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<h2 id="author-laurent-fainsintitle-tav-tp1-lfdescription-rapport-de-traitement-des-données-audio-visuelles-travail-pratique-1-laurent-fainsin">author: Laurent Fainsin
title: TAV, TP1, LF
description: Rapport de Traitement des données audio-visuelles, Travail Pratique 1, Laurent Fainsin</h2>
<h1 id="fonctions-auxilliaires">Fonctions auxilliaires</h1>
<pre><code class="language-matlab"><div><span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">y</span> = <span class="hljs-title">bezier</span><span class="hljs-params">(beta_0,beta,beta_d,x)</span></span>
d = <span class="hljs-built_in">length</span>(<span class="hljs-built_in">beta</span>)+<span class="hljs-number">1</span>;
y = beta_0 * (<span class="hljs-number">1</span>-x).^d + beta_d * x.^d;
<span class="hljs-keyword">for</span> <span class="hljs-built_in">i</span> = <span class="hljs-number">1</span>:d<span class="hljs-number">-1</span>
y = y + <span class="hljs-built_in">beta</span>(<span class="hljs-built_in">i</span>) * <span class="hljs-built_in">nchoosek</span>(d,<span class="hljs-built_in">i</span>) * x.^<span class="hljs-built_in">i</span> .* (<span class="hljs-number">1</span>-x).^(d-<span class="hljs-built_in">i</span>);
<span class="hljs-keyword">end</span>
<span class="hljs-keyword">end</span>
</div></code></pre>
<pre><code class="language-matlab"><div><span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">y</span> = <span class="hljs-title">bezier_bruitee</span><span class="hljs-params">(beta_0,beta,beta_d,x,sigma)</span></span>
y = bezier(beta_0,<span class="hljs-built_in">beta</span>,beta_d,x)+sigma*<span class="hljs-built_in">randn</span>(<span class="hljs-built_in">size</span>(x));
<span class="hljs-keyword">end</span>
</div></code></pre>
<h1 id="exercice-1">Exercice 1</h1>
<pre><code class="language-matlab"><div><span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">estimation</span> = <span class="hljs-title">moindres_carres</span><span class="hljs-params">(D_app, beta_0, beta_d, d)</span></span>
X = D_app(<span class="hljs-number">1</span>,:)&#x27;;
Y = D_app(<span class="hljs-number">2</span>,:)&#x27;;
B = Y - beta_0*(<span class="hljs-number">1</span>-X).^d - beta_d*(X.^d);
A = <span class="hljs-built_in">zeros</span>(<span class="hljs-built_in">length</span>(X), d<span class="hljs-number">-1</span>);
<span class="hljs-keyword">for</span> <span class="hljs-built_in">i</span>=<span class="hljs-number">1</span>:(d<span class="hljs-number">-1</span>)
A(:,<span class="hljs-built_in">i</span>) = <span class="hljs-built_in">nchoosek</span>(d, <span class="hljs-built_in">i</span>) .* X.^<span class="hljs-built_in">i</span> .* (<span class="hljs-number">1</span>-X).^(d-<span class="hljs-built_in">i</span>);
<span class="hljs-keyword">end</span>
estimation = A \ B;
<span class="hljs-keyword">end</span>
</div></code></pre>
<pre><code class="language-matlab"><div>donnees_apprentissage;
close all;
<span class="hljs-comment">% DegrŽé de la courbe de BŽezier </span>
d = <span class="hljs-number">8</span>;
<span class="hljs-comment">% Estimation des paramètres de la courbe de BŽezier (sauf beta_0 et beta_d) :</span>
beta_estime = moindres_carres(D_app,beta_0,beta_d,d);
<span class="hljs-comment">% TracŽ de la courbe de BŽezier estimŽe, de degréŽ d (trait rouge) :</span>
y_estime = bezier(beta_0,beta_estime,beta_d,x);
