From 17671e583af1ad548c30163af809b5259a6ad601 Mon Sep 17 00:00:00 2001 From: Polarolouis Date: Tue, 5 Mar 2024 11:25:55 +0100 Subject: [PATCH] Removing unnecessary files --- Rnw/dummy-main.Rnw | 189 --------------------------------------------- test.png | Bin 103398 -> 0 bytes 2 files changed, 189 deletions(-) delete mode 100644 Rnw/dummy-main.Rnw delete mode 100644 test.png diff --git a/Rnw/dummy-main.Rnw b/Rnw/dummy-main.Rnw deleted file mode 100644 index c59252b..0000000 --- a/Rnw/dummy-main.Rnw +++ /dev/null @@ -1,189 +0,0 @@ -\documentclass[a4paper, 12pt]{report} - -% Packages -\usepackage[utf8]{inputenc} -\usepackage[T1]{fontenc} -\usepackage[french]{babel} -\usepackage{geometry} - -%Images -\usepackage{graphicx} -\graphicspath{{img/}} - -\usepackage{caption} -\usepackage{subcaption} -\usepackage{amsmath} -\usepackage{amsfonts} -\usepackage{amssymb} -\usepackage{hyperref} -\usepackage{natbib} -\usepackage{listings} -\usepackage{xcolor} -\usepackage{amsthm} -\usepackage{cancel} - -\usepackage[style=authoryear-comp,backend=biber]{biblatex} -%== use and define color ==% -\AtEveryCite{\color{blue}} -\addbibresource{references.bib} - -% Configurations -\geometry{a4paper, margin=2.5cm} -\graphicspath{ {img/} } - -% Macros utiles -\newcommand{\Normal}{\mathcal{N}} - - -% Titre du document -\title{Rapport de Projet : ANOVA Phylogénétique} -\author{Alizée Geffroy \and Louis Lacoste} -\date{\today} - -\begin{document} - -\maketitle - -\tableofcontents -\listoffigures -\listoftables - -<>= - knitr::opts_chunk$set(echo = FALSE) - require("knitr", quietly = TRUE) - options(knitr.table.format = "latex") -@ - - -\chapter{Introduction} -\label{chap:intro} -% Introduction au projet, contexte, objectifs. -Ici contexte biologique, les données de \cite{gomez-mestrePhylogeneticAnalysesReveal2012}, les données de Paul et Mélina etc. - -Avec l'avènement des données massives de génomiques, transcriptomiques, protéomiques etc, il y a besoin de techniques statistiques robustes et passant à l'échelle permettant de mener à bien l'analyse. - -\chapter{État de l'Art} -\label{chap:etat_art} -% Revue de la littérature sur l'ANOVA phylogénétique. -Ici les rappels sur l'ANOVA, l'explication de l'ANOVA phylogénétique. La démonstration des limites de l'ANOVA phylogénétique par des simulations - -\section{L'ANOVA} - -L'ANOVA est un cas classique du modèle linéaire, nous utilisons ici les notations et le formalisme de \cite{belModeleLineaireSes}. - -Le principe de l'ANOVA est d'expliciter le lien entre une variable quantitative et une ou plusieurs variables qualitatives. - -La forme usuelle de l'ANOVA à 1 facteur est la suivante : - -\begin{align} -Y_{ik} = \mu_i + E_{ik}, & &i = 1,\dots I, k = 1,\dots n_i, E_{ik} \sim \Normal (0, \sigma^2) -\end{align} - -où dans cette équation, reprise du livre \parencite{belModeleLineaireSes}, $i$ représente le niveau du facteur et $k$ indique le numéro de l'observation dans ce niveau. $I$ est le nombre total de niveaux du facteur, $n_i$ le nombre d'observation du niveau $i$. - -L'ANOVA se généralise à deux facteurs, plus facilement compréhensible avec cette forme, non identifiable : -\begin{align} -Y_{ijk} = \mu + \alpha_i + \beta_j + E_{ijk}, & &i = 1,\dots I, j = 1,\dots J, k = 1,\dots n_ij, E_{ijk} \sim \Normal (0, \sigma^2) -\end{align} - -où $\mu$ représente un effet moyen de la population (\emph{intercept}), $\alpha_i$ l'effet du premier facteur de niveau $i$, $\beta_j$ l'effet du second facteur de niveau $j$. - -Les paramètres de l'ANOVA sont estimables, grâce par exemple à la méthode du maximum de vraisemblance et ont des formules bien connues. - -% ICI LES FORMULES - -% LIMITES de l'ANOVA classique sur les données phylo - -\section{L'ANOVA phylogénétique} - -\chapter{Méthodologie} -\label{chap:metho} -\section{Approximation de Satterthwaite} -Pourquoi vouloir l'utiliser ? Réduire nbre de degrés de liberté utilisés dans la stat de test. -Le but est d'approximé le nbre de degré de Liberté. -On se basera sur la documentation du package lmer \cite{lmerPackage} pour ensuite implémenter une approximation de Satterthwaite. -\begin{equation} - Y = X\beta + u + \epsilon -\end{equation} -\[ -\text{où} \quad \mathbf{Y} = \begin{bmatrix} Y_1 \\ Y_2 \\ \vdots \\ Y_n \end{bmatrix}, \mathbf{\beta} = \begin{bmatrix} \beta_1 \\ \beta_2 \end{bmatrix}\text{,} \quad u \sim \mathcal{N}_n(0, \sigma^2_{phy}K) \text{,} \quad \epsilon \sim \mathcal{N}_n(0, \sigma^2_{err}I_n) -\] -\newline -\[ - \text{Alors} \quad Y \sim \mathcal{N}_n(X\beta, \sigma^2_{phy}K + \sigma^2_{err}I_n) \quad \text{et} \quad Var_\theta(Y) = V(\theta) = \sigma^2_{phy}K + \sigma^2_{err}I_n -\] -De là on obtient: -\begin{equation} - C(\theta) = (Cov(\beta_i , \beta_j))_{i,j} = (X^TV(\theta)^{-1}X)^{-1} = (X^T(\sigma^2_{phy}K + \sigma^2_{err}I_n)^{-1}X)^{-1} -\end{equation} - -Toujours en suivant la documentation \cite{lmerPackage} on obtient une expression pour les degrés de liberté $df$ ainsi qu'une approximation. Ce qui nous donne : -\begin{equation} - df = \frac{2(l^T\hat{C}l)^2}{[Var(l^T\hat{C}l)]}=\frac{2(f(\hat{\theta}))^2}{[Var(f(\hat{\theta})]}\approx \frac{2(f(\hat{\theta}))^2}{[\nabla f(\hat{\theta})]^T A[\nabla f(\hat{\theta})]} -\end{equation} -\[\text{où} \quad \hat{C} = C(\hat\theta) \quad \text{et} \quad f(\theta) = l^TC(\theta)l\] -On va donc dans la suite calculer $\nabla f(\theta)$ puis l'appliquer en $\hat{\theta}$ et $A$ la matrice de variance-covariance de $\hat{\theta}=(\hat{\sigma}^2_{phy}, \hat{\sigma}^2_{err})$ -\begin{proof}[Calcul du gradient] -Nous voulons calculer les dérivées partielles $\partial_{\sigma^2_{phy}}f(\theta)$ et $\partial_{\sigma^2_{err}}f(\theta)$. Pour les premières étapes de calculs, on écrira seulement $\partial$ sans distinction car ce sont les mêmes expressions pour les 2 dérivées. -On utilisera dans la suite les formules de \cite{matrixcookbook2012} pour les dérivées de matrice -\[ -\partial f(\theta)=l^T\partial C(\theta)l -\] -\[ -\partial C(\theta)=\partial (X^TV(\theta)^{-1}X)^{-1} = -C(\theta) \partial (X^TV(\theta)^{-1}X)C(\theta) -\] - -\[ - \partial (X^TV(\theta)^{-1}X) = \partial (X^TV(\theta)^{-1})X + \cancel{X^TV(\theta)^{-1})\partial(X)} -\] - -% Commençons par utiliser la définition des fonctions trigonométriques : -% \[ -% \sin(x) = \frac{e^{ix} - e^{-ix}}{2i} \quad \text{et} \quad \cos(x) = \frac{e^{ix} + e^{-ix}}{2} -% \] - -% En substituant ces expressions dans l'identité, nous obtenons : -% \begin{align*} -% \sin^2(x) + \cos^2(x) &= \left(\frac{e^{ix} - e^{-ix}}{2i}\right)^2 + \left(\frac{e^{ix} + e^{-ix}}{2}\right)^2 \\ -% &= \frac{(e^{ix} - e^{-ix})^2}{4i^2} + \frac{(e^{ix} + e^{-ix})^2}{4} \\ -% &= \frac{(e^{ix} - e^{-ix})(e^{ix} - e^{-ix})}{-4} + \frac{(e^{ix} + e^{-ix})(e^{ix} + e^{-ix})}{4} \\ -% &= \frac{e^{2ix} - 2e^{ix}e^{-ix} + e^{-2ix}}{-4} + \frac{e^{2ix} + 2e^{ix}e^{-ix} + e^{-2ix}}{4} \\ -% &= \frac{e^{2ix} - 2 + e^{-2ix}}{-4} + \frac{e^{2ix} + 2 + e^{-2ix}}{4} \\ -% &= \frac{e^{2ix} + e^{-2ix}}{4} + \frac{e^{2ix} + e^{-2ix}}{4} - \frac{2}{4} + \frac{2}{4} \\ -% &= \frac{2(e^{2ix} + e^{-2ix})}{4} \\ -% &= \frac{2 \cdot 2\cos(2x)}{4} \quad \text{(par la formule d'Euler)} \\ -% &= \cos(2x) -% \end{align*} - -% Ainsi, nous avons montré que $\sin^2(x) + \cos^2(x) = 1$. -\end{proof} -Supposons que nous avons une expression $x^2 - 2x + \cancel{3} - 3$. Comme la partie $\cancel{3}$ est nulle, nous pouvons la barrer. - -\section{Simulations} -% On importe le fichier -<>= -@ - -\chapter{Données} -\label{chap:data} -% Présentation des données utilisées. -<>= -@ - -\chapter{Résultats} -\label{chap:results} -% Présentation des résultats obtenus. - -\chapter{Discussion} -\label{chap:discuss} -% Analyse critique des résultats, limites, perspectives. - -\chapter{Conclusion} -\label{chap:conclu} -% Résumé des principales conclusions du projet. - -% Bibliographie -\printbibliography -\nocite{*} - -\end{document} diff --git a/test.png b/test.png deleted file mode 100644 index 06987548584e8baab2242aee87b2d4878c09cd3c..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 103398 zcmdSBgFV+0$JxQTxtyG}HCw)qj7ikRi_BPt#-BYNi{TSUVv5eFvs0Jkrd^~p_~mdRMgfz^ zMCGOUWznA(88Z-HE+gCNmG@#NC78;}9FsaW!(BZVTx?7Q6in$E5)y{Di|7^X6(1}n z*G45LKV}Rzx&+trzhBfRj}uca{O{k<-*tLSW 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