688 lines
No EOL
30 KiB
TeX
688 lines
No EOL
30 KiB
TeX
\begin{frame}[noframenumbering]
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\vfill
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\centering
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\begin{beamercolorbox}[sep=8pt,center,shadow=true,rounded=true]{title}
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\usebeamerfont{title}Annexes\par%
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\end{beamercolorbox}
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\vfill
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\end{frame}
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\section{Modèles à variables latentes pour collection de réseaux bipartites}
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\begin{frame}
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\frametitle{Latent Block Model (LBM)}
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%DONE remplacer i \in bullet par Zi = \bullet
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Proposé par~\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{figures/lbm.tex}
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\end{tikzpicture}
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\caption{Exemple de LBM\footnotemark}
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\label{fig:LBMvisu}
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\end{figure}
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\end{column}
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\begin{column}{0.51\linewidth}
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Pour \begin{itemize}
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\item $Q_1 =
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|\{{\color{blueind}\bullet},{\color{cyanind}\bullet},{\color{electricblue}\bullet}\}|$
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blocs fixés en ligne
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\item $Q_2 =
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|\{{\color{burntorange}\bullet},{\color{goldenyellow}\bullet},{\color{peach}\bullet}\}|$
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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
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et $\rho_{\bullet} = \mathbb{P}(W_j = \bullet)$ en colonne
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\item
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$\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}} =
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\mathbb{P}(X_{ij} = 1 | Z_i = {\color{blueind}\bullet}, W_j =
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{\color{burntorange}\bullet})$
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\end{itemize}
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\end{block}
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\end{column}
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\end{columns}
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\footnotetext{Que j'appellerai par la suite BiSBM}
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\end{frame}
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\begin{frame}
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\frametitle{\emph{colSBM}}
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Le modèle \emph{colSBM}
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\parencite{chabert-liddellLearningCommonStructures2023}.\\
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% Difficulté estimer les parametres
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% DONE Modifier les realisations pour variabilite, mettre iid au dessus du sim et inverser modele et realisations
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\smallskip
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\definecolor{yellow}{RGB}{255,190,60}
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\begin{center}
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\begin{adjustbox}{trim=0 0 0 1cm}
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\begin{tikzpicture}[scale=.32]
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\tikzstyle{every edge}=[-,>=stealth',shorten
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>=1pt,auto,draw,line width=.5pt, bend left]
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\tikzstyle{every state}=[draw, text=black,scale=0.95, transform
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shape]
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\tikzset{edge_proba/.style={draw=white, fill=none, text=black}}
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\tikzstyle{every node}=[fill=yellow]
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\node[state, draw=black!50] (A1) at (0,2) {\textbf{A1}};
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\node[state, draw=black!50] (A2) at (1.5, 2) {\textbf{A2}};
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\node[state, draw=black!50] (A3) at (0.75,3.25) {\textbf{A3}};
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\tikzstyle{every node}=[fill=blueind]
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\node[state, draw=black!50] (B1) at (4.5,3) {\textbf{B1}};
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\node[state, draw=black!50] (B2) at (4,4.75) {\textbf{B2}};
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\node[state, draw=black!50] (B3) at (5.5,6) {\textbf{B3}};
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\node[state, draw=black!50] (B4) at (7,4.75) {\textbf{B4}};
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\node[state, draw=black!50] (B5) at (6.5,3) {\textbf{B5}};
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\tikzstyle{every node}=[fill=greenind]
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\node[state, draw=black!50] (C1) at (5,0) {\textbf{C1}};
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\node[state, draw=black!50] (C2) at (7,1) {\textbf{C2}};
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\path (A1) edge[bend right] (A2);
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\path (A1) edge node[midway, left, fill=none]
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{$\alpha_{{\color{yellow}\bullet}{\color{yellow}\bullet}}$} (A3);
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\path (A3) edge (A2);
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\path (A3) edge node[midway, above, fill=none]
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{$\alpha_{{\color{yellow}\bullet}{\color{blueind}\bullet}}$} (B3);
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\path (B1) edge (B2);
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\path (B2) edge (B3);
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\path (B3) edge (B4);
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\path (B4) edge (B5);
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\path (B5) edge (B1);
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\path (B1) edge[bend left=0] (B4);
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\path (B5) edge[bend left=0] (B2);
