soutenance : vfinale
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.woodpecker.yaml
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31
.woodpecker.yaml
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when:
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branch: main
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steps:
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- name: build rapport
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image: texlive/texlive:latest
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commands:
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- cd rapport
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- make
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- name: build presentation
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image: texlive/texlive:latest
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commands:
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- cd presentation
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- make
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- name: build soutenance
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image: texlive/texlive:latest
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commands:
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- cd soutenance
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- make
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- name: publish
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image: woodpeckerci/plugin-release
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settings:
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files:
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# Could also be "hello-world*" to match both
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- "hello-world"
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- "hello-world.exe"
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api_key:
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from_secret: ACCESS_TOKEN
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@ -1,5 +1,78 @@
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\section{Notes supplémentaires}
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\printappxnotes
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\section{VEM}
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\begin{frame}
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\frametitle{Pourquoi VE minimise KL ?}
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\begin{align*}
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\ell_c(\bY,\bZ,\bW;\theta) & = \log \Prob(\bZ, \bW|\bY;\theta) + \ell(\bY;\theta) \\
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\Leftrightarrow \ell(\bY;\theta) & = \ell_c(\bY,\bZ,\bW;\theta) - \log \Prob(\bZ, \bW|\bY;\theta) \\
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\Leftrightarrow \Esp_{\Ryt}[\ell(\bY;\theta)] & = \Esp_{\Ryt}[\ell_c(\bY,\bZ,\bW;\theta)] - \Esp_{\Ryt}[\log \Prob(\bZ,\bW|\bY;\theta)] \\
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\Leftrightarrow \ell(\bY;\theta) & = \Esp_{\Ryt}[\ell_c(\bY,\bZ,\bW;\theta)] - \Esp_{\Ryt}[\log \Prob(\bZ,\bW|\bY;\theta)] \\
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\end{align*}
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\begin{align*}
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\text{Or }\KL{\Ryt}{\log \Prob(\bZ,\bW|\bY;\theta)} & = - \Esp_{\Ryt} [\log \frac{\Prob(\bZ,\bW|\bY;\theta)}{\Ryt}] \\
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= - \Esp_{\Ryt} [\log \Prob(\bZ,\bW|\bY;\theta)] + & \underbrace{\Esp_{\Ryt[\log \Ryt]}}_{-\Hshannon(\Ryt)} \\
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\Leftrightarrow \KL{\Ryt}{\log \Prob(\bZ,\bW|\bY;\theta)} + \Hshannon(\Ryt) & = - \Esp_{\Ryt} [\log \Prob(\bZ,\bW|\bY;\theta)]
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\end{align*}
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D'où $\ell(\bY;\theta) - \KL{\Ryt}{\log \Prob(\bZ,\bW|\bY;\theta)} = \mathcal{J}(\tau;\theta) \qed$
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\end{frame}
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\section{Sélection de modèle}
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\section{Résultats~\cite{baldockSystemsApproachReveals2019a,baldockDailyTemporalStructure2011}}
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\begin{frame}[allowframebreaks]
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\begin{figure}[ht]
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\centering
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\begin{subfigure}[t]{0.5\textwidth}
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\centering
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\includestandalone[width=0.45\textwidth]{tikz/applications/baldock/mat-Baldock2019_Bristol}
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\caption{Donnée}
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\end{subfigure}\hfil
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\begin{subfigure}[t]{0.5\textwidth}
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\centering
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\includestandalone[width=0.45\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Bristol}
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\caption{Réordonnée}
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\end{subfigure}
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\caption{Bristol}
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\end{figure}
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\begin{figure}[ht]
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\centering
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\begin{subfigure}[t]{0.5\textwidth}
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\centering
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\includestandalone[width=0.45\textwidth]{tikz/applications/baldock/mat-Baldock2019_Edinburgh}
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\caption{Donnée}
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\end{subfigure}\hfil
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\begin{subfigure}[t]{0.5\textwidth}
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\centering
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\includestandalone[width=0.5\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Edinburgh}
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\caption{Réordonnée}
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\end{subfigure}
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\caption{Edinburgh}
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\end{figure}
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\begin{figure}
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\begin{subfigure}[ht]{0.5\textwidth}
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\centering
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\includestandalone[width=0.5\textwidth]{tikz/applications/baldock/mat-Baldock2019_Leeds}
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\caption{Donnée}
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\end{subfigure}\hfil
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\begin{subfigure}[ht]{0.5\textwidth}
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\centering
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\includestandalone[width=0.5\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Leeds}
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\caption{Réordonnée}
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\end{subfigure}
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\caption{Leeds}
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\end{figure}
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\begin{figure}
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\begin{subfigure}[ht]{0.5\textwidth}
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\centering
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\includestandalone[width=0.5\textwidth]{tikz/applications/baldock/mat-Baldock2019_Reading}
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\caption{Donnée}
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\end{subfigure}\hfil
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\begin{subfigure}[ht]{0.5\textwidth}
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\centering
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\includestandalone[width=0.5\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Reading}
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\caption{Réordonnée}
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\end{subfigure}
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\caption{Reading}
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\end{figure}
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\end{frame}
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@ -2,28 +2,21 @@
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\label{sec:contexte-du-modele}
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\begin{frame}
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\frametitle{Pourquoi un réseau ?}
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\begin{columns}
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\begin{column}{0.5\textwidth}
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\begin{block}{Contexte écologique}
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\begin{itemize}
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\small
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\item Nombreux réseaux disponibles pour
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interactions similaires.
