From dcaa51b50d1966229638f912d1619a9b2fcf0156 Mon Sep 17 00:00:00 2001 From: Louis Lacoste Date: Thu, 29 Aug 2024 09:30:29 +0200 Subject: [PATCH] soutenance : vfinale --- .woodpecker.yaml | 31 + soutenance/annexe.tex | 79 +- soutenance/principal.tex | 460 +- soutenance/soutenance.bcf-SAVE-ERROR | 3106 + soutenance/soutenance.pdf | Bin 510409 -> 5094769 bytes soutenance/soutenance.tex | 9 + soutenance/tikz | 1 + ...isbm-mat-Baldock2011_TB+Baldock2011_JN.pdf | Bin 0 -> 93011 bytes ...isbm-mat-Baldock2011_TB+Baldock2011_JN.tex | 11634 ++- .../baldock/bisbm-mat-Baldock2019_Bristol.pdf | Bin 510478 -> 510478 bytes .../baldock/bisbm-mat-Baldock2019_Bristol.tex | 2 +- .../bisbm-mat-Baldock2019_Edinburgh.pdf | Bin 268591 -> 268591 bytes .../bisbm-mat-Baldock2019_Edinburgh.tex | 2 +- .../baldock/bisbm-mat-Baldock2019_Leeds.pdf | Bin 292973 -> 292973 bytes .../baldock/bisbm-mat-Baldock2019_Leeds.tex | 2 +- .../baldock/bisbm-mat-Baldock2019_Reading.tex | 54096 ++++++++++++ ...isbm-mat-Baldock2011_TB+Baldock2011_JN.pdf | 0 ...isbm-mat-Baldock2011_TB+Baldock2011_JN.tex | 70644 ++++++++++++++++ .../colbisbm-mat-Baldock2019_Bristol.pdf | Bin 504003 -> 504024 bytes .../colbisbm-mat-Baldock2019_Bristol.tex | 2182 +- .../colbisbm-mat-Baldock2019_Edinburgh.pdf | Bin 263834 -> 264016 bytes .../colbisbm-mat-Baldock2019_Edinburgh.tex | 1462 +- .../colbisbm-mat-Baldock2019_Leeds.pdf | Bin 288186 -> 288213 bytes .../colbisbm-mat-Baldock2019_Leeds.tex | 1960 +- .../colbisbm-mat-Baldock2019_Reading.pdf | Bin 380620 -> 380554 bytes .../colbisbm-mat-Baldock2019_Reading.tex | 1954 +- .../mat-Baldock2011_TB+Baldock2011_JN.pdf | Bin 0 -> 92982 bytes .../mat-Baldock2011_TB+Baldock2011_JN.tex | 2 +- .../baldock/mat-Baldock2019_Bristol.pdf | Bin 510829 -> 510829 bytes .../baldock/mat-Baldock2019_Bristol.tex | 2 +- .../baldock/mat-Baldock2019_Edinburgh.pdf | Bin 268776 -> 268776 bytes .../baldock/mat-Baldock2019_Edinburgh.tex | 2 +- .../baldock/mat-Baldock2019_Leeds.pdf | Bin 293228 -> 293228 bytes .../baldock/mat-Baldock2019_Leeds.tex | 2 +- .../baldock/mat-Baldock2019_Reading.pdf | Bin 391056 -> 391056 bytes .../baldock/mat-Baldock2019_Reading.tex | 2 +- ...isbm-mat-Baldock2011_TB+Baldock2011_JN.pdf | Bin 0 -> 85825 bytes ...isbm-mat-Baldock2011_TB+Baldock2011_JN.tex | 11186 +++ ...pirho-colbisbm-mat-Baldock2019_Bristol.tex | 70618 +++++++++++++++ ...rho-colbisbm-mat-Baldock2019_Edinburgh.tex | 36100 ++++++++ .../pirho-colbisbm-mat-Baldock2019_Leeds.tex | 39798 +++++++++ ...pirho-colbisbm-mat-Baldock2019_Reading.tex | 53102 ++++++++++++ 42 files changed, 354501 insertions(+), 3937 deletions(-) create mode 100644 .woodpecker.yaml create mode 100644 soutenance/soutenance.bcf-SAVE-ERROR create mode 120000 soutenance/tikz create mode 100644 tikz/applications/baldock/bisbm-mat-Baldock2011_TB+Baldock2011_JN.pdf create mode 100644 tikz/applications/baldock/bisbm-mat-Baldock2019_Reading.tex create mode 100644 tikz/applications/baldock/colbisbm-mat-Baldock2011_TB+Baldock2011_JN.pdf create mode 100644 tikz/applications/baldock/colbisbm-mat-Baldock2011_TB+Baldock2011_JN.tex