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\path (A2) edge[bend right] node[midway, below, fill=none]
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{$\alpha_{{\color{yellow}\bullet}{\color{greenind}\bullet}}$} (C1);
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\path (C1) edge[bend right] node[midway, below, fill=none]
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{$\alpha_{{\color{greenind}\bullet}{\color{greenind}\bullet}}$} (C2);
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\path (C2) edge[bend right] node[midway, right, fill=none]
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{$\alpha_{{\color{greenind}\bullet}{\color{blueind}\bullet}}$} (B4);
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\node[font=\small, text justified,draw=none, fill=none] at
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(4.5,-1.5) {SBM};
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% Sampled network
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\begin{scope}[xshift=-16cm,yshift=4cm]
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\node[font=\small, text justified, fill=none] at (10, -2.5)
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{$\overset{iid}{\sim}$};
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\tikzstyle{every node}=[fill=gray, scale=0.95]
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\tikzstyle{every edge}=[-,>=stealth',shorten
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>=1pt,auto,draw,line width=.5pt, bend left]
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\tikzstyle{every state}=[draw, text=black,scale=0.95,
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transform shape]
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\node[state, draw=black!50] (A1) at (0,0) {\textbf{10}};
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\node[state, draw=black!50] (A2) at (1, 0) {\textbf{2}};
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\node[state, draw=black!50] (A3) at (0.5,1) {\textbf{5}};
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\node[state, draw=black!50] (B2) at (2,2.75) {\textbf{9}};
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\node[state, draw=black!50] (B3) at (3.5,4) {\textbf{6}};
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\node[state, draw=black!50] (B4) at (5,2.75) {\textbf{3}};
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\node[state, draw=black!50] (B5) at (4.5,1) {\textbf{7}};
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\node[state, draw=black!50] (C1) at (3,-0.5) {\textbf{4}};
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\path (A1) edge[bend right] (A2);
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\path (A1) edge (A3);
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\path (A3) edge (A2);
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\path (A3) edge (B3);
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\path (B2) edge (B3);
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\path (B3) edge (B4);
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\path (B4) edge (B5);
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\path (B5) edge[bend left=0] (B2);
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\path (A2) edge[bend right] (C1);
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\node[text width=3cm,font=\small, text justified,
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rotate=90, fill=none, below = -0.8cm of C1] (dots) {\dots};
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\end{scope}
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\begin{scope}[xshift=-16cm,yshift=-4cm]
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\tikzstyle{every node}=[fill=gray, scale=0.95]
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\tikzstyle{every edge}=[-,>=stealth',shorten
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>=1pt,auto,draw,line width=.5pt, bend left]
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\tikzstyle{every state}=[draw, text=black,scale=0.95,
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transform shape]
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\node[state, draw=black!50] (A2) at (1, 0) {\textbf{2}};
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\node[state, draw=black!50] (A3) at (0.5,1) {\textbf{1}};
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\node[state, draw=black!50] (B1) at (2.5,1) {\textbf{5}};
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\node[state, draw=black!50] (B2) at (2,2.75) {\textbf{10}};
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\node[state, draw=black!50] (B4) at (5,2.75) {\textbf{8}};
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\node[state, draw=black!50] (B5) at (4.5,1) {\textbf{7}};
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\node[state, draw=black!50] (C2) at (5,0) {\textbf{3}};
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\path (A3) edge (A2);
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\path (B1) edge (B2);
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\path (B4) edge (B5);
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\path (B5) edge (B1);
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\path (B1) edge[bend left=0] (B4);
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\path (B5) edge[bend left=0] (B2);
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\path (C2) edge[bend right] (B4);
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\end{scope}
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\end{tikzpicture}
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\end{adjustbox}
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\end{center}
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Pour $Q =
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|\{{\color{yellow}\bullet},{\color{blueind}\bullet},{\color{greenind}\bullet}\}|$
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blocs fixés :
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\begin{block}{Paramètres}
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\begin{itemize}
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\item $\pi_{\bullet} = \mathbb{P}(Z_i =\bullet)$
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\item $\alpha_{{\color{greenind}\bullet}{\color{blueind}\bullet}} =
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\mathbb{P}(X_{ij} = 1 | Z_i = {\color{greenind}\bullet}, Z_j =
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{\color{blueind}\bullet})$
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\end{itemize}
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\end{block}
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\end{frame}
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\begin{frame}
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\frametitle{Collections bipartites}
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\begin{center}
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\begin{adjustbox}{trim=0 0 1 1.5cm}
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\begin{tikzpicture}[scale=.33]
