{\frac{\widehat{\VVar}(\hat{\theta})}{n}}} \underset{H_0}{\sim} t
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\]
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where
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