From 4de134c21f0dbf11fea0afcff05665a9a8948f1e Mon Sep 17 00:00:00 2001 From: Pierre-Edouard Portier Date: Wed, 9 Jun 2021 08:57:56 +0200 Subject: [PATCH] =?UTF-8?q?Progr=C3=A8s...?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- ML1_regression.Rmd | 323 +++++++++++++++++++++++++++++++++++---------- ML1_regression.pdf | Bin 0 -> 509441 bytes 2 files changed, 250 insertions(+), 73 deletions(-) create mode 100644 ML1_regression.pdf diff --git a/ML1_regression.Rmd b/ML1_regression.Rmd index 8509e48..dc9396d 100644 --- a/ML1_regression.Rmd +++ b/ML1_regression.Rmd @@ -1,22 +1,25 @@ --- -title: "Régression" +title: "Régression et classification" output: + pdf_document: + number_section: yes html_notebook: number_sections: yes - pdf_document: default +toc: yes +classoption: fleqn --- -\renewcommand{\vec}[1]{\mathbf{#1}} +# Introduction -# Concepts de base +## Premiers concepts -$\vec{x}^{(1)} \dots \vec{x}^{(P)}$ sont des vecteurs de $\mathbb{R}^N$ associés aux valeurs, aussi appelées étiquettes, $y^{(1)} \dots y^{(P)}$ de $\mathbb{R}$. On cherche une fonction $f(\vec{x}) : \mathbb{R}^N \rightarrow \mathbb{R}$ qui modélise la relation entre les observations $\vec{x}$ et les étiquettes $y$. +$\mathbf{x}^{(1)} \dots \mathbf{x}^{(P)}$ sont des vecteurs de $\mathbb{R}^N$ associés aux valeurs, aussi appelées étiquettes, $y^{(1)} \dots y^{(P)}$ de $\mathbb{R}$. Nous cherchons une fonction $f(\mathbf{x}) : \mathbb{R}^N \rightarrow \mathbb{R}$ qui modélise la relation entre les observations $\mathbf{x}$ et les étiquettes $y$. La fonction $f$ peut avoir une forme paramétrique, comme par exemple : $$ -f(\vec{x}) = a_0 + a_1x_1 + a_2x_2 + \dots + a_Nx_N +f(\mathbf{x}) = a_0 + a_1x_1 + a_2x_2 + \dots + a_Nx_N $$ -Si $P=N+1$, les paramètres $a_0, a_1, \dots a_N$ sont solutions d'un système linéaire : +Si $P=N+1$, les paramètres $a_0, a_1, \dots a_N$ sont solution d'un système linéaire : $$ \begin{cases} y^{(1)} &= a_0 + a_1 x_1^{(1)} + a_2 x_2^{(1)} + \dots + a_N x_N^{(1)} \\ @@ -41,15 +44,15 @@ a_0 \\ a_1 \\ \dots \\ a_N y^{(1)} \\ y^{(2)} \\ \dots \\ y^{(P)} \end{array} \right) $$ -Chaque ligne $i$ de la matrice du terme de gauche de l'égalité ci-dessus est le vecteur ligne $\vec{x}^{(i)T}$ avec l'addition d'un premier terme constant qui correspond au paramètre $a_0$. En nommant cette matrice $\vec{X}^T$, le système linéaire ci-dessus s'écrit : +Chaque ligne $i$ de la matrice du terme de gauche de l'égalité ci-dessus est le vecteur ligne $\mathbf{x}^{(i)T}$ avec l'addition d'un premier terme constant qui correspond au paramètre $a_0$. En nommant cette matrice $\mathbf{X}^T$, le système linéaire ci-dessus s'écrit : $$ -\vec{X}^T \vec{a} = \vec{y} +\mathbf{X}^T \mathbf{a} = \mathbf{y} $$ -On considère le cas particulier où $x$ est un scalaire et $f$ est un polynome de degré $N$ : +Soit le cas particulier où $x$ est un scalaire et $f$ est un polynome de degré $N$ : $$ f(x) = a_0 + a_1x + a_2x^2 + \dots + a_Nx^N $$ -Avec $P=N+1$ observations et les étiquettes associées $\left( x^{(k)}, y^{(k)} \right)$, les coefficients de ce polynome sont solution d'un système linéaire : +Avec $P=N+1$ observations et les étiquettes associées $\left( x^{(k)}, y^{(k)} \right)$, les coefficients de ce polynôme sont solution d'un système linéaire : $$ \left( \begin{array}{ccccc} @@ -69,13 +72,13 @@ $$ La matrice du terme de gauche de l'égalité ci-dessus est traditionnellement appelée "matrice de Vandermonde". -# Exemple sur un jeu de données synthétique +## Interpolation polynomiale sur un jeu de données synthétique -On considère un exemple de fonction non-linéaire. +Soit un exemple de fonction non-linéaire : ```{r} f <- function(x) {exp(x) * cos(2*pi*sin(pi*x))} ``` -On utilise cette fonction pour générer un jeu de données avec l'ajout d'un bruit gaussien. +Cette fonction est utilisée pour générer un jeu de données avec l'ajout d'un bruit gaussien. ```{r} gendat <- function(N, sd) { # N: nombre d'observations à générer @@ -89,7 +92,7 @@ gendat <- function(N, sd) { list(X = X, Y = Y) } ``` -On se donne une fonction pour afficher simultanément un jeu de données et la fonction utilisée pour le générer. +Voici une fonction pour afficher simultanément un jeu de données et la fonction utilisée pour le générer : ```{r} plt <- function(data, f, ...) { xs = seq(0,1,length.out=100) @@ -97,7 +100,7 @@ plt <- function(data, f, ...) { points(data$X, data$Y) } ``` -On affiche un exemple de jeu de données. +Nous affichons un exemple de jeu de données. ```{r} set.seed(1123) data = gendat(10,0.2) @@ -127,27 +130,27 @@ pltpoly <- function(coef) { lines(xs, polyeval(coef,xs), lty="dotted") } ``` -On affiche la fonction génératrice, le jeu de donnée et le polynôme qui passe par chaque point du jeu de données. +Nous affichons la fonction génératrice, le jeu de donnée et le polynôme qui passe par chaque point du jeu de données. ```{r} coef = polyreg1(data) plt(data,f) pltpoly(coef) ``` -Ce polynôme qui passe exactement par chaque observation a peu de chance d'offrir de bonnes capacités prédictives. Vérifier par exemple que, sur notre exemple synthétique, pour cinq points générés à partir de la fonction $f$ et avec l'ajout d'un bruit gaussien (par exemple d'écart type $0.2$), le polynôme découvert, de degré quatre, peut être très éloigné de la fonction génératrice. C'est un exemple du phénomène de sur-apprentissage. Pour limiter ce problème, on cherche à découvrir un polynôme de degré plus faible, qui ne passera donc pas exactement par toutes les observations, mais qui aura probablement une meilleure capacité à prédire les étiquettes de nouvelles observations. +Il est improbable que ce polynôme, passant exactement par chaque observation, puisse offrir de bonnes capacités prédictives. Vérifier par exemple que, sur notre exemple synthétique, pour cinq points générés à partir de la fonction $f$ et avec l'ajout d'un bruit gaussien (par exemple d'écart type $0.2$), le polynôme découvert, de degré quatre, peut être très éloigné de la fonction génératrice. C'est un exemple du phénomène de sur-apprentissage. Pour limiter ce problème, nous cherchons à découvrir un polynôme de degré plus faible. Il ne passera pas exactement par toutes les observations, mais il prédira probablement mieux les étiquettes associées à de nouvelles observations. # Généralisation à un espace de fonctions -On considère un espace vectoriel composé