From d35d4c964b695ffdc730b41d46642b2ad0faf13e Mon Sep 17 00:00:00 2001 From: Alexis BAYLET Date: Sat, 21 Mar 2026 23:45:07 +0100 Subject: [PATCH] Add initial chapters on statistical estimation and regression - Introduced Chapter 0: Introduction to Statistical Estimation with foundational concepts and methods. - Added Chapter 1: Non-Parametric Density Estimation covering kernel methods and performance analysis. - Included Chapter 2: Theory of Regression focusing on non-parametric methods and regularization techniques. - Implemented Chapter 3: Neural Networks as Approximators discussing the limitations of linear approximation methods. - Added corresponding PDF files for each chapter. --- ch0.pdf | Bin 0 -> 291931 bytes ch0.tex | 142 ++++++++++++++++++++++++++++++++++++ ch1.pdf | Bin 0 -> 306145 bytes ch1.tex | 223 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ch2.pdf | Bin 0 -> 261813 bytes ch2.tex | 133 +++++++++++++++++++++++++++++++++ ch3.pdf | Bin 0 -> 236012 bytes ch3.tex 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zMfc@+W`XNN3Rb38CnrJXNqhXPe$@{8ZJ5WoF)VTYnBSc)}W5sDy-fpcRb+$iW~19 lT5S-sud_({cVmpk_(fsjg(sIJ)EEL4+8Pje-pNMte*j%(&1V1r literal 0 HcmV?d00001 diff --git a/ch0.tex b/ch0.tex new file mode 100644 index 0000000..679eeab --- /dev/null +++ b/ch0.tex @@ -0,0 +1,142 @@ +\documentclass[11pt,a4paper]{article} + +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage[french]{babel} +\usepackage{amsmath, amssymb, amsthm} +\usepackage{geometry} +\usepackage{booktabs} +\usepackage{xcolor} +\usepackage{mdframed} +\usepackage{hyperref} + +\geometry{top=2.5cm, bottom=2.5cm, left=2.5cm, right=2.5cm} + +\title{\textbf{Cours 0 : Introduction à l'Estimation Statistique}} +\author{Apprentissage Statistique / Statistiques Avancées} +\date{} + +\begin{document} + +\maketitle + +\begin{abstract} +Ce cours introductif pose les fondations mathématiques des statistiques paramétriques et non paramétriques. Nous rappelons les principaux paradigmes de l'inférence statistique, passons en revue les méthodes classiques d'estimation paramétrique (Maximum de Vraisemblance et Méthode des Moments), et introduisons les concepts centraux des statistiques non paramétriques et de la théorie de l'approximation. +\end{abstract} + +\section{Rappels et Cadre Général} + +Soit $(X_i)_{1 \le i \le N}$ un ensemble de variables aléatoires indépendantes et identiquement distribuées (i.i.d.). Nous supposons que les données sont générées par un processus dont la densité de probabilité (ou fonction de masse) est notée $p_\theta$. + +Ici, le paramètre d'intérêt est $\theta \in \Theta$, où $\Theta \subseteq \mathbb{R}^d$ ($d < \infty$). +Le problème fondamental de l'estimation statistique est le suivant : \textit{Étant donné l'observation du jeu de données $\{x_1, \dots, x_N\}$, comment trouver un estimateur de $\theta$, noté $\widehat{\theta}$ ?