From 14928631eca645d109aa5ebca41554b403b4ffaf Mon Sep 17 00:00:00 2001 From: Samuel Ortion Date: Tue, 19 Sep 2023 20:13:48 +0200 Subject: [PATCH 1/9] Initial commit --- .gitignore | 304 +++++++++++++++++++++++++++++++++++ content/chapters/include.tex | 18 +++ content/chapters/part1/0.tex | 0 content/chapters/part1/1.tex | 0 content/conclusion.tex | 0 content/introduction.tex | 0 definitions.tex | 0 glossary.tex | 0 main.tex | 70 ++++++++ preamble.tex | 0 references.bib | 0 11 files changed, 392 insertions(+) create mode 100755 .gitignore create mode 100755 content/chapters/include.tex create mode 100644 content/chapters/part1/0.tex create mode 100644 content/chapters/part1/1.tex create mode 100644 content/conclusion.tex create mode 100644 content/introduction.tex create mode 100644 definitions.tex create mode 100644 glossary.tex create mode 100644 main.tex create mode 100644 preamble.tex create mode 100644 references.bib diff --git a/.gitignore b/.gitignore new file mode 100755 index 0000000..76f44d5 --- /dev/null +++ b/.gitignore @@ -0,0 +1,304 @@ +build/ +.auctex-auto + +## Core latex/pdflatex auxiliary files: +*.aux +*.lof +*.log +*.lot +*.fls +*.out +*.toc +*.fmt +*.fot +*.cb +*.cb2 +.*.lb + +## Intermediate documents: +*.dvi +*.xdv +*-converted-to.* +# these rules might exclude image files for figures etc. +# *.ps +# *.eps +# *.pdf + +## Generated if empty string is given at "Please type another file name for output:" +.pdf + +## Bibliography auxiliary files (bibtex/biblatex/biber): +*.bbl +*.bcf +*.blg +*-blx.aux +*-blx.bib +*.run.xml + +## Build tool auxiliary files: +*.fdb_latexmk +*.synctex +*.synctex(busy) +*.synctex.gz +*.synctex.gz(busy) +*.pdfsync + +## Build tool directories for auxiliary files +# latexrun +latex.out/ + +## Auxiliary and intermediate files from other packages: +# algorithms +*.alg +*.loa + +# achemso +acs-*.bib + +# amsthm +*.thm + +# beamer +*.nav +*.pre +*.snm +*.vrb + +# changes +*.soc + +# comment +*.cut + +# cprotect +*.cpt + +# elsarticle (documentclass of Elsevier journals) +*.spl + +# endnotes +*.ent + +# fixme +*.lox + +# feynmf/feynmp +*.mf +*.mp +*.t[1-9] +*.t[1-9][0-9] +*.tfm + +#(r)(e)ledmac/(r)(e)ledpar +*.end +*.?end +*.[1-9] +*.[1-9][0-9] +*.[1-9][0-9][0-9] +*.[1-9]R +*.[1-9][0-9]R +*.[1-9][0-9][0-9]R +*.eledsec[1-9] +*.eledsec[1-9]R +*.eledsec[1-9][0-9] +*.eledsec[1-9][0-9]R +*.eledsec[1-9][0-9][0-9] +*.eledsec[1-9][0-9][0-9]R + +# glossaries +*.acn +*.acr +*.glg +*.glo +*.gls +*.glsdefs +*.lzo +*.lzs +*.slg +*.slo +*.sls + +# uncomment this for glossaries-extra (will ignore makeindex's style files!) +# *.ist + +# gnuplot +*.gnuplot +*.table + +# gnuplottex +*-gnuplottex-* + +# gregoriotex +*.gaux +*.glog +*.gtex + +# htlatex +*.4ct +*.4tc +*.idv +*.lg +*.trc +*.xref + +# hyperref +*.brf + +# knitr +*-concordance.tex +# TODO Uncomment the next line if you use knitr and want to ignore its generated tikz files +# *.tikz +*-tikzDictionary + +# listings +*.lol + +# luatexja-ruby +*.ltjruby + +# makeidx +*.idx +*.ilg +*.ind + +# minitoc +*.maf +*.mlf +*.mlt +*.mtc[0-9]* +*.slf[0-9]* +*.slt[0-9]* +*.stc[0-9]* + +# minted +_minted* +*.pyg + +# morewrites +*.mw + +# newpax +*.newpax + +# nomencl +*.nlg +*.nlo +*.nls + +# pax +*.pax + +# pdfpcnotes +*.pdfpc + +# sagetex +*.sagetex.sage +*.sagetex.py +*.sagetex.scmd + +# scrwfile +*.wrt + +# svg +svg-inkscape/ + +# sympy +*.sout +*.sympy +sympy-plots-for-*.tex/ + +# pdfcomment +*.upa +*.upb + +# pythontex +*.pytxcode +pythontex-files-*/ + +# tcolorbox +*.listing + +# thmtools +*.loe + +# TikZ & PGF +*.dpth +*.md5 +*.auxlock + +# titletoc +*.ptc + +# todonotes +*.tdo + +# vhistory +*.hst +*.ver + +# easy-todo +*.lod + +# xcolor +*.xcp + +# xmpincl +*.xmpi + +# xindy +*.xdy + +# xypic precompiled matrices and outlines +*.xyc +*.xyd + +# endfloat +*.ttt +*.fff + +# Latexian +TSWLatexianTemp* + +## Editors: +# WinEdt +*.bak +*.sav + +# Texpad +.texpadtmp + +# LyX +*.lyx~ + +# Kile +*.backup + +# gummi +.*.swp + +# KBibTeX +*~[0-9]* + +# TeXnicCenter +*.tps + +# auto folder when using emacs and auctex +./auto/* +*.el + +# expex forward references with \gathertags +*-tags.tex + +# standalone packages +*.sta + +# Makeindex log files +*.lpz + +# xwatermark package +*.xwm + +# REVTeX puts footnotes in the bibliography by default, unless the nofootinbib +# option is specified. Footnotes are the stored in a file with suffix Notes.bib. +# Uncomment the next line to have this generated file ignored. +#*Notes.bib diff --git a/content/chapters/include.tex b/content/chapters/include.tex new file mode 100755 index 0000000..7922d7d --- /dev/null +++ b/content/chapters/include.tex @@ -0,0 +1,18 @@ +%---------------------------------------- +% CHAPTERS +%---------------------------------------- + +\newcommand{\includechapters}[2]{% + \foreach \i in {0, ..., #2} {% + \edef\FileName{content/chapters/#1/\i}% + \IfFileExists{\FileName}{% + \input{\FileName}% + } + } +} + +\includechapters{part1}{2} + +% \includechapters{part2}{2} + +% \includechapters{part3}{1} \ No newline at end of file diff --git a/content/chapters/part1/0.tex b/content/chapters/part1/0.tex new file mode 100644 index 0000000..e69de29 diff --git a/content/chapters/part1/1.tex b/content/chapters/part1/1.tex new file mode 100644 index 0000000..e69de29 diff --git a/content/conclusion.tex b/content/conclusion.tex new file mode 100644 index 0000000..e69de29 diff --git a/content/introduction.tex b/content/introduction.tex new file mode 100644 index 0000000..e69de29 diff --git a/definitions.tex b/definitions.tex new file mode 100644 index 0000000..e69de29 diff --git a/glossary.tex b/glossary.tex new file mode 100644 index 0000000..e69de29 diff --git a/main.tex b/main.tex new file mode 100644 index 0000000..6fe434b --- /dev/null +++ b/main.tex @@ -0,0 +1,70 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Course -- Multivariate Statistics --- GENIOMHE --- M1 - S1 +% +% Author: Samuel Ortion +% Version: 0.1.0 +% Date: 2023 +% License: CC-By-SA 4.0+ International +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\documentclass[ + a4paper, + fontsize=10pt, + fleqn, + oneside +]{scrbook} + +\usepackage{mus} + +\titlehead{GENIOMHE} +\title{Multivariate Statistics} +\author{Samuel Ortion} +\teacher{Cyril Dalmasso} +\cursus{GENIOMHE} +\university{Université Paris-Saclay, Université d'Évry val d'Essonne} +\semester{M1 - S1} +\date{} + +\input{definitions} + +\hypersetup{ + pdftitle={ + Course - Multivariate Statistics + }, + pdfauthor={ + Samuel Ortion + }, + pdfsubject={}, + pdfkeywords={}, + pdfcreator={LaTeX} +} + +\addbibresource{references} + +\usepackage[ + type={CC}, + modifier={by-sa}, + version={4.0}, +]{doclicense} + +\input{preamble} +\input{glossary} +\input{definitions} + +\makeindex% +\makeglossary% +\begin{document} + +\maketitlefullpage + +\tableofcontents + +\doclicenseThis% + +\input{content/introduction} + +\input{content/chapters/include} + +\input{content/conclusion} + +\end{document} diff --git a/preamble.tex b/preamble.tex new file mode 100644 index 0000000..e69de29 diff --git a/references.bib b/references.bib new file mode 100644 index 0000000..e69de29 From a92a13d3547315766e30d3446125dedd3103ee1f Mon Sep 17 00:00:00 2001 From: Samuel Ortion Date: Fri, 22 Sep 2023 17:32:56 +0200 Subject: [PATCH 2/9] 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 From ae51323390f23862b7d92515afd0d82a1657c66d Mon Sep 17 00:00:00 2001 From: Samuel Ortion Date: Fri, 22 Sep 2023 23:48:46 +0200 Subject: [PATCH 3/9] feat: Add some basic (but probably uncomplete) schemes --- content/chapters/part1/1.tex | 62 ++++++++---- content/chapters/part1/2.tex | 111 ++++++++++------------ figures/schemes/base_plan.pdf | Bin 0 -> 3108 bytes figures/schemes/base_plan.tex | 16 ++++ figures/schemes/coordinates_systems.pdf | Bin 8829 -> 9339 bytes figures/schemes/coordinates_systems.tex | 23 +++-- figures/schemes/orthogonal_projection.pdf | Bin 0 -> 17473 bytes figures/schemes/orthogonal_projection.tex | 32 +++++++ figures/schemes/vector_orthogonality.pdf | Bin 0 -> 16486 bytes figures/schemes/vector_orthogonality.tex | 27 ++++++ main.pdf | Bin 242670 -> 247349 bytes preamble.tex | 5 +- 12 files changed, 191 insertions(+), 85 deletions(-) create mode 100644 figures/schemes/base_plan.pdf create mode 100644 figures/schemes/base_plan.tex create mode 100644 figures/schemes/orthogonal_projection.pdf create mode 100644 figures/schemes/orthogonal_projection.tex create mode 100644 figures/schemes/vector_orthogonality.pdf create mode 100644 figures/schemes/vector_orthogonality.tex diff --git a/content/chapters/part1/1.tex b/content/chapters/part1/1.tex index 8ac4a3f..a83f2c5 100644 --- a/content/chapters/part1/1.tex +++ b/content/chapters/part1/1.tex @@ -13,7 +13,7 @@ with $g$ being \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. +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. @@ -21,8 +21,7 @@ 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 \\ + \Y &= \X \beta + \varepsilon \\ \begin{pmatrix} Y_1 \\ Y_2 \\ @@ -35,11 +34,11 @@ Lasso regression, Elastic Net, etc. \vdots & \vdots \\ 1 & X_n \end{pmatrix} - & \begin{pmatrix} + \begin{pmatrix} \beta_0 \\ \beta_1 \end{pmatrix} - & + & + + \begin{pmatrix} \varepsilon_1 \\ \varepsilon_2 \\ @@ -61,8 +60,35 @@ Lasso regression, Elastic Net, etc. \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} @@ -87,26 +113,26 @@ We want to minimize the distance between $\X\beta$ and $\Y$: \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& {\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 subspace define by the explicative variables $\X$ - - - +This formula comes from the orthogonal projection of $\Y$ on the subspace define 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: $R^2$} +\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 $\Y$ explicated by the model. + \] 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} \ No newline at end of file diff --git a/content/chapters/part1/2.tex b/content/chapters/part1/2.tex index eb391fe..f614b6a 100644 --- a/content/chapters/part1/2.tex +++ b/content/chapters/part1/2.tex @@ -15,15 +15,28 @@ Let $u = \begin{pmatrix} v_n \end{pmatrix}$ -\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}[Scalar Product (Dot Product)] + \begin{align*} + \scalar{u, v} & = \begin{pmatrix} + u_1, \ldots, u_v + \end{pmatrix} + \begin{pmatrix} + v_1 \\ + \vdots \\ + v_n + \end{pmatrix} \\ + & = u_1 v_1 + u_2 v_2 + \ldots + u_n v_n + \end{align*} + We may use $\scalar{u, v}$ or $u \cdot v$ notations. +\end{definition} +\paragraph{Dot 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{definition}[Norm] Length of the vector. @@ -41,9 +54,7 @@ Let $u = \begin{pmatrix} \end{definition} \begin{definition}[Orthogonality] - \[ - u \perp v \Leftrightarrow \scalar{u, v} = 0 - \] + \end{definition} \begin{remark} @@ -55,18 +66,17 @@ Let $u = \begin{pmatrix} \] \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{figure} + \centering + \includestandalone{figures/schemes/vector_orthogonality} + \caption{Illustration for the scalar product of two orthogonal vectors.} + \label{fig:scheme-orthogonal-scalar-product} +\end{figure} \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} + & = \norm{v}^2 + \norm{u}^2 \\ + & = -2 \scalar{u, v} \end{align*} \begin{align*} @@ -74,10 +84,14 @@ Scalar product properties: \norm{u + v}^2 & = \norm{u}^2 + \norm{v}^2 + 2 \scalar{u,v} \end{align*} +\begin{proposition}[Scalar product of orthogonal vectors] +\[ + u \perp v \Leftrightarrow \scalar{u, v} = 0 +\] +\end{proposition} -If $u \perp v$, then $\scalar{u, v} = 0$ \begin{proof}[Indeed] - $\norm{u-v}^2 = \norm{u+v}^2$, + $\norm{u-v}^2 = \norm{u+v}^2$, as illustrated in \autoref{fig:scheme-orthogonal-scalar-product}. \begin{align*} \Leftrightarrow & -2 \scalar{u, v} = 2 \scalar{u, v} \\ \Leftrightarrow & 4 \scalar{u, v} = 0 \\ @@ -138,17 +152,17 @@ The scalar product between $z$ and (?) is zero. Then, \begin{align*} Ax & = \begin{pmatrix} - a & b \\ - c & d - \end{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} + & = \begin{pmatrix} + a x_1 + b x_2 \\ + c x_1 + d x_2 + \end{pmatrix} \end{align*} Similarly, @@ -164,9 +178,9 @@ The scalar product between $z$ and (?) is zero. x_3 \\ x_4 \end{pmatrix} - & = \begin{pmatrix} - a x_1 + b x_2 + c x_3 \ldots - \end{pmatrix} + & = \begin{pmatrix} + a x_1 + b x_2 + c x_3 \ldots + \end{pmatrix} \end{align*} \end{example} @@ -182,31 +196,8 @@ The number of columns has to be the same as the dimension of the vector to which \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} - 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Z8a8xt@o+LVL*Qg%Vn-k&6H^dJ_&+oq0W1Ij diff --git a/figures/schemes/coordinates_systems.tex b/figures/schemes/coordinates_systems.tex index 2671264..4b59680 100644 --- a/figures/schemes/coordinates_systems.tex +++ b/figures/schemes/coordinates_systems.tex @@ -1,12 +1,23 @@ \documentclass[tikz]{standalone} \usepackage{tikz} -\usepackage{tkz-euclide} \begin{document} -\begin{tikzpicture} - \tkzInit[xmax=5,ymax=5,xmin=-5,ymin=-5] - \tkzGrid - \tkzAxeXY - \draw[thick, latex-latex] (-1,4) -- (4,-6) node[anchor=south west] {$a$}; +\usetikzlibrary{3d} +% 1D axis +\begin{tikzpicture}[->] + \begin{scope}[xshift=0] + \draw (0, 0, 0) -- (xyz cylindrical cs:radius=1) node[right] {$x$}; + \end{scope} +% 2D coordinate system + \begin{scope}[xshift=50] + \draw (0, 0, 0) -- (xyz cylindrical