From 99a39f48d7eda1f818d7476c52d766f14f70189e Mon Sep 17 00:00:00 2001 From: Pierre-antoine Comby Date: Wed, 26 Dec 2018 11:46:15 +0100 Subject: [PATCH] fixed untracked files --- 421-Controle_processus/Cours/Cours.dvi | Bin 243180 -> 0 bytes 421-Controle_processus/Cours/Cours.pdf | Bin 415785 -> 416623 bytes 421-Controle_processus/TP2/TP2.bbl | 0 421-Controle_processus/TP2/TP2.blg | 48 -- 431-Electronique_transmission/TP1/auto/TP1.el | 13 - 431-Electronique_transmission/TP2/auto/TP2.el | 20 - 441-Info_indus/TD1/auto/td.el | 39 -- 451-Signal_Image/Cours/estimateur.pdf | Bin 0 -> 211379 bytes 451-Signal_Image/Cours/estimateur.tex | 507 ++++++++++++++++++ 451-Signal_Image/Cours/filtrage.pdf | Bin 0 -> 132475 bytes 451-Signal_Image/Cours/filtrage.tex | 166 ++++++ 451-Signal_Image/Cours/main.pdf | Bin 0 -> 150134 bytes 451-Signal_Image/Cours/main.tex | 52 ++ 13 files changed, 725 insertions(+), 120 deletions(-) delete mode 100644 421-Controle_processus/Cours/Cours.dvi delete mode 100644 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+ +\begin{figure}[H] + \centering + \begin{tikzpicture} + \node[draw, ellipse] (P) at (0,0) { + \begin{tabular}{c} +paramètres \\$\theta = \vect{\theta_1\\ \vdots\\\theta_n}$ + \end{tabular}}; + \node[draw, ellipse] (O) at (5,4) { + \begin{tabular}{c} + Observation \\ + Y=$g(\theta)$ + \end{tabular}}; + \node[draw, ellipse] (E) at (10,0){ + \begin{tabular}{c} + Estimée\\ + $\hat{\theta} = h(y)$ + \end{tabular}}; + \draw[->,>=latex] (P) to[out=90, in = 180] (O); + \draw[->,>=latex] (O) to[out=0, in=90] node[near end,left]{ + \begin{tabular}{c} + Information à priori\\ + + Critère + \end{tabular}} + (E); +\end{tikzpicture} +\caption{Méthode d'estimation classique} +\end{figure} + +Le raisonnement se transpose alors sur la figure suivante: +\begin{figure}[H] + \centering + \begin{tikzpicture} + \draw[->,>=latex] (0,2) node{$\bullet$}node[right](theta){$\theta$} -- node[midway,left]{$\tilde{\theta}$}(-0.5,0) node{$\bullet$}node[right](hat){$\hat{\theta}$} ; + \node[draw,ellipse,fit= (theta) (hat)](par) {}; + \node[below=5em] at (par) {\emph{Espace des paramètres}}; + \node (y) at (5,2) {$\bullet$~$y$}; + \node[draw,ellipse,minimum height=4cm,minimum width=2cm] (obs) at (5,1){}; + \node[below=5em] at (obs){\emph{Espace des observations}}; + \draw[->,>=latex] (theta) to[out=60, in=120] node[midway,above]{\emph{observation}} (y); + \draw[->,>=latex] (y) to[out=-120,in=30,bend left] node[midway,below=0.5em]{\emph{estimation}}(hat); + + \end{tikzpicture} + \caption{Raisonnement en espace algébrique} +\end{figure} + + +On défini les index suivants: +\begin{description} +\item[m] nombre d'expérience réalisée (taille de $y$) +\item[n] nombre de paramètres (taille de $\theta$) +\end{description} +\paragraph{Estimateurs statistiques} +On observe une réalisation $y= g(\theta)$ où $\theta$ est une VA. et on détermine $\hat{\theta} = h(Y)$ estimée. + +\paragraph{Exemple} +\subparagraph{Exemple 1}$\Theta$ tension constante.