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\item Suivi biodiversité, robustesse et risque
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d'effondrement \dots
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\end{itemize}
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\begin{columns}
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\begin{column}{0.5\textwidth}
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\begin{figure}[ht]
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\centering
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\begin{tikzpicture}[scale=.45,rotate=270]
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\input{../tikz/plantpollinatornetwork.tex}
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\begin{tikzpicture}[scale=.6,rotate=270]
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\input{tikz/plantpollinatornetwork.tex}
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\end{tikzpicture}
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\caption{Exemple d'un réseau plantes-pollinisateurs}
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\caption{Exemple d'un réseau}
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\label{fig:plantes-pollin}
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\end{figure}
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\end{column}
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\begin{column}{0.4\textwidth}
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\begin{column}{0.3\textwidth}
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\centering
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\begin{align*}
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\begin{pmatrix}
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@ -37,170 +30,250 @@
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Matrice d'adjacence associée
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\end{column}
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\end{columns}
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\end{block}
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\begin{figure}[ht]
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\centering
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\includestandalone[width=0.7\textwidth]{tikz/applications/baldock/graph-Baldock2019_Bristol}
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\caption{Réseau plante-pollinisateur de
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Bristol\newline\cite{baldockSystemsApproachReveals2019a}}
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\label{fig:label}
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\end{figure}
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\end{column}
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\onslide<2>{
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\begin{column}{0.45\textwidth}
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\begin{block}{Contexte mathématique}
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Pour un unique réseau~: variables latentes,
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\emph{embedding}, \dots
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Motivations pour proposer des méthodes adaptées aux collections
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de réseaux~:
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\begin{column}{0.5\textwidth}
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\begin{itemize}
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\item Espèces différentes, rôles analogues.
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\item Transfert d'informations grands vers petits réseaux.
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\item Regrouper les réseaux selon leur similarité (\emph{clustering}
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de réseaux).
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\item Modélisation d'interactions variées, ici d'écosystèmes
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\item Structure nécessaire pour~: suivi biodiversité,
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robustesse, risque d'effondrement
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\item De plus en plus disponibles
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\end{itemize}
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\end{block}
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\end{column}
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}
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\end{columns}
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\end{frame}
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\begin{frame}{Méthodes d'analyse pour un réseau}
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Plusieurs méthodes~:
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\begin{itemize}
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\item Métriques~: degré, centralité, emboîtement \dots
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\item Plongement des réseaux avec GNN
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\item \textbf<2>{\emph{Clustering} des n\oe uds avec modèles à variables latentes}
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\end{itemize}
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\end{frame}
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\begin{frame}
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\addtocounter{footnote}{1}
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\frametitle{Latent Block Model (LBM\footnotemark[\thefootnote])}
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%DONE remplacer i \in bullet par Zi = \bullet
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Proposé par~\cite{govaertEMAlgorithmBlock2005}.
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\cite{govaertEMAlgorithmBlock2005}.
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\begin{columns}
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\begin{column}{0.40\linewidth}
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\begin{figure}[H]
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\center
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\begin{tikzpicture}[scale=0.35]
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\input{../tikz/lbm.tex}
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\input{tikz/lbm.tex}
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\end{tikzpicture}
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\caption{Exemple de LBM\footnotemark[\thefootnote]}
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\label{fig:LBMvisu}
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\end{figure}
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\end{column}
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\only<1>{
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\begin{column}{0.51\linewidth}
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Pour \begin{itemize}
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\begin{block}{Modèle hiérarchique}
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\vspace{-\baselineskip}
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\begin{align*}
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\forall q\in[\![ 1, Q_1]\!],~ & \mathbb{P}(Z_i = q) = \pi_q \\
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\forall r\in[\![ 1, Q_2]\!],~ & \mathbb{P}(W_j = r) = \rho_r \\
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& Y_{ij} | Z_i, W_j \sim \mathcal{F}(\alpha_{Z_i,W_j})
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\end{align*}
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où $|\pi| = Q_1, |\rho| = Q_2, |\alpha| = Q_1 \times Q_2$
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\end{block}
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\begin{block}{Formule concise LBM}
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$Y \sim \mathcal{F}\text{-BiSBM}_{n_1,n_2}(Q_1, Q_2, \pi, \rho, \alpha)$
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\end{block}
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\end{column}}
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\only<2>{
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\begin{column}{0.51\linewidth}
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Avec \begin{itemize}
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\item $Q_1 = |\{{\color{blueind}\bullet},{\color{cyanind}\bullet},{\color{electricblue}\bullet}\}|$ blocs fixés en ligne
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\item $Q_2 = |\{{\color{burntorange}\bullet},{\color{goldenyellow}\bullet},{\color{peach}\bullet}\}|$ blocs fixés en colonne
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\end{itemize}
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\begin{block}{Paramètres}