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\begin{scope}[xshift=18cm, yshift=2cm]
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\input{figures/lbm.tex}
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\end{scope}
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\begin{scope}[xshift=3cm, yshift = 1cm]
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\node[text justified, fill=none] at (10, 3.5)
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{$\overset{iid}{\sim}$};
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\begin{scope}[yshift = 6cm]
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\tikzstyle{every state}=[draw, text=black,scale=0.75,
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transform shape]
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\tikzstyle{every node}=[fill=gray]
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\node[state, draw=black!50] (R11) at (0,1.25)
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{\textbf{1}};
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\node[state, draw=black!50] (R12) at (1,1.25)
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{\textbf{2}};
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\node[state, draw=black!50] (R13) at (2,1.25)
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{\textbf{3}};
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\node[state, draw=black!50] (R21) at (3,1.25)
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{\textbf{4}};
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\node[state, draw=black!50] (R31) at (5,1.25)
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{\textbf{6}};
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\tikzstyle{every
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state}=[draw=none,text=black,scale=0.75, transform shape, shape=rectangle]
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\node[state, draw=black!50] (B1) at (0.5,-1)
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{\textbf{1}};
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\node[state, draw=black!50] (B31) at (2.5,-1)
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{\textbf{3}};
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\node[state, draw=black!50] (B4) at (3.5,-1)
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{\textbf{4}};
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\node[state, draw=black!50] (B5) at (4.5,-1)
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{\textbf{5}};
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\tikzstyle{every edge}=[-,>=stealth',shorten
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>=1pt,auto,draw,line width=1pt, draw=gray, fill=gray]
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\path (R11) edge (B1);
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\path (R11) edge (B31);
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\path (R11) edge (B4);
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\path (R12) edge [] (B1);
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\path (R12) edge (B31);
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\path (R12) edge (B4);
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\path (R13) edge [] (B1);
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\path (R13) edge (B31);
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\path (R13) edge (B4);
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\path (R21) edge (B31);
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\path (R21) edge (B4);
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\path (R21) edge (B5);
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\path (R31) edge (B5);
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\end{scope}
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\node[text width=3cm,font=\small, text justified,
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rotate=90, fill=none] (dots) at (2.5, 7.5){\dots};
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\begin{scope}[yshift = 0cm]
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\tikzstyle{every state}=[draw, text=black,scale=0.75,
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transform shape]
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\tikzstyle{every node}=[fill=gray]
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\node[state, draw=black!50] (R11) at (0,2.25)
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{\textbf{4}};
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\node[state, draw=black!50] (R13) at (2,2.25)
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{\textbf{6}};
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\node[state, draw=black!50] (R21) at (3,2.25)
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{\textbf{3}};
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\node[state, draw=black!50] (R22) at (4,2.25)
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{\textbf{5}};
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\node[state, draw=black!50] (R31) at (5,2.25)
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{\textbf{2}};
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|
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\tikzstyle{every
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state}=[draw=none,text=black,scale=0.75, transform shape, shape=rectangle]
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\node[state, draw=black!50] (B1) at (0.5,0)
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{\textbf{5}};
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\node[state, draw=black!50] (B2) at (1.5,0)
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{\textbf{1}};
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\node[state, draw=black!50] (B4) at (3.5,0)
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{\textbf{2}};
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\node[state, draw=black!50] (B5) at (4.5,0)
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{\textbf{4}};
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\tikzstyle{every edge}=[-,>=stealth',shorten
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>=1pt,auto,draw,line width=1pt, draw=gray, fill=gray]
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\path (R11) edge (B1);
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\path (R11) edge (B2);
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\path (R11) edge (B4);
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\path (R13) edge [] (B1);
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\path (R13) edge (B2);
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\path (R13) edge (B4);
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\path (R21) edge (B4);
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\path (R21) edge (B5);
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\path (R22) edge (B4);
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\path (R22) edge (B5);
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\path (R31) edge (B5);