\\ +$y(t) = \theta +b(t)$. soit $y_i = \theta + b_i$\\ +On défini donc $Y$ et $\Theta$ VA et on a $Y = A\Theta + B$ -> régression linéaire. +\subparagraph{Exemple 2} filtre $RC$ $y(t) = (1-e^{-t/\tau})u(t)+b(t)$ , $\Theta=\tau$. modèle non linéaire, traité en TD. + +\subsection{Performance-Qualité d'une estimation} +\begin{prop}[Grandeurs utiles] + \begin{itemize} + \item erreur d'estimation + \[ + \tilde{\theta} = \hat{\theta}-\theta + \] + \item moment d'ordre 1: +\[ +E_{Y|\Theta}[\tilde{\theta}]= E_{Y|\Theta}[\hat{\theta}]-\theta +\] +\item Biais moyen : + \[ + E[\tilde{\theta}] = E_{Y\Theta}[\tilde{\theta}] = E[\hat{\theta}]-\theta + \] +\item moment d'ordre 2: + \begin{itemize} + \item covariance de l'erreur d'estimation + \[ + C_{\tilde{\theta}\tilde{\theta}} = E[(\tilde{\theta}-m_{\tilde{\theta}})(.)^T] + \] + \item Corrélation de l'erreur d'estimation + \[ + \Gamma_{\tilde{\theta}\tilde{\theta}} = E[\tilde{\theta}\tilde{\theta}^T] + \] + \item Puissance :(Estimateur Quadratique moyen) + \[ + P_{\tilde{\theta}} = E[\| \tilde{\theta}\|^2] = tr(\Gamma_{\tilde{\theta}\tilde{\theta}}) + \] + \end{itemize} + \end{itemize} +\end{prop} +\subsection{Caractérisation des estimateurs} +\begin{defin} + \begin{itemize} + \item Borne de Cramer Rao: + borne minimale du biais de variance (qui dépend de l'estimateur choisi) + \item Estimateur non biaisé: $E[\tilde{\theta}] = 0$ + \item Estimateur efficace: Borne de Cramer-Rao atteinte. + \item Estimateur consistent: $E[\tilde{\theta}]\xrightarrow[N_{obs}\to\infty]{}0$ et $V[\tilde{\theta}]\xrightarrow[N_{obs}\to\infty]{}0$ + \item Estimateur robuste:\\ Les performances de l'estimateur ne sont pas trop dégradé si on s'écarte un peu des hypothèses sous laquelle l'estimateur a été établi. + \item Complexité de l'estimateur:\\ + sur l'o btention des connaissances et mise en oeuvre de l'estimateur. + \end{itemize} +\end{defin} +\section{Théorie classique de l'estimation} +\subsection{Estimateur des moindres carrés} +\begin{defin} + Pour $Y$ une VA de moyenne $m_y =m_{Y|\theta}$ on défini le critère : + \[ + J_{MC} = (Y-m_y)^TM(Y-m_y) + \] + Avec $M$ matrice symétrique définie positive + et alors: + \[ +\hat{\theta}_{MC} = \arg\min_{\theta} J_{MC}(Y,\theta) + \] +\end{defin} + +\subsubsection{Condition nécessaire d'existance} + +Si $J_{MC}(y,\theta)$ est dérivable et pas de contrainte sur $\theta$. +\[ + \left.\nabla_J(\theta)\right|_{\hat{\theta}_{MC}} = \derivp[J_{MC}]{\theta} = 0 \quad \text{ Gradien} +\] + +Il faut ensuite vérifié que c'est un minimum absolu: + +\[ + \nabla^2_{J}(\theta) = \derivp[{}^2J_{MC}]{\theta\partial\theta^T} > 0 \quad \text{Hessien} +\] + + +\paragraph{Application} $Y = A\theta{} + B$, avec $B$ une VA. +le critère des moindres carrés est alors : +\[ +J_{MC} = (Y-A\theta-m_B)^TM (Y-A\theta-m_B) +\] +On a une forme quadratique positive car $A^TMA \geq0 $. (dans le cas $>0$ on a une CNS sur ce qui suit) +\subparagraph{Méthode 1} +\[ + \left.