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\begin{itemize}
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\item $\pi_{\bullet} = \mathbb{P}(Z_i = \bullet)$ en ligne et $\rho_{\bullet} = \mathbb{P}(W_j = \bullet)$ en colonne
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\item $\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}} = \mathbb{P}(X_{ij} = 1 | Z_i = {\color{blueind}\bullet}, W_j = {\color{burntorange}\bullet})$
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\item $\pi_{{\color{blueind}\bullet}} = \mathbb{P}(Z_i = {\color{blueind}\bullet})$
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\item $\rho_{{\color{burntorange}\bullet}} = \mathbb{P}(W_j = {\color{burntorange}\bullet})$
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\item $\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}} = \mathbb{P}(Y_{ij} = 1 | Z_i = {\color{blueind}\bullet}, W_j = {\color{burntorange}\bullet})$
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\end{itemize}
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\end{block}
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\end{column}
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\end{column}}
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\end{columns}
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\footnotetext[\thefootnote]{Que j'appellerai par la suite BiSBM}
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\end{frame}
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\section{Modèle de collection de réseaux bipartites}
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\begin{frame}
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\frametitle{Plusieurs réseaux}
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\begin{figure}[ht]
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\centering
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\begin{subfigure}[ht]{0.3\textwidth}
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\includestandalone[width=1.1\textwidth]{tikz/applications/baldock/mat-Baldock2019_Bristol}
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\caption{Bristol}
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\end{subfigure}
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\begin{subfigure}[ht]{0.3\textwidth}
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\includestandalone[width=1.1\textwidth]{tikz/applications/baldock/mat-Baldock2019_Edinburgh}
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\caption{Edinburgh}
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\end{subfigure}
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\begin{subfigure}[ht]{0.3\textwidth}
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\includestandalone[width=1.1\textwidth]{tikz/applications/baldock/mat-Baldock2019_Leeds}
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\caption{Leeds}
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\end{subfigure}
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\caption{Matrices d'adjacence,~\cite{baldockSystemsApproachReveals2019a}}
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\label{fig:adj}
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\end{figure}
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\end{frame}
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\section[Modèles collection bipartites]{Modèles de collection de réseaux bipartites}
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\label{sec:extension-de-colsbm-aux-reseaux-bipartites}
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\begin{frame}
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\frametitle{Collections bipartites}
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\begin{tikzpicture}[scale=0.33]
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\input{../tikz/collbm-iid.tex}
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\end{tikzpicture}
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\begin{itemize}
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\item $Q_1 = |\{{\color{blueind}\bullet},{\color{cyanind}\bullet},{\color{electricblue}\bullet}\}|$ blocs fixés en ligne
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\item $Q_2 = |\{{\color{burntorange}\bullet},{\color{goldenyellow}\bullet},{\color{peach}\bullet}\}|$ blocs fixés en colonne
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\end{itemize}
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\begin{block}{Paramètres}
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\begin{itemize}
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\item $\pi_{\bullet} = \mathbb{P}(Z_i =\bullet)$ en ligne et $\rho_{\bullet} = \mathbb{P}(W_j = \bullet)$ en colonne
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\item $\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}} = \mathbb{P}(X_{ij} = 1 | Z_i = {\color{blueind}\bullet}, W_j = {\color{burntorange}\bullet})$
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\end{itemize}
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\end{block}
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\[
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\forall m \in \{1\dots M\}, Y^m \overset{ind}{\sim} \mathcal{F}\text{-BiSBM}_{n_1^m,n_2^m}(Q_1^m, Q_2^m, \pi^m, \rho^m, \alpha^m)
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\]
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\onslide<2>{
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\begin{figure}[ht]
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\centering
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\begin{subfigure}[ht]{0.3\textwidth}
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\includestandalone[width=1.1\textwidth]{tikz/applications/baldock/bisbm-mat-Baldock2019_Bristol}
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\caption{Bristol}
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\end{subfigure}
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\begin{subfigure}[ht]{0.3\textwidth}
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\includestandalone[width=1.1\textwidth]{tikz/applications/baldock/bisbm-mat-Baldock2019_Edinburgh}
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\caption{Edinburgh}
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\end{subfigure}
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\begin{subfigure}[ht]{0.3\textwidth}
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\includestandalone[width=1.1\textwidth]{tikz/applications/baldock/bisbm-mat-Baldock2019_Leeds}
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\caption{Leeds}
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\end{subfigure}
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\caption{Matrices d'adjacence réordonnées, grâce au LBM}
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\label{fig:adj-reord}
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\end{figure}
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}
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\end{frame}
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\begin{frame}
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\frametitle{Différents modèles}
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\only<1>{
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\begin{tikzpicture}[scale=0.33]
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\input{../tikz/collbm-iid.tex}
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\end{tikzpicture}
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\begin{block}{\emph{iid-colBiSBM}}
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$\bm{\pi} = (\pi_1, \dots \pi_{Q_1})$ et $\bm{\rho} = (\rho_1, \dots \rho_{Q_2})$
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\onslide<1->{ \begin{block}{\emph{iid}-colBiSBM}
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\[
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\forall m \in \{1\dots M\}, Y^m \overset{iid}{\sim}
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\mathcal{F}\text{-BiSBM}_{n_1^m,n_2^m}(Q_1, Q_2, \pi, \rho, \alpha)
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\]
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avec $\theta = (\pi, \rho, \alpha)$.
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\end{block}}
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\onslide<2>{ \begin{block}{$\pi\rho$-colBiSBM}
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\[
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\forall m \in \{1\dots M\}, Y^m \overset{ind}{\sim}
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\mathcal{F}\text{-BiSBM}_{n_1^m,n_2^m}(Q_1, Q_2, \pi^m, \rho^m, \alpha)
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\]
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avec $\theta = ((\pi^m)_{m=1,\dots, M}, (\rho^m)_{m=1,\dots, M}, \alpha)$.