<span class="hljs-built_in">plot</span>(x,y_estime,<span class="hljs-string">&#x27;-r&#x27;</span>,<span class="hljs-string">&#x27;MarkerSize&#x27;</span>,<span class="hljs-number">10</span>,<span class="hljs-string">&#x27;LineWidth&#x27;</span>,<span class="hljs-number">3</span>);
lg = <span class="hljs-built_in">legend</span>(<span class="hljs-string">&#x27; Modele exact&#x27;</span>,<span class="hljs-string">&#x27; Donnees d&#x27;&#x27;apprentissage&#x27;</span>,[<span class="hljs-string">&#x27; Modele estime ($d=&#x27;</span> num2str(d) <span class="hljs-string">&#x27;$)&#x27;</span>],<span class="hljs-string">&#x27;Location&#x27;</span>,<span class="hljs-string">&#x27;Best&#x27;</span>);
set(lg,<span class="hljs-string">&#x27;Interpreter&#x27;</span>,<span class="hljs-string">&#x27;Latex&#x27;</span>);
</div></code></pre>
<img src="file:////home/lfainsin/2A/TDAV/TP1/rapport/exercice_1.svg">
<h1 id="exercice-2">Exercice 2</h1>
<pre><code class="language-matlab"><div><span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">erreur</span> = <span class="hljs-title">erreur_apprentissage</span><span class="hljs-params">(D_app,beta_0,beta_d,d)</span></span>
X = D_app(<span class="hljs-number">1</span>,:)&#x27;;
Y = D_app(<span class="hljs-number">2</span>,:)&#x27;;
<span class="hljs-built_in">beta</span> = moindres_carres(D_app, beta_0, beta_d, d);
estimation = bezier(beta_0, <span class="hljs-built_in">beta</span>, beta_d, X);
erreur = <span class="hljs-built_in">mean</span>((estimation - Y).^<span class="hljs-number">2</span>);
<span class="hljs-keyword">end</span>
</div></code></pre>
<pre><code class="language-matlab"><div>donnees_apprentissage;
close all;
<span class="hljs-comment">% Calcul de l&#x27;erreur d&#x27;apprentissage en fonction de d :</span>
liste_d = <span class="hljs-number">2</span>:<span class="hljs-built_in">length</span>(D_app);
liste_erreurs_apprentissage = [];
<span class="hljs-keyword">for</span> d = liste_d
erreur = erreur_apprentissage(D_app,beta_0,beta_d,d);
liste_erreurs_apprentissage = [liste_erreurs_apprentissage erreur];
<span class="hljs-keyword">end</span>
<span class="hljs-comment">% TracŽ de l&#x27;erreur d&#x27;apprentissage en fonction de d :</span>
<span class="hljs-built_in">figure</span>(<span class="hljs-string">&#x27;Name&#x27;</span>,<span class="hljs-string">&#x27;Erreur d&#x27;&#x27;apprentissage&#x27;</span>,<span class="hljs-string">&#x27;Position&#x27;</span>,[<span class="hljs-number">0.4</span>*L,<span class="hljs-number">0.05</span>*H,<span class="hljs-number">0.6</span>*L,<span class="hljs-number">0.7</span>*H]);
<span class="hljs-built_in">plot</span>(liste_d,liste_erreurs_apprentissage,<span class="hljs-string">&#x27;sb-&#x27;</span>,<span class="hljs-string">&#x27;LineWidth&#x27;</span>,<span class="hljs-number">2</span>);
set(gca,<span class="hljs-string">&#x27;FontSize&#x27;</span>,<span class="hljs-number">20</span>);
xlabel(<span class="hljs-string">&#x27;$d$&#x27;</span>,<span class="hljs-string">&#x27;Interpreter&#x27;</span>,<span class="hljs-string">&#x27;Latex&#x27;</span>,<span class="hljs-string">&#x27;FontSize&#x27;</span>,<span class="hljs-number">30</span>);
ylabel(<span class="hljs-string">&#x27;Erreur&#x27;</span>,<span class="hljs-string">&#x27;FontSize&#x27;</span>,<span class="hljs-number">30</span>);
<span class="hljs-built_in">legend</span>(<span class="hljs-string">&#x27; Erreur d&#x27;&#x27;apprentissage&#x27;</span>,<span class="hljs-string">&#x27;Location&#x27;</span>,<span class="hljs-string">&#x27;Best&#x27;</span>);
