From 1ccc750802b0f3b14c593c6564146565b2d9eb45 Mon Sep 17 00:00:00 2001 From: Alexis BAYLET Date: Wed, 8 Apr 2026 15:56:29 +0200 Subject: [PATCH] add cheatsheet --- README.md | 2 + cheatsheet.md | 379 +++++++++++++++++++++++++++++++++++++++++++ learning in RKHS.pdf | Bin 0 -> 1284472 bytes 3 files changed, 381 insertions(+) create mode 100644 cheatsheet.md create mode 100644 learning in RKHS.pdf diff --git a/README.md b/README.md index ef7c58e..d0c19a7 100644 --- a/README.md +++ b/README.md @@ -1,5 +1,7 @@ # APM_4AI09 - Statistiques +Cheatsheet [cheatsheet.md](./cheatsheet.md) + | Chapitre | Titre du cours | Sources | | :--- | :--- | :--- | | **Ch. 0** | Introduction à l'Estimation Statistique | [ch0.md](./ch0.md) | diff --git a/cheatsheet.md b/cheatsheet.md new file mode 100644 index 0000000..e59f74e --- /dev/null +++ b/cheatsheet.md @@ -0,0 +1,379 @@ +# Cours de Révision — SD-TSIA205 +*Apprentissage Statistique Non-Paramétrique — Synthèse Examen* + +> **Lecture des examens :** Les deux examens passés (2024, 2025) portent quasi-exclusivement sur **l'estimateur à noyau** (densité et CDF), la **décomposition biais-variance** et l'**optimisation du bandwidth**. Ces thèmes sont à maîtriser parfaitement. Les autres chapitres (RKHS, régression, réseaux) sont du cours vu mais moins testés directement. + +--- + +## 0. Panorama des Paradigmes Statistiques + +| | $\theta$ **déterministe** | $\theta$ **aléatoire** | +|---|---|---| +| **Ensemble discret/fini** | Tests d'hypothèses — Neyman-Pearson | Théorie de la décision — MAP | +| **Dimension finie** (paramétrique) | Estimation — Cramér-Rao | Bayésien — MMSE $= \mathbb{E}[\theta\mid X]$ | +| **Dimension infinie** (non-param.) | Minimax $\inf_{\hat f}\sup_{f\in\mathcal{F}}\mathbb{E}[L(\hat f,f)]$ | — | + +**Compromis Biais-Variance fondamental :** pour tout estimateur $\hat f$, +$$\mathbb{E}[(\hat f(x) - f(x))^2] = \underbrace{(\mathbb{E}[\hat f(x)] - f(x))^2}_{\text{Biais}^2} + \underbrace{\text{Var}(\hat f(x))}_{\text{Variance}}$$ + +--- + +## 1. Estimation de Densité par Noyau ⭐ (Cœur de l'examen) + +### 1.1 Définition de l'estimateur + +Soit $X_1,\dots,X_N \overset{\text{iid}}{\sim} f$. L'**estimateur de Parzen-Rosenblatt** est : + +$$\boxed{\hat{f}_h(x) = \frac{1}{Nh}\sum_{n=1}^N K\!