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\end{block}
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}
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\only<2>{
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\begin{tikzpicture}[scale=0.33]
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\input{../tikz/collbm-pirho.tex}
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\end{tikzpicture}
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\begin{block}{\emph{$\pi\rho$-colBiSBM}}
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$\bm{\pi} = ((\pi_{\color{black}1}^{\color{red}m}, \dots \pi_{\color{black}Q_1}^{\color{red}m}))_{m=1,\dots M}$ et $\bm{\rho} = ((\rho_{\color{black}1}^{\color{red}m}, \dots \rho_{\color{black}Q_2}^{\color{red}m}))_{m=1,\dots M}$ %{$\forall q \in \llbracket 1, Q_1 - 1\rrbracket, \pi_q > 0$ et $\forall r \in \llbracket 1, Q_2 - 1\rrbracket, \rho_r > 0$}
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\small \\
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avec $\forall q,m \in \llbracket 1, Q_1 \rrbracket \times \llbracket 1, M \rrbracket, \pi_q^m \in \left[ 0,1 \right]$
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et $\forall r,m \in \llbracket 1, Q_2 \rrbracket \times \llbracket 1, M \rrbracket, \rho_r^m \in \left[ 0,1 \right]$
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\end{block}
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}
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Dans tous les modèles la structure de connectivité ($\bm{\alpha}$) est supposée identique au sein de la collection.
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\end{frame}
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\begin{frame}
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\frametitle{Estimation des paramètres}
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% DONE dire que tau i q m c' est la proba que Zim = q, approximation de la proba variationnelle. Parce qu on impose lindependance
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% Par maximisation d'une borne inférieure variationnelle de la
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% log-vraisemblance des données observées.
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En adaptant \cite{chabert-liddellLearningCommonStructures2024a} qui se base
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sur la méthode proposée par \cite{daudinMixtureModelRandom2008} utilisant
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l'algorithme \emph{Variational EM}.
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Maximisation de la log-vraisemblance ?
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\begin{block}{log-vraisemblance et log-vraisemblance complète}
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\[
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\ell(\bm{Y};\theta) = \sum_{\bm{Z,W}\in \bm{\mathcal{Z}\times\mathcal{W}}} \ell_c(\bm{Y}, \bm{Z}, \bm{W};\theta)
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\]
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avec $\bm{\mathcal{Z}} = \{1,\dots,\alert<2>{Q_1}\}^{\alert<2>{n}}, \bm{\mathcal{W}} = \{1,\dots,\alert<2>{Q_2}\}^{\alert<2>{n}}$
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\end{block}
|
||||
\uncover<3>{Donc, algorithme classique $\Rightarrow$
|
||||
\emph{Expectation-Maximization} (EM).}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{Par EM classique}
|
||||
A l'itération $(t)$ :
|
||||
\begin{itemize}
|
||||
\item[$\bullet$]\textbf{Étape E}: calculer
|
||||
$$ \mathcal{Q}(\theta | \theta^{(t-1)}) = \mathbb E_{\alert<2>{\bm Z, \bm W | \bm Y, \theta^{(t-1)}} } \left[\ell_c(\bm Y, \bm W, \bm Z; \theta) \right] $$
|
||||
\item[$\bullet$]\textbf{Étape M}:
|
||||
$$ \theta^{(t)} = \arg \max_{\theta} \mathcal{Q}(\theta | \theta^{(t-1)})$$
|
||||
\end{itemize}
|
||||
\uncover<2>{
|
||||
\begin{alertblock}{Problème pour l'EM classique}
|
||||
Loi de $\bm{Z,W|Y},\theta^{(t-1)}$ inaccessible
|
||||
\end{alertblock}}
|
||||
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}
|
||||
Par \emph{Variational EM}, comme proposé
|
||||
par~\cite{daudinMixtureModelRandom2008,
|
||||
chabert-liddellLearningCommonStructures2024a}.
|
||||
\begin{block}{Approximation variationnelle de $\bm{Z,W|Y},\theta^{(t-1)}$}
|
||||
$\mathcal{R}_{Y^m,\tau}(\mathbf{Z}^m, \mathbf{W}^m) =
|
||||
\mathcal{R}^1_{Y^m,\tau}(\mathbf{Z}^m)
|
||||
{\color{red}\times}
|
||||
\mathcal{R}^2_{Y^m,\tau}(\mathbf{W}^m) \Rightarrow$ indépendance lignes, colonnes.