} + +\subsection{Résumé des Paradigmes Statistiques} +Selon la nature de $\Theta$ et selon que $\theta$ est considéré comme déterministe ou comme une variable aléatoire, différents cadres mathématiques s'appliquent. Ces notions sont généralement abordées dans les cours d'introduction aux statistiques et à l'analyse numérique. Les principaux résultats sont résumés dans le Tableau \ref{tab:paradigmes}. + +\begin{table}[h!] + \centering + \renewcommand{\arraystretch}{1.5} + \begin{tabular}{p{3.5cm} p{5.5cm} p{5.5cm}} + \toprule + & \textbf{$\theta$ est déterministe} & \textbf{$\theta$ est aléatoire} (possède une distribution a priori) \\ + \midrule + \textbf{$\theta$ dans un ensemble discret ou fini}\newline (ex : $\theta \in \{0, 1\}$) & + \textbf{Tests d'hypothèses}\newline Lemme de Neyman-Pearson \newline (Maximiser $P_D$ sous la contrainte $P_{FA} \le \alpha$) & + \textbf{Théorie de la décision}\newline - Maximum de Vraisemblance (MV) si $\theta$ est équiprobable.\newline - Maximum A Posteriori (MAP) si les a priori sont inégaux. \\ + \midrule + \textbf{$\theta$ dans un ensemble continu, dimension finie}\newline (ex : $\theta \in [0, 1]$) & + \textbf{Théorie de l'estimation}\newline - Aucun estimateur universellement optimal n'existe.\newline - Les performances sont bornées inférieurement par la \textit{Borne de Cramér-Rao} (BCR). & + \textbf{Approche bayésienne}\newline - Estimateur de l'Erreur Quadratique Moyenne Minimale (MMSE) : $\widehat{\theta} = \mathbb{E}[\theta | X]$\newline - Les performances sont bornées par la BCR bayésienne. \\ + \bottomrule + \end{tabular} + \caption{Aperçu des cadres statistiques} + \label{tab:paradigmes} +\end{table} + +\section{Statistiques Paramétriques} + +En statistiques paramétriques, nous supposons que la distribution sous-jacente appartient à une famille connue régie par un paramètre déterministe de dimension finie $\theta \in \mathbb{R}^d$. + +\subsection{Estimation par le Maximum de Vraisemblance (MV)} +La fonction de vraisemblance $L(\theta; x_1, \dots, x_N)$ représente la probabilité conjointe d'observer les données sachant le paramètre $\theta$. En raison de l'hypothèse i.i.d., elle se factorise ainsi : +\begin{equation} + L(\theta; x_1, \dots, x_N) = p_\theta(x_1, \dots, x_N) = \prod_{n=1}^N p_\theta(x_n) +\end{equation} + +L'Estimateur du Maximum de Vraisemblance (EMV) est la valeur du paramètre qui maximise cette fonction : +\begin{equation} + \widehat{\theta}_{\text{MV}} = \arg\max_{\theta \in \mathbb{R}^d} L(\theta; x_1, \dots, x_N) +\end{equation} +En pratique, il est strictement équivalent et numériquement beaucoup plus stable de maximiser la log-vraisemblance, $l(\theta) = \log L(\theta)$ (en utilisant le logarithme népérien, de base $e$) : +\begin{equation} + \widehat{\theta}_{\text{MV}} = \arg\max_{\theta \in \mathbb{R}^d} l(\theta; x_1, \dots, x_N) \quad \text{où} \quad l(\theta; x_1, \dots, x_N) = \sum_{n=1}^N \log p_\theta(x_n) +\end{equation} + +\begin{mdframed}[backgroundcolor=gray!10] +\textbf{Exemple : Loi de Bernoulli} \\ +Soit $p_\theta$ une loi de Bernoulli de paramètre $\theta \in [0,1]$. Soient $x_1, \dots, x_N \in \{0, 1\}$ des réalisations i.i.d. La fonction de masse pour une observation unique est $p_\theta(x_n) = \theta^{x_n} (1-\theta)^{1-x_n}$. + +La fonction de vraisemblance est : +\[ L(\theta; x_1, \dots, x_N) = \prod_{n=1}^N \theta^{x_n} (1-\theta)^{1-x_n} = \theta^{S_N} (1-\theta)^{N-S_N} \] +où $S_N = \sum_{n=1}^N x_n$ est le nombre de succès. + +La log-vraisemblance est : +\[ l(\theta) = S_N \log(\theta) + (N - S_N) \log(1-\theta) \] +Pour trouver le maximum, nous dérivons par rapport à $\theta$ et annulons la dérivée : +\[ \frac{\partial l}{\partial \theta} = \frac{S_N}{\theta} - \frac{N - S_N}{1-\theta} = 0 \implies S_N(1-\theta) = \theta(N - S_N) \implies S_N = N\theta \] +Ainsi, l'estimateur du MV est la moyenne empirique : $\widehat{\theta}_{\text{MV}} = \frac{1}{N} S_N$. +\end{mdframed} + +\subsection{Méthode des Moments} +La méthode des moments consiste à exprimer les moments théoriques de la distribution (qui sont des fonctions de $\theta$) et à les égaler aux moments empiriques de l'échantillon. + +\begin{mdframed}[backgroundcolor=gray!10] +\textbf{Exemple utilisant la mesure empirique} \\ +Soit $X$ une variable aléatoire dont la distribution de probabilité peut s'écrire à l'aide de masses de Dirac : $p_\theta(x) = \theta \delta_1(x) + (1-\theta)\delta_0(x)$. +Le premier moment théorique (l'espérance) est : +\[ \mathbb{E}_{p_\theta}[X] = \int x p_\theta(x) dx = 1 \cdot \theta + 0 \cdot (1-\theta) = \theta \] +La distribution empirique basée sur $N$ échantillons est donnée par $\widehat{p}(x) = \frac{1}{N} \sum_{n=1}^N \delta_{x_n}(x)$. +L'espérance empirique est : +\[ \widehat{\theta} = \mathbb{E}_{\widehat{p}}[X] = \int x \left( \frac{1}{N} \sum_{n=1}^N \delta_{x_n}(x) \right) dx = \frac{1}{N} \sum_{n=1}^N x_n \] +ce qui représente intuitivement le ratio du nombre de $1$ sur le nombre total d'échantillons. +\end{mdframed} + +\section{Statistiques Non Paramétriques} + +En statistiques non paramétriques, l'hypothèse selon laquelle le mécanisme générateur des données appartient à une famille paramétrique de dimension finie est abandonnée. L'objet d'intérêt est plutôt une fonction $f$ appartenant à un espace fonctionnel de dimension infinie $\mathcal{F}$ (par ex., $f \in \mathcal{F}$). + +\subsection{Travailler dans des Espaces de Dimension Infinie} +Pour manipuler rigoureusement les espaces de dimension infinie, nous restreignons généralement $\mathcal{F}$ à un \textbf{Espace de Hilbert}. Un espace de Hilbert généralise la notion d'espace euclidien ; il est muni d'un produit scalaire $\langle \cdot, \cdot \rangle$ qui induit une distance, et il possède une base dénombrable. Cela permet de représenter les fonctions via des décompositions sur une base (ex : séries de Fourier, ondelettes). + +\subsection{Théorie de l'Approximation et Compromis d'Erreur} +Lorsque l'on tente d'estimer une fonction $f \in \mathcal{F}$ en utilisant un espace d'hypothèses restreint ou fini $\mathcal{H}$ à partir de $N$ échantillons finis, nous rencontrons deux sources principales d'erreur : + +\begin{enumerate} + \item \textbf{L'erreur d'approximation (Biais) :} L'erreur introduite en restreignant notre recherche à un espace plus petit et plus simple $\mathcal{H}$ plutôt qu'au véritable espace de dimension infinie $\mathcal{F}$. Elle mesure à quel point le meilleur modèle possible dans $\mathcal{H}$ peut approcher la vraie fonction $f$. + \item \textbf{L'erreur d'estimation (Variance) :} L'erreur provenant du fait que nous ne disposons que d'un nombre fini $N$ d'échantillons pour trouver la fonction optimale au sein de $\mathcal{H}$. +\end{enumerate} + +Ces deux erreurs conduisent au fondamental \textbf{compromis Biais-Variance}. Augmenter la complexité de l'espace d'hypothèses diminue l'erreur d'approximation mais augmente l'erreur d'estimation, et inversement. + +\subsection{Approche Minimax} +Pour dériver des bornes sur ces erreurs