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\end{scope}
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\end{scope}
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\end{tikzpicture}
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\end{adjustbox}
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\end{center}
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Pour
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\begin{itemize}
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\item $Q_1 =
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|\{{\color{blueind}\bullet},{\color{cyanind}\bullet},{\color{electricblue}\bullet}\}|$
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blocs fixés en ligne
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\item $Q_2 =
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|\{{\color{burntorange}\bullet},{\color{goldenyellow}\bullet},{\color{peach}\bullet}\}|$
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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
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$\rho_{\bullet} = \mathbb{P}(W_j = \bullet)$ en colonne
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\item
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$\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}} =
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\mathbb{P}(X_{ij} = 1 | Z_i = {\color{blueind}\bullet}, W_j =
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{\color{burntorange}\bullet})$
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\end{itemize}
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\end{block}
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\end{frame}
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\begin{frame}
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\frametitle{Différents modèles}
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\begin{block}{\emph{iid-colBiSBM}}
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$\bm{\pi} = (\pi_1, \dots \pi_{Q_1})$ et $\bm{\rho} = (\rho_1, \dots
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\rho_{Q_2})$ %{$\forall q \in \llbracket 1, Q_1 - 1\rrbracket, \pi_q > 0$ et $\forall r \in \llbracket 1, Q_2 - 1\rrbracket, \rho_r > 0$}
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, tous les réseaux partagent les mêmes paramètres\footnotemark
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\end{block}
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\begin{block}{\emph{$\pi\rho$-colBiSBM}}
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$\bm{\pi} = ((\pi_{\color{black}1}^{\color{red}m}, \dots
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\pi_{\color{black}Q_1}^{\color{red}m}))_{m=1,\dots M}$ et $\bm{\rho} =
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((\rho_{\color{black}1}^{\color{red}m}, \dots
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\rho_{\color{black}Q_2}^{\color{red}m}))_{m=1,\dots M}$
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%{$\forall q \in \llbracket 1, Q_1 - 1\rrbracket, \pi_q > 0$ et $\forall r \in \llbracket 1, Q_2 - 1\rrbracket, \rho_r > 0$}
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\small \\
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avec $\forall q,m \in \llbracket 1, Q_1 \rrbracket \times \llbracket 1,
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M \rrbracket, \pi_q^m \in \left[ 0,1 \right]$
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et $\forall r,m \in \llbracket 1, Q_2 \rrbracket \times \llbracket 1, M
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\rrbracket, \rho_r^m \in \left[ 0,1 \right]$
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\end{block}
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Et également deux autres modèles ($\pi$-colBiSBM et $\rho$-colBiSBM) où
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seulement une des deux dimensions est libre.
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\footnotetext{Dans tous les modèles la structure de connectivité est
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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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Maximisation d'une borne inférieure de la log-vraisemblance des données
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observées.
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\begin{multline*}
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\ell (\bm{X};\bm{\theta}) \geq \color{red}\sum_{m=1}^{M} \bigg(
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\color{black} \sum_{i = 1}^{n_1^m}\sum_{j=1}^{n_2^m}\sum_{q \in
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\mathcal{Q}_{1,m}} \sum_{r \in \mathcal{Q}_{2,m}} \tau^{1,m}_{i,q}
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\tau^{2,m}_{j,r} \log f(X^{m}_{ij}; \alpha_{qr}) \\
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+ \sum_{i=1}^{n_1^m} \sum_{q \in \mathcal{Q}_{1,m}} \tau^{1,m}_{i,q}
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\log \pi_{\color{black}q}^{\color{gray}m} + \sum_{j=1}^{n_2^m} \sum_{r \in
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\mathcal{Q}_{2,m}} \tau^{2,m}_{j,r} \log \rho_{\color{black}r}^{\color{gray}m}
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\\
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- \sum_{i=1}^{n_1} \tau^{1,m}_{i,q} \log \tau^{1,m}_{i,q} -
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\sum_{j=1}^{n_2} \tau^{2,m}_{j,r} \log \tau^{2,m}_{j,r} \color{red}\bigg)
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\color{black} =: J(\bm{\tau};\bm{\theta}) $$
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\end{multline*}
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\begin{block}{Approximation variationnelle}
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$\tau_{i,q}^{1,m} = P(Z_i = q | X^m_{ij})$ et $\tau_{j,r}^{2,m} = P(W_j
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= r | X^m_{ij})$ tels que $P(Z_i = q, W_j = r | X^m_{ij}) =
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\tau_{i,q}^{1,m}\times\tau_{j,r}^{2,m}$
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\end{block}
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\end{frame}
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\begin{frame}
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\frametitle{Sélection de modèle : choix de $(Q_1,Q_2)$ - Approche
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gloutonne}
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% DONE But maximiser un critere le BICL, deplacer voir St Clair dans la note
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% VEM a Q1 Q2 fixer
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% Choix de Q1 Q2 par maximisation du BICL
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% Itemize dans la box : init, explo voisin, arrets
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\underline{Le VEM se fait à $Q_1, Q_2$ fixés}, il faut donc déterminer les
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\enquote*{meilleures} coordonnées.