</div></code></pre>
<img src="file:////home/lfainsin/2A/TDAV/TP1/rapport/exercice_2.svg">
<h1 id="données-de-test">Données de test</h1>
<h2 id="exercice-1-1">Exercice 1</h2>
<pre><code class="language-matlab"><div>donnees_test;
<span class="hljs-comment">% % DegrŽ de la courbe de BŽzier utilisŽe comme modle (testez plusieurs valeurs de d entre 2 et 20) :</span>
degres = <span class="hljs-number">2</span>:<span class="hljs-number">5</span>:<span class="hljs-number">20</span>;
<span class="hljs-keyword">for</span> d=degres
beta_estime = moindres_carres(D_test,beta_0,beta_d,d);
y_estime = bezier(beta_0,beta_estime,beta_d,x);
<span class="hljs-built_in">plot</span>(x,y_estime,<span class="hljs-string">&#x27;MarkerSize&#x27;</span>,<span class="hljs-number">10</span>,<span class="hljs-string">&#x27;LineWidth&#x27;</span>,<span class="hljs-number">3</span>, <span class="hljs-string">&#x27;DisplayName&#x27;</span>, [<span class="hljs-string">&#x27;d=&#x27;</span>,num2str(d)]);
<span class="hljs-keyword">end</span>
</div></code></pre>
<img src="file:////home/lfainsin/2A/TDAV/TP1/rapport/exercice_1_test.svg">
<h2 id="exercice-2-1">Exercice 2</h2>
<pre><code class="language-matlab"><div>donnees_test;
close all;
liste_d = <span class="hljs-number">2</span>:<span class="hljs-number">10</span>:<span class="hljs-built_in">length</span>(D_test);
liste_erreurs_apprentissage = [];
<span class="hljs-keyword">for</span> d = liste_d
erreur = erreur_apprentissage(D_test,beta_0,beta_d,d);
liste_erreurs_apprentissage = [liste_erreurs_apprentissage erreur];
<span class="hljs-keyword">end</span>
<span class="hljs-comment">% TracŽ de l&#x27;erreur d&#x27;apprentissage en fonction de d :</span>
<span class="hljs-built_in">figure</span>(<span class="hljs-string">&#x27;Name&#x27;</span>,<span class="hljs-string">&#x27;Erreur d&#x27;&#x27;apprentissage&#x27;</span>,<span class="hljs-string">&#x27;Position&#x27;</span>,[<span class="hljs-number">0.4</span>*L,<span class="hljs-number">0.05</span>*H,<span class="hljs-number">0.6</span>*L,<span class="hljs-number">0.7</span>*H]);
<span class="hljs-built_in">plot</span>(liste_d,liste_erreurs_apprentissage,<span class="hljs-string">&#x27;sb-&#x27;</span>,<span class="hljs-string">&#x27;LineWidth&#x27;</span>,<span class="hljs-number">2</span>);
set(gca,<span class="hljs-string">&#x27;FontSize&#x27;</span>,<span class="hljs-number">20</span>);
xlabel(<span class="hljs-string">&#x27;$d$&#x27;</span>,<span class="hljs-string">&#x27;Interpreter&#x27;</span>,<span class="hljs-string">&#x27;Latex&#x27;</span>,<span class="hljs-string">&#x27;FontSize&#x27;</span>,<span class="hljs-number">30</span>);
ylabel(<span class="hljs-string">&#x27;Erreur&#x27;</span>,<span class="hljs-string">&#x27;FontSize&#x27;</span>,<span class="hljs-number">30</span>);
<span class="hljs-built_in">legend</span>(<span class="hljs-string">&#x27; Erreur d&#x27;&#x27;apprentissage&#x27;</span>,<span class="hljs-string">&#x27;Location&#x27;</span>,<span class="hljs-string">&#x27;Best&#x27;</span>);
</div></code></pre>
<img src="file:////home/lfainsin/2A/TDAV/TP1/rapport/exercice_2_test.svg">
<h1 id="exercice-3">Exercice 3</h1>
<pre><code class="language-matlab"><div><span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">erreur</span> = <span class="hljs-title">erreur_generalisation</span><span class="hljs-params">(D_test,D_app,beta_0,beta_d,d)</span></span>