lorsque $N$ est fini et que la dimensionnalité est potentiellement élevée, les statisticiens utilisent souvent l'\textbf{approche Minimax}. Cela implique de trouver l'estimateur qui minimise le risque maximum possible (erreur espérée) sur la pire distribution possible dans la classe $\mathcal{F}$. +\[ \inf_{\widehat{f}} \sup_{f \in \mathcal{F}} \mathbb{E}[L(\widehat{f}, f)] \] + +\section{Problèmes Classiques en Non Paramétrique} + +\subsection{Estimation de Densité} +Soit $X$ une variable aléatoire ayant une Fonction de Répartition (FR) notée $F$ et une Densité de Probabilité (DP) notée $f$. +Par définition : +\[ F(A) = \mathbb{P}(X \in A) = \int_A f(x) dx \] +Pour $A = (-\infty, x]$, la fonction de répartition est : +\[ F(x) = \mathbb{P}(X \le x) = \int_{-\infty}^x f(u) du \] +\textbf{Objectif :} Étant donné des échantillons $\{x_1, \dots, x_N\}$, trouver un estimateur $\widehat{f}(x; x_1, \dots, x_N)$ qui approche la vraie densité $f(x)$ partout. (Les méthodes classiques incluent les histogrammes et l'estimation par noyaux). + +\subsection{Régression Non Paramétrique} +Considérons des observations appariées $(X, Y)$ où la relation est régie par : +\[ Y = f(X) + \varepsilon \] +Ici, $\varepsilon$ est un bruit aléatoire centré (de moyenne nulle) et indépendant de $X$, tel que $\mathbb{E}[\varepsilon] = 0$. +\textbf{Objectif :} Étant donné un jeu de données de paires $\{(X_n, Y_n)\}_{1 \le n \le N}$, estimer la fonction inconnue $f(x)$ pour prédire les valeurs futures : $\widehat{y} = \widehat{f}(x)$. + +\end{document} \ No newline at end of file diff --git a/ch1.pdf b/ch1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..a962ebc1858526641552e81a177f6c837fd13baf GIT binary patch literal 306145 zcma%jLy#cAwq@D2ZQHhO+qP}1%eHOX?6Pg!Rns#ufA!zuVwN^8rcMO(Vm5{@rXr@s z_9muKe0)&OE>5O~woo1$%Uasb*lnnOt91t7g~v-G4t;Bfd?a#aNgTCV?aUdYH`7r% znwtfwlTH<#zrCjbYtU->A|+DXz6(OoG<2sAIghw^Q?KUwKEzu=%7v`z(XVOHqZIpM 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On définit la classe $\Fcal_C$ comme l'ensemble des fonctions $f \in L^2([0,1]^d)$ dont la transformée de Fourier $F(\vec{\omega})$ vérifie : +\begin{equation} +\Fcal_C = \left\{ f \mid f(\vec{x}) = \int_{\R^d} F(\vec{\omega}) e^{2\pi i \vec{\omega} \cdot \vec{x}} \dw \text{ et } \int_{\R^d} \|\vec{\omega}\|_1 |F(\vec{\omega})| \dw \le C \right\} +\end{equation} +où $\|\vec{\omega}\|_1 = \sum_{j=1}^d |\omega_j|$. +\end{definition} + +\subsection{Écart de Kolmogorov} + +\begin{definition}[Écart de Kolmogorov] +Pour une classe $K \subset L^2([0,1]^d)$, l'écart de dimension $N$ est : +\begin{equation} +w_N(K) = \inf_{H_N, \dim(H_N) \le N} \sup_{f \in K} \norm{f - \proj_{H_N}f}_{L^2} +\end{equation} +\end{definition} + +\section{Résultat Principal : Fléau de la Dimension} + +\begin{theorem} +Il existe $\kappa > 0$ tel que pour tout $N \ge 1$ et $d \ge 1$ : +\begin{equation} +\label{eq:lower_bound} +w_N(\Fcal_C) \ge \kappa \frac{C}{d} \frac{1}{N^{1/d}} +\end{equation} +\end{theorem} + +[Image of curse of dimensionality in function approximation] + +\section{Preuve du Théorème} + +\subsection{Étape 1 : Fonctions de test} +Soient $\{\kvec_j\}_{j=1}^{2N} \subset \N^d$, ordonnés par $\|\kvec_1\|_1 \le \dots \le \|\kvec_{2N}\|_1$. On définit : +\[ h_j^*(\x) = \cos(2\pi \kvec_j \cdot \x), \quad j=1, \dots, 