|
||||
\end{block}
|
||||
\begin{multline*}
|
||||
\ell (\bm{X};\bm{\theta}) \geq \color{red}\sum_{m=1}^{M} \bigg(
|
||||
\color{black} Q^m(\boldsymbol\theta\mid\boldsymbol\theta^{(t)}) +
|
||||
\mathcal{H}(\mathcal{R}_{\mathbf{X}^m,\boldsymbol\theta^{(t)}}
|
||||
\ell (\bm{Y};\theta) \geq \color{red}\sum_{m=1}^{M} \bigg(
|
||||
\color{black} \mathcal{Q}^m(\theta\mid\theta^{(t)}) +
|
||||
\mathcal{H}(\mathcal{R}_{Y^m,\theta^{(t)}}
|
||||
(\mathbf{Z}^m, \mathbf{W}^m))
|
||||
\color{red}\bigg) \color{black}
|
||||
=: J(\bm{\tau};\bm{\theta})
|
||||
\eqcolon \mathcal{J}(\tau;\theta)
|
||||
\end{multline*}
|
||||
où $Q^m(\boldsymbol\theta\mid\boldsymbol\theta^{(t)}) =
|
||||
\operatorname{E}_{\mathbf{Z}^m,\mathbf{W}^m
|
||||
\sim \mathcal{R}_{\mathbf{X}^m,\boldsymbol\theta^{(t)}}(.)}
|
||||
\left[ \log p (\mathbf{X}^m,\mathbf{Z}^m,\mathbf{W}^m | \boldsymbol\theta) \right] \,$
|
||||
|
||||
\begin{block}{Approximation variationnelle}
|
||||
$\mathcal{R}_{\mathbf{X}^m,\boldsymbol\theta^{(t)}}(\mathbf{Z}^m, \mathbf{W}^m) =
|
||||
P(\mathbf{Z}^m | \mathbf{X}^m,\boldsymbol\theta^{(t)}) P(\mathbf{W}^m | \mathbf{X}^m,\boldsymbol\theta^{(t)})$, c'est à dire avoir
|
||||
une indépendance lignes, colonnes.
|
||||
\end{block}
|
||||
où $\mathcal{Q}^m(\theta\mid\theta^{(t)}) =
|
||||
\mathbb{E}_{\mathbf{Z}^m,\mathbf{W}^m
|
||||
\sim \mathcal{R}_{Y^m,\tau}(.)}
|
||||
\left[ \ell_c(Y^m,\mathbf{Z}^m,\mathbf{W}^m | \theta) \right] \,$
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}{Formule développée de l'EM variationnel}
|
||||
\begin{multline*}
|
||||
\ell (\bm{X};\bm{\theta}) \geq \color{red}\sum_{m=1}^{M} \bigg( \color{black} \sum_{i = 1}^{n_1^m}\sum_{j=1}^{n_2^m}\sum_{q \in \mathcal{Q}_{1,m}} \sum_{r \in \mathcal{Q}_{2,m}} \tau^{1,m}_{i,q} \tau^{2,m}_{j,r} \log f(X^{m}_{ij}; \alpha_{qr}) \\
|
||||
\ell (\bm{Y};\theta) \geq \color{red}\sum_{m=1}^{M} \bigg( \color{black} \sum_{i = 1}^{n_1^m}\sum_{j=1}^{n_2^m}\sum_{q \in \mathcal{Q}_{1,m}} \sum_{r \in \mathcal{Q}_{2,m}} \tau^{1,m}_{i,q} \tau^{2,m}_{j,r} \log f(Y^{m}_{ij}; \alpha_{qr}) \\
|
||||
+ \sum_{i=1}^{n_1^m} \sum_{q \in \mathcal{Q}_{1,m}} \tau^{1,m}_{i,q} \log \pi_{\color{black}q}^{\color{gray}m} + \sum_{j=1}^{n_2^m} \sum_{r \in \mathcal{Q}_{2,m}} \tau^{2,m}_{j,r} \log \rho_{\color{black}r}^{\color{gray}m} \\
|
||||
- \sum_{i=1}^{n_1} \tau^{1,m}_{i,q} \log \tau^{1,m}_{i,q} - \sum_{j=1}^{n_2} \tau^{2,m}_{j,r} \log \tau^{2,m}_{j,r} \color{red}\bigg) \color{black} =: J(\bm{\tau};\bm{\theta}),
|
||||
- \sum_{i=1}^{n_1} \tau^{1,m}_{i,q} \log \tau^{1,m}_{i,q} - \sum_{j=1}^{n_2} \tau^{2,m}_{j,r} \log \tau^{2,m}_{j,r} \color{red}\bigg) \color{black} \eqcolon
|
||||
\mathcal{J}(\tau;\theta),
|
||||
\end{multline*}
|
||||
où $\ell$ désigne la $\log$ vraisemblance.