X = D_test(<span class="hljs-number">1</span>,:)&#x27;;
Y = D_test(<span class="hljs-number">2</span>,:)&#x27;;
<span class="hljs-built_in">beta</span> = moindres_carres(D_app, beta_0, beta_d, d);
estimation = bezier(beta_0, <span class="hljs-built_in">beta</span>, beta_d, X);
erreur = <span class="hljs-built_in">mean</span>((estimation - Y).^<span class="hljs-number">2</span>);
<span class="hljs-keyword">end</span>
</div></code></pre>
<pre><code class="language-matlab"><div><span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-params">[d_estime, sigma_estime]</span> = <span class="hljs-title">estimation_d_sigma</span><span class="hljs-params">(liste_d, liste_erreurs_generalisation)</span></span>
[~, index] = <span class="hljs-built_in">min</span>(liste_erreurs_generalisation);
d_estime = liste_d(index);
sigma_estime = std(liste_erreurs_generalisation);
<span class="hljs-keyword">end</span>
</div></code></pre>
<pre><code class="language-matlab"><div>donnees_test;
close all;
<span class="hljs-comment">% Calcul de l&#x27;erreur d&#x27;apprentissage (risque empirique) :</span>
liste_d = <span class="hljs-number">2</span>:<span class="hljs-number">20</span>;
liste_erreurs_apprentissage = [];
<span class="hljs-keyword">for</span> d = liste_d
erreur = erreur_apprentissage(D_app,beta_0,beta_d,d);
liste_erreurs_apprentissage = [liste_erreurs_apprentissage erreur];
<span class="hljs-keyword">end</span>
<span class="hljs-comment">% Tracé de l&#x27;erreur d&#x27;apprentissage en fonction de d :</span>
<span class="hljs-built_in">figure</span>(<span class="hljs-string">&#x27;Name&#x27;</span>,<span class="hljs-string">&#x27;Erreur d&#x27;&#x27;apprentissage et erreur de generalisation&#x27;</span>,<span class="hljs-string">&#x27;Position&#x27;</span>,[<span class="hljs-number">0.4</span>*L,<span class="hljs-number">0.05</span>*H,<span class="hljs-number">0.6</span>*L,<span class="hljs-number">0.7</span>*H]);
<span class="hljs-built_in">plot</span>(liste_d,liste_erreurs_apprentissage,<span class="hljs-string">&#x27;sb-&#x27;</span>,<span class="hljs-string">&#x27;LineWidth&#x27;</span>,<span class="hljs-number">2</span>);
set(gca,<span class="hljs-string">&#x27;FontSize&#x27;</span>,<span class="hljs-number">20</span>);
xlabel(<span class="hljs-string">&#x27;$d$&#x27;</span>,<span class="hljs-string">&#x27;Interpreter&#x27;</span>,<span class="hljs-string">&#x27;Latex&#x27;</span>,<span class="hljs-string">&#x27;FontSize&#x27;</span>,<span class="hljs-number">30</span>);
ylabel(<span class="hljs-string">&#x27;Erreur&#x27;</span>,<span class="hljs-string">&#x27;FontSize&#x27;</span>,<span class="hljs-number">30</span>);
<span class="hljs-built_in">hold</span> on;
<span class="hljs-comment">% Calcul de l&#x27;erreur de généralisation (risque espéré) :</span>
liste_erreurs_generalisation = [];
<span class="hljs-keyword">for</span> d = liste_d
erreur = erreur_generalisation(D_test,D_app,beta_0,beta_d,d);
liste_erreurs_generalisation = [liste_erreurs_generalisation erreur];
<span class="hljs-keyword">end</span>
<span class="hljs-comment">% Tracé de l&#x27;erreur de généralisation en fonction de d :</span>
<span class="hljs-built_in">plot</span>(liste_d,liste_erreurs_generalisation,<span class="hljs-string">&#x27;sg-&#x27;</span>,<span class="hljs-string">&#x27;LineWidth&#x27;</span>,<span class="hljs-number">2</span>);