2N \] + +\subsection{Étape 2 : Normalisation} +\begin{lemma} +La fonction $f_{\kvec}(\x) = \frac{C}{2\|\kvec\|_1} \cos(2\pi \kvec \cdot \x)$ appartient à $\Fcal_C$. +\end{lemma} +\begin{proof} +La transformée de Fourier de $\cos(2\pi \kvec \cdot \x)$ est $\frac{1}{2} (\delta_{\kvec} + \delta_{-\kvec})$. Ainsi : +\[ \int \|\w\|_1 |F_{f_{\kvec}}(\w)| \dw = \frac{C}{4\|\kvec\|_1} (\|\kvec\|_1 + \|-\kvec\|_1) = \frac{C}{2} \le C \] +\end{proof} + +\subsection{Étape 3 : Borne sur l'erreur} +Pour tout sous-espace $H_N$ de dimension $N$, il existe une combinaison des $2N$ fonctions de test qui est orthogonale à $H_N$. L'erreur est alors minorée par : +\begin{equation} +\label{eq:gen_bound} +w_N(\Fcal_C) \ge \min_{j \in \{1, \dots, 2N\}} \frac{C}{2\sqrt{2}\|\kvec_j\|_1} = \frac{C}{2\sqrt{2}\|\kvec_{2N}\|_1} +\end{equation} + +\subsection{Étape 4 : Combinatoire} +Le nombre de vecteurs $\kvec \in \N^d$ tels que $\|\kvec\|_1 \le m$ est $\binom{m+d}{d}$. On cherche $m$ tel que : +\begin{equation} +\label{eq:comb_condition} +\binom{m+d}{d} \ge 2N +\end{equation} +En utilisant l'inégalité $\binom{m+d}{d} \ge (\frac{m}{d})^d$, la condition est satisfaite si : +\begin{equation} +\label{eq:m_choice} +m \ge d (2N)^{1/d} +\end{equation} + +\subsection{Étape 5 : Conclusion} +En injectant \eqref{eq:m_choice} dans \eqref{eq:gen_bound}, on obtient : +\begin{equation} +w_N(\Fcal_C) \ge \frac{C}{2\sqrt{2} \cdot d (2N)^{1/d}} \ge \kappa \frac{C}{d} \frac{1}{N^{1/d}} +\end{equation} +Ceci démontre que pour les méthodes linéaires, l'erreur décroît de plus en plus lentement à mesure que $d$ augmente. +\end{document} \ No newline at end of file

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+\newtheorem{theorem}{Théorème}[section] +\newtheorem{remark}{Remarque}[section] +\newtheorem{example}{Exemple}[section] +\newtheorem{proposition}{Proposition}[section] + +% --- Commandes personnalisées --- +\newcommand{\E}{\mathbb{E}} +\renewcommand{\P}{\mathbb{P}} +\newcommand{\R}{\mathbb{R}} +\newcommand{\Ltwo}{L^2([0,1])} +\newcommand{\ind}{\mathbb{1}} + +\title{Cours de Statistique : Estimation de Densité Non-Paramétrique} +\author{Note de cours réorganisée} +\date{Mars 2026} + +\begin{document} + +\maketitle + +\section{Introduction et Motivation} + +L'objectif de l'estimation de densité est de reconstruire une fonction de densité de probabilité $f$ à partir d'un échantillon de variables aléatoires $X_1, \dots, X_N$ indépendantes et identiquement distribuées (i.i.d.) selon $f$. + +\begin{definition}[Densité de probabilité] +Une fonction $f : \R \to \R$ est une densité de probabilité si elle vérifie : +\begin{enumerate} + \item $f(x) \geq 0$ pour presque tout $x$. + \item $f \in L^1(\R)$ et $\int_{\R} f(x) dx = 1$. +\end{enumerate} +\end{definition} + +\begin{remark} +La valeur ponctuelle $f(x_0)$ n'est pas informative par elle-même car la mesure de Lebesgue d'un singleton $\{x_0\}$ est nulle. Pour estimer $f$, nous devons imposer une \textbf{contrainte de régularité} (smoothness). On suppose généralement que $f$ appartient à un sous-espace de fonctions régulières $\mathcal{F} \subsetneq L^1$. +\end{remark} + +\section{L'approche "Naïve" : L'estimateur par intervalle} + +Considérons une réalisation i.i.d. $x_1, \dots, x_N$ d'une loi de densité $f$. Pour un point $x$ donné