|
||||
|
||||
\begin{block}{Approximation variationnelle}
|
||||
$\tau_{iq}^{1,m} = P_{\mathcal{R}_m}(Z_{iq}^m = 1|X_{i\bullet}^m)$
|
||||
et $\tau_{jr}^{2,m} = P_{\mathcal{R}_m}(W_{jr}^m = 1|X_{\bullet j}^m)$
|
||||
$\tau_{iq}^{1,m} = \mathcal{R}^1_{Y^m,\tau}(Z_{iq}^m = 1)$
|
||||
et $\tau_{jr}^{2,m} = \mathcal{R}^2_{Y^m,\tau}(W_{jr}^m = 1)$
|
||||
\end{block}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}{Étape \emph{Variational Expectation}}
|
||||
$$\widehat{\bm{\tau}}^{(t+1)} = \arg \max_{\bm{\tau}}
|
||||
\mathcal{J}(\mathcal{\bm{\tau}},\bm{\widehat{\theta}}^{(t)})$$
|
||||
\[
|
||||
\widehat{\tau}^{(t+1)} = \arg \max_{\tau}
|
||||
\mathcal{J}(\mathcal{\tau},\bm{\widehat{\theta}}^{(t)})
|
||||
\Leftrightarrow \arg\min_{\tau\in\mathcal{T}} \mathbf{KL}[\mathcal{R}_{\mathbf{Y},\tau}, \mathbb{P}(.|\mathbf{Y})]
|
||||
\]
|
||||
|
||||
\begin{equation*}
|
||||
\begin{cases}
|
||||
\widehat{\tau}_{iq}^{1,m} \propto \widehat{\pi}_{q}^{m(t)} \prod_{j=1}^{n_2^m}\prod_{r\in\mathcal{Q}_2^m} f(X_{ij}^m;\widehat{\alpha}_{qr}^{(t)})^{\widehat{\tau}_{jr}^{2,m(t+1)}} & \forall i = 1, \dots , n_1^m, q \in \mathcal{Q}_1^m \\
|
||||
\widehat{\tau}_{jr}^{2,m} \propto \widehat{\rho}_{r}^{m(t)} \prod_{i=1}^{n_1^m}\prod_{q\in\mathcal{Q}_1^m} f(X_{ij}^m;\widehat{\alpha}_{qr}^{(t)})^{\widehat{\tau}_{iq}^{1,m(t+1)}} & \forall j = 1, \dots , n_2^m, r \in \mathcal{Q}_2^m
|
||||
\widehat{\tau}_{iq}^{1,m} \propto \widehat{\pi}_{q}^{m(t)} \prod_{j=1}^{n_2^m}\prod_{r\in\mathcal{Q}_2^m} f(Y_{ij}^m;\widehat{\alpha}_{qr}^{(t)})^{\widehat{\tau}_{jr}^{2,m(t+1)}} & \forall i = 1, \dots , n_1^m, q \in \mathcal{Q}_1^m \\
|
||||
\widehat{\tau}_{jr}^{2,m} \propto \widehat{\rho}_{r}^{m(t)} \prod_{i=1}^{n_1^m}\prod_{q\in\mathcal{Q}_1^m} f(Y_{ij}^m;\widehat{\alpha}_{qr}^{(t)})^{\widehat{\tau}_{iq}^{1,m(t+1)}} & \forall j = 1, \dots , n_2^m, r \in \mathcal{Q}_2^m
|
||||
\end{cases}
|
||||
\end{equation*}
|
||||
\footnotetext[2]{Initialisation des $\widehat{\bm{\tau}}$ avec un
|
||||
\footnotetext[2]{Initialisation des $\widehat{\tau}$ avec un
|
||||
\emph{spectral clustering} sur les réseaux.}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}{Étape \emph{Maximization}}
|
||||
\[
|
||||
\widehat{\bm{\theta}}^{(t+1)} = \arg \max_{\bm{\theta}} \mathcal{J}(\mathcal{\bm{\widehat{\tau}}}^{(t+1)},\bm{\theta})
|
||||
\widehat{\theta}^{(t+1)} = \arg \max_{\theta} \mathcal{J}(\mathcal{\bm{\widehat{\tau}}}^{(t+1)},\theta)
|
||||
\]
|
||||
|
||||
\begin{block}{Paramètres de connectivité}
|
||||
\begin{align*}
|
||||
\widehat{\alpha}_{qr} = \frac{\sum_{m=1}^{M} \sum_{i=1}^{n_1^m} \sum_{j=1}^{n_2^m} \tau_{iq}^{1,m} \tau_{jr}^{2,m} X_{ij}^m}{\sum_{m=1}^{M} \sum_{i=1}^{n_1^m} \sum_{j=1}^{n_2^m} \tau_{iq}^{1,m} \tau_{jr}^{2,m}}
|
||||
\widehat{\alpha}_{qr} = \frac{\sum_{m=1}^{M} \sum_{i=1}^{n_1^m} \sum_{j=1}^{n_2^m} \tau_{iq}^{1,m} \tau_{jr}^{2,m} \alert<2>{Y_{ij}^m}}{\sum_{m=1}^{M} \sum_{i=1}^{n_1^m} \sum_{j=1}^{n_2^m} \tau_{iq}^{1,m} \tau_{jr}^{2,m}}
|
||||
\end{align*}
|
||||
\end{block}
|
||||
\only<1>{
|
||||
|
|
@ -224,19 +297,20 @@
|
|||
|
||||
\section{Sélection de modèle}
|
||||
\begin{frame}
|
||||
\frametitle{Problème de choix de $(Q_1, Q_2)$}
|
||||
\underline{L'estimation de paramètres se fait à $Q_1, Q_2$ blocs fixés}, il faut donc déterminer les \enquote*{meilleures} coordonnées.
|
||||
Nous maximisons un critère, le \emph{Bayesian Information Criterion - Like}
|
||||
(BIC-L), de vraisemblance pénalisée en adaptant les formules
|
||||
de~\cite{chabert-liddellLearningCommonStructures2024a}.
|
||||
\frametitle{Problème choix de $(Q_1, Q_2)$}
|
||||
Besoin sélectionner $Q_1$ et $Q_2$. Critère BIC-Like\footnote{ICL + Entropie + pénalité}
|
||||
|
||||
\begin{align*}
|
||||
\text{BIC-L}(\bm{Y}, Q_1, Q_2) & = \max_{\theta} \mathbb{E}_{\mathcal{R}_{\mathbf{Y},\hat{\tau}}} [\ell_c(\bm{Y,Z,W};\theta)] + \mathcal{H(\mathcal{R}_{\mathbf{Y},\hat{\tau}})} - \frac{1}{2}\text{pen}(\theta, Q_1, Q_2) \\
|
||||
& = \max_{\theta} \mathcal{J(\mathcal{R}_{\mathbf{Y},\hat{\tau}}, \theta)} - \frac{1}{2}\text{pen}(\theta, Q_1, Q_2)
|
||||
\end{align*}
|
||||
|
||||
|
||||
\begin{alertblock}{Problèmes de l'exploration}
|
||||
\begin{itemize}
|
||||
\item Exploration de l'espace $\mathbb{N}^2$ coûteux, besoin d'une
|
||||
stratégie.