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Nous maximisons un BIC-L\footnote{\emph{Bayesian Information Criterion -
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Like}, en adaptant les formules
|
|
de~\cite{chabert-liddellLearningCommonStructures2023}}.
|
|
|
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Détermination d'un premier mode par approche \emph{gloutonne} \smallskip
|
|
\begin{columns}
|
|
\begin{column}{0.5\linewidth}
|
|
\begin{tikzpicture}
|
|
\draw[step=1cm, help lines] (-2,-2) grid (2,2);
|
|
\draw[fill=gray, draw=gray] (0,0) circle [radius=0.225cm];
|
|
\draw[fill=blueind, draw=blueind] (1,0) circle
|
|
[radius=0.225cm];
|
|
\draw[fill=blueind, draw=blueind] (0,1) circle
|
|
[radius=0.225cm];
|
|
\draw[fill=red, draw=red] (-1,0) circle [radius=0.225cm];
|
|
\draw[fill=red, draw=red] (0,-1) circle [radius=0.225cm];
|
|
|
|
% Légende
|
|
\node[font=\tiny, text justified,fill=none, rotate=-45]
|
|
(Splits) at (0.5,0.5){{\color{blueind} Splits}};
|
|
\node[font=\tiny, text justified,fill=none, rotate=-45]
|
|
(Merges) at (-0.5,-0.5){{\color{red} Merges}};
|
|
|
|
% Splitting
|
|
\draw[>=stealth,->,thick, draw=blueind] (0.225,0) -- +(0.55,0);
|
|
\draw[>=stealth,->,thick, draw=blueind] (0,0.225) -- +(0,0.55);
|
|
|
|
% Merging
|
|
\draw[>=stealth,->,thick, draw=red] (-0.225,0) -- +(-0.55,0);
|
|
\draw[>=stealth,->,thick, draw=red] (0,-0.225) -- +(0,-0.55);
|
|
|
|
% Axes
|
|
\draw[>=to,->,thick] (-2,-2) -- +(1,0);
|
|
\node[font=\small, fill=none] (Q_1) at (-0.75,-2) {$Q_1$};
|
|
\draw[>=to,->,thick] (-2,-2) -- +(0,1);
|
|
\node[font=\small, fill=none] (Q_2) at (-2,-0.75) {$Q_2$};
|
|
|
|
\end{tikzpicture}
|
|
\end{column}
|
|
\begin{column}{0.5\linewidth}
|
|
\begin{block}{Exploration gloutonne}
|
|
\begin{itemize}
|
|
\item Initialisation sur $(1,2)$ et $(2,1)$
|
|
\item Exploration des 4 voisins et déplacement sur le
|
|
meilleur des 4
|
|
\item Arrêt après 2 étapes successives sans augmentation du
|
|
BIC-L
|
|
\end{itemize}
|
|
\end{block}
|
|
\end{column}
|
|
\end{columns}
|
|
\end{frame}
|
|
\begin{frame}
|
|
\frametitle{Sélection de modèle : choix de $(Q_1,Q_2)$ - Fenêtre glissante}
|
|
\begin{columns}
|
|
\begin{column}{0.60\linewidth}
|
|
\begin{figure}
|
|
\includegraphics[scale=0.18]{img/moving_window.png}
|
|
\caption{Exemple de parcours de fenêtre glissante}
|
|
\end{figure}
|
|
\end{column}
|
|
\begin{column}{0.4\linewidth}
|
|
\definecolor{mypurple}{RGB}{128,0,128}
|
|
\begin{tikzpicture}
|
|
|
|
\tikzstyle{model}=[circle,draw=none,fill=gray]
|
|
\tikzstyle{split}=[>=stealth,->,thick, draw=blueind]
|
|
\tikzstyle{merge}=[>=stealth,->,thick, draw=red]
|
|
\draw[step=1cm, help lines] (-2,-2) grid (2,2);
|
|