et un petit paramètre $h > 0$, la probabilité que $X$ appartienne à l'intervalle $[x-h, x+h]$ est : +\[ \P(X \in [x-h, x+h]) = \int_{x-h}^{x+h} f(u) du \] + +Si $f$ est continue en $x$ et $h$ est petit, alors $\int_{x-h}^{x+h} f(u) du \approx 2h f(x)$. On peut estimer cette probabilité par la proportion empirique de données tombant dans cet intervalle : +\[ \hat{p} = \frac{1}{N} \sum_{n=1}^N \ind_{[x-h, x+h]}(x_n) \] + +D'où l'estimateur "naïf" de la densité : +\[ \hat{f}(x) = \frac{1}{2Nh} \sum_{n=1}^N \ind_{[x-h, x+h]}(x_n) = \frac{1}{Nh} \sum_{n=1}^N K\left(\frac{x-x_n}{h}\right) \] +où $K(u) = \frac{1}{2}\ind_{[-1, 1]}(u)$ est appelé le \textbf{noyau rectangulaire}. + +\section{L'approche par Projection (Espaces de Hilbert)} + +Cette méthode repose sur la décomposition de $f$ dans une base orthonormée de l'espace de Hilbert $\Ltwo$. + +\subsection{Espace de Hilbert et Bases Orthonormées} +On munit $\Ltwo$ du produit scalaire : +\[ \langle f, g \rangle = \int_0^1 f(x)\overline{g(x)} dx \] +Soit $\{e_k\}_{k \in \mathbb{Z}}$ une base orthonormée de $\Ltwo$ (par exemple la base de Fourier $e_k(x) = e^{i 2\pi k x}$). Toute fonction $f \in \Ltwo$ peut s'écrire : +\[ f(x) = \sum_{k \in \mathbb{Z}} \alpha_k e_k(x) \quad \text{avec} \quad \alpha_k = \langle f, e_k \rangle \] + +\subsection{Régularité et Espaces de Sobolev} +Pour quantifier la "douceur" de $f$, on utilise les espaces de Sobolev $W^s$. +\begin{definition}[Espace de Sobolev] +Pour $s > 0$, on définit $W^s([0,1])$ comme l'espace des fonctions $f \in \Ltwo$ telles que : +\[ \sum_{k \in \mathbb{Z}} |\alpha_k|^2 (1 + |k|)^{2s} < +\infty \] +On définit l'ellipsoïde de Sobolev $B(s, R) = \{ f \in W^s : \|f\|_{W^s}^2 \leq R^2 \}$. +\end{definition} + +\subsection{Construction de l'estimateur par projection} +On approche d'abord $f$ par une version tronquée à l'ordre $M$ : +\[ \tilde{f}(x) = \sum_{|k| \leq M} \alpha_k e_k(x) \] +Cependant, les coefficients $\alpha_k$ sont inconnus. On remarque que : +\[ \alpha_k = \int_0^1 e_k(x) f(x) dx = \E[e_k(X)] \] +Par la méthode des moments, on estime $\alpha_k$ par la moyenne empirique : +\[ \hat{\alpha}_k = \frac{1}{N} \sum_{n=1}^N e_k(X_n) \] +L'estimateur final est : +\[ \hat{f}(x) = \sum_{|k| \leq M} \hat{\alpha}_k e_k(x) \] + +\section{Analyse des Performances : Risque Quadratique} + +Le risque est mesuré par l'Erreur Quadratique Moyenne Intégrée (MISE). +\[ \text{Risk}(\hat{f}, f) = \E \left[ \int_0^1 |\hat{f}(x) - f(x)|^2 dx \right] \] + +\subsection{Décomposition Biais-Variance} +En utilisant l'orthogonalité de la base, le risque se décompose en : +\[ \text{Risk} = \underbrace{\|\tilde{f} - f\|^2}_{\text{Biais}^2 \text{ (Troncature)}} + \underbrace{\E[\|\hat{f} - \tilde{f}\|^2]}_{\text{Variance} \text{ (Estimation)}} \] + +\begin{enumerate} + \item \textbf{Le Biais (Erreur d'approximation)} : Pour $f \in B(s, R)$, on montre que : + \[ \text{Biais}^2 = \sum_{|k| > M} |\alpha_k|^2 = \sum_{|k| > M} |\alpha_k|^2 \frac{(1+|k|)^{2s}}{(1+|k|)^{2s}} \leq \frac{R^2}{M^{2s}} = O(M^{-2s}) \] + + \item \textbf{La Variance (Erreur statistique)} : + \[ \text{Variance} = \sum_{|k| \leq M} \E[|\hat{\alpha}_k - \alpha_k|^2] = \sum_{|k| \leq M} \frac{\text{Var}(e_k(X))}{N} \] + Si les fonctions de base sont bornées ($|e_k| \leq C$), alors $\text{Var}(e_k(X)) \leq