Retours Pierre
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90
annexe.tex
90
annexe.tex
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@ -1,3 +1,16 @@
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\section{Clustering}
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\begin{frame}{Clustering algorithm}
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\centering
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\vspace{0.25\baselineskip}
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\begin{tikzpicture}[scale=0.85]
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\input{tikz/clustering.tex}
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\end{tikzpicture}
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\[
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D_{\mathcal{M}}(m,m') = \sum_{q = 1}^{Q_1} \sum_{r = 1}^{Q_2} \max(\widetilde{\pi}_{q}^{m}, \widetilde{\pi}_{q}^{m'}) \left( \widetilde{\alpha}_{qr}^{m} - \widetilde{\alpha}_{qr}^{m'}\right)^{2} \max(\widetilde{\rho}_{r}^{m}, \widetilde{\rho}_{r}^{m'})
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\]
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\end{frame}
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\section{VEM}
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\begin{frame}{Developed formula of variational EM}
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@ -75,6 +88,7 @@
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\end{align*}
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Thus $\ell(\bY;\theta) - \KL{\Ryt}{\log \Prob(\bZ,\bW|\bY;\theta)} = \mathcal{J}(\tau;\theta) \qed$
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\end{frame}
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\section{Model selection}
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\begin{frame}
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\frametitle{On the BIC-L}
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@ -97,8 +111,6 @@
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\]
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\end{frame}
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\section{Model selection}
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\begin{frame}
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\frametitle{Choice of $(Q_1,Q_2)$ - Greedy approach}
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\begin{columns}
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@ -156,77 +168,3 @@
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\end{column}
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\end{columns}
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\end{frame}
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\section{Clustering}
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\begin{frame}{Clustering algorithm}
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\centering
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\vspace{0.25\baselineskip}
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\begin{tikzpicture}[scale=0.85]
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\input{tikz/clustering.tex}
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\end{tikzpicture}
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\[
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D_{\mathcal{M}}(m,m') = \sum_{q = 1}^{Q_1} \sum_{r = 1}^{Q_2} \max(\widetilde{\pi}_{q}^{m}, \widetilde{\pi}_{q}^{m'}) \left( \widetilde{\alpha}_{qr}^{m} - \widetilde{\alpha}_{qr}^{m'}\right)^{2} \max(\widetilde{\rho}_{r}^{m}, \widetilde{\rho}_{r}^{m'})
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\]
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\end{frame}
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\section{Results~\cite{baldockSystemsApproachReveals2019,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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\includegraphics[width=0.45\textwidth]{tikz/applications/baldock/mat-Baldock2019_Bristol.pdf}
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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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\includegraphics[width=0.45\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Bristol.pdf}
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\caption{Reordered}
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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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\includegraphics[width=0.45\textwidth]{tikz/applications/baldock/mat-Baldock2019_Edinburgh.pdf}
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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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\includegraphics[width=0.5\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Edinburgh.pdf}
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\caption{Reordered}
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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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\includegraphics[width=0.5\textwidth]{tikz/applications/baldock/mat-Baldock2019_Leeds.pdf}
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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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\includegraphics[width=0.5\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Leeds.pdf}
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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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\includegraphics[width=0.5\textwidth]{tikz/applications/baldock/mat-Baldock2019_Reading.pdf}
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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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\includegraphics[width=0.5\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Reading.pdf}
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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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@ -112,10 +112,10 @@
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\subtitle{JdS 2025}
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\title[Bipartite networks collection]{Joint analysis of bipartite networks collection}
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\title[Bipartite networks collection]{Joint estimation of bipartite network collections. Application to plant-pollinator networks.}