\node[model] (mode) at (0,0) {{\color{red}X}};
|
|
|
|
\onslide<2->{
|
|
\draw[color=red, line width=1pt] (-1.5,-1.5) rectangle
|
|
++(3,3);
|
|
}
|
|
\onslide<2-2>{
|
|
|
|
\node[model] (bottom_left) at (-1,-1) {};
|
|
\node[model, opacity=0.6] (row_1) at (0,-1) {};
|
|
\node[model, opacity=0.6] (col_1) at (-1,0) {};
|
|
|
|
\draw[split] (bottom_left) -- (col_1);
|
|
\draw[split] (-1.75,0) -- (col_1);
|
|
\draw[split] (bottom_left) -- (row_1);
|
|
\draw[split] (0,-1.75) -- (row_1);
|
|
|
|
\node[model] (bottom_left) at (-1,-1) {};
|
|
\node[model, draw=blue] (row_1) at (0,-1) {};
|
|
\node[model, draw=blue] (col_1) at (-1,0) {};
|
|
|
|
\node[model, opacity=0.6] (row_2) at (1,-1) {};
|
|
\node[model, opacity=0.6] (col_2) at (-1,1) {};
|
|
|
|
\draw[split] (col_1) -- (col_2);
|
|
\draw[split] (-1.75,1) -- (col_2);
|
|
\draw[split] (row_1) -- (row_2);
|
|
\draw[split] (1,-1.75) -- (row_2);
|
|
\draw[split] (row_1) -- (mode);
|
|
\draw[split] (col_1) -- (mode);
|
|
|
|
\node[model, draw=blue] (row_2) at (1,-1) {};
|
|
\node[model, draw=blue] (col_2) at (-1,1) {};
|
|
\node[model, draw=blue] (mode) at (0,0) {{\color{red}X}};
|
|
|
|
\node[model, opacity=0.6] (row_3) at (1,0) {};
|
|
\node[model, opacity=0.6] (col_3) at (0,1) {};
|
|
|
|
\draw[split] (col_2) -- (col_3);
|
|
\draw[split] (row_2) -- (row_3);
|
|
\draw[split] (mode) -- (row_3);
|
|
\draw[split] (mode) -- (col_3);
|
|
|
|
\node[model, draw=blue] (row_3) at (1,0) {};
|
|
\node[model, draw=blue] (col_3) at (0,1) {};
|
|
|
|
\node[model, opacity=0.6] (top_right) at (1,1) {};
|
|
\draw[split] (col_3) -- (top_right);
|
|
\draw[split] (row_3) -- (top_right);
|
|
|
|
\node[model, draw=blue] (top_right) at (1,1) {};
|
|
}
|
|
\onslide<3->{
|
|
\node[model, draw=mypurple] (top_right) at (1,1) {};
|
|
\node[model, draw=mypurple] (row_3) at (1,0) {};
|
|
\node[model, draw=mypurple] (col_3) at (0,1) {};
|
|
\node[model, draw=mypurple] (row_2) at (1,-1) {};
|
|
\node[model, draw=mypurple] (col_2) at (-1,1) {};
|
|
\node[model, draw=mypurple] (mode) at (0,0)
|
|
{{\color{red}X}};
|
|
\node[model, draw=red] (bottom_left) at (-1,-1) {};
|
|
\node[model, draw=mypurple] (row_1) at (0,-1) {};
|
|
\node[model, draw=mypurple] (col_1) at (-1,0) {};
|
|
|
|
\draw[merge] (1,1.75) -- (top_right);
|
|
\draw[merge] (1.75,1) -- (top_right);
|
|
\draw[merge] (0,1.75) -- (col_3);
|
|
\draw[merge] (1.75,0) -- (row_3);
|
|
\draw[merge] (1.75,-1) -- (row_2);
|
|
\draw[merge] (-1,1.75) -- (col_2);
|
|
|
|
\draw[merge] (top_right) -- (col_3);
|
|
\draw[merge] (top_right) -- (row_3);
|
|
\draw[merge] (col_3) -- (col_2);
|
|
\draw[merge] (row_3) -- (row_2) ;
|
|
\draw[merge] (row_3) -- (mode);
|
|
\draw[merge] (col_3) -- (mode);
|
|
\draw[merge] (col_2) --(col_1);
|
|