de fonctions. Une base de cet espace est un ensemble de fonctions ($f_1, f_2, \dots f_N$) tel que toute fonction de l'espace s'exprime comme combinaison linéaire des fonctions de base. +Soit un espace vectoriel composé de fonctions. Une base de cet espace est un ensemble de fonctions ($f_1, f_2, \dots f_N$) tel que toute fonction de l'espace s'exprime comme combinaison linéaire des fonctions de base. $$ -f(\vec{x}) = a_1 f_1(\vec{x}) + a_2 f_2(\vec{x}) + \dots + a_N f_N(\vec{x}) +f(\mathbf{x}) = a_1 f_1(\mathbf{x}) + a_2 f_2(\mathbf{x}) + \dots + a_N f_N(\mathbf{x}) $$ -Pour un jeu de données $\left\{ \vec{x^{(k)}},y^{(k)}\right\}_{k=1}^{N}$ de taille $N$, les coefficients $a_i$ sont solutions d'un système linéaire. +Pour un jeu de données $\left\{ \mathbf{x^{(k)}},y^{(k)}\right\}_{k=1}^{N}$ de taille $N$, les coefficients $a_i$ sont solution d'un système linéaire. $$ \left( \begin{array}{ccccc} -f_1(\vec{x^{(1)}}) & f_2(\vec{x^{(1)}}) & \dots & f_N(\vec{x^{(1)}}) \\ -f_1(\vec{x^{(2)}}) & f_2(\vec{x^{(2)}}) & \dots & f_N(\vec{x^{(2)}}) \\ +f_1(\mathbf{x^{(1)}}) & f_2(\mathbf{x^{(1)}}) & \dots & f_N(\mathbf{x^{(1)}}) \\ +f_1(\mathbf{x^{(2)}}) & f_2(\mathbf{x^{(2)}}) & \dots & f_N(\mathbf{x^{(2)}}) \\ \dots & \dots & \dots & \dots \\ -f_1(\vec{x^{(N)}}) & f_2(\vec{x^{(N)}}) & \dots & f_N(\vec{x^{(N)}}) +f_1(\mathbf{x^{(N)}}) & f_2(\mathbf{x^{(N)}}) & \dots & f_N(\mathbf{x^{(N)}}) \end{array} \right) \left( \begin{array}{c} a_1 \\ a_2 \\ \dots \\ a_N @@ -157,44 +160,44 @@ a_1 \\ a_2 \\ \dots \\ a_N y^{(1)} \\ y^{(2)} \\ \dots \\ y^{(N)} \end{array} \right) $$ -On note ce système linéaire $\vec{Ax} = \vec{b}$. +Nous notons ce système linéaire $\mathbf{Ax} = \mathbf{b}$. # Approche des moindres carrés -Le système linéaire $\vec{Ax} = \vec{b}$ avec $\vec{A} \in \mathbb{R}^{M \times N}$ n'a pas de solution quand le nombre d'observations dépasse le nombre de fonctions de base (c'est-à-dire, $M>N$). Une approche est alors de chercher une approximation $\vec{Ax} \approx \vec{b}$ qui minimise la somme des carrés des erreurs : $\|\vec{A}\vec{x}-\vec{b}\|^2_2$. +## Expression matricielle + +Le système linéaire $\mathbf{Ax} = \mathbf{b}$ avec $\mathbf{A} \in \mathbb{R}^{M \times N}$ n'a pas de solution quand le nombre d'observations dépasse le nombre de fonctions de base (c'est-à-dire, $M>N$). Une approche possible est alors de chercher une approximation $\mathbf{Ax} \approx \mathbf{b}$ qui minimise la somme des carrés des erreurs : $\|\mathbf{Ax}-\mathbf{b}\|^2_2$. -$$ \begin{align*} - & \|\vec{A}\vec{x}-\vec{b}\|^2_2 \\ -= \{ & \|\vec{x}\|_2 = \sqrt{\vec{x}\cdot\vec{x}} \} \\ - & \left(\vec{A}\vec{x}-\vec{b}\right) \cdot \left(\vec{A}\vec{x}-\vec{b}\right) \\ + & \|\mathbf{A}\mathbf{x}-\mathbf{b}\|^2_2 \\ += \{ & \|\mathbf{x}\|_2 = \sqrt{\mathbf{x}\cdot\mathbf{x}} \} \\ + & \left(\mathbf{A}\mathbf{x}-\mathbf{b}\right) \cdot \left(\mathbf{A}\mathbf{x}-\mathbf{b}\right) \\ = \{ & \text{Par définition du produit scalaire euclidien} \} \\ - & \left(\vec{A}\vec{x}-\vec{b}\right)^T \left(\vec{A}\vec{x}-\vec{b}\right) \\ + & \left(\mathbf{A}\mathbf{x}-\mathbf{b}\right)^T \left(\mathbf{A}\mathbf{x}-\mathbf{b}\right) \\ = \{ & \text{propriété de la