<span class="hljs-built_in">legend</span>(<span class="hljs-string">&#x27; Erreur d&#x27;&#x27;apprentissage&#x27;</span>,<span class="hljs-string">&#x27; Erreur de generalisation&#x27;</span>,<span class="hljs-string">&#x27;Location&#x27;</span>,<span class="hljs-string">&#x27;Best&#x27;</span>);
<span class="hljs-comment">% Estimation du degré d et de l&#x27;écart-type sigma :</span>
[d_estime,sigma_estime] = estimation_d_sigma(liste_d,liste_erreurs_generalisation);
fprintf(<span class="hljs-string">&#x27;Estimation du degre : d = %d\n&#x27;</span>,d_estime);
fprintf(<span class="hljs-string">&#x27;Estimation de l&#x27;&#x27;ecart-type du bruit sur les donnees : %.3f\n&#x27;</span>,sigma_estime);
</div></code></pre>
<pre><code><code><div>Estimation du degre : d = 5
Estimation de l'ecart-type du bruit sur les donnees : 0.477
</div></code></code></pre>
<img src="file:////home/lfainsin/2A/TDAV/TP1/rapport/exercice_3.svg">
<h1 id="exercice-4">Exercice 4</h1>
<pre><code class="language-matlab"><div><span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">VC</span> = <span class="hljs-title">calcul_VC</span><span class="hljs-params">(D_app, beta_0, beta_d, d)</span></span>
X = D_app(<span class="hljs-number">1</span>,:)&#x27;;
Y = D_app(<span class="hljs-number">2</span>,:)&#x27;;
n = <span class="hljs-built_in">length</span>(X);
VC = <span class="hljs-number">0</span>;
<span class="hljs-keyword">for</span> <span class="hljs-built_in">j</span>=<span class="hljs-number">1</span>:n
D_app_loo = [D_app(:,<span class="hljs-number">1</span>:<span class="hljs-built_in">j</span><span class="hljs-number">-1</span>) , D_app(:,<span class="hljs-built_in">j</span>+<span class="hljs-number">1</span>:n)];
<span class="hljs-built_in">beta</span> = moindres_carres(D_app_loo, beta_0, beta_d, d);
estimation = bezier(beta_0, <span class="hljs-built_in">beta</span>, beta_d, X(<span class="hljs-built_in">j</span>));
VC = VC + (Y(<span class="hljs-built_in">j</span>) - estimation).^<span class="hljs-number">2</span>;
<span class="hljs-keyword">end</span>
VC = VC/n;
<span class="hljs-keyword">end</span>
</div></code></pre>
<pre><code class="language-matlab"><div><span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-params">[d_estime,sigma_estime]</span> = <span class="hljs-title">estimation_d_sigma_bis</span><span class="hljs-params">(liste_d,liste_VC)</span></span>
[~, index] = <span class="hljs-built_in">min</span>(liste_VC);
d_estime = liste_d(index);
sigma_estime = std(liste_VC);
<span class="hljs-keyword">end</span>
</div></code></pre>
<pre><code class="language-matlab"><div>donnees_test;
close all;
<span class="hljs-comment">% Calcul de la validation croisée Leave-one-out :</span>
liste_d = <span class="hljs-number">2</span>:<span class="hljs-number">20</span>;
liste_VC = [];
tic;
<span class="hljs-keyword">for</span> d = liste_d
VC = calcul_VC(D_app,beta_0,beta_d,d);
liste_VC = [liste_VC VC];
<span class="hljs-keyword">end</span>
toc;
<span class="hljs-comment">% Tracé de la validation croisée Leave-one-out en fonction de d :</span>
<span class="hljs-built_in">figure</span>(<span class="hljs-string">&#x27;Name&#x27;</span>,<span class="hljs-string">&#x27;Validation croisee&#x27;</span>,<span class="hljs-string">&#x27;Position&#x27;</span>,[<span class="hljs-number">0.4</span>*L,<span class="hljs-number">0.05</span>*H,<span class="hljs-number">0.6</span>*L,<span class="hljs-number">0.7</span>*H]);
