From a92a13d3547315766e30d3446125dedd3103ee1f Mon Sep 17 00:00:00 2001 From: Samuel Ortion Date: Fri, 22 Sep 2023 17:32:56 +0200 Subject: [PATCH] cm1: Base introduction and some elements of linear algebra --- Makefile | 6 + content/chapters/part1/0.tex | 1 + content/chapters/part1/1.tex | 112 +++++++++++++ content/chapters/part1/2.tex | 212 ++++++++++++++++++++++++ content/chapters/part1/3.tex | 0 content/introduction.tex | 25 +++ figures/schemes/coordinates_systems.pdf | Bin 0 -> 8829 bytes figures/schemes/coordinates_systems.tex | 12 ++ main.pdf | Bin 0 -> 242670 bytes main.tex | 36 ++-- preamble.tex | 3 + 11 files changed, 389 insertions(+), 18 deletions(-) create mode 100644 Makefile create mode 100644 content/chapters/part1/2.tex create mode 100644 content/chapters/part1/3.tex create mode 100644 figures/schemes/coordinates_systems.pdf create mode 100644 figures/schemes/coordinates_systems.tex create mode 100644 main.pdf diff --git a/Makefile b/Makefile new file mode 100644 index 0000000..a84950a --- /dev/null +++ b/Makefile @@ -0,0 +1,6 @@ +options=-shell-escape -file-line-error + +all: main.pdf + +%.pdf: %.tex + lualatex $(options) $< diff --git a/content/chapters/part1/0.tex b/content/chapters/part1/0.tex index e69de29..ebca180 100644 --- a/content/chapters/part1/0.tex +++ b/content/chapters/part1/0.tex @@ -0,0 +1 @@ +\part{} \ No newline at end of file diff --git a/content/chapters/part1/1.tex b/content/chapters/part1/1.tex index e69de29..8ac4a3f 100644 --- a/content/chapters/part1/1.tex +++ b/content/chapters/part1/1.tex @@ -0,0 +1,112 @@ + + +\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 Poission 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 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. +In order to estimate the parameters, we can use penalties (additional terms). + +Lasso regression, Elastic Net, etc. + +\subsection{Simple Linear Model} + +\begin{align*} + \Y &= \X & \beta & + & \varepsilon.\\ + n \times 1 & n \times 2 & 2 \times 1 & + & n \times 1 \\ + \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} +\end{align*} + +\subsection{Assumptions} + +\begin{itemize} + \item +\end{itemize} + + +\subsection{Statistical Analysis Workflow} + +\begin{enumerate}[label={\bfseries\color{primary}Step \arabic*.}] + \item Graphical representation; + \item ... +\end{enumerate} + + +\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{red}(\X^T \X)^{-1}} \X^\T \Y = {\color{red}(\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 subspace define by the explicative 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: $R^2$} +\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 $\Y$ explicated by the model. +\end{definition} diff --git a/content/chapters/part1/2.tex b/content/chapters/part1/2.tex new file mode 100644 index 0000000..eb391fe --- /dev/null +++ b/content/chapters/part1/2.tex @@ -0,0 +1,212 @@ +\chapter{Elements of Linear Algebra} +\label{ch:elements-of-linear-algebra} + +\begin{remark}[vector] + Let $u$ a vector, we will use interchangeably the following notations: $u$ and $\vec{u}$ +\end{remark} + +Let $u = \begin{pmatrix} + u_1 \\ + \vdots \\ + u_n + \end{pmatrix}$ and $v = \begin{pmatrix} + v_1 \\ + \vdots \\ + v_n + \end{pmatrix}$ + +\begin{align*} + \langle u, v\rangle & = \left(u_1, \ldots, u_v\right) \begin{pmatrix} + v_1 \\ + \vdots \\ + v_n + \end{pmatrix} \\ + & = u_1 v_1 + u_2 v_2 + \ldots + u_n v_n +\end{align*} + + +\begin{definition}[Norm] + Length of the vector. + \[ + \norm{u} = \sqrt{\scalar{u, v}} + \] + + $\norm{u, v} > 0$ +\end{definition} + +\begin{definition}[Distance] + \[ + dist(u, v) = \norm{u-v} + \] +\end{definition} + +\begin{definition}[Orthogonality] + \[ + u \perp v \Leftrightarrow \scalar{u, v} = 0 + \] +\end{definition} + +\begin{remark} + \[ + (dist(u, v))^2 = \norm{u - v}^2, + \] and + \[ + \scalar{v-u, v-u} + \] +\end{remark} + +Scalar product properties: +\begin{itemize} + \item $\scalar{u, v} = \scalar{v, u}$ + \item $\scalar{(u+v), w} = \scalar{u, w} + \scalar{v, w}$ + \item $\scalar{u, v}$ + \item $\scalar{\vec{u}, \vec{v}} = \norm{\vec{u}} \times \norm{\vec{v}} \times \cos(\widehat{\vec{u}, \vec{v}})$ +\end{itemize} + +\begin{align*} + \scalar{v-u, v-u} & = \scalar{v, v} + \scalar{u, u} - 2 \scalar{u, v} \\ + & = \norm{v}^2 + \norm{u}^2 \\ + & = -2 \scalar{u, v} +\end{align*} + +\begin{align*} + \norm{u - v}^2 & = \norm{u}^2 + \norm{v}^2 - 2 \scalar{u,v} \\ + \norm{u + v}^2 & = \norm{u}^2 + \norm{v}^2 + 2 \scalar{u,v} +\end{align*} + + +If $u \perp v$, then $\scalar{u, v} = 0$ +\begin{proof}[Indeed] + $\norm{u-v}^2 = \norm{u+v}^2$, + \begin{align*} + \Leftrightarrow & -2 \scalar{u, v} = 2 \scalar{u, v} \\ + \Leftrightarrow & 4 \scalar{u, v} = 0 \\ + \Leftrightarrow & \scalar{u, v} = 0 + \end{align*} +\end{proof} + +\begin{theorem}{Pythagorean theorem} + If $u \perp v$, then $\norm{u+v}^2 = \norm{u}^2 + \norm{v}^2$ . +\end{theorem} + +\begin{definition}[Orthogonal Projection] + +\end{definition} +Let $y = \begin{pmatrix} + y_1 \\ + . \\ + y_n + \end{pmatrix} \in \RR[n]$ and $w$ a subspace of $\RR[n]$ +$\mathcal{Y}$ can be written as the orthogonal projection of $y$ on $w$: +\[ + \mathcal{Y} = proj^w(y) + z, +\] +where +\[ + \begin{cases} + z \in w^\perp \\ + proj^w(y) \in w + \end{cases} +\] +There is only one vector $\mathcal{Y}$ that ? + +The scalar product between $z$ and (?) is zero. + +\begin{property} + $proj^w(y)$ is the closest vector to $y$ that belongs to $w$. +\end{property} + +\begin{definition}[Matrix] + A matrix is an application, that is, a function that transform a thing into another, it is a linear function. +\end{definition} + +\begin{example}[Matrix application] + + Let $A$ be a matrix: + \[ + A = \begin{pmatrix} + a & b \\ + c & d + \end{pmatrix} + \] and + \[ + x = \begin{pmatrix} + x_1 \\ + x_2 + \end{pmatrix} + \] + Then, + \begin{align*} + Ax & = \begin{pmatrix} + a & b \\ + c & d + \end{pmatrix} + \begin{pmatrix} + x_1 \\ + x_2 + \end{pmatrix} \\ + & = \begin{pmatrix} + a x_1 + b_x2 \\ + c x_1 + d x_2 + \end{pmatrix} + \end{align*} + + Similarly, + \begin{align*} + \begin{pmatrix} + a & b & c & d \\ + e & f & g & h \\ + i & j & k & l + \end{pmatrix} + \begin{pmatrix} + x_1 \\ + x_2 \\ + x_3 \\ + x_4 + \end{pmatrix} + & = \begin{pmatrix} + a x_1 + b x_2 + c x_3 \ldots + \end{pmatrix} + \end{align*} +\end{example} + +The