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Woodpecker CI 2026-02-17 10:14:55 +00:00
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@ -224,14 +224,14 @@ window.Quarto = {
<div>
<div class="quarto-title-meta-heading">Date de publication</div>
<div class="quarto-title-meta-contents">
<p class="date">10 février 2026</p>
<p class="date">17 février 2026</p>
</div>
</div>
<div>
<div class="quarto-title-meta-heading">Modifié</div>
<div class="quarto-title-meta-contents">
<p class="date-modified">10 février 2026</p>
<p class="date-modified">17 février 2026</p>
</div>
</div>
@ -263,7 +263,7 @@ Agenda
<h2 class="anchored" data-anchor-id="journaux">Journaux</h2>
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@ -297,7 +297,7 @@ Louis Lacoste
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@ -331,7 +331,7 @@ Louis Lacoste
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<p class="card-img-top"><img src="suivi/2025-51/figs/tendance_temps.png" class="thumbnail-image card-img"/></p>
@ -365,7 +365,7 @@ Louis Lacoste
</a>
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@ -399,7 +399,7 @@ Louis Lacoste
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<div class="listing-item-img-placeholder card-img-top" >&nbsp;</div>
@ -433,7 +433,7 @@ Louis Lacoste
</a>
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<div class="listing-item-img-placeholder card-img-top" >&nbsp;</div>
@ -467,7 +467,7 @@ Louis Lacoste
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@ -501,7 +501,7 @@ Louis Lacoste
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<div class="listing-item-img-placeholder card-img-top" >&nbsp;</div>
@ -535,7 +535,7 @@ Louis Lacoste
</a>
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<div class="listing-item-img-placeholder card-img-top" >&nbsp;</div>
@ -569,7 +569,7 @@ Louis Lacoste
</a>
</div>
</div>
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<p class="card-img-top"><img src="suivi/2025-33/figs/auc-model.png" class="thumbnail-image card-img"/></p>
@ -604,7 +604,7 @@ Louis Lacoste
</a>
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<div class="listing-item-img-placeholder card-img-top" >&nbsp;</div>
@ -639,7 +639,7 @@ Louis Lacoste
</a>
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@ -674,7 +674,7 @@ Louis Lacoste
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<div class="listing-item-img-placeholder card-img-top" >&nbsp;</div>
@ -709,7 +709,7 @@ Louis Lacoste
</a>
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<div class="quarto-post image-right" data-index="13" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1750377600000" data-listing-file-modified-sort="1770736624946" data-listing-date-modified-sort="1770736624000" data-listing-reading-time-sort="4" data-listing-word-count-sort="691">
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@ -744,7 +744,7 @@ Louis Lacoste
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<div class="thumbnail"><a href="./suivi/2025-24/2025-24.html" class="no-external">
<p class="card-img-top"><img src="suivi/2025-24/figs/ari-clustering-desc&asc9.png" alt="9 réseaux - ARI pour le clustering avec modèles iid, procédure descendante et descendante&ascendante" class="thumbnail-image card-img"/></p>
@ -778,7 +778,7 @@ Louis Lacoste
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@ -813,7 +813,7 @@ Louis Lacoste
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@ -847,7 +847,7 @@ Louis Lacoste
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@ -881,7 +881,7 @@ Louis Lacoste
</a>
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</div>
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@ -913,7 +913,7 @@ Louis Lacoste
</a>
</div>
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<p class="card-img-top"><img src="suivi/2025-18/figs/density-subdore.png" class="thumbnail-image card-img"/></p>