<span class="hljs-built_in">plot</span>(liste_d,liste_VC,<span class="hljs-string">&#x27;sr-&#x27;</span>,<span class="hljs-string">&#x27;LineWidth&#x27;</span>,<span class="hljs-number">2</span>);
set(gca,<span class="hljs-string">&#x27;FontSize&#x27;</span>,<span class="hljs-number">20</span>);
xlabel(<span class="hljs-string">&#x27;$d$&#x27;</span>,<span class="hljs-string">&#x27;Interpreter&#x27;</span>,<span class="hljs-string">&#x27;Latex&#x27;</span>,<span class="hljs-string">&#x27;FontSize&#x27;</span>,<span class="hljs-number">30</span>);
ylabel(<span class="hljs-string">&#x27;$VC$&#x27;</span>,<span class="hljs-string">&#x27;Interpreter&#x27;</span>,<span class="hljs-string">&#x27;Latex&#x27;</span>,<span class="hljs-string">&#x27;FontSize&#x27;</span>,<span class="hljs-number">30</span>);
<span class="hljs-comment">% Estimation du degré d et de l&#x27;écart-type sigma :</span>
[d_estime,sigma_estime] = estimation_d_sigma_bis(liste_d,liste_VC);
fprintf(<span class="hljs-string">&#x27;Estimation du degre : d = %d\n&#x27;</span>,d_estime);
fprintf(<span class="hljs-string">&#x27;Estimation de l&#x27;&#x27;ecart-type du bruit sur les donnees : %.3f\n&#x27;</span>,sigma_estime);
</div></code></pre>
<pre><code><code><div>Estimation du degre : d = 5
Estimation de l'ecart-type du bruit sur les donnees : 0.563
</div></code></code></pre>
<img src="file:////home/lfainsin/2A/TDAV/TP1/rapport/exercice_4.svg">
<h1 id="exercice-5-optionnel">Exercice 5 (Optionnel)</h1>
<pre><code class="language-matlab"><div>clear;
close all;
<span class="hljs-comment">% constantes</span>
beta_0 = <span class="hljs-number">115</span>;
beta_d = <span class="hljs-number">123</span>;
<span class="hljs-built_in">beta</span> = [<span class="hljs-number">133</span>,<span class="hljs-number">96</span>,<span class="hljs-number">139</span>,<span class="hljs-number">118</span>];
n_app = <span class="hljs-number">100</span>;
pas_app = <span class="hljs-number">1</span>/(n_app<span class="hljs-number">-1</span>);
x_j = <span class="hljs-number">0</span>:pas_app:<span class="hljs-number">1</span>;
sigma = <span class="hljs-number">0.5</span>;
d = <span class="hljs-number">5</span>;
n = <span class="hljs-number">10000</span>;
beta_moyen = <span class="hljs-built_in">zeros</span>(<span class="hljs-number">1</span>, d<span class="hljs-number">-1</span>)&#x27;;
sigma_moyen = <span class="hljs-number">0</span>;
<span class="hljs-keyword">for</span> <span class="hljs-built_in">i</span>=<span class="hljs-number">1</span>:n
<span class="hljs-comment">% génération de nouveaux points d&#x27;apprentissage</span>
y_j = bezier_bruitee(beta_0,<span class="hljs-built_in">beta</span>,beta_d,x_j,sigma);
D_app = [x_j ; y_j];
beta_estime = moindres_carres(D_app,beta_0,beta_d,d);
beta_moyen = beta_moyen + beta_estime/n;
<span class="hljs-keyword">end</span>
[beta&#x27; , beta_moyen]
</div></code></pre>
<pre><code class="language-matlab"><div>&gt;&gt; exercice_5
<span class="hljs-built_in">ans</span> =
<span class="hljs-number">133.0000</span> <span class="hljs-number">133.0007</span>
<span class="hljs-number">96.0000</span> <span class="hljs-number">96.0013</span>
<span class="hljs-number">139.0000</span> <span class="hljs-number">138.9989</span>
<span class="hljs-number">118.0000</span> <span class="hljs-number">118.0024</span>
</div></code></pre>
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