\left(\frac{x - X_n}{h}\right)}$$ + +avec $K : \mathbb{R}\to\mathbb{R}$ un **noyau** vérifiant $\int K(u)\,du = 1$, et $h > 0$ la **fenêtre** (*bandwidth*). + +> **Intuition :** On "lisse" la mesure empirique $\hat p = \frac{1}{N}\sum \delta_{X_n}$ par le noyau $K_h(u) = \frac{1}{h}K(u/h)$ : $\hat f_h = K_h * \hat p$. + +### 1.2 Calcul du Biais — Technique Taylor + +L'espérance de l'estimateur est la **convolution** : +$$\mathbb{E}[\hat f_h(x)] = (K_h * f)(x) = \int K_h(x - t)\,f(t)\,dt = \int K(u)\,f(x - hu)\,du$$ + +Le biais est donc : +$$\text{Biais}(\hat f_h(x)) = \mathbb{E}[\hat f_h(x)] - f(x) = \int K(u)\,[f(x-hu) - f(x)]\,du$$ + +**Développement de Taylor** à l'ordre $k$ de $f(x-hu)$ en $x$ : + +$$f(x - hu) - f(x) = \sum_{l=1}^k \frac{(-hu)^l}{l!} f^{(l)}(x) + R_k(x, hu)$$ + +Si le noyau est **d'ordre $k$** (i.e. $\int u^l K(u)\,du = 0$ pour $l = 1,\dots,k$), tous les termes en $h^l$ disparaissent après intégration. Il reste : + +$$\text{Biais}(\hat f_h(x)) = \int K(u)\, R_k(x, hu)\,du$$ + +**Cas de l'examen 2025** ($k = 2$, $f''$ est $L$-lipschitzienne) : + +Les hypothèses sont $K$ paire (donc $\int uK = 0$), $\int u^2 K(u)\,du = 0$, $\int |u^3 K(u)|\,du = C_1$. + +Le reste de Taylor d'ordre 2 vérifie (avec $|f''(x-\theta hu) - f''(x)| \le L|\theta hu|$) : +$$|R_2(x, hu)| = \left|\int_0^{-hu} f''(x+t)\,(-hu - t)\,dt - \frac{(hu)^2}{2}f''(x)\right| \le \frac{L}{6}|hu|^3$$ + +Donc : +$$\boxed{|\text{Biais}(\hat f_h(x))| \le \frac{L\,C_1}{6}\,h^3 =: \gamma_1 h^3}$$ + +**Cas général** (f est $s$-höldérienne, noyau d'ordre $k = \lfloor s\rfloor$) : $|\text{Biais}| = O(h^s)$. + +### 1.3 Calcul de la Variance + +Par indépendance des $X_n$ : +$$\text{Var}(\hat f_h(x)) = \frac{1}{N}\,\text{Var}\!\left(K_h(x - X_1)\right) \le \frac{1}{N}\,\mathbb{E}\!\left[K_h(x-X_1)^2\right]$$ + +Par changement de variable $u = (x - t)/h$ : +$$\mathbb{E}[K_h(x-X_1)^2] = \int \frac{1}{h^2}K\!\left(\frac{x-t}{h}\right)^2 f(t)\,dt = \frac{1}{h}\int K(u)^2\,f(x-hu)\,du \le \frac{\|f\|_\infty}{h}\int K(u)^2\,du$$ + +**Cas de l'examen 2025** ($\|f\|_\infty \le f_{\max}$, $\int K^2 = C_2$) : + +$$\boxed{\text{Var}(\hat f_h(x)) \le \frac{f_{\max}\,C_2}{Nh} =: \frac{\gamma_2}{Nh}}$$ + +### 1.4 Borne sur le MSE + +$$\text{MSE}(\hat f_h(x)) = \text{Biais}^2 + \text{Variance} \le \underbrace{\gamma_1^2\,h^{2k+2}}_{\text{Biais}^2} + \underbrace{\frac{\gamma_2}{Nh}}_{\text{Variance}} =: g(h)$$ + +Pour l'examen 2025 ($k=2$) : +$$g(h) = \gamma_1^2\, h^6 + \frac{\gamma_2}{Nh}$$ + +### 1.5 Fenêtre Optimale $h^*$ + +On minimise $g(h)$ par rapport à $h > 0$ : +$$g'(h) = 0 \iff 6\,\gamma_1^2\,h^5 - \frac{\gamma_2}{Nh^2} = 0 \iff h^7 = \frac{\gamma_2}{6N\gamma_1^2}$$ + +$$\boxed{h^* = \left(\frac{\gamma_2}{6N\gamma_1^2}\right)^{1/7}}$$ + +**Cas général** (biais en $h^{2s}$, variance en $\frac{1}{Nh}$) : +$$h^* \asymp N^{-\frac{1}{2s+1}}$$ + +### 1.6 Vitesse de Convergence + +En substituant $h^*$ dans $g(h^*)$ : +$$g(h^*) = \gamma_1^2\left(\frac{\gamma_2}{6N\gamma_1^2}\right)^{6/7} + \frac{\gamma_2}{N}\left(\frac{\gamma_2}{6N\gamma_1^2}\right)^{-1/7} \sim N^{-6/7}$$ + +**Examen 2025 :** vitesse $N^{-6/7}$ car $s=3$ → $\frac{2s}{2s+1} = \frac{6}{7}$. + +**Cas général :** vitesse **minimax** $N^{-\frac{2s}{2s+1}}$. + +| Régularité $s$ | Vitesse | Commentaire | +|---|---|---| +| $s = 1$ | $N^{-2/3}$ | Hölder 1, biais en $h^2$ | +| $s = 2$ | $N^{-4/5}$ | biais en $h^4$ (noyau ordre 2) | +| $s = 3$ | $N^{-6/7}$ | biais en $h^6$ (**examen 2025**) | +| $s \to \infty$ | $\to N^{-1}$ | vitesse paramétrique | + +> **Interprétation :** Plus $f$ est régulière (grand $s$), plus on approche la vitesse paramétrique $1/N$. La fenêtre optimale $h^*$ décroît avec $N$ (on "zoome" en affinant avec plus de données). + +### 1.7 Résumé : Propriétés des Noyaux + +Un noyau $K$ **d'ordre $k$** vérifie : +1. $\int K(u)\,du = 1$ +2. $\int u^l K(u)\,du = 0$ pour $l = 1,\dots,k$ +3. $\int |u^{k+1} K(u)|\,du < +\infty$ + +**Astuce :** Si $K$ est pair, la condition $\int u^l K = 0$ pour $l$ impair est **automatique**. Il suffit d'annuler les moments pairs. + +**Exemples :** +- Noyau rectangulaire : $K(u) = \frac{1}{2}\mathbf{1}_{[-1,1]}(u)$ — ordre 0 +- Noyau gaussien : $K(u) = \frac{1}{\sqrt{2\pi}}e^{-u^2/2}$ — ordre 1 (pair, mais $\int u^2 K \ne 0$) +- Noyau d'Epanechnikov : $K(u) = \frac{3}{4}(1-u^2)\mathbf{1}_{[-1,1]}(u)$ — ordre 1 + +--- + +## 2. Estimation de la CDF par Noyau ⭐ (Examen 2024) + +### 2.1 L'estimateur + +Soit $k$ un noyau de densité et $K(x) = \int_{-\infty}^x k(t)\,dt$ sa **primitive** (une CDF). L'estimateur de $F$ est : + +$$\hat{F}_N(x) = \frac{1}{N}\sum_{n=1}^N K\!\left(\frac{x - x_n}{h}\right)$$ + +> **Attention :** Ici $K$ est la **primitive** du noyau $k$, pas le noyau lui-même. + +### 2.2 Biais comme convolution + +**Théorème :** +$$\mathbb{E}[\hat F_N(x)] = (k_h * F)(x) = \int k_h(x-u)\,F(u)\,du$$ + +*Preuve :* Par intégration par parties (IBP) : +$$\int \frac{1}{h}k\!\left(\frac{x-u}{h}\right)F(u)\,du \overset{\text{IBP}}{=} \underbrace{\left[-K\!\left(\frac{x-u}{h}\right)F(u)\right]_{-\infty}^{+\infty}}_{=0} + \int K\!\left(\frac{x-u}{h}\right)f(u)\,du = \mathbb{E}[K((x-X)/h)]$$ + +Les termes au bord sont nuls car : $K(-\infty)F(+\infty) = 0\cdot 1 = 0$ et $K(+\infty)F(-\infty) = 1\cdot 0 = 0$. + +### 2.3 Décomposition MSE + +$$\mathbb{E}\!