cm1: Base introduction and some elements of linear algebra
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options=-shell-escape -file-line-error
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all: main.pdf
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%.pdf: %.tex
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lualatex $(options) $<
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\part{}
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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 Poission 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 explicative variable is above the number of observations, if $p >> n$ ($p$: the number of explicative 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{Simple Linear Model}
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\begin{align*}
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\Y &= \X & \beta & + & \varepsilon.\\
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n \times 1 & n \times 2 & 2 \times 1 & + & n \times 1 \\
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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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&= \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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\end{align*}
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\subsection{Assumptions}
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\begin{itemize}
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\item
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\end{itemize}
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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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\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{red}(\X^T \X)^{-1}} \X^\T \Y = {\color{red}(\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 subspace define by the explicative 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{Coefficient of Determination: $R^2$}
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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 $\Y$ explicated by the model.
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\end{definition}
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\chapter{Elements of Linear Algebra}
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\label{ch:elements-of-linear-algebra}
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\begin{remark}[vector]
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Let $u$ a vector, we will use interchangeably the following notations: $u$ and $\vec{u}$
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\end{remark}
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Let $u = \begin{pmatrix}
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u_1 \\
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\vdots \\
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u_n
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\end{pmatrix}$ and $v = \begin{pmatrix}
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v_1 \\
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\vdots \\
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v_n
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\end{pmatrix}$
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\begin{align*}
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\langle u, v\rangle & = \left(u_1, \ldots, u_v\right) \begin{pmatrix}
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v_1 \\
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\vdots \\
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v_n
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\end{pmatrix} \\
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& = u_1 v_1 + u_2 v_2 + \ldots + u_n v_n
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\end{align*}
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\begin{definition}[Norm]
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Length of the vector.
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\[
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\norm{u} = \sqrt{\scalar{u, v}}
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\]
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$\norm{u, v} > 0$
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\end{definition}
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\begin{definition}[Distance]
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\[
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dist(u, v) = \norm{u-v}
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\]
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\end{definition}
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\begin{definition}[Orthogonality]
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\[
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u \perp v \Leftrightarrow \scalar{u, v} = 0
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\]
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\end{definition}
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\begin{remark}
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\[
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(dist(u, v))^2 = \norm{u - v}^2,
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\] and
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\[
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\scalar{v-u, v-u}
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\]
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\end{remark}
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Scalar product properties:
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\begin{itemize}
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\item $\scalar{u, v} = \scalar{v, u}$
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\item $\scalar{(u+v), w} = \scalar{u, w} + \scalar{v, w}$
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\item $\scalar{u, v}$
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\item $\scalar{\vec{u}, \vec{v}} = \norm{\vec{u}} \times \norm{\vec{v}} \times \cos(\widehat{\vec{u}, \vec{v}})$
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\end{itemize}
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\begin{align*}
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\scalar{v-u, v-u} & = \scalar{v, v} + \scalar{u, u} - 2 \scalar{u, v} \\
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& = \norm{v}^2 + \norm{u}^2 \\
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& = -2 \scalar{u, v}
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\end{align*}
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\begin{align*}
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\norm{u - v}^2 & = \norm{u}^2 + \norm{v}^2 - 2 \scalar{u,v} \\
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\norm{u + v}^2 & = \norm{u}^2 + \norm{v}^2 + 2 \scalar{u,v}
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\end{align*}
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If $u \perp v$, then $\scalar{u, v} = 0$
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\begin{proof}[Indeed]
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$\norm{u-v}^2 = \norm{u+v}^2$,
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\begin{align*}
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\Leftrightarrow & -2 \scalar{u, v} = 2 \scalar{u, v} \\
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\Leftrightarrow & 4 \scalar{u, v} = 0 \\
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\Leftrightarrow & \scalar{u, v} = 0
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\end{align*}
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\end{proof}
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\begin{theorem}{Pythagorean theorem}
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If $u \perp v$, then $\norm{u+v}^2 = \norm{u}^2 + \norm{v}^2$ .
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\end{theorem}
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\begin{definition}[Orthogonal Projection]
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\end{definition}
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Let $y = \begin{pmatrix}
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y_1 \\
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. \\
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y_n
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\end{pmatrix} \in \RR[n]$ and $w$ a subspace of $\RR[n]$
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$\mathcal{Y}$ can be written as the orthogonal projection of $y$ on $w$:
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\[
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\mathcal{Y} = proj^w(y) + z,
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\]
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where
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\[
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\begin{cases}
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z \in w^\perp \\
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proj^w(y) \in w
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\end{cases}
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\]
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There is only one vector $\mathcal{Y}$ that ?