|
||||
\item Sensibilité aux initialisations et à l'aléatoire.
|
||||
\item Exploration de $\mathbb{N}^2$ coûteux.
|
||||
\item Sensibilité initialisations.
|
||||
\end{itemize}
|
||||
|
||||
\end{alertblock}
|
||||
\end{frame}
|
||||
|
||||
|
|
@ -245,7 +319,7 @@
|
|||
\begin{columns}
|
||||
\begin{column}{0.5\linewidth}
|
||||
\begin{tikzpicture}
|
||||
\input{../tikz/greedy-exploration.tex}
|
||||
\input{tikz/greedy-exploration.tex}
|
||||
\end{tikzpicture}
|
||||
\end{column}
|
||||
\begin{column}{0.35\linewidth}
|
||||
|
|
@ -279,7 +353,7 @@
|
|||
\begin{columns}
|
||||
\begin{column}{0.6\textwidth}
|
||||
\begin{figure}
|
||||
\input{../tikz/moving-window.tex}
|
||||
\input{tikz/moving-window}
|
||||
\caption{Fenêtre glissante}
|
||||
\end{figure}
|
||||
\end{column}
|
||||
|
|
@ -301,11 +375,72 @@
|
|||
\label{sec:application}
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{Clustering de réseaux}
|
||||
\frametitle{Résultats~\cite{baldockSystemsApproachReveals2019a}}
|
||||
|
||||
\only<1>{
|
||||
\begin{figure}[ht]
|
||||
\centering
|
||||
\begin{tikzpicture}
|
||||
\input{../tikz/clustering.tex}
|
||||
\end{tikzpicture}
|
||||
\begin{subfigure}[t]{0.5\textwidth}
|
||||
\centering
|
||||
\includestandalone[width=0.45\textwidth]{tikz/applications/baldock/mat-Baldock2019_Bristol}
|
||||
\caption{Bristol}
|
||||
\end{subfigure}\hfil
|
||||
\begin{subfigure}[t]{0.5\textwidth}
|
||||
\centering
|
||||
\includestandalone[width=0.45\textwidth]{tikz/applications/baldock/mat-Baldock2019_Edinburgh}
|
||||
\caption{Edinburgh}
|
||||
\end{subfigure}
|
||||
\newline
|
||||
\begin{subfigure}[ht]{0.5\textwidth}
|
||||
\centering
|
||||
\includestandalone[width=0.5\textwidth]{tikz/applications/baldock/mat-Baldock2019_Leeds}
|
||||
\caption{Leeds}
|
||||
\end{subfigure}\hfil
|
||||
\begin{subfigure}[ht]{0.5\textwidth}
|
||||
\centering
|
||||
\includestandalone[width=0.5\textwidth]{tikz/applications/baldock/mat-Baldock2019_Reading}
|
||||
\caption{Reading}
|
||||
\end{subfigure}
|
||||
\caption{Matrices d'adjacence,~\cite{baldockSystemsApproachReveals2019a}}
|
||||
\end{figure}
|
||||
}
|
||||
\only<2>{
|
||||
\begin{figure}[ht]
|
||||
\centering
|
||||
\begin{subfigure}[t]{0.5\textwidth}
|
||||
\centering
|
||||
\includestandalone[width=0.45\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Bristol}
|
||||
\caption{Bristol}
|
||||
\end{subfigure}\hfil
|
||||
\begin{subfigure}[t]{0.5\textwidth}
|
||||
\centering
|
||||
\includestandalone[width=0.45\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Edinburgh}
|
||||
\caption{Edinburgh}
|
||||
\end{subfigure}
|
||||
\newline
|
||||
\begin{subfigure}[ht]{0.5\textwidth}
|
||||
\centering
|
||||
\includestandalone[width=0.5\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Leeds}
|
||||
\caption{Leeds}
|
||||
\end{subfigure}\hfil
|
||||
\begin{subfigure}[ht]{0.5\textwidth}
|
||||
\centering
|
||||
\includestandalone[width=0.5\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Reading}
|
||||
\caption{Reading}
|
||||
\end{subfigure}
|
||||
\caption{Matrices d'adjacence réordonnée par \emph{iid}-colBiSBM,~\cite{baldockSystemsApproachReveals2019a}}
|
||||
\end{figure}
|
||||
}
|
||||
\end{frame}
|
||||
|
||||
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{Clustering de réseaux}
|
||||
\begin{figure}[ht]
|
||||
\includestandalone[width=0.45\textwidth]{tikz/applications/baldock/mat-Baldock2011_TB+Baldock2011_JN}
|
||||
\caption{Matrice d'adjacence,~\cite{baldockDailyTemporalStructure2011}}
|
||||
\end{figure}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}[allowframebreaks]
|
||||
|
|
@ -316,15 +451,13 @@
|
|||
\begin{subfigure}{0.5\textwidth}
|
||||
\centering
|
||||
\includegraphics[scale=0.2,angle=-90]{backup-app-iid.png}
|
||||
\caption{Modèle $iid$,\\
|
||||
séparent réseau africain et réseaux anglais}
|
||||
\caption{Modèle $iid$}
|
||||
\end{subfigure}%
|
||||
~
|
||||
\begin{subfigure}{0.5\textwidth}