\left[(F(x) - \hat F_N(x))^2\right] = \text{Var}(\hat F_N(x)) + \underbrace{((k_h * F)(x) - F(x))^2}_{\text{Biais}^2}$$ + +### 2.4 Borne sur la Variance + +Comme $0 \le K(t) \le 1$, on a $K(t)^2 \le K(t)$, donc : +$$\mathbb{E}[K((x-X_1)/h)^2] \le \mathbb{E}[K((x-X_1)/h)] \le 1$$ + +Par indépendance : $\text{Var}(\hat F_N(x)) \le \frac{1}{N}$. + +### 2.5 Borne sur le Biais + +$$(k_h * F)(x) - F(x) = \int k(t)\,(F(x - th) - F(x))\,dt$$ + +*Preuve :* Changement de variable $u = th$ dans $\int k_h(x-u)F(u)du$, puis soustraction de $F(x)\int k = F(x)$. + +Par le théorème des accroissements finis $|F(x-th) - F(x)| \le |f|_\infty\cdot|th|$ : +$$|(k_h * F)(x) - F(x)| \le \sup_t f(t)\cdot h \int |t|\,k(t)\,dt$$ + +### 2.6 MSE optimal en $O(1/N)$ + +$\text{MSE} \le \frac{1}{N} + C^2 h^2$ où $C = \|f\|_\infty \int |t|k(t)\,dt$. + +En choisissant $h = N^{-1/2}$ : $\text{MSE} \le \frac{1}{N} + \frac{C^2}{N} = \frac{1+C^2}{N} = O(1/N)$. + +> **Remarque clé :** La vitesse $1/N$ pour la CDF est meilleure que pour la densité ($N^{-2s/(2s+1)} < 1/N$ pour tout $s$ fini). Estimer la CDF est un problème "plus facile" que d'estimer la densité. + +--- + +## 3. Espaces de Régularité + +### 3.1 Espace de Hölder $\Lambda(s, L)$ + +Pour $s = k + \beta$ avec $k \in \mathbb{N}$, $\beta \in (0, 1]$ : +$$\Lambda(s, L) = \left\{f : f \text{ est } k \text{ fois dérivable et } |f^{(k)}(x) - f^{(k)}(y)| \le L|x-y|^\beta\right\}$$ + +**Cas importants :** +- $s = 1$ ($k=0$, $\beta=1$) : $f$ est $L$-lipschitzienne +- $s = 2$ ($k=1$, $\beta=1$) : $f'$ est $L$-lipschitzienne +- $s = 3$ ($k=2$, $\beta=1$) : $f''$ est $L$-lipschitzienne ← **examen 2025** + +### 3.2 Espace de Sobolev $W^s([0,1])$ + +$$W^s([0,1]) = \left\{f \in L^2([0,1]) : \sum_{k \in \mathbb{Z}} |\alpha_k|^2(1+|k|)^{2s} < +\infty\right\}$$ + +où $\alpha_k = \langle f, e_k\rangle$ sont les coefficients de Fourier. + +**Ellipsoïde de Sobolev :** $B(s, R) = \{f \in W^s : \|f\|_{W^s} \le R\}$. + +Le biais de l'estimateur par projection tronqué à l'ordre $M$ vérifie : +$$\text{Biais}^2 = \sum_{|k| > M}|\alpha_k|^2 \le \frac{R^2}{M^{2s}}$$ + +--- + +## 4. Estimation par Projection (Fourier) + +### 4.1 Estimateur + +Dans la base de Fourier $e_k(x) = e^{2\pi ikx}$ : +$$\hat f(x) = \sum_{|k| \le M} \hat\alpha_k\, e_k(x), \qquad \hat\alpha_k = \frac{1}{N}\sum_{n=1}^N e_k(X_n)$$ + +### 4.2 Analyse MISE + +$$\text{MISE} = \underbrace{\sum_{|k|>M}|\alpha_k|^2}_{\text{Biais}^2 = O(M^{-2s})} + \underbrace{\sum_{|k|\le M}\frac{\text{Var}(e_k(X))}{N}}_{\text{Variance} = O(M/N)}$$ + +**Ordre optimal :** $M^* \asymp N^{1/(2s+1)}$ → MISE $\asymp N^{-2s/(2s+1)}$ (même vitesse que le noyau). + +| Méthode | Paramètre | Rôle du paramètre | +|---|---|---| +| Projection | $M$ (nb de modes) | $\uparrow M$ → moins de biais, plus de variance | +| Noyau | $h$ (bandwidth) | $\downarrow h$ → moins de biais, plus de variance | + +--- + +## 