C^2$. + D'où : $\text{Variance} = O\left(\frac{M}{N}\right)$. +\end{enumerate} + +Le risque total est donc de l'ordre de : +\[ \text{Risk} \approx \frac{M}{N} + \frac{1}{M^{2s}} \] + +\section{Optimisation du paramètre de troncature} + +Pour minimiser le risque, nous cherchons le $M$ optimal ($M^*$) qui équilibre le biais et la variance. +\[ \frac{\partial}{\partial M} \left( \frac{M}{N} + M^{-2s} \right) = 0 \implies \frac{1}{N} - 2s M^{-(2s+1)} = 0 \] +On obtient : +\[ M^* \sim N^{\frac{1}{2s+1}} \] + +\textbf{Vitesse de convergence :} +En réinjectant $M^*$ dans l'expression du risque, on trouve : +\[ \text{Risk}^* \sim N^{-\frac{2s}{2s+1}} \] +\begin{itemize} + \item Si $s \to \infty$ (fonction très lisse), la vitesse approche $N^{-1}$ (vitesse paramétrique). + \item Si $s \to 0$ (fonction peu régulière), $M^* \sim N$ et la vitesse est très lente. +\end{itemize} + +\section{Introduction à l'Approche par Noyau} + +L'approche par noyau généralise l'idée de la section 2. +\begin{definition}[Noyau] +Un noyau $K : \R \to \R$ est une fonction intégrable telle que $\int_{\R} K(u) du = 1$. On définit le noyau translaté et mis à l'échelle par la fenêtre (bandwidth) $h$ : +\[ K_h(u) = \frac{1}{h} K\left(\frac{u}{h}\right) \] +\end{definition} + +L'estimateur à noyau est défini par le produit de convolution entre la mesure empirique et le noyau : +\[ \hat{f}(x) = (K_h * f_{emp})(x) = \frac{1}{N} \sum_{n=1}^N K_h(x - x_n) \] + +\section{Analyse de l'Estimateur à Noyau} + +L'estimateur à noyau repose sur deux idées fondamentales (souvent appelées "Tricks" dans les notes) : +\begin{enumerate} + \item \textbf{L'approximation :} La convolution $(K_h * f)(x)$ converge vers $f(x)$ quand $h \to 0$. + \item \textbf{L'estimation :} L'espérance de l'estimateur $\hat{f}(x)$ est précisément cette convolution. +\end{enumerate} + +\subsection{Cadre d'analyse : Espaces de Hölder} + +Pour garantir une certaine vitesse de convergence, nous supposons que la densité $f$ appartient à un espace de régularité fonctionnelle. + +\begin{definition}[Espace de Hölder $\Lambda(s, L)$] +Soit $s > 0$. On écrit $s = k + \beta$ avec $k \in \mathbb{N}$ et $\beta \in ]0, 1]$. Une fonction $f$ appartient à l'espace de Hölder $\Lambda(s, L)$ si : +\begin{enumerate} + \item $f$ est $k$ fois dérivable. + \item La $k$-ième dérivée est $\beta$-Höldérienne : + $\forall x, y \in \R, |f^{(k)}(x) - f^{(k)}(y)| \leq L |x-y|^\beta$. +\end{enumerate} +\end{definition} + +\subsection{Hypothèses sur le noyau (Noyaux de Parzen-Rosenblatt)} + +Pour exploiter la régularité d'ordre $s$, le noyau $K$ doit posséder des propriétés de moments : +\begin{enumerate} + \item $\int K(u) du = 1$. + \item $\int |u|^s |K(u)| du < +\infty$. + \item $\int u^l K(u) du = 0$ pour tout $l \in \{1, \dots, k\}$. On dit alors que $K$ est un \textbf{noyau d'ordre $k$}. +\end{enumerate} + +\subsection{Résultat 1 : Contrôle du Biais} + +\begin{theorem}[Borne sur le biais] +Si $f \in \Lambda(s, L)$ et $K$ est un noyau d'ordre $k$, alors le biais de l'estimateur vérifie : +\[ \sup_x | \E[\hat{f}(x)] - f(x) | = \sup_x | (K_h * f)(x) - f(x) | \leq \frac{L \cdot C}{k!