\draw[merge] (row_2) -- (row_1);
|
|
\draw[merge] (mode) -- (row_1);
|
|
\draw[merge] (mode) -- (col_1);
|
|
\draw[merge] (col_1) -- (bottom_left);
|
|
\draw[merge] (row_1) -- (bottom_left);
|
|
}
|
|
\end{tikzpicture}
|
|
\end{column}
|
|
\end{columns}
|
|
\end{frame}
|
|
|
|
\begin{frame}
|
|
\frametitle{Clustering de réseaux}
|
|
\begin{columns}
|
|
\begin{column}{0.2\linewidth}
|
|
\begin{block}{Objectif}
|
|
Déterminer une partition qui maximise la somme du BICL de ses
|
|
sous-collections.
|
|
\end{block}
|
|
\end{column}
|
|
\begin{column}{0.78\linewidth}
|
|
\begin{tikzpicture}[scale=0.6, every node/.style={scale=0.75}]
|
|
\tikzstyle{instruct}=[font=\small, text justified,
|
|
rectangle,draw,fill=yellow!50]
|
|
\tikzstyle{first_col}=[rectangle, text justified,
|
|
draw,fill=gray!50]
|
|
\tikzstyle{second_col}=[scale=0.55, circle, draw,fill=red!50]
|
|
\tikzstyle{test}=[font=\small, text justified, diamond,
|
|
aspect=2.5,thick,
|
|
draw=blue,fill=yellow!50,text=blue]
|
|
\tikzstyle{es}=[font=\small, text justified,
|
|
rectangle,draw,rounded corners=4pt,fill=cyanind!25]
|
|
|
|
\node[es] (liste) at (0,4) {Donner une collection à
|
|
partitionner};
|
|
\node[instruct, text width=5cm, below = 0.45cm of liste]
|
|
(1-collection) {Ajuster \emph{colBiSBM}};
|
|
\node[first_col, right = 0.5cm of 1-collection] (1-col-obj) {};
|
|
\node[instruct, text width=5cm, below = 0.45cm of 1-collection]
|
|
(dissimi) {Calculer une matrice de dissimilarité de la collection};
|
|
\node[instruct, text width=5cm, below = 0.45cm of dissimi]
|
|
(2-sous-collection) {Séparer la \emph{collection en 2 sous-collections} et
|
|
ajuster les \emph{colBiSBM}};
|
|
\node[second_col, right = 0.25cm of 2-sous-collection]
|
|
(1-sec-col-obj) {1};
|
|
\node[second_col, right = 0.25cm of 1-sec-col-obj]
|
|
(1-sec-col-obj) {2};
|
|
\node[test,below = 0.45cm of 2-sous-collection, scale=0.5]
|
|
(BICL-test) {$\sum_{i=1}^{2}
|
|
(\text{BIC-L}(\tikz[baseline=-0.25cm]{\node[second_col] {i};} )) >
|
|
\text{BIC-L}(\tikz[baseline=-0.25cm]{\node[first_col] {};})$?};
|
|
\node[es, right = 0.55cm of BICL-test] (sortie) {Renvoyer
|
|
\tikz{\node[rectangle, draw, fill=gray!50, rounded corners=0pt] {};}};
|
|
\node[es, left = 0.45cm of dissimi, text width = 2cm]
|
|
(recursion) {Recommencer sur \tikz{\node[second_col] {1};} et
|
|
\tikz{\node[second_col] {2};} };
|
|
|
|
\tikzstyle{suite}=[->,>=stealth,thick,rounded corners=4pt]
|
|
\draw[suite] (liste) -- (1-collection);
|
|
\draw[suite] (1-collection) -- (dissimi);
|
|
\draw[suite] (dissimi) -- (2-sous-collection);
|
|
\draw[suite] (2-sous-collection) -- (BICL-test);
|
|
\draw[suite] (BICL-test) -| node[near start, above, fill=none]
|
|
{Oui} (recursion);
|
|