transposition} \} \\ - & \left(\vec{x}^T\vec{A}^T - \vec{b}^T \right) \left(\vec{A}\vec{x}-\vec{b}\right) \\ + & \left(\mathbf{x}^T\mathbf{A}^T - \mathbf{b}^T \right) \left(\mathbf{A}\mathbf{x}-\mathbf{b}\right) \\ = \{ & \text{multiplication} \} \\ - & \vec{x}^T\vec{A}^T\vec{A}\vec{x} - \vec{x}^T\vec{A}^T\vec{b} - \vec{b}^T\vec{A}\vec{x} + \vec{b}^T\vec{b} \\ -= \{ & \vec{b}^T\vec{A}\vec{x} \text{ étant une valeur scalaire, } \vec{b}^T\vec{A}\vec{x} = \left(\vec{b}^T\vec{A}\vec{x}\right)^T = \vec{x}^T\vec{A}^T\vec{b} \} \\ - & \vec{x}^T\vec{A}^T\vec{A}\vec{x} - 2\vec{x}^T\vec{A}^T\vec{b} + \vec{b}^T\vec{b} + & \mathbf{x}^T\mathbf{A}^T\mathbf{A}\mathbf{x} - \mathbf{x}^T\mathbf{A}^T\mathbf{b} - \mathbf{b}^T\mathbf{A}\mathbf{x} + \mathbf{b}^T\mathbf{b} \\ += \{ & \mathbf{b}^T\mathbf{A}\mathbf{x} \text{ étant une valeur scalaire, } \mathbf{b}^T\mathbf{A}\mathbf{x} = \left(\mathbf{b}^T\mathbf{A}\mathbf{x}\right)^T = \mathbf{x}^T\mathbf{A}^T\mathbf{b} \} \\ + & \mathbf{x}^T\mathbf{A}^T\mathbf{A}\mathbf{x} - 2\mathbf{x}^T\mathbf{A}^T\mathbf{b} + \mathbf{b}^T\mathbf{b} \end{align*} -$$ -Cette dernière expression quadratique en $\vec{x}$ correspond à une surface convexe. Donc, le minimum de cette expression peut être calculé en annulant sa dérivée (penser à une courbe $y = a+bx+cx^2$ dont l'unique extremum est atteint lorsque la pente est nulle). -$$ +Cette dernière expression quadratique en $\mathbf{x}$ correspond à une surface convexe. Donc, le minimum de cette expression peut être calculé en annulant sa dérivée (penser à une courbe $y = a+bx+cx^2$ dont l'unique extremum est atteint lorsque la pente est nulle). + \begin{align*} - & \vec{0} = 2\vec{A}^T\vec{A}\vec{x} - 2\vec{A}^T\vec{b} \\ + & \mathbf{0} = 2\mathbf{A}^T\mathbf{A}\mathbf{x} - 2\mathbf{A}^T\mathbf{b} \\ =& \\ - & \vec{A}^T\vec{A}\vec{x} = \vec{A}^T\vec{b} + & \mathbf{A}^T\mathbf{A}\mathbf{x} = \mathbf{A}^T\mathbf{b} \end{align*} + +Ainsi, quand $M>N$, la solution approximée $\mathbf{x}$, telle que $\mathbf{Ax} \approx \mathbf{b}$ par minimisation de la somme des carrés des erreurs, est la solution du système linéaire suivant où $\mathbf{A}^T\mathbf{A}$ est appelée la matrice de Gram. $$ -Ainsi, quand $M>N$, la solution approximée $\vec{x}$, telle que $\vec{Ax} \approx \vec{b}$ en minimisant la somme des carrés des erreurs, est la solution du système linéaire suivant où $\vec{A}^T\vec{A}$ est appelée la matrice de Gram. -$$ -\vec{A}^T\vec{A} \vec{x} = \vec{A}^T\vec{b} +\mathbf{A}^T\mathbf{A} \mathbf{x} = \mathbf{A}^T\mathbf{b} $$ ## Approche des moindres carrés appliquée à la régression polynomiale -Pour un polynôme de degré $N-1$, les fonctions de bases mentionnées ci-dessus sont : $f_1(x)=1$, $f_2(x)=x$, $f_3(x)=x^2$,..., $f_N(x)=x^{N-1}$. Elles permettent de définir la matrice $\vec{A}$ et la matrice de Gram $\vec{A}^T\vec{A}$. +Pour un polynôme de degré $N-1$, les fonctions de bases mentionnées ci-dessus sont : $f_1(x)=1$, $f_2(x)=x$, $f_3(x)=x^2$,..., $f_N(x)=x^{N-1}$. Elles permettent de définir la matrice des données $\mathbf{A}$ et la matrice de Gram $\mathbf{A}^T\mathbf{A}$. La fonction ci-dessous résout le système linéaire correspondant à la matrice de Gram pour