create mode 100644 tikz/applications/baldock/mat-Baldock2011_TB+Baldock2011_JN.pdf create mode 100644 tikz/applications/baldock/pirho-colbisbm-mat-Baldock2011_TB+Baldock2011_JN.pdf create mode 100644 tikz/applications/baldock/pirho-colbisbm-mat-Baldock2011_TB+Baldock2011_JN.tex create mode 100644 tikz/applications/baldock/pirho-colbisbm-mat-Baldock2019_Bristol.tex create mode 100644 tikz/applications/baldock/pirho-colbisbm-mat-Baldock2019_Edinburgh.tex create mode 100644 tikz/applications/baldock/pirho-colbisbm-mat-Baldock2019_Leeds.tex create mode 100644 tikz/applications/baldock/pirho-colbisbm-mat-Baldock2019_Reading.tex diff --git a/.woodpecker.yaml b/.woodpecker.yaml new file mode 100644 index 0000000..46e17bc --- /dev/null +++ b/.woodpecker.yaml @@ -0,0 +1,31 @@ +when: + branch: main + +steps: + - name: build rapport + image: texlive/texlive:latest + commands: + - cd rapport + - make + + - name: build presentation + image: texlive/texlive:latest + commands: + - cd presentation + - make + + - name: build soutenance + image: texlive/texlive:latest + commands: + - cd soutenance + - make + + - name: publish + image: woodpeckerci/plugin-release + settings: + files: + # Could also be "hello-world*" to match both + - "hello-world" + - "hello-world.exe" + api_key: + from_secret: ACCESS_TOKEN diff --git a/soutenance/annexe.tex b/soutenance/annexe.tex index c13fc16..3bff47a 100644 --- a/soutenance/annexe.tex +++ b/soutenance/annexe.tex @@ -1,5 +1,78 @@ -\section{Notes supplémentaires} -\printappxnotes +\section{VEM} +\begin{frame} + \frametitle{Pourquoi VE minimise KL ?} + \begin{align*} + \ell_c(\bY,\bZ,\bW;\theta) & = \log \Prob(\bZ, \bW|\bY;\theta) + \ell(\bY;\theta) \\ + \Leftrightarrow \ell(\bY;\theta) & = \ell_c(\bY,\bZ,\bW;\theta) - \log \Prob(\bZ, \bW|\bY;\theta) \\ + \Leftrightarrow \Esp_{\Ryt}[\ell(\bY;\theta)] & = \Esp_{\Ryt}[\ell_c(\bY,\bZ,\bW;\theta)] - \Esp_{\Ryt}[\log \Prob(\bZ,\bW|\bY;\theta)] \\ + \Leftrightarrow \ell(\bY;\theta) & = \Esp_{\Ryt}[\ell_c(\bY,\bZ,\bW;\theta)] - \Esp_{\Ryt}[\log \Prob(\bZ,\bW|\bY;\theta)] \\ + \end{align*} + \begin{align*} + \text{Or }\KL{\Ryt}{\log \Prob(\bZ,\bW|\bY;\theta)} & = - \Esp_{\Ryt} [\log \frac{\Prob(\bZ,\bW|\bY;\theta)}{\Ryt}] \\ + = - \Esp_{\Ryt} [\log \Prob(\bZ,\bW|\bY;\theta)] + & \underbrace{\Esp_{\Ryt[\log \Ryt]}}_{-\Hshannon(\Ryt)} \\ + \Leftrightarrow \KL{\Ryt}{\log \Prob(\bZ,\bW|\bY;\theta)} + \Hshannon(\Ryt) & = - \Esp_{\Ryt} [\log \Prob(\bZ,\bW|\bY;\theta)] + \end{align*} + D'où $\ell(\bY;\theta) - \KL{\Ryt}{\log \Prob(\bZ,\bW|\bY;\theta)} = \mathcal{J}(\tau;\theta) \qed$ +\end{frame} -\section{Sélection de modèle} +\section{Résultats~\cite{baldockSystemsApproachReveals2019a,baldockDailyTemporalStructure2011}} +\begin{frame}[allowframebreaks] + \begin{figure}[ht] + \centering + \begin{subfigure}[t]{0.5\textwidth} + \centering + \includestandalone[width=0.45\textwidth]{tikz/applications/baldock/mat-Baldock2019_Bristol} + \caption{Donnée} + \end{subfigure}\hfil + \begin{subfigure}[t]{0.5\textwidth} + \centering + \includestandalone[width=0.45\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Bristol} + \caption{Réordonnée} + \end{subfigure} + \caption{Bristol} + \end{figure} + + \begin{figure}[ht] + \centering + \begin{subfigure}[t]{0.5\textwidth} + \centering + \includestandalone[width=0.45\textwidth]{tikz/applications/baldock/mat-Baldock2019_Edinburgh} + \caption{Donnée} + \end{subfigure}\hfil + \begin{subfigure}[t]{0.5\textwidth} + \centering + \includestandalone[width=0.5\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Edinburgh} + \caption{Réordonnée} + \end{subfigure} + \caption{Edinburgh} + \end{figure} + + \begin{figure} + \begin{subfigure}[ht]{0.5\textwidth} + \centering + \includestandalone[width=0.5\textwidth]{tikz/applications/baldock/mat-Baldock2019_Leeds} + \caption{Donnée} + \end{subfigure}\hfil + \begin{subfigure}[ht]{0.5\textwidth} + \centering + \includestandalone[width=0.5\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Leeds} + \caption{Réordonnée} + \end{subfigure} + \caption{Leeds} + \end{figure} + + \begin{figure} + \begin{subfigure}[ht]{0.5\textwidth} + \centering + \includestandalone[width=0.5\textwidth]{tikz/applications/baldock/mat-Baldock2019_Reading} + \caption{Donnée} + \end{subfigure}\hfil + \begin{subfigure}[ht]{0.5\textwidth} + \centering + \includestandalone[width=0.5\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Reading} + \caption{Réordonnée} + \end{subfigure} + \caption{Reading} + \end{figure} +\end{frame} \ No newline at end of file diff --git a/soutenance/principal.tex b/soutenance/principal.tex index 5ad7d26..d3b0330 100644 --- a/soutenance/principal.tex +++ b/soutenance/principal.tex @@ -2,205 +2,278 @@ \label{sec:contexte-du-modele} \begin{frame} + \frametitle{Pourquoi un réseau ?} \begin{columns} \begin{column}{0.5\textwidth} - \begin{block}{Contexte écologique} - \begin{itemize} - \small - \item Nombreux réseaux disponibles pour - interactions similaires. - \item Suivi biodiversité, robustesse et risque - d'effondrement \dots - \end{itemize} - \begin{columns} - \begin{column}{0.5\textwidth} - \begin{figure}[ht] - \centering - \begin{tikzpicture}[scale=.45,rotate=270] - \input{../tikz/plantpollinatornetwork.tex} - \end{tikzpicture} - \caption{Exemple d'un réseau plantes-pollinisateurs} - \label{fig:plantes-pollin} - \end{figure} - \end{column} - \begin{column}{0.4\textwidth} + \begin{columns} + \begin{column}{0.5\textwidth} + \begin{figure}[ht] \centering - \begin{align*} - \begin{pmatrix} - 1 & 0 & 1 \\ - 1 & 0 & 0 \\ - 1 & 0 & 0 \\ - 1 & 1 & 0 - \end{pmatrix} - \end{align*} - \footnotesize - Matrice d'adjacence associée - \end{column} - \end{columns} - \end{block} - + \begin{tikzpicture}[scale=.6,rotate=270] + \input{tikz/plantpollinatornetwork.tex} + \end{tikzpicture} + \caption{Exemple d'un réseau} + \label{fig:plantes-pollin} + \end{figure} + \end{column} + \begin{column}{0.3\textwidth} + \centering + \begin{align*} + \begin{pmatrix} + 1 & 0 & 1 \\ + 1 & 0 & 0 \\ + 1 & 0 & 0 \\ + 1 & 1 & 0 + \end{pmatrix} + \end{align*} + \footnotesize + Matrice d'adjacence associée + \end{column} + \end{columns} + \begin{figure}[ht] + \centering + \includestandalone[width=0.7\textwidth]{tikz/applications/baldock/graph-Baldock2019_Bristol} + \caption{Réseau plante-pollinisateur de + Bristol\newline\cite{baldockSystemsApproachReveals2019a}} + \label{fig:label} + \end{figure} + \end{column} + \begin{column}{0.5\textwidth} + \begin{itemize} + \item Modélisation d'interactions variées, ici d'écosystèmes + \item Structure nécessaire pour~: suivi biodiversité, + robustesse, risque d'effondrement + \item De plus en plus disponibles + \end{itemize} \end{column} - \onslide<2>{ - \begin{column}{0.45\textwidth} - \begin{block}{Contexte mathématique} - Pour un unique réseau~: variables latentes, - \emph{embedding}, \dots - - Motivations pour proposer des méthodes adaptées aux collections - de réseaux~: - \begin{itemize} - \item Espèces différentes, rôles analogues. - \item Transfert d'informations grands vers petits réseaux. - \item Regrouper les réseaux selon leur similarité (\emph{clustering} - de réseaux). - \end{itemize} - \end{block} - \end{column} - } \end{columns} \end{frame} +\begin{frame}{Méthodes d'analyse pour un réseau} + Plusieurs méthodes~: + \begin{itemize} + \item Métriques~: degré, centralité, emboîtement \dots + \item Plongement des réseaux avec GNN + \item \textbf<2>{\emph{Clustering} des n\oe uds avec modèles à variables latentes} + \end{itemize} +\end{frame} + \begin{frame} \addtocounter{footnote}{1} \frametitle{Latent Block Model (LBM\footnotemark[\thefootnote])} %DONE remplacer i \in bullet par Zi = \bullet - Proposé par~\cite{govaertEMAlgorithmBlock2005}. + \cite{govaertEMAlgorithmBlock2005}. \begin{columns} \begin{column}{0.40\linewidth} \begin{figure}[H] \center \begin{tikzpicture}[scale=0.35] - \input{../tikz/lbm.tex} + \input{tikz/lbm.tex} \end{tikzpicture} \caption{Exemple de LBM\footnotemark[\thefootnote]} \label{fig:LBMvisu} \end{figure} \end{column} - \begin{column}{0.51\linewidth} - Pour \begin{itemize} - \item $Q_1 = |\{{\color{blueind}\bullet},{\color{cyanind}\bullet},{\color{electricblue}\bullet}\}|$ blocs fixés en ligne - \item $Q_2 = |\{{\color{burntorange}\bullet},{\color{goldenyellow}\bullet},{\color{peach}\bullet}\}|$ blocs fixés en colonne - \end{itemize} - \begin{block}{Paramètres} - \begin{itemize} - \item $\pi_{\bullet} = \mathbb{P}(Z_i = \bullet)$ en ligne et $\rho_{\bullet} = \mathbb{P}(W_j = \bullet)$ en colonne - \item $\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}} = \mathbb{P}(X_{ij} = 1 | Z_i = {\color{blueind}\bullet}, W_j = {\color{burntorange}\bullet})$ + \only<1>{ + \begin{column}{0.51\linewidth} + \begin{block}{Modèle hiérarchique} + \vspace{-\baselineskip} + \begin{align*} + \forall q\in[\![ 1, Q_1]\!],~ & \mathbb{P}(Z_i = q) = \pi_q \\ + \forall r\in[\![ 1, Q_2]\!],~ & \mathbb{P}(W_j = r) = \rho_r \\ + & Y_{ij} | Z_i, W_j \sim \mathcal{F}(\alpha_{Z_i,W_j}) + \end{align*} + où $|\pi| = Q_1, |\rho| = Q_2, |\alpha| = Q_1 \times Q_2$ + \end{block} + \begin{block}{Formule concise LBM} + $Y \sim \mathcal{F}\text{-BiSBM}_{n_1,n_2}(Q_1, Q_2, \pi, \rho, \alpha)$ + \end{block} + \end{column}} + \only<2>{ + \begin{column}{0.51\linewidth} + Avec \begin{itemize} + \item $Q_1 = |\{{\color{blueind}\bullet},{\color{cyanind}\bullet},{\color{electricblue}\bullet}\}|$ blocs fixés en ligne + \item $Q_2 = |\{{\color{burntorange}\bullet},{\color{goldenyellow}\bullet},{\color{peach}\bullet}\}|$ blocs fixés en colonne \end{itemize} - \end{block} - \end{column} + \begin{block}{Paramètres} + \begin{itemize} + \item $\pi_{{\color{blueind}\bullet}} = \mathbb{P}(Z_i = {\color{blueind}\bullet})$ + \item $\rho_{{\color{burntorange}\bullet}} = \mathbb{P}(W_j = {\color{burntorange}\bullet})$ + \item $\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}} = \mathbb{P}(Y_{ij} = 1 | Z_i = {\color{blueind}\bullet}, W_j = {\color{burntorange}\bullet})$ + \end{itemize} + \end{block} + \end{column}} \end{columns} \footnotetext[\thefootnote]{Que j'appellerai par la suite BiSBM} - \end{frame} -\section{Modèle de collection de réseaux bipartites} +\begin{frame} + \frametitle{Plusieurs réseaux} + \begin{figure}[ht] + \centering + \begin{subfigure}[ht]{0.3\textwidth} + \includestandalone[width=1.1\textwidth]{tikz/applications/baldock/mat-Baldock2019_Bristol} + \caption{Bristol} + \end{subfigure} + \begin{subfigure}[ht]{0.3\textwidth} + \includestandalone[width=1.1\textwidth]{tikz/applications/baldock/mat-Baldock2019_Edinburgh} + \caption{Edinburgh} + \end{subfigure} + \begin{subfigure}[ht]{0.3\textwidth} + \includestandalone[width=1.1\textwidth]{tikz/applications/baldock/mat-Baldock2019_Leeds} + \caption{Leeds} + \end{subfigure} + \caption{Matrices d'adjacence,~\cite{baldockSystemsApproachReveals2019a}} + \label{fig:adj} + \end{figure} +\end{frame} + + +\section[Modèles collection bipartites]{Modèles de collection de réseaux bipartites} \label{sec:extension-de-colsbm-aux-reseaux-bipartites} \begin{frame} \frametitle{Collections bipartites} - \begin{tikzpicture}[scale=0.33] - \input{../tikz/collbm-iid.tex} - \end{tikzpicture} - - \begin{itemize} - \item $Q_1 = |\{{\color{blueind}\bullet},{\color{cyanind}\bullet},{\color{electricblue}\bullet}\}|$ blocs fixés en ligne - \item $Q_2 = |\{{\color{burntorange}\bullet},{\color{goldenyellow}\bullet},{\color{peach}\bullet}\}|$ blocs fixés en colonne - \end{itemize} - \begin{block}{Paramètres} - \begin{itemize} - \item $\pi_{\bullet} = \mathbb{P}(Z_i =\bullet)$ en ligne et $\rho_{\bullet} = \mathbb{P}(W_j = \bullet)$ en colonne - \item $\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}} = \mathbb{P}(X_{ij} = 1 | Z_i = {\color{blueind}\bullet}, W_j = {\color{burntorange}\bullet})$ - \end{itemize} - \end{block} + \[ + \forall m \in \{1\dots M\}, Y^m \overset{ind}{\sim} \mathcal{F}\text{-BiSBM}_{n_1^m,n_2^m}(Q_1^m, Q_2^m, \pi^m, \rho^m, \alpha^m) + \] + \onslide<2>{ + \begin{figure}[ht] + \centering + \begin{subfigure}[ht]{0.3\textwidth} + \includestandalone[width=1.1\textwidth]{tikz/applications/baldock/bisbm-mat-Baldock2019_Bristol} + \caption{Bristol} + \end{subfigure} + \begin{subfigure}[ht]{0.3\textwidth} + \includestandalone[width=1.1\textwidth]{tikz/applications/baldock/bisbm-mat-Baldock2019_Edinburgh} + \caption{Edinburgh} + \end{subfigure} + \begin{subfigure}[ht]{0.3\textwidth} + \includestandalone[width=1.1\textwidth]{tikz/applications/baldock/bisbm-mat-Baldock2019_Leeds} + \caption{Leeds} + \end{subfigure} + \caption{Matrices