\draw[suite] (recursion.north) |- (1-collection.west);
|
|
\draw[suite] (BICL-test) -- node[near start, above, fill=none]
|
|
{Non} (sortie);
|
|
|
|
\end{tikzpicture}
|
|
\end{column}
|
|
\end{columns}
|
|
\blfootnote{Même approche
|
|
que~\cite{chabert-liddellLearningCommonStructures2023}}
|
|
\end{frame}
|
|
|
|
|
|
\begin{frame}
|
|
\frametitle{Application, données plantes pollinisateurs}
|
|
\small
|
|
Voici des résultats du modèle \emph{iid-colBiSBM} sur des données
|
|
plantes-pollinisateurs (\cite{doreRelativeEffectsAnthropogenic2021}
|
|
et~\cite{thebaultDatabasePlantpollinatorNetworks2020})
|
|
% DONE Ajouter un tableau avec le nombre de réseaux dans chaque sous-collection
|
|
\begin{columns}
|
|
\begin{column}{0.49\linewidth}
|
|
\includegraphics[scale=0.30]{img/annual_time_span_vs_iid.png}
|
|
|
|
\begin{center}
|
|
\begin{table}
|
|
\tiny
|
|
\begin{tabular}{ |c|c|c|c|c|c| }
|
|
\hline
|
|
\thead{N°de \\collection} & 1 & 2 & 3 & 4 & 5 \\
|
|
\hline
|
|
\thead{Nombre de \\réseaux} & 38 & 45 & 1 & 20 & 19 \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{table}
|
|
|
|
\end{center}
|
|
\end{column}
|
|
\begin{column}{0.49\linewidth}
|
|
\begin{figure}[H]
|
|
\includegraphics[width=0.45\textwidth]{img/iid-meso-1.png}
|
|
\includegraphics[width=0.45\textwidth]{img/iid-meso-2.png}
|
|
\includegraphics[width=0.45\textwidth]{img/iid-meso-3.png}
|
|
\includegraphics[width=0.45\textwidth]{img/iid-meso-4.png}
|
|
\includegraphics[width=0.30\textwidth]{img/iid-meso-5.png}
|
|
\caption{Connectivités de la partition}
|
|
\end{figure}
|
|
\end{column}
|
|
\end{columns}
|
|
\end{frame}
|
|
|
|
\begin{frame}
|
|
\frametitle{Perspectives sur \emph{colSBM}}
|
|
% DONE Ajouter une slide conclusion perspective
|
|
% Rappeler les modeles avec clustering
|
|
% Evoquer l'analyse de reseaux corrigés pour l'échantillonnage
|
|
% Lien vers le package
|
|
|
|
\begin{itemize}
|
|
\item 4 modèles dont 3 qui ont une flexibilité sur au moins une des
|
|
dimensions (adaptabilité aux données)
|
|
\item Partitionner un ensemble de réseaux selon leurs structures
|
|
\item Comparer les \emph{clusterings} de réseaux obtenus entre données
|
|
brutes et données corrigées (par exemple par la méthode
|
|
\emph{CoOPLBM}\footnote{~\cite{anakokDisentanglingStructureEcological2022}})
|
|
\end{itemize}
|
|
|
|
\bigskip
|
|
\centering
|
|
Le package est disponible sur GitHub : \faGithub
|
|
\url{https://github.com/Chabert-Liddell/colSBM}
|
|
|
|
\bigskip
|
|
|
|
\end{frame}
|
|
|
|
\section{Autres questions}
|
|
\begin{frame}{\emph{Message passing} et \emph{Graph Convolutional Network}}
|
|
TODO Formule
|
|
Fonction de perte possible
|
|
\end{frame}
|
|
|
|
\begin{frame}{Distance de Wasserstein}
|
|
TODO
|
|
\end{frame} |