un polynôme de degré fixé. Elle retourne les coefficients du polynôme résultat. ```{r} polyreg2 <- function(data, degre) { @@ -204,7 +207,7 @@ polyreg2 <- function(data, degre) { solve(gram, as.vector(t(A) %*% data$Y)) } ``` -Sur notre exemple synthétique, on affiche la fonction génératrice, le jeu de donnée et le "meilleur" polynôme de degré 3. +Sur notre exemple synthétique, nous affichons la fonction génératrice, le jeu de donnée et le "meilleur" polynôme de degré 3. ```{r} coef = polyreg2(data,3) plt(data,f) @@ -214,7 +217,7 @@ Ce polynôme de degré trois modélise mieux la fonction génératrice inconnue # Régularisation de Tikhonov -Avec moins d'observations que de fonctions de base ($M_{\mathbf{M}} = \mathbf{x}^T\mathbf{M}\mathbf{y}$. Pour que ce produit scalaire soit valide, $\mathbf{M}$ doit respecter la contrainte $\mathbf{x}^T\mathbf{M}\mathbf{y} \geq 0$ pour tout $\mathbf{x}$ et $\mathbf{y}$. + +Une fois donné un produit scalaire, les notions liées de norme et d'angle suivent~: $\|\mathbf{x}\|_{\mathbf{M}} = \sqrt{<\mathbf{x},\mathbf{x}>_{\mathbf{M}}}$ et $cos^2(\mathbf{x},\mathbf{y}) = <\mathbf{x},\mathbf{y}>_{\mathbf{M}}/\|\mathbf{x}\|_{\mathbf{M}}\|\mathbf{y}\|_{\mathbf{M}}$. + +La géométrie euclidienne "classique" (associée à la base canonique) correspond à l'emploi de la matrice identité pour métrique, $\mathbf{M}=\mathbf{I}$. + +Aussi, les valeurs propres d'une matrice PSD sont toutes positives~: + +\begin{align*} + & \mathbf{x}^T\mathbf{M}\mathbf{x} \geq 0 \\ += \{ & \lambda \text{ est une valeur propre de } \mathbf{M} \} \\ + & \mathbf{x}^T\lambda\mathbf{x} \geq 0 \\ += \{ &\text{utilisation de la métrique identité} \} \\ + & \lambda \|x\|^2_{\mathbf{I}} \geq 0 \\ +\Rightarrow \{ &\|x\|^2_{\mathbf{I}} \geq 0 \} \\ + & \lambda \geq 0 +\end{align*} + +Soit $\mathbf{V}$ la matrice des vecteurs propres de $\mathbf{M}$ et $\mathbf{L}$ la matrice diagonale de ses valeurs propres positives. + +\begin{align*} + & \mathbf{M}\mathbf{V} = \mathbf{V}\mathbf{L} \\ += \phantom{\{} & \\ + & \mathbf{M} = \mathbf{V}\mathbf{L}\mathbf{V}^{-1} \\ += \{ & \mathbf{S} = \sqrt{\mathbf{L}}\} \\ +& \mathbf{M} = \mathbf{V}\mathbf{S}\mathbf{S}\mathbf{V}^{-1} \\ += \{ & \quad \text{Les vecteurs propres distincts sont orthogonaux deux à deux : } \mathbf{V}^{-1} = \mathbf{V}^T. \\ +\phantom{=}\phantom{ }\phantom{\{} & \quad \mathbf{S}=\mathbf{S}^T. \text{On pose } \mathbf{T}=\mathbf{V}\mathbf{S}. \} \\ +& \mathbf{M} = \mathbf{T}\mathbf{T}^T \\ +\end{align*} + +Ainsi, toute matrice définie non négative $\mathbf{M}$ peut être décomposée sous la forme $\mathbf{M}=\mathbf{T}\mathbf{T}^T$ où $\mathbf{T}$ est une transformation linéaire composée d'un changement d'échelle des axes du repère ($\mathbf{S}$) et d'une rotation ($\mathbf{V}$). + +Nous remarquons par exemple que l'équation d'un cercle selon la métrique $\mathbf{M}$ correspond à celle d'une ellipse selon la métrique identité $\mathbf{I}$ : + +\begin{align*} + & \|\mathbf{x}\|_{\mathbf{M}}^2 = c \\ += \{ &\text{C'est l'équation d'un cercle, c est une constante.}\} \\ + & < \mathbf{x}, \mathbf{x}>_{\mathbf{M}} = c \\ += \{ &\text{Par définition du produit scalaire.