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The scalar product between $z$ and (?) is zero.
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\begin{property}
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$proj^w(y)$ is the closest vector to $y$ that belongs to $w$.
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\end{property}
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\begin{definition}[Matrix]
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A matrix is an application, that is, a function that transform a thing into another, it is a linear function.
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\end{definition}
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\begin{example}[Matrix application]
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Let $A$ be a matrix:
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\[
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A = \begin{pmatrix}
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a & b \\
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c & d
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\end{pmatrix}
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\] and
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\[
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x = \begin{pmatrix}
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x_1 \\
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x_2
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\end{pmatrix}
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\]
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Then,
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\begin{align*}
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Ax & = \begin{pmatrix}
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a & b \\
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c & d
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\end{pmatrix}
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\begin{pmatrix}
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x_1 \\
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x_2
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\end{pmatrix} \\
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& = \begin{pmatrix}
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a x_1 + b_x2 \\
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c x_1 + d x_2
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\end{pmatrix}
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\end{align*}
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Similarly,
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\begin{align*}
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\begin{pmatrix}
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a & b & c & d \\
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e & f & g & h \\
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i & j & k & l
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\end{pmatrix}
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\begin{pmatrix}
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x_1 \\
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x_2 \\
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x_3 \\
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x_4
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\end{pmatrix}
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& = \begin{pmatrix}
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a x_1 + b x_2 + c x_3 \ldots
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\end{pmatrix}
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\end{align*}
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\end{example}
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The number of columns has to be the same as the dimension of the vector to which the matrix is applied.
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\begin{definition}[Tranpose of a Matrix]
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Let $A = \begin{pmatrix}
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a & b \\
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c & d
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\end{pmatrix}$, then $A^\T = \begin{pmatrix}
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a & c \\
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b & d
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\end{pmatrix}$
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\end{definition}
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\begin{example}
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\begin{align*}
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Y & = X \beta + \varepsilon \\
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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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\end{align*}
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\end{example}
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\chapter{Introduction}
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\begin{definition}[Long Term Nonprocessor (LTNP)]
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Patient who will remain a long time in good health condition, even with a large viral load (cf. HIV).
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\end{definition}
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\begin{example}[Genotype: Qualitative or Quantitative?]
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\[
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\text{SNP}:
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\begin{cases}
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\text{AA} \\
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\text{AB} \\
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\text{BB}
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\end{cases}
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\rightarrow
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\begin{pmatrix}
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0 \\
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1 \\
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2
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\end{pmatrix},
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\]
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thus we might consider genotype either as a qualitative variable or quantitative variable.
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\end{example}
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When the variable are quantitative, we use regression, whereas for qualitative variables, we use an analysis of variance.
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Binary file not shown.
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\documentclass[tikz]{standalone}
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\usepackage{tikz}
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\usepackage{tkz-euclide}
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\begin{document}
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\begin{tikzpicture}
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\tkzInit[xmax=5,ymax=5,xmin=-5,ymin=-5]
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\tkzGrid
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\tkzAxeXY
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\draw[thick, latex-latex] (-1,4) -- (4,-6) node[anchor=south west] {$a$};
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\end{tikzpicture}
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\end{document}
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16
main.tex
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main.tex
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\usepackage{mus}
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\titlehead{GENIOMHE}
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\title{Multivariate Statistics}
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\title{Multivariate\newline{}Statistics}
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\author{Samuel Ortion}
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\teacher{Cyril Dalmasso}
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\cursus{GENIOMHE}
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\university{Université Paris-Saclay, Université d'Évry val d'Essonne}
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\semester{M1 - S1}
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\date{}
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\date{Fall 2023}
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\definecolor{myblue}{HTML}{5654fa}
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\colorlet{primary}{myblue}
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\input{definitions}
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\input{preamble}
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\hypersetup{
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pdftitle={
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Course - Multivariate Statistics
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},
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pdfauthor={
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Samuel Ortion
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},
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pdftitle={Course - Multivariate Statistics},
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pdfauthor={Samuel Ortion},
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pdfsubject={},
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pdfkeywords={},
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pdfcreator={LaTeX}
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\usepackage{pgffor}
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\usetikzlibrary{math}
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\usepackage{standalone}
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