} h^s = O(h^s) \] +où $C = \int |y|^s |K(y)| dy$. +\end{theorem} + +\begin{proof}[Esquisse de preuve] +Par changement de variable $y = \frac{x-u}{h}$ : +\[ (K_h * f)(x) - f(x) = \int K(y) [f(x - hy) - f(x)] dy \] +En utilisant un développement de Taylor-Young de $f$ à l'ordre $k$ en $x$ : +\[ f(x-hy) = f(x) - h y f'(x) + \dots + \frac{(-hy)^k}{k!} f^{(k)}(x - \epsilon hy) \] +Grâce aux propriétés d'annulation des moments du noyau (ordre $k$), les termes de dérivées s'annulent à l'intégration, ne laissant que le reste de Taylor qui est borné par la condition de Hölder en $h^s$. +\end{proof} + +\subsection{Résultat 2 : Contrôle de la Variance} + +\begin{theorem}[Borne sur la variance] +Si $K$ est de carré intégrable ($K \in L^2$) et $f$ est bornée, alors : +\[ \text{Var}(\hat{f}(x)) \leq \frac{C'}{Nh} = O\left(\frac{1}{Nh}\right) \] +où $C' = \|f\|_\infty \int K^2(u) du$. +\end{theorem} + +\begin{proof} +Puisque les $X_n$ sont i.i.d. : +\[ \text{Var}(\hat{f}(x)) = \frac{1}{N} \text{Var}(K_h(x - X_1)) \leq \frac{1}{N} \E[K_h(x - X_1)^2] \] +\[ \E[K_h(x - X_1)^2] = \int \frac{1}{h^2} K\left(\frac{x-u}{h}\right)^2 f(u) du \] +Par changement de variable $u' = \frac{x-u}{h}$, on obtient : +\[ \frac{1}{h} \int K(u')^2 f(x - hu') du' \leq \frac{\|f\|_\infty}{h} \int K^2(u') du' \] +\end{proof} + +\section{Conclusion : Le compromis Biais-Variance} + +Le risque quadratique total se comporte comme : +\[ \text{Risque}(h) \approx \underbrace{h^{2s}}_{\text{Biais}^2} + \underbrace{\frac{1}{Nh}}_{\text{Variance}} \] + +\subsection{Fenêtre optimale $h^*$} +En minimisant cette expression par rapport à $h$, on trouve : +\[ h^{2s+1} \sim \frac{1}{N} \implies h^* = N^{-\frac{1}{2s+1}} \] + +\subsection{Vitesse de convergence minimax} +En remplaçant $h^*$ dans l'expression du risque, on obtient la vitesse de convergence : +\[ \text{Risque}^* \sim N^{-\frac{2s}{2s+1}} \] + +\begin{remark}[Synthèse entre les deux approches] +\begin{table}[h] +\centering +\begin{tabular}{|l|c|c|} +\hline +\textbf{Méthode} & \textbf{Paramètre de lissage} & \textbf{Rôle} \\ \hline +Projection & $M$ (nombre de modes) & Régularisation par troncature \\ \hline +Noyau & $1/h$ (inverse de la fenêtre) & Régularisation par lissage \\ \hline +\end{tabular} +\end{table} +Dans les deux cas, on retrouve la même vitesse de convergence. Le paramètre $M$ en projection joue un rôle inversement proportionnel à $h$ dans l'approche par noyau. Plus $s$ (la régularité) est grand, plus la vitesse approche $1/N$, la vitesse "standard" de la statistique paramétrique. +\end{remark} + +\end{document} \ No newline at end of file diff --git a/ch2.pdf b/ch2.pdf new file mode 100644 index 0000000000000000000000000000000000000000..33fc47bbf37ac50bf8ba6f044807f99be87a27a4 GIT binary patch literal 261813 zcma&NQ5TA8HsBBHd6 zbgWRM^UEXaP^<(D1a^j&P&_NhQzv*^BAm1cFxbbfrPe5PE%v#a^}P%gnP zL7GY0+V0w^S0W``#F%OjrL1pXv-gS4byeWEHjFQt>m5`^p5Lg)zA&uz$))(*FURhA zyOX0_eRe*cWG(*wW#H|-tS~I=bp70%iedD#MU#doX2ZyP6hb#dWQap$Y<=k2I^nt8 z_+mGeZ#SOHw`kQCHDvSg2WW2h-fq6Dzn%mv$MrBrnJKVtR^_-^PWIyIcjRDK(IN$W z7$pn>hveL-v|yL*_RQ79U|J_5%tXg9%6TM0wnUhuLfm!_>*Gl+o#WrA)|+2*o9i$5 zZf0E6dz#6V7M-2!O8XbE`?zWCuEJ((Y;=ggbtMwfrf|~H9k`bXcjn4_vH;4>M$sXs#S7_-&Zh4yB zy%@XkqJfSThd?rwTyy8O^2F`WS`s-7+&tC?SgFCdn)}HBrW?ggkf)^hrByn=l2-_3 zzXF44j5G6~LY|FS-}eRO9!NBB#Mv3uI*E@aN=cIi)*>NUzGPGxw=PH&C=nMVX`^G~ z(47QY;6Ij0J93ceMv$($;2_R5x7KDIJ=mzl$ws}ZMkp1^Q4@fNcmpPp>hT~tA^OA> 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