\nabla_J(\theta)\right|_{\hat{\theta}_{MC}} = 0 = -2 A^TM(Y-A\theta-m_B) +\] +Donc +\[ +A^TMA \theta = A^TM(Y-m_B) +\] +Soit \[ + \boxed{\hat{\theta}_{MC} = \underbrace{(A^TMA)^{-1}AM}_{D}(Y-m_B)} +\] +On remarque que $DA = I_n$. +\subparagraph{Méthode 2} Pour $A^TMA>0$. + +\[ +J_{MC} = \underbracket{(D(Y-m_B)-\Theta)^TA^TMA(D(Y-m_B)-\theta)}_ {J_1(Y,\theta)} + \underbracket{(Y-m_B)^T(M-D^TA^TMAD)(Y-m_B)}_{J_2(Y)} +\] +Alors $\nabla J_{MC} = 0 \implies J_1 = 0 \implies D(Y-m_B) = \hat{\theta}_{MC}$ + + +\subsubsection{Caractéristique de l'estimateur} +\begin{itemize} +\item Estimateur non biaisé + + \begin{align*} + \tilde{\theta}_{MC} &=\hat{\Theta}-\theta\\ + &= D(Y-m_B)-\theta \\ + &= D(B-m_B) + \end{align*} +Donc $E[\hat{\theta_{MC}}] = 0 $ +\item moment d'ordre 2 : + \[ + C_{\tilde{\theta}\tilde{\theta}} = E[(\tilde{\theta}-m_{\tilde{\theta}})(.)^T] = D E[(B-m_B)(B-m_B)^T]D^T = D C_{BB}D^T + \] + \begin{itemize} + \item Cas MC ordinaire ($M=I_n$) + \[ + C_{\tilde{\theta}\tilde{\theta}} = (A^TA)^{-1}A^TC_{BB}A(A^TA)^{-1} + \] + \item Cas MC pondéré ($M = C_{BB}^{-1}$) + \[ + C_{\tilde{\theta}\tilde{\theta}} = (A^TC_{BB}^{-1}A)^{-1} + \] + \end{itemize} + \item Cas $\theta$ scalaire $Y_i = \theta +B_i$ donc : + \[ + C_{BB} = + \begin{bmatrix} + \sigma_1^2 & &0 \\ + & \ddots & \\ + 0 & & \sigma_m^2 + \end{bmatrix} \text{ et }A = \vect{1\\ \vdots \\ 1} + \] + \begin{itemize} + \item Cas MCO : $A^TA = m $ + \[ + \hat{\theta_{MC}} =\frac{\Sigma(y_i-m_{bi})}{m} \quad \text{ et } \quad \sigma_{\tilde{\theta}}^2 = \frac{\Sigma\sigma_i^2}{m^2} + \] + \item cas MCP pour $M = C_{BB}^{-1} = diag(\sigma_1^{-2}, \dots, \sigma_m^{-2})$ +\[ + A^TC_{BB}A = \sum_{i=1}^m \frac{1}{\sigma_i^2} \quad \text{ donc } \quad \hat{\theta}_{MCP} = \frac{1}{\sum \frac{1}{\sigma_i^2}}\sum_{}^{}\frac{Y_i-mB_i}{\sigma_i^2} +\] +\begin{itemize} +\item $\hat{\theta_{MCP}}$ défini un barycentre +\item Pour $\sigma_i = \sigma$ on a $M=\sigma I \implies MCO =MCP $ +\end{itemize} + +\end{itemize} +\item Comparaison MCO et MCP (avec $M = C_{BB}$) + \begin{align*} + \sigma_{MCO}^2 &\leq \sigma_{MCP}^2\\ + \frac{1}{\sum\sigma_i^{-2}} & \leq \frac{1}{m^2}\sum\sigma_i^2\\ + m ^2 &\leq \frac{1}{\sum\sigma_i^{-2}} \sum\sigma_i^2 + \end{align*} + +\end{itemize} + +\subsection{Estimateur du maximum de vraisemblance} +\begin{defin} +On considère $f_{Y}(y)$ ddp de $y$ paramétrée par $\theta$. On a $f_{Y|\theta}(y) = V(Y,\theta)$. on pose également $L(Y,\theta) = \ln(V(Y,\theta))$. + +on défini alors: +\[ +\hat{\theta}_{MV} = \arg\min f_{Y|\theta}(y) = \arg\min L(Y,\theta) +\] +\end{defin} + +GRAPHE + +\paragraph{Exemple} Modèle avec bruit additif gaussien. + + +\begin{prop} + Dans le cas d'un brui Gaussien et pour $M = C_{BB}^{-1}$ + \[ + \hat{\theta}_{MCP}=\hat{\theta}_{MV} + \] + +\end{prop} + +\paragraph{Remarque} +L'estimateur