5. Régression Non-Paramétrique + +### 5.1 Solution de Bayes + +Problème : minimiser $R(f) = \mathbb{E}[(Y - f(X))^2]$. La solution est : +$$m(x) = \mathbb{E}[Y \mid X = x] \quad \text{(espérance conditionnelle)}$$ + +*Preuve :* $\mathbb{E}[(Y-f(X))^2] = \mathbb{E}_X[\mathbb{E}_Y[(Y-f(X))^2\mid X]]$. Pour tout $x$ fixé, le minimum en $c$ de $\mathbb{E}[(Y-c)^2\mid X=x]$ est $c = \mathbb{E}[Y\mid X=x]$. + +### 5.2 Estimateur de Nadaraya-Watson + +$$\hat m(x) = \frac{\displaystyle\sum_{n=1}^N Y_n\, K_h(x - X_n)}{\displaystyle\sum_{n=1}^N K_h(x - X_n)} = \sum_{n=1}^N w_n(x)\, Y_n$$ + +Les poids $w_n(x) = K_h(x-X_n)/\sum_i K_h(x-X_i)$ sont positifs et somment à 1. + +**Cas limites :** +- $h \to 0$ : interpolation exacte (overfitting) +- $h \to \infty$ : $\hat m(x) = \bar Y$ (underfitting) + +### 5.3 Splines de Lissage + +Minimisation dans $C^2([a,b])$ : +$$J(f) = \frac{1}{N}\sum_{n=1}^N(Y_n - f(X_n))^2 + \lambda\int_a^b |f''(t)|^2\,dt$$ + +**Résultat fondamental :** la solution est une **spline cubique naturelle** avec nœuds en $X_1,\dots,X_N$. Malgré l'espace infini-dimensionnel, la solution vit dans un espace de dimension $N$ → problème matriciel. + +--- + +## 6. RKHS et Méthodes à Noyau + +### 6.1 Noyau Défini Positif (PDS) + +$k : \mathcal{X}\times\mathcal{X} \to \mathbb{R}$ est **symétrique défini positif** si : +$$\forall (x_i)_{i=1}^n,\; \forall c \in \mathbb{R}^n,\quad \sum_{i,j} c_i c_j k(x_i, x_j) \ge 0$$ + +⟺ la **matrice de Gram** $K_{ij} = k(x_i, x_j)$ est semi-définie positive (SDP). + +**Exemples :** +- Gaussien (RBF) : $k(x,x') = e^{-\gamma\|x-x'\|^2}$ +- Polynomial : $k(x,x') = (1 + \langle x,x'\rangle)^p$ +- Linéaire : $k(x,x') = x^T x'$ + +**Inégalité de Cauchy-Schwartz :** $k(x,z)^2 \le k(x,x)\cdot k(z,z)$. + +### 6.2 RKHS et Propriété de Reproduction + +Un espace de Hilbert $\mathcal{H}$ est un **RKHS** de noyau $k$ si : +1. $k(\cdot, x) \in \mathcal{H}$ pour tout $x$ +2. $f(x) = \langle f, k(\cdot, x)\rangle_{\mathcal{H}}$ pour tout $f \in \mathcal{H}$ (**propriété de reproduction**) + +La **feature map** naturelle est $\varphi(x) = k(\cdot, x)$, de sorte que $k(x,x') = \langle\varphi(x), \varphi(x')\rangle_{\mathcal{H}}$. + +### 6.3 Théorème de Moore-Aronszajn + +> Pour tout noyau PDS $k$, il existe un **unique** RKHS $\mathcal{H}_k$ dont $k$ est le noyau reproduisant. + +*Construction :* $H_0 = \text{span}\{k(\cdot, x_i)\}$, produit scalaire $\langle \sum_i \alpha_i k(\cdot,x_i), \sum_j \beta_j k(\cdot,z_j)\rangle = \sum_{i,j}\alpha_i\beta_j k(x_i,z_j)$, puis complétion. + +### 6.4 Théorème du Représentant (Wahba 1978) + +Soit $g : [0,\infty)\to\mathbb{R}$ strictement croissante. Toute solution de : +$$\min_{f\in\mathcal{H}_k} c(f(x_1),\dots,f(x_n)) + \lambda\,g(\|f\|_{\mathcal{H}_k})$$ + +admet une représentation finie : +$$\boxed{f_n(\cdot) = \sum_{i=1}^n \alpha_i\, k(\cdot, x_i)}$$ + +*Preuve (esquisse) :* Décomposer $f = f_1 + f_1^\perp$ avec $f_1 \in \text{span}\{k(\cdot,x_i)\}$. Par reproduction, $f(x_i) = f_1(x_i)$. Par Pythagore, $\|f\| \ge \|f_1\|$. Donc $J(f_1) \le J(f)$ → le terme $f_1^\perp$ est nul à l'optimum. + +### 6.5 Kernel Ridge Regression (KRR) + +Problème : +$$\min_{f\in\mathcal{H}_k} \frac{1}{n}\sum_i (y_i - f(x_i))^2 + \lambda\|f\|^2_{\mathcal{H}_k}$$ + +Par le théorème du représentant $f(x) = \sum_i \alpha_i k(x,x_i)$. En notant $K$ la matrice de Gram $n\times n$ : + +$$\boxed{\boldsymbol\alpha = (K + n\lambda I)^{-1} Y}$$ + +*Dérivation :* La loss s'écrit $\frac{1}{n}\|Y - K\boldsymbol\alpha\|^2 + \lambda\boldsymbol\alpha^T K\boldsymbol\alpha$. La dérivée par rapport à $\boldsymbol\alpha$ donne $K(K + n\lambda I)\boldsymbol\alpha = KY$. + +**Cas linéaire** $k(x,x')=x^Tx'$ → $K = XX^T$ → régression Ridge classique. + +### 6.6 SVM à Noyau + +Perte hinge : $\ell(y, h(x)) = \max(1 - yh(x), 0)$ + +Problème : $\min_{h\in\mathcal{H}} \frac{1}{n}\sum_i \max(1-y_i h(x_i),0) + \lambda\|h\|^2_{\mathcal{H}}$ + +Par le théorème du représentant : $h(x) = \sum_i \alpha_i k(x,x_i)$. Le problème se réécrit en QP dans $\mathbb{R}^n$ via variables d'écart $\xi_i \ge 0$. + +**Avec biais** : théorème du représentant semi-paramétrique → $h(x) = \sum_i \alpha_i k(x,x_i) + b$. + +### 6.7 Propriétés de Clôture + +Les noyaux PDS sont stables par : somme, produit scalaire positif, produit, composition par une fonction positive, limite ponctuelle. + +--- + +## 7. Fléau de la Dimension et Réseaux de Neurones + +### 7.1 La Classe $\mathcal{F}_C$ + +$$\mathcal{F}_C = \left\{f \;\middle|\; \int_{\mathbb{R}^d}\|\vec\omega\|_1|F(\vec\omega)|\,d\vec\omega \le C\right\}$$ + +où $F$ est la transformée de Fourier de $f$. + +### 7.2 Écart de Kolmogorov + +$$w_N(\mathcal{F}_C) = \inf_{\substack{H_N \text{ s.e.v.} \\ \dim H_N \le N}} \sup_{f\in\mathcal{F}_C} \|f - \text{proj}_{H_N}f\|_{L^2}$$ + +C'est l'erreur minimale d'approximation par **tout** sous-espace linéaire de dimension $N$. + +### 7.3 Théorème (Fléau de la Dimension) + +$$\exists\,\kappa > 0,\quad w_N(\mathcal{F}_C) \ge \kappa\,\frac{C}{d}\,N^{-1/d}$$ + +**Conséquence :** Pour toute méthode **linéaire** (polynômes, Fourier tronqué, ondelettes...), l'erreur en dimension $d$ décroît comme $N^{-1/d}$. Pour $d = 100$, il faudrait $N = 10^{100}$ échantillons pour atteindre