cs:radius=1) node[right] {$x$}; + \draw (0, 0, 0) -- (xyz cylindrical cs:radius=1,angle=90) node[above] {$y$}; + \end{scope} +% 3D coordinate systems +\begin{scope}[xshift=100] + \draw (0, 0, 0) -- (xyz cylindrical 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+\usepackage{tikz-3dplot} +\usepackage{tkz-euclide} +\usepackage{mathtools} +\begin{document} +\tdplotsetmaincoords{50}{0} +\begin{tikzpicture}[tdplot_main_coords,bullet/.style={circle,inner + sep=1pt,fill=black,fill opacity=1}] + \begin{scope}[canvas is xy plane at z=0] + \tkzDefPoints{-2/-1/A,3/-1/B,4/2/C} + \tkzDefParallelogram(A,B,C) + \tkzGetPoint{D} + \tkzDrawPolygon[fill=gray!25!white](A,B,C,D) + \draw[decorate,decoration={brace, + amplitude=8pt},xshift=0pt,very thin,gray] (2,0) -- ++(-1,-0.5) node [black,midway,xshift=0.5em,yshift=-1em] {\color{blue}$a$}; + \end{scope} + \begin{scope}[canvas is xz plane at y=0] + \draw[thick,fill=white,fill opacity=0.7,nodes={opacity=1}] + (2,0) node[bullet,label=below right:{$\mathbf{X}$}] {} + -- (0,0) node[bullet] {} + -- (0,3) node[bullet,label=above:{$\mathbf{Y}$}] {} -- cycle; + \draw (0.25,0) -- (0.25,0.25) -- (0,0.25); + \draw[decorate,decoration={brace, + amplitude=8pt},xshift=0pt,very thin,gray] (0,0) -- (0,3) node [black,midway,xshift=-1.25em,yshift=0em] {\color{blue}$b$}; + \end{scope} + \begin{scope}[canvas is xy plane at z=0] + \draw[->] (2,0) -- ++(-0.75,0.75) node [left] {$\mathbf{1}$}; + \draw[->] (2,0) -- ++(-1,-0.5); + \end{scope} +\end{tikzpicture} +\end{document} \ No newline at end of file diff --git a/figures/schemes/vector_orthogonality.pdf b/figures/schemes/vector_orthogonality.pdf new file mode 100644 index 0000000000000000000000000000000000000000..5ff62b4d70865d27dbbffd3c19f79817ee1e9bfe GIT binary patch literal 16486 zcma)@18^rz+wP-{ZQIU2wryu)TN~T9v$1V&Y}>YN>+JKP-nZ&}2U9aOjk~Mwo|>y_ zs;~KxDTs*CG10R^lT9r4E)1^B3=BcD5i%0m8CpW~@DMVHSy(%nIQ+d^8#tMWm>Ag^ zn-DU{nAn;*nG>?JGZXUh5&qW!?darSVqgR9wzjAyYlp&w*m5oa6&P*%v=yfsXX2}8C_Ls6eELx^5zYgi@jKm zmk4Ysi<>fYrgARiOC%u2yfaG!`ABC ziN+ug)iKH?0hTMw7PHnZf|H;iiX>-IZy_Qbu`Dgy(8>4BqfK!nfH4W@+|@YkW^nt( zmB6!ISGV`drjY+-zR)ApxYVw?`k#wDQ^plF=o}$P^(;)My|vNn3OSD2X(fgOrsU)P ztME+4oJ%C|^N6aS?+QyP=VLLXNq~8DraL1-uhw&Yza&^o@}sr%A8STR2hF1ur|XWPUj( 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-0,0 +1,27 @@ +\documentclass[margin=0.5cm]{standalone} +\usepackage{tikz} +\usepackage{tkz-euclide} +\usepackage{mathtools} + +\begin{document} +\begin{tikzpicture} + \coordinate (A) at (0.5, 1) {}; + \coordinate (B) at (-0.5, -1) {}; + \coordinate (C) at (1.25, -0.70) {}; + \coordinate (0) at (0, 0) {}; + + % left angle + \tkzMarkRightAngle[draw=black,size=0.1](A,0,C); + \draw[lightgray,very thin] (A) -- (C); + % Curly brace annotation for ||u-v|| + \draw[decorate,decoration={brace, + amplitude=10pt},xshift=0pt,yshift=4pt,very thin] (A) -- (C) node [black,midway,xshift=27pt,yshift=0.5em] {$\lVert u-v \rVert$}; + \draw[lightgray,very thin] (B) -- (C); + + % axis lines + \draw[->] (0) -- (A) node[above] {$u$}; + \draw[->] (0) -- (B) node[below] {$-u$}; + \draw[->] (0) -- (C) node[right] {$v$}; + +\end{tikzpicture} +\end{document} \ No newline at end of file diff --git a/main.pdf b/main.pdf index 2a1a123c44c4ae1e670fa28739848969a474629d..a0ebaace980c8d030daa224108604951bf467e3d 100644 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