@ -945,7 +945,7 @@ Louis Lacoste
</a>
</div>
</div>
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<p class="card-img-top"><img src="suivi/2025-17/figs/density-subdore.png" class="thumbnail-image card-img"/></p>
@ -975,7 +975,7 @@ Louis Lacoste
</a>
</div>
</div>
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@ -1005,7 +1005,7 @@ Louis Lacoste
</a>
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@ -355,21 +355,16 @@ V \Gamma &amp; \approx \log((\pmb{\pi}^i)_{i=1,\dots,n_1}) = \log(\pmb{\Pi})
</section>
<section id="note-sur-lidentifiabilité-à-partir-jbl-et-réunion-ja-pb-sd" class="level4">
<h4 class="anchored" data-anchor-id="note-sur-lidentifiabilité-à-partir-jbl-et-réunion-ja-pb-sd">Note sur l’identifiabilité (à partir JBL et réunion JA, PB, SD)</h4>
<p>Soient <span class="math inline">B,B^{\prime}</span> avec <span class="math inline">B_{\bullet,R} = B^{\prime}_{\bullet,R} = \vec{0}_{p+1}</span> et <span class="math inline">X</span> de rang plein tel que <span class="math inline">X^{\top}X</span> soit inversible. Soit <span class="math inline">j=1,\dots,n_2</span> alors <span class="math display">\begin{align*}
&amp;\sigma(XB)_{j} = \sigma(XB^{\prime})_{j}\\
&amp;\implies \exists c \in \mathbb{R}, X_{j,\bullet}B = X_{j,\bullet}B^{\prime} + c \\
&amp;\implies \exists c \in \mathbb{R}, \begin{pmatrix} X_{j,\bullet} \beta_1 &amp; \dots &amp; X_{j,\bullet} \beta_{R-1} &amp; \vec{0}_{p+1} \end{pmatrix} = X_{j,\bullet}B^{\prime} + c
\end{align*}</span></p>
<p>Soient <span class="math inline">X : (p+1, n_2), B : (p+1, R)</span> avec <span class="math inline">X</span> de plein rang, i.e., <span class="math inline">rg(X) = p+1\implies XX^{\top}</span> est inversible.</p>
<p>On veut qu’il existe <span class="math inline">B^{\prime}</span> et <span class="math inline">B</span> avec <span class="math inline">B_{:,R} = \vec 0_p</span>, par les propriétés de la fonction softmax, <span class="math inline">\sigma(.)</span> :</p>
<p>Soient <span class="math inline">B,B^{\prime}</span> avec <span class="math inline">B_{\bullet,R} = B^{\prime}_{\bullet,R} = \vec{0}_{p+1}</span> <del>et <span class="math inline">X</span> de rang plein tel que <span class="math inline">X^{\top}X</span> soit inversible</del> pas nécessaire.</p>
<p><span class="math display">\begin{align*}
&amp; \sigma(B^{\top}X) = \sigma({B^{\prime}}^{\top}X)\\
&amp; \iff \exists C \in \mathbb{R}^{n_2}, B^{\top} X = {B^{\prime}}^{\top} X + \pmb{1}_R C^{\top}\\
&amp; \iff \exists C \in \mathbb{R}^{n_2}, B^{\top} X - \pmb{1}_R C^{\top} = {B^{\prime}}^{\top} X\\
&amp; \iff \exists C \in \mathbb{R}^{n_2}, (B^{\top} X - \pmb{1}_R C^{\top}) X^{\top} = {B^{\prime}}^{\top} X X^{\top}\\
&amp; \iff \exists C \in \mathbb{R}^{n_2}, (B^{\top} X - \pmb{1}_R C^{\top}) X^{\top}(X X^{\top})^{-1} = {B^{\prime}}^{\top}\\
&amp;\sigma(XB) = \sigma(XB^{\prime})\\
&amp;\implies \exists C = \begin{pmatrix}c_1 \\ \vdots \\ c_j \\ \vdots \\ c_{n_2}\end{pmatrix} \in \mathbb{R}^{n_2}, X B = X B^{\prime} + C \pmb{1}_{R}^{\top} \\
&amp;\implies \exists C \in \mathbb{R}^{n_2}, (X B)_{j,r} = (X B^{\prime})_{j,r} + (C \pmb{1}_{R}^{\top})_{j,r} \\
&amp;\implies \exists C \in \mathbb{R}^{n_2}, \forall r\in\{1\dots,R\}, \forall j\in\{1,\dots,n_2\}, \sum_{k=1}^{p+1} x_{j,k} \beta_{k,r} = \sum_{k=1}^{p+1} x_{j,k} \beta^{\prime}_{k,r} + c_j\\
&amp;\implies \exists C \in \mathbb{R}^{n_2}, \forall j\in\{1,\dots,n_2\}, \sum_{k=1}^{p+1} x_{j,k} \beta_{k,R} = \sum_{k=1}^{p+1} x_{j,k} \beta^{\prime}_{k,R} + c_j \\
&amp;\implies \exists C \in \mathbb{R}^{n_2}, \forall j\in\{1,\dots,n_2\}, \sum_{k=1}^{p+1} x_{j,k} \times 0 = \sum_{k=1}^{p+1} x_{j,k} \times 0 + c_j \\
&amp;\implies \exists C \in \mathbb{R}^{n_2}, \forall j\in\{1,\dots,n_2\}, 0 = 0 + c_j \implies c_j = 0 \\
&amp;\implies C = \begin{pmatrix} 0 \\ \vdots \\ 0 \end{pmatrix} \text{and thus}, B = B^{\prime} \\
\end{align*}</span></p>
</section>