une erreur de $10^{-2}$. + +*Preuve (idée) :* On construit $2N$ fonctions de test $h_j^* = \cos(2\pi \vec k_j \cdot \vec x)$ normalisées pour appartenir à $\mathcal{F}_C$. Tout sous-espace $H_N$ ne peut être orthogonal à toutes, donc l'une d'elles a une grande composante hors de $H_N$. L'argument combinatoire montre que $\|\vec k_{2N}\|_1 \ge d(2N)^{1/d}$. + +### 7.4 Pourquoi les Réseaux de Neurones ? + +Un réseau avec $N$ neurones est une **somme de fonctions non-linéaires** : +$$f(x) = \sum_{j=1}^N c_j\,\sigma\!\left(\vec w_j^T \vec x + b_j\right)$$ + +Ce n'est **pas** un sous-espace linéaire fixé : les $\vec w_j, b_j$ s'adaptent aux données. Les réseaux peuvent briser le fléau de la dimension pour certaines classes de régularité. + +--- + +## 8. Récapitulatif Formules Clés à Retenir + +| Quantité | Formule | Ordre | +|---|---|---| +| Estimateur noyau densité | $\hat f_h(x) = \frac{1}{Nh}\sum K\!\left(\frac{x-X_i}{h}\right)$ | — | +| Espérance de $\hat f_h$ | $\mathbb{E}[\hat f_h(x)] = (K_h * f)(x)$ | — | +| Biais (noyau ordre $k$, $f \in \Lambda(s,L)$) | $|\text{Biais}| \le \gamma_1 h^s$ | $O(h^s)$ | +| Variance | $\text{Var}(\hat f_h(x)) \le \frac{\gamma_2}{Nh}$ | $O(1/(Nh))$ | +| MSE = Biais² + Var | $\le \gamma_1^2 h^{2s} + \frac{\gamma_2}{Nh}$ | — | +| Bandwidth optimal | $h^* \asymp N^{-1/(2s+1)}$ | — | +| Vitesse minimax densité | $\text{MSE}(h^*) \asymp N^{-2s/(2s+1)}$ | — | +| Vitesse estimateur CDF | $\text{MSE} = O(1/N)$ avec $h = N^{-1/2}$ | — | +| KRR | $\boldsymbol\alpha = (K + n\lambda I)^{-1}Y$ | — | + +--- + +## 9. Méthodologie pour l'Examen + +**Étapes systématiques pour un problème d'estimateur à noyau :** + +1. **Écrire l'estimateur** — vérifier les hypothèses sur le noyau (parité, moments) +2. **Calculer le biais** : + - Écrire $\mathbb{E}[\hat f(x)] - f(x) = \int K(u)[f(x-hu) - f(x)]du$ + - Faire le développement de Taylor jusqu'à l'ordre où les moments de $K$ annulent les termes + - Borner le reste par la condition de Hölder/Lipschitz +3. **Calculer la variance** : + - Utiliser $\text{Var}(\hat f) = \frac{1}{N}\text{Var}(K_h(x-X_1)) \le \frac{1}{N}\mathbb{E}[K_h^2(x-X_1)]$ + - Changement de variable $u=(x-t)/h$, puis borner $f$ par $f_{\max}$ +4. **MSE = Biais² + Variance** +5. **Optimiser en $h$** : dériver et annuler $\frac{d}{dh}(\text{Biais}^2 + \text{Variance}) = 0$ +6. **Analyser** $g(h^*)$ en fonction de $N$ (vitesse de convergence) + +**Piège fréquent :** ne pas oublier le facteur $\frac{1}{N}$ devant la variance (indépendance des $X_i$). diff --git a/learning in RKHS.pdf b/learning 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