|
||||
\centering
|
||||
\includegraphics[scale=0.2,angle=-90]{backup-app-pirho.png}
|
||||
\caption{Modèle $\pi\rho$,\\
|
||||
fusionnent réseaux africain et anglais}
|
||||
\caption{Modèle $\pi\rho$}
|
||||
\end{subfigure}%
|
||||
\caption{Partitionnement des réseaux
|
||||
de~\cite{baldockDailyTemporalStructure2011,
|
||||
|
|
@ -353,6 +486,36 @@
|
|||
\end{figure}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}{Algorithme du clustering}
|
||||
\centering
|
||||
\vspace{0.25\baselineskip}
|
||||
\begin{tikzpicture}[scale=0.85]
|
||||
\input{tikz/clustering.tex}
|
||||
\end{tikzpicture}
|
||||
\[
|
||||
D_{\mathcal{M}}(m,m') = \sum_{q = 1}^{Q_1} \sum_{r = 1}^{Q_2} \max(\widetilde{\pi}_{q}^{m}, \widetilde{\pi}_{q}^{m'}) \left( \widetilde{\alpha}_{qr}^{m} - \widetilde{\alpha}_{qr}^{m'}\right)^{2} \max(\widetilde{\rho}_{r}^{m}, \widetilde{\rho}_{r}^{m'})
|
||||
\]
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}{Résultats}
|
||||
\begin{figure}[ht]
|
||||
\centering
|
||||
\begin{subfigure}{0.5\textwidth}
|
||||
\centering
|
||||
\includestandalone[width=1\textwidth]{tikz/applications/baldock/bisbm-mat-Baldock2011_TB+Baldock2011_JN}
|
||||
\caption{Réordonnée par LBM}
|
||||
\end{subfigure}\hfil
|
||||
\begin{subfigure}{0.5\textwidth}
|
||||
\centering
|
||||
\includestandalone[width=1\textwidth]{tikz/applications/baldock/pirho-colbisbm-mat-Baldock2011_TB+Baldock2011_JN}
|
||||
\caption{Réordonnée par $\pi\rho$-colBiSBM}
|
||||
\end{subfigure}
|
||||
|
||||
\caption{Matrice d'adjacence réordonnée par $\pi\rho$-colBiSBM,~\cite{baldockDailyTemporalStructure2011}}
|
||||
\end{figure}
|
||||
\end{frame}
|
||||
|
||||
|
||||
\section{Conclusion}
|
||||
\label{sec:conclusion}
|
||||
\begin{frame}
|
||||
|
|
@ -378,16 +541,15 @@
|
|||
\begin{itemize}
|
||||
\item Investiguer stabilité face à l'aléatoire et aux \emph{optima} locaux.
|
||||
\item Preuve d'identifiabilité du modèle $\pi\rho$.
|
||||
\item
|
||||
\end{itemize}
|
||||
|
||||
\begin{block}{Package et applications}
|
||||
\begin{itemize}
|
||||
\item Intégration au package \texttt{colSBM} et publication CRAN
|
||||
\item Intégration au package \texttt{colSBM}, amélioration interface utilisateur et ajout retours écologues
|
||||
\item Publication CRAN
|
||||
\item Intégrer possibilité d'un critère supplémentaire pour le
|
||||
clustering
|
||||
\item Appliquer clustering données de \cite{pichonTellingMutualisticAntagonistic2024,doreRelativeEffectsAnthropogenic2021}
|
||||
\item
|
||||
\end{itemize}
|
||||
|
||||
\end{block}
|
||||
|
|
|
|||
3106
soutenance/soutenance.bcf-SAVE-ERROR
Normal file
3106
soutenance/soutenance.bcf-SAVE-ERROR
Normal file
File diff suppressed because it is too large
Load diff
Binary file not shown.
|
|
@ -50,6 +50,15 @@
|
|||
%% Biblio
|
||||
\input{../shared/biblio.tex}
|
||||
|
||||
\newcommand{\bZ}{\bm{Z}}
|
||||
\newcommand{\bY}{\bm{Y}}
|
||||
\newcommand{\bW}{\bm{W}}
|
||||
\newcommand{\Prob}{\mathbb{P}}
|
||||
\newcommand{\Ryt}{\mathcal{R}_{\bY,\tau}}
|
||||
\newcommand{\KL}[2]{\mathbf{KL}[#1,#2]}
|
||||
\newcommand{\Esp}{\mathbb{E}}
|
||||
\newcommand{\Hshannon}{\mathcal{H}}
|
||||
|
||||
% Footnote
|
||||
\makeatletter
|
||||
\newcommand\blfootnote[1]{%
|
||||
|
|
|
|||
1
soutenance/tikz
Symbolic link
1
soutenance/tikz
Symbolic link
|
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70644
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tikz/applications/baldock/pirho-colbisbm-mat-Baldock2019_Bristol.tex
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36100
tikz/applications/baldock/pirho-colbisbm-mat-Baldock2019_Edinburgh.tex
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39798
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53102
tikz/applications/baldock/pirho-colbisbm-mat-Baldock2019_Reading.tex
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53102
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Loading…
Add table
Reference in a new issue