d'adjacence réordonnées, grâce au LBM} + \label{fig:adj-reord} + \end{figure} + } \end{frame} \begin{frame} \frametitle{Différents modèles} - \only<1>{ - \begin{tikzpicture}[scale=0.33] - \input{../tikz/collbm-iid.tex} - \end{tikzpicture} - \begin{block}{\emph{iid-colBiSBM}} - $\bm{\pi} = (\pi_1, \dots \pi_{Q_1})$ et $\bm{\rho} = (\rho_1, \dots \rho_{Q_2})$ + \onslide<1->{ \begin{block}{\emph{iid}-colBiSBM} + \[ + \forall m \in \{1\dots M\}, Y^m \overset{iid}{\sim} + \mathcal{F}\text{-BiSBM}_{n_1^m,n_2^m}(Q_1, Q_2, \pi, \rho, \alpha) + \] + + avec $\theta = (\pi, \rho, \alpha)$. + \end{block}} + \onslide<2>{ \begin{block}{$\pi\rho$-colBiSBM} + \[ + \forall m \in \{1\dots M\}, Y^m \overset{ind}{\sim} + \mathcal{F}\text{-BiSBM}_{n_1^m,n_2^m}(Q_1, Q_2, \pi^m, \rho^m, \alpha) + \] + + avec $\theta = ((\pi^m)_{m=1,\dots, M}, (\rho^m)_{m=1,\dots, M}, \alpha)$. \end{block} } - \only<2>{ - \begin{tikzpicture}[scale=0.33] - \input{../tikz/collbm-pirho.tex} - \end{tikzpicture} - \begin{block}{\emph{$\pi\rho$-colBiSBM}} - $\bm{\pi} = ((\pi_{\color{black}1}^{\color{red}m}, \dots \pi_{\color{black}Q_1}^{\color{red}m}))_{m=1,\dots M}$ 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 $\forall r \in \llbracket 1, Q_2 - 1\rrbracket, \rho_r > 0$} - \small \\ - avec $\forall q,m \in \llbracket 1, Q_1 \rrbracket \times \llbracket 1, M \rrbracket, \pi_q^m \in \left[ 0,1 \right]$ - et $\forall r,m \in \llbracket 1, Q_2 \rrbracket \times \llbracket 1, M \rrbracket, \rho_r^m \in \left[ 0,1 \right]$ - \end{block} - } - Dans tous les modèles la structure de connectivité ($\bm{\alpha}$) est supposée identique au sein de la collection. \end{frame} \begin{frame} \frametitle{Estimation des paramètres} % DONE dire que tau i q m c' est la proba que Zim = q, approximation de la proba variationnelle. Parce qu on impose lindependance % Par maximisation d'une borne inférieure variationnelle de la % log-vraisemblance des données observées. - En adaptant \cite{chabert-liddellLearningCommonStructures2024a} qui se base - sur la méthode proposée par \cite{daudinMixtureModelRandom2008} utilisant - l'algorithme \emph{Variational EM}. + Maximisation de la log-vraisemblance ? + \begin{block}{log-vraisemblance et log-vraisemblance complète} + \[ + \ell(\bm{Y};\theta) = \sum_{\bm{Z,W}\in \bm{\mathcal{Z}\times\mathcal{W}}} \ell_c(\bm{Y}, \bm{Z}, \bm{W};\theta) + \] + + avec $\bm{\mathcal{Z}} = \{1,\dots,\alert<2>{Q_1}\}^{\alert<2>{n}}, \bm{\mathcal{W}} = \{1,\dots,\alert<2>{Q_2}\}^{\alert<2>{n}}$ + \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} @@ -300,12 +374,73 @@ \section{Application} \label{sec:application} +\begin{frame} + \frametitle{Résultats~\cite{baldockSystemsApproachReveals2019a}} + + \only<1>{ + \begin{figure}[ht] + \centering + \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} - \centering - \begin{tikzpicture} - \input{../tikz/clustering.tex} - \end{tikzpicture} + \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} diff --git a/soutenance/soutenance.bcf-SAVE-ERROR b/soutenance/soutenance.bcf-SAVE-ERROR new file mode 100644 index 0000000..8cabf0a --- /dev/null +++ b/soutenance/soutenance.bcf-SAVE-ERROR @@ -0,0 +1,3106 @@ + + + + + + output_encoding + utf8 + + + input_encoding + utf8 + + + debug + 0 + + + mincrossrefs + 999 + + + minxrefs + 2 + + + sortcase + 1 + + + sortupper + 1 + + + + + + + alphaothers + + + + + extradatecontext + labelname + labeltitle + + + labelalpha + 0 + + + labelnamespec + shortauthor + author + shorteditor + editor + + + labeltitle + 0 + + + labeltitlespec + shorttitle + title + maintitle + + + labeltitleyear + 0 + + + labeldateparts + 1 + + + labeldatespec + pubstate + date + eventdate + year + nodate + + + julian + 0 + + + gregorianstart + 1582-10-15 + + + maxalphanames + 3 + + + maxbibnames + 20 + + + maxcitenames + 2 + + + maxsortnames + 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inbook + bookinbook + suppbook + + author + title + booktitle + + + + booklet + + + author + editor + + title + + + + collection + reference + mvcollection + mvreference + + editor + title + + + + incollection + suppcollection + inreference + + author + editor + title + booktitle + + + + dataset + + title + + + + manual + + title + + + + misc + software + + title + + + + online + + title + + url + doi + eprint + + + + + patent + + author + title + number + + + + periodical + + editor + title + + + + proceedings + mvproceedings + + title + + + + inproceedings + + author + title + booktitle + + + + report + + author + title + type + institution + + + + thesis + + author + title + type + institution + + + + unpublished + + author + title + + + + + isbn + + + issn + + + ismn + + + gender + + + + book + inbook + article + report + + author + title + + + + + + + ../references.bib + + + + + + ../references.bib + + + baldockSystemsApproachReveals2019a + baldockSystemsApproachReveals2019a + baldockSystemsApproachReveals2019a + baldockSystemsApproachReveals2019a + govaertEMAlgorithmBlock2005 + govaertEMAlgorithmBlock2005 + daudinMixtureModelRandom2008 + chabert-liddellLearningCommonStructures2024a + baldockSystemsApproachReveals2019a + baldockSystemsApproachReveals2019a + baldockSystemsApproachReveals2019a + baldockSystemsApproachReveals2019a + baldockSystemsApproachReveals2019a + baldockSystemsApproachReveals2019a + baldockDailyTemporalStructure2011 + baldockDailyTemporalStructure2011 + baldockDailyTemporalStructure2011 + baldockSystemsApproachReveals2019a + baldockDailyTemporalStructure2011 + baldockSystemsApproachReveals2019a + baldockDailyTemporalStructure2011 + baldockSystemsApproachReveals2019a + baldockDailyTemporalStructure2011 + baldockSystemsApproachReveals2019a + baldockDailyTemporalStructure2011 + baldockSystemsApproachReveals2019a + baldockDailyTemporalStructure2011 + baldockSystemsApproachReveals2019a diff --git a/soutenance/soutenance.pdf 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