de MV n'est pas nécessairement efficace mais si un estimateur sans biais existe et est efficace c'est celui-ci. + +Si $m \to\infty $ on montre que le MV est asymptotiquement efficace. (loi des grands nombres) +\section{Théorie générale de l'estimation} +\subsection{Estimateur linéaire en moyenne quadratique (ELMQ)} + +\begin{defin} + Un ELMQ fourni une estimée de la forme + \[ + \hat{\theta} = HY +C + \] + à partir de l'erreur quadratique moyenne $E[\|\tilde{\theta}\|^2] = E[\tilde{\theta}\tilde{\theta}^T] =P_{\tilde{\theta}}$ +\end{defin} +\paragraph{Concept} $H$ et $C$ tel que $P_{\tilde{\theta}}$ minimal. +\[ +(1) \quad \derivp[P_{\tilde{\theta}}]{H} = 0 \quad\text{ et }\quad (2)\quad \derivp[P_{\tilde{\theta}}]{C} = 0 +\] +\begin{enumerate}[label=\arabic*)] +\item + + \begin{prop} + \[ + \derivp[P_{\tilde{\theta}}]{H} =2E[HY+C-\theta] = 2E[\tilde{\theta}] = 0 + \] + L'ELMQ est un estimateur non biaisé. + \end{prop} + et donc : + \begin{align*} + C &= -Hm_Y+m_\theta\\ + \hat{\theta} &= H(Y-m_y)+m_\theta \\ + \tilde{\theta} &= H(Y-m_y) - (\theta-m_\theta) + \end{align*} + +\item + \begin{prop} + \[ + \derivp[P_{\tilde{\theta}}]{C} =2E[(HY+C-\theta)Y^T] = 2E[\tilde{\theta}Y^T] = 0 + \] + $\tilde{\theta} \perp Y $ quand la puissance est minimale, $\tilde{\theta}$ et $Y$ sont décorrélées, on a extrait toute l'information commune. + \end{prop} + + De plus : + \begin{align*} + E[\tilde{\theta}Y^T]& =E[\tilde{\theta}(Y-m_Y)^T] \\ + &= E[(H(Y-m_Y)-\theta-m_\theta)(Y-m_y)^T]\\ + &= HC_{yy}-C_{\theta Y} = 0 \implies H = C_{\theta Y}C_{YY}^{-1} + \end{align*} + + on a donc + \[ + \boxed{\hat{\theta}=C_{\theta Y}C_{YY}^{-1}(Y-m_Y)+m_\theta} + \] + + \paragraph{Remarque} L'ELMQ nécessite des connaissances du premier et du second ordre sur $\theta$ et $Y$. + + \begin{prop} + \[ + C_{\tilde{\theta}\tilde{\theta}} = C_{\theta\theta}-C_{\theta Y}C_{YY}^{-1}C_{Y\theta} + \] + La corrélation entre $\theta$ et $Y$ permet de diminuer l'ELMQ. + \end{prop} +\end{enumerate} + + +\subsection{Estimateur Bayésiens} +\subsubsection{Fonction coût/pénalité} +\begin{defin} + On appelle fonction de coût ou fonction de pénalité une fonction qui mesure l'erreur entrainée par la prise de la valeur $\hat{\theta}$ pour $\theta$. + \[ + C(\hat{\theta},\theta) \geq 0 \quad \text{ ou encore }\quad C(\tilde{\theta}) \ge 0 + \] + On prendra le plus souvent une \og bonne \fg{} fonction (continue, paire , croissante ...) +\end{defin} + +\paragraph{Exemple de coût} +\begin{itemize} +\item quadratique +\item Valeur absolue +\item uniforme +\end{itemize} +% Insert graphique + +\begin{defin} + On appelle estimateur bayésiens l'estimateur qui minimise le coût moyen : + \begin{align*} + E_{\theta,Y}[C(\hat{\theta},\theta)] &= \int_{\R^{m+n}}C(\hat{\theta},\theta)f_{\theta Y}(\theta,y)d\theta dy\\ &=\int_{\R^m}\left(\underbrace{\int_{\R^n}C(\hat{\theta},\theta)f_{\theta|Y=y}(\theta)d\theta}_{E_{\theta|Y}[C(\hat{\theta},\theta)]}\right) f_{Y}(y)dy + \end{align*} +On minimise donc $ E_{\theta|Y}[C(\hat{\theta},\theta)]$ à coût conditionnel donné + \[ + \hat{\theta}_{B} = \arg\min_{\hat{\theta}}E_{\theta|Y}[C(\hat{\theta},\theta)] + \] +\end{defin} + +\subsubsection{Estimateur du maximum a posteriori (MAP)} + +On considère un cout uniforme. +\begin{defin} + En prenant: + \begin{align*} + E_{\theta|Y}[C(\hat{\theta},\theta)] &= \int_{\R^m}(1-\Pi_{\Delta}(\tilde{\theta}))f_{\theta|Y=y}(\theta)d\theta + &= 1 - \int_{\hat{\theta}-\Delta/2}^{{\hat{\theta}+\Delta/2}}f_{\theta|Y=y}(\theta)d\theta + &\simeq 1- \Delta^nf_{\theta|Y=y}(\hat{\theta}) + \end{align*} + Soit \[ + \hat{\theta}_{MAP}=\arg\max_{\theta} f_{\theta|Y=y}(\theta) + \] +\end{defin} + +\paragraph{Lien MAP-MV} + +on a $f_{\theta|Y=y}(\theta) f_{Y}(y) = f_{\theta Y}(\theta,y)$. Avec $f_\theta(\theta) = C^{ste}$ quand $f_{\theta Y}(\theta,y)$ à une valeur significative (ie $C_{\theta\theta}$ grand / $\sigma_\theta$ grand ) alors : + \[ +\arg\max f_{\theta|Y=y}(\theta) \simeq \arg\max f_{Y|\Theta=\theta}(y) + \] + + On considère alors que $\theta$ est un paramètre aléatoire mais très mal connu. (ddp uniforme sur un interval tres grand, peu d'infos sur $\theta$). + +\emph{cf. TD \og file d'attente\fg{}} + +\paragraph{Exemple et Application} + +On considère $\theta$ scalaire aléatoire avec: $Y_i = \theta +B_i$ Avec : + +$ +\begin{cases} +B \hookrightarrow \mathcal{N}(0,C_{BB})\\ +\Theta \hookrightarrow\mathcal{N}(m_\theta,\sigma_\theta^2) \\ +B \perp \Theta +\end{cases}$ + +\subparagraph{Rappel} MC=MV avec: +$\begin{cases} +m_B=0\\ +\hat{\theta}_{MV} =\hat{\theta}_{MC} = \frac{\sum_{i=1}^{m}Y_i}{m}\\ + +E[\hat{\theta}_{MV}] = E[\theta]=m_\theta \text{ et } \sigma_{\tilde{\theta}_{MV}}=\frac{\sigma_B}{m}\\ +\end{cases}$ + +On a donc: +\[ + f_{Y|\theta}(y)=f_{B}(Y-A\theta) = \prod_{i=1}^{m}f_{B_i}(Y_i-\theta) = C_1 \exp\left(-\frac{1}{2}\frac{\sum(Y_i-\theta)^2}{\sigma_B^2}\right) +\] +Or +\[ + f_{\theta|Y=y}(\theta) = \frac{f_{Y|\theta}(y)f_\theta(\theta)}{f_Y(y)} = C_2 \exp\left(-\frac{1}{2}\underbrace{\left[\frac{\sum(Y_i-\theta)^2}{\sigma_B^2}+\frac{(\theta-m_\theta)^2}{\sigma_\theta^2}\right]}_{J_{MAP}}\right) +\] +Le critère est ici une forme quadratique, donc : + +\[ + \hat{\theta}_{MAP} = \arg\max f_{\theta|Y=y}(\theta) = \arg\min J_{MAP}(\theta,Y) +\] +Alors on a la CNS : +\[ +\deriv[J_{MAP}]{\theta} = 0 = 2 \left[ -\sum_{i=1}^{m}\frac{Y_i-\theta}{\sigma_b^2}+\frac{(\theta-m_\theta)^2}{\sigma_\theta^2}\right] +\] +Soit une expression barycentrique : +\[ + \hat{\theta}_{MAP} = \frac{\frac{m}{\sigma_B^2}\sum_{}^{}\frac{Y_i}{m}+\frac{m_\theta}{\sigma_\theta^2}}{\frac{m}{\sigma_B^2}+\frac{1}{\sigma_\theta^2}} +\] +Donc : +\begin{prop} +\[ +E[\hat{\theta}_{MAP}] = m_\theta +\] +L'estimateur est non biaisé. De plus : +\[ + \sigma_{\tilde{\theta}_{MAP}}^2= \frac{1}{\frac{1}{\sigma_{MV}}+\frac{1}{\sigma_\theta^2}} < + \begin{cases} + \sigma_\theta^2 \\ + \sigma_{MV}^2 + \end{cases} +\] +On a fait mieux en prenant en compte toutes les sources d'informations. +\end{prop} + +\paragraph{Remarque} +\begin{itemize} +\item Si $\sigma_\theta>>\sigma_{MV}$ alors $\hat{\theta}_{MAP}\simeq \hat{\theta}_{MV}$ (ce qui arrive pour $\sigma_B$ ou $m$ grand) +\item Si $\sigma_\theta<<\sigma_{MV}$ et $\hat{\theta}_{MAP} \simeq m_\theta$ (l'obersavation apporte peu d'info) +\end{itemize} + +\subsubsection{Estimateur en moyenne quadratique (EQM)} + +\begin{defin} + On le cout moyen de l'EQM: + \[ + C(\hat{\theta},\theta) = (\hat{\theta}-\theta)^T M (\hat{\theta}-\theta) + \] + Avec $M>0$. + On cherche a minimiser le cout moyen mais sans contrainte de linéarité avec une matrice de pondération qui peux prendre en compte des facteurs d'echelles ou des unités différentes. +\end{defin} + + +\paragraph{Etude de l'estimateur} On veut minimiser $E_{\Theta|Y}[C(\hat{\theta},\theta)]$ +\begin{align*} + \nabla_{\hat{\theta}}E_{\Theta|Y}[C(\hat{\theta},\theta)] &= 0 \\ + E_{\theta|Y}[2M(\hat{\theta}-\theta)] &= 0 \\ + 2M E_{\theta|Y}[\underbracket{\hat{\theta}}_{h(y)}]-E_{\theta|Y}[\theta]&=0 \\ + 2M(\hat{\theta}-E_{\theta|Y}[\theta]) &= 0 \\ + \Aboxed{ \hat{\theta}_{MQ} &=E_{\theta|Y}[\theta]} \\ + &= \int_{\R^n}\theta f_{\theta|y}(\theta)d\theta = h(Y=y) +\end{align*} + +Par conséquent: $E[\hat{\theta}_{MQ}]=E[\theta]$. on a un estimateur non biaisé. +\paragraph{Remarque} +Si $f_{\theta|Y}$ possède un axe de symétrie (ex: gaussienne) : + + +FIGURE . ($\hat{\theta}_{MQ}=\hat{\theta}_{MAP}$ dans le cas gaussien. Différent avec deux bosses.) + + +Dans le cas général la contrainte de linéarité pour l'ELMQ conduit à une valeur plus grande qu'avec l'EQM. Dans le cas gaussien: $\hat{\theta}_{ELMQ}=\hat{\theta}_{MQ}$, mais $\hat{\theta}_{MQ}$ nécessite plus de connaissance (ddp). + +\subsubsection{Estimateur en valeur absolu} + +\begin{defin} + on s'interesse au cas $n=1$ (un paramètre) + On choisit le cout moyen : + \[ + C(\hat{\theta},\theta) = |\hat{\theta}-\theta| + \] + Alors : + \[ + E_{\theta|Y}[C(\hat{\theta},\theta)] = \int_{-\infty}^{\hat{\theta}}(\hat{\theta}-\theta)f_{\theta|Y=y}(\theta)d\theta-\int_{\hat{\theta}}^{+\infty}(\hat{\theta}-\theta)f_{\theta|Y=y}(\theta)d\theta + \] +\end{defin} + +Donc : +\begin{align*} + 0 =& \nabla_{\hat{\theta}}E_{\theta|Y}[C(\hat{\theta},\theta)] \\ + =& \dots \\ + =&\int_{-\infty}^{\hat{\theta}}f_{\theta|Y=y}(\theta)d\theta-\int_{\hat{\theta}}^{+\infty}f_{\theta|Y=y}(\theta)d\theta +\end{align*} + +\begin{prop} + L'estimée est alors $\hat{\theta}_{VA}$ tel que : + \[ + \int_{-\infty}^{\hat{\theta}}f_{\theta|Y=y}(\theta)d\theta = \int_{\hat{\theta}}^{+\infty}f_{\theta|Y=y}(\theta)d\theta + \] + On parle de médiane a posteriori. Le résultat se généralise pour tout $n$. +\end{prop} + +\paragraph{Remarque} Dans le cas où $f_{\theta|Y=y}(\theta)$ possède un axe de symétrie (ex gaussienne) on a : +\[ + \hat{\theta}_{VA} =\hat{\theta}_{MV} \equals^{\stackrel{\max}{\downarrow}} \hat{\theta}_{MAP} +\] + +\paragraph{Exemple} Localisation d'un véhicule / Ellipsoïde de confiance (cf poly). +\section{Conclusion} +\begin{itemize} +\item L'estimateur statistique dépend des connaissances a priori, de la complexité des calculs et de la robustesse attendue. +\item Dans certains cas particuliers/ limites on retrouve des estimateurs intuitifs /empirique. +\item La loi normale joue un rôle important (hypothèses qui se justifie par la loi des grands nombres): les calculs sont simplifiés et conduisent au même résultat. + +\end{itemize} + + + +\end{document} diff --git a/451-Signal_Image/Cours/filtrage.pdf b/451-Signal_Image/Cours/filtrage.pdf new file mode 100644 index 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j2Ip`Y{`*68aMZJRbp4r$fTU-nXJ==JBqkP?5rO<)Kqrsm literal 0 HcmV?d00001 diff --git a/451-Signal_Image/Cours/filtrage.tex b/451-Signal_Image/Cours/filtrage.tex new file mode 100644 index 0000000..692707c --- /dev/null +++ b/451-Signal_Image/Cours/filtrage.tex @@ -0,0 +1,166 @@ +\documentclass[main.tex]{subfiles} + +\begin{document} + +\section{Formule des moments et des interférences} +On considère les filtres: +{\huge +\begin{center} + \begin{tikzpicture} + \node (e) at (0,0) {$e$}; + \node[rectangle,draw] (f) at (2,0) {$\mathcal{FL}$}; + \node (s) at (4,0) {$s$}; + \draw[->] (e) -- (f) -- (s); + \end{tikzpicture} +\end{center}} + +On s'interesse aux filtre linéaires: + +\begin{defin} + \begin{itemize} + \item Un fltre linéaire conservent la linéarité des systèmes auxquels il est appliqué. + \item Il est temps-invariant. + \item et stationnaire. + \end{itemize} + On peux caractériser un filtre linéaire par: + \begin{itemize} + \item sa réponse impulsionnelle $h$ + \item sa réponse fréquentielle $H= TF[h]$ + \item sa fonction de transfert $H_{II}$. + \end{itemize} +\end{defin} + +\begin{prop}[Moyenne] + \[ +m_s = H(0) m_e + \] +\end{prop} + +Pour deux filtres on a : +\begin{center} + \begin{tikzpicture} + \node (e) at (0,0) {$e_1$}; + \node[rectangle,draw] (f) at (2,0) {$\mathcal{H}_1$}; + \node (s) at (4,0) {$s_1$}; + \draw[->] (e) -- (f) -- (s); + \end{tikzpicture}\\[1.5em] + \begin{tikzpicture} + \node (e) at (0,0) {$e_2$}; + \node[rectangle,draw] (f) at (2,0) {$\mathcal{H}_2$}; + \node (s) at (4,0) {$s_2$}; + \draw[->] (e) -- (f) -- (s); + \end{tikzpicture} +\end{center} + +\begin{prop}[Formule des interférences] + + \[ +\Gamma_{s_1,s_2}(f)=H_1(f)\cdot H_2(f)^*\cdot \Gamma_{e_1,e_2}(f) + \] +\end{prop} +\section{Application} +\subsection{Blanchiement d'un signal} + +Pour générer un bruit blanc $s(t)$ on veux : +\[ + \Gamma_0 = |H(f)|^2\Gamma_{ee}(f)\implies |H(f)|^2 = \frac{\Gamma_0}{\Gamma_{ee}(f)} +\] + + +\subsection{Identification d'un filtre linéaire} + +On applique en entrée un bruit blanc tel que $\Gamma_{ee}(f)=\Gamma_0$. +Alors: +\[ + \Gamma_{se}=H(f)\Gamma_e(f) \implies H(f)=\frac{\Gamma_{se}(f)}{\Gamma_0} \propto \text{intercorrélation entrée sortie} +\] + + +\subsection{Signaux ARMA} + +On peux utilise un Filtre Linéaire (FL) pour définir un Signal Aléatoire. +(SA). +Le SA sera la sortie d'un filtre dynamique (Fonction de transfert rationnel ,stable ,causal) excité par un bruit blanc. + + +\subsubsection{AR : autoregressif} + + +\[ + \boxed{ + H_{II}(z) = \frac{1}{D(z)}=\frac{1}{1-\sum_{i=1}^qa_iz^{-i}} + } +\] +Alors on aura en sortie du filtre: +\[ + s_k= e_k + \sum_{i=1}^qa_is_{k-i} +\] + +on parle aussi de filtre \og tout pôle\fg{} + +\subsubsection{MA : Moyenne ajustée} + +\[ + \boxed{ + H_{II}(z) = N(z)= 1+\sum_{i=1}^qb_iz^{-i} + } +\] + +Alors on aura en sortie du filtre: +\[ + s_k= e_k + \sum_{i=1}^qb_ie_{k-i} +\] + + +\subsubsection{ARMA} + +\[ +H_{II}(z) = \frac{N(z)}{D(z)} +\] + +On connait alors $\Gamma_{ss}$ et le modèle AR. (Équation de Yule WAlker, cf TP2) +\subsection{Signaux AR : illustration} + + +pour une entrée en bruit blanc , les poles proches du cercle unités sont dominant +(approche géométrique , joli dessin) + + + +\subsection{Filtre Adapté (FA)} + +\paragraph{Contexte} Problème de transmission numérique (tout ou rien) d'un signal déterministe, connu avec bruit additif. +\paragraph{Objectif} déterminer le meilleur traitement linéaire pour décider de la présence ou non d'un signal. + +Exemple en TD +\paragraph{Méthode} :Maximiser le RSB à l'instant de décision : avec $|s_{n_0}^f|^2$ puissance instantanée à l'instant de décision. +\[ + \boxed{ + \frac{|s_{n_0}^f|^2}{E[|b_n^f|^2]} + } +\] +\begin{prop}[Application au bruit blanc] + \begin{align*} + E[|b_n^f|^2] &=\int_{-\frac{1}{2}}^{\frac{1}{2}} |H(f)|^2\Gamma_{bb}(f)df = \Gamma_0\int_{-\frac{1}{2}}^{\frac{1}{2}} |H(f)|^2df \\ +|S_{n_0}^f|^2 &= \left| \int_{-\frac{1}{2}}^{\frac{1}{2}} |H(f)|^2 S(f) e^{j2\pi n_0f}df\right| \leq \int_{-\frac{1}{2}}^{\frac{1}{2}} |H(f)|^2df \int_{-\frac{1}{2}}^{\frac{1}{2}} |S(f)|^2df + \end{align*} + + On a égalité si $H(f)\propto S^{*}(f)e^{-j2\pi n_0f} \iff h_n \propto s_{n_0-n}^f$ + +LA RI du filtre est donc + \begin{itemize} + \item un retour temporel + \item translaté autour de l'instant de décision (attention a la causalité) + \item conjugué. + \end{itemize} +\end{prop} + + +\paragraph{Remarque} +\begin{itemize} +\item Le FA peut être non causal, la RI est alors tronqué et le filtre + sous-optimal. +\item Le FA est un corrélateur (d'énergie), l'objectif n'est pas de restituer le signal utile mais d'avoir le meilleur RSB à l'instant de décision. +\end{itemize} + +\end{document} diff --git a/451-Signal_Image/Cours/main.pdf b/451-Signal_Image/Cours/main.pdf new file mode 100644 index 0000000000000000000000000000000000000000..1b2c8cfb97a70ffcf9f1405d0dab7bdb3dfe2768 GIT binary patch literal 150134 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