} \} \\ + & \mathbf{x}^T \mathbf{M} \mathbf{x} = c \\ += \{ &\mathbf{M} = \mathbf{T}^T\mathbf{T}\} \\ + & \mathbf{x}^T \mathbf{T}^T \mathbf{T} \mathbf{x} = c \\ += \{ &\text{Par définition du produit scalaire, on retrouve le cercle déformé en une ellipse par } \mathbf{T} \} \\ + & \|\mathbf{T}\mathbf{x}\|_{\mathbf{I}}^2 = c \\ +\end{align*} + +Ainsi, une matrice symétrique définie non négative projette le cercle unité sur une ellipse dont les axes orthonormaux sont les vecteurs propres et les longueurs des axes sont les valeurs propres. + +Enfin, l'angle formé entre $\mathbf{x}$ et $\mathbf{Mx}$ est inférieur ou égal à $\pi/2$ radians : + +\begin{align*} + & \mathbf{x}^T \mathbf{M} \mathbf{x} \geq 0 \\ += \{&\text{Géométrie du produit scalaire. On note } \theta \text{ l'angle entre les deux vecteurs.}\} \\ + & \|\mathbf{x}\|_\mathbf{I} \; \|\mathbf{Mx}\|_\mathbf{I} \; cos(\theta) \geq 0 \\ +\Rightarrow \phantom{\{}&\\ + & |\theta| \leq \pi/2 +\end{align*} + +Ainsi, le vecteur résultat $\mathbf{Mx}$ est contraint de rester du même côté que $\mathbf{x}$ de l'hyperplan perpendiculaire à $\mathbf{x}$ qui sépare l'espace en deux. Nous comprenons que le concept de matrice symétrique définie non négative est une généralisation pour un espace multidimensionnel du concept de nombre positif sur la droite réelle. + +## Dérivation de la décomposition en valeur singulière + +Après cette parenthèse sur la notion de métrique associée à une matrice définie non négative, nous reprenons la dérivation de la décomposition en valeurs singulières. + +Nous notons $\mathbf{v_1}$ le vecteur directeur du meilleur sous-espace de dimension 1 pour le nuage des $N$ observations de $\mathcal{R}^P$. Le meilleur sous-espace de dimension $2$ contient $\mathbf{v_1}$ et il faut le compléter avec un second vecteur unitaire $\mathbf{v_2}$, orthogonal à $\mathbf{v_1}$ et qui maximise $\mathbf{v_2}^T \mathbf{X}^T \mathbf{X} \mathbf{v_2}$. + +Nous montrons maintenant que $\mathbf{v_1}$ est le vecteur propre de $\mathbf{X}^T\mathbf{X}$ associé à la plus grande valeur propre $\lambda_1$. + +Le résultat est sans difficulté plus général. Nous le prouvons pour toute matrice symétrique $\mathbf{A}$ (non seulement $\mathbf{X}^T\mathbf{X}$) et toute métrique de $\mathcal{R}^P$ représentée par une matrice symétrique définie non négative $\mathbf{M}$ (non seulement la martice identité $\mathbf{I}$). + +Nous rappelons que $\mathbf{v}^T\mathbf{A}\mathbf{v} = \Sigma a_{i,j}v_iv_j$. Ainsi, comme $\mathbf{A}$ et $\mathbf{M}$ sont symétriques (i.e., $a_{i,j}=a_{j,i}$ et $m_{i,j}=m_{j,i}$), les dérivées partielles des formes quadratiques ont les formes suivantes : +$$ +\frac{\partial \mathbf{v}^T\mathbf{A}\mathbf{v}}{\partial \mathbf{v}}=2\mathbf{A}\mathbf{v} \quad \text{et} \quad \frac{\partial \mathbf{v}^T\mathbf{M}\mathbf{v}}{\partial \mathbf{v}}=2\mathbf{M}\mathbf{v} +$$ +Pour trouver le maximum de $\mathbf{v}^T\mathbf{A}\mathbf{v}$ en intégrant la contrainte unitaire sur $\mathbf{v}$ (i.e., $\mathbf{v}^T\mathbf{M}\mathbf{v}=1$), nous annulons les dérivées du langrangien $\mathcal{L}$ : + +$$ +\mathcal{L} = \mathbf{v}^T\mathbf{A}\mathbf{v} - \lambda (\mathbf{v}^T\mathbf{M}\mathbf{v} - 1) +$$ +\begin{align*} + & \frac{\partial \mathcal{L}}{\partial \mathbf{v}} = 0 \\ += \{& \text{voir calcul des dérivées ci-dessus} \} \\ + & 2\mathbf{A}\mathbf{v} - 2\lambda\mathbf{M}\mathbf{v} = 0 \\ += \phantom{\{} & \\ + & \mathbf{A}\mathbf{v} = \lambda\mathbf{M}\mathbf{v} \\ += \{& \mathbf{v}^T\mathbf{M}\mathbf{v}=1 \} \\ + & \lambda = \mathbf{v}^T\mathbf{A}\mathbf{v} +\end{align*} + +Nous découvrons ainsi que la valeur du multiplicateur de Lagrange $\lambda$ est le maximum recherché. Par ailleurs, nous avons : + +\begin{align*} + & \mathbf{A}\mathbf{v} = \lambda\mathbf{M}\mathbf{v} \\ += \{& \mathbf{M} \text{ est définie non négative et donc inversible} \} \\ + & \mathbf{M}^{-1}\mathbf{A}\mathbf{v} = \lambda\mathbf{v} +\end{align*} + +Donc $\mathbf{v}$ est le vecteur propre de $\mathbf{M}^{-1}\mathbf{A}$ associé à la plus grande valeur propre $\lambda$. Nous notons cette solution $\mathbf{v_1}$ et la valeur propre correspondante $\lambda_1$. + +Nous cherchons ensuite $\mathbf{v_2}$, unitaire ($\mathbf{v_2}^T\mathbf{M}\mathbf{v_2}=1$), orthogonal à $\mathbf{v_1}$ ($\mathbf{v_1}^T\mathbf{M}\mathbf{v_2}=0$) et qui maximise $\mathbf{v_2}^T\mathbf{A}\mathbf{v_2}$. Pour ce faire, nous annulons les dérivées du Langrangien ci-après. + +$$ +\mathcal{L} = \mathbf{v_2}^T\mathbf{A}\mathbf{v_2} - \lambda_2 (\mathbf{v_2}^T\mathbf{M}\mathbf{v_2} - 1) - \mu_2\mathbf{v_2}^T\mathbf{M}\mathbf{v_1} +$$ +En cours de rédaction... + +# Régression avec régularisation de Tikhonov et SVD + +# Validation croisée "un contre tous" (leave-one-out-cross-validation, LOOCV) et régression régularisée + +# Interprétation d'un modèle de régression linéraire + +## Signification des coefficients + +## Exemple sur un dataset + +## Intervalles de confiances et approche par bootstrap + +# Estimateurs par maximum de vraissemblance + +## Application aux paramètres d'une loi normale uni-dimensionnelle + +Permet d'introduire les notions de biais et de variance minimale. + +## Notation généralisée pour toute distribution + +## Vraissemblance conditionnelle pour la régression linéaire + +## Interprétation bayésienne de la régularisation commme distribution a priori des paramètres + +# Régularisation et sélection de variable avec l'approche LASSO + +# Régression logistique + +## Vraissemblance conditionnelle pour la régression logistique + +## Interprétation des coefficients d'une régression logistique + +## Descente de gradient (stochastique) appliquée à la régression logistique + +# Relations générales entre biais, variance, nombre d'observations et complexité du modèle + +# Ensembles de modèles + +## Effets sur le biais et la variance d'une approche par ensembles de modèles + +## Bagging illustré avec le modèle de forêt aléatoire + +## Boosting ? + +# Réseaux de neurones + +## Neurone + +## Architecture en couches + +## Rétropropagation du gradient + +## Régularisation implicite par descente de gradient stochastique + +## Régularisation par dropout + +## Régularisation par batch normalization + diff --git a/ML1_regression.pdf b/ML1_regression.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0d19d345b689d58aab1f377e280c195b344db5eb GIT 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