number of columns has to be the same as the dimension of the vector to which the matrix is applied. + +\begin{definition}[Tranpose of a Matrix] + Let $A = \begin{pmatrix} + a & b \\ + c & d + \end{pmatrix}$, then $A^\T = \begin{pmatrix} + a & c \\ + b & d + \end{pmatrix}$ +\end{definition} + +\begin{example} + \begin{align*} + Y & = X \beta + \varepsilon \\ + \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} + \end{align*} +\end{example} \ No newline at end of file diff --git a/content/chapters/part1/3.tex b/content/chapters/part1/3.tex new file mode 100644 index 0000000..e69de29 diff --git a/content/introduction.tex b/content/introduction.tex index e69de29..775a081 100644 --- a/content/introduction.tex +++ b/content/introduction.tex @@ -0,0 +1,25 @@ +\chapter{Introduction} + +\begin{definition}[Long Term Nonprocessor (LTNP)] + Patient who will remain a long time in good health condition, even with a large viral load (cf. HIV). +\end{definition} + +\begin{example}[Genotype: Qualitative or Quantitative?] + \[ + \text{SNP}: + \begin{cases} + \text{AA} \\ + \text{AB} \\ + \text{BB} + \end{cases} + \rightarrow + \begin{pmatrix} + 0 \\ + 1 \\ + 2 + \end{pmatrix}, + \] + thus we might consider genotype either as a qualitative variable or quantitative variable. +\end{example} + +When the variable are quantitative, we use regression, whereas for qualitative variables, we use an analysis of variance. \ No newline at end of file diff --git a/figures/schemes/coordinates_systems.pdf b/figures/schemes/coordinates_systems.pdf new file mode 100644 index 0000000000000000000000000000000000000000..953e6c419ffff92413e2c4f3e0126a14eb5fe09a GIT binary patch literal 8829 zcma)?1yCJLx2}V`1xfG?1hKSrHQE_@E1~x?U(YdbKzQxJjenfTv5MXCyiO93x3b**Wnh3_ea=0;Lvg2k$jAazDkxr+^+&{aS)EXymf5 zZOG;1s`K5C+^_zEgn(a|zi#Av^h2(g)^<3LMSm0;?VFnwkixW_-NyS@_*g({=7QB% zVX)c|?D433M?*3IU4P$5*<)7=kO@<)53)#!KY8QucC17cj3U_T{*pZ 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\documentclass[ - a4paper, - fontsize=10pt, - fleqn, - oneside + a4paper, + fontsize=10pt, + fleqn, + oneside ]{scrbook} \usepackage{mus} \titlehead{GENIOMHE} -\title{Multivariate Statistics} +\title{Multivariate\newline{}Statistics} \author{Samuel Ortion} \teacher{Cyril Dalmasso} \cursus{GENIOMHE} \university{Université Paris-Saclay, Université d'Évry val d'Essonne} \semester{M1 - S1} -\date{} +\date{Fall 2023} + +\definecolor{myblue}{HTML}{5654fa} +\colorlet{primary}{myblue} \input{definitions} +\input{preamble} \hypersetup{ - pdftitle={ - Course - Multivariate Statistics - }, - pdfauthor={ - Samuel Ortion - }, - pdfsubject={}, - pdfkeywords={}, - pdfcreator={LaTeX} + pdftitle={Course - Multivariate Statistics}, + pdfauthor={Samuel Ortion}, + pdfsubject={}, + pdfkeywords={}, + pdfcreator={LaTeX} } \addbibresource{references} \usepackage[ - type={CC}, - modifier={by-sa}, - version={4.0}, + type={CC}, + modifier={by-sa}, + version={4.0}, ]{doclicense} \input{preamble} diff --git a/preamble.tex b/preamble.tex index e69de29..f2c2d8f 100644 --- a/preamble.tex +++ b/preamble.tex @@ -0,0 +1,3 @@ +\usepackage{pgffor} +\usetikzlibrary{math} +\usepackage{standalone} \ No newline at end of file