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\author[L. Lacoste]{\underline{Louis Lacoste}, Pierre Barbillon and
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Sophie Donnet\newline Laboratoire MIA Paris-Saclay\newline\ccbysa}
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\date{\today}
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Sophie Donnet\newline UMR MIA Paris-Saclay, AgroParisTech, INRAE, Université Paris-Saclay\newline\ccbysa}
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\date{03 Juin 2025}
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\begin{document}
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@ -56,14 +56,14 @@
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\begin{column}{0.4\textwidth}
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\only<1>{
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\begin{itemize}
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\item A bipartite graph $G = (U,V,E)$
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\item Can be encoded by a bi-adjacency matrix $Y \in \{0,1\}^{n_1 \times n_2}$
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\item Bipartite graph $G = (U,V,E)$
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\item Encoded in bi-adjacency matrix $Y \in \{0,1\}^{n_1 \times n_2}$
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\end{itemize}}
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\only<2>{
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\begin{itemize}
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\item Increasingly available
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\item Modeling of various interactions, here ecosystems
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\item Structure necessary for: biodiversity monitoring, robustness, risk of collapse
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\item Ecosystems described by their interactions
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\item Functional structure for: biodiversity monitoring, robustness, risk of collapse
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\end{itemize}}
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\end{column}
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\end{columns}
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@ -139,7 +139,7 @@
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\begin{tikzpicture}[scale=0.35]
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\input{tikz/lbm.tex}
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\end{tikzpicture}
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\caption{Example of LBM\footnotemark[\thefootnote]}
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\caption{Example of BiSBM}
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\label{fig:LBMvisu}
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\end{figure}
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\end{column}
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@ -192,8 +192,7 @@
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\includegraphics[width=0.5\textwidth]{tikz/applications/baldock/bisbm-mat-Baldock2019_Reading.pdf}
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\caption{Reading, $Q_1 = 3, Q_2 = 3$}
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\end{subfigure}
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\vspace{-\baselineskip}
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\caption{Reordered adjacency matrices, using BiSBM for each network}
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\caption{Separate BiSBM fit for each network}
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\label{fig:adj-reord}
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\end{figure}
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}
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\onslide<2>{ \begin{block}{$\pi\rho$-colBiSBM}
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\[
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\forall m \in \{1\dots M\}, Y^m \overset{ind}{\sim}
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\mathcal{B}ern\text{-BiSBM}_{n_1^m,n_2^m}(Q_1, Q_2, \pi^m, \rho^m, \alpha)
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\mathcal{B}ern\text{-BiSBM}_{n_1^m,n_2^m}(Q_1, Q_2, \pi\alert{^m}, \rho\alert{^m}, \alpha)
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\]
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with $\theta = ((\pi\alert{^m})_{m=1,\dots, M}, (\rho\alert{^m})_{m=1,\dots,
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\begin{frame}{Parameter estimation}{How ?}
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\begin{align*}
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\ell(\mathbf{Y};\theta) = & \sum_{m=1}^{M} \ell(Y^m;\theta) \\
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= & \sum_{m=1}^{M} \log \int_{\alert<2->{\mathcal{Z}^m\times\mathcal{W}^m}} \exp\{\ell_c(Y^m,Z^m,W^m;\theta)\} dZ^m dW^m \\
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= & \sum_{m=1}^{M} \log\int_{\alert<2->{\mathcal{Z}^m\times\mathcal{W}^m}}\exp\{\ell(Y^m | Z^m,W^m;\alpha) + \\
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& \ell(Z^m;\pi) + \ell(W^m;\rho)\} dZ^m dW^m
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= & \sum_{m=1}^{M} \log \sum_{\alert<2->{Z^m \in \mathcal{Z}^m,W^m\in\mathcal{W}^m}} \exp\{\ell_c(Y^m,Z^m,W^m;\theta)\} \\
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= & \sum_{m=1}^{M} \log\sum_{\alert<2->{Z^m \in \mathcal{Z}^m,W^m\in\mathcal{W}^m}}\exp\{\ell(Y^m | Z^m,W^m;\alpha) + \\
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& \ell(Z^m;\pi) + \ell(W^m;\rho)\}
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% & = \sum_{m=1}^{M} \sum_{i=1}^{n_1^m} \sum_{q=1}^{Q_1} Z_{iq} \log(\pi_q) + \sum_{j=1}^{n_2^m}\sum_{r=1}^{Q_2} W_{jr} \log(\rho_r) \\
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% & + \sum_{i,j}\sum_{q,r} Z_{iq}W_{jr} \log \mathcal{B}ern(Y_{ij};\alpha_{qr})
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\end{align*}
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\onslide<3>{
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We would like to use Expectation-Maximization (EM) algorithm~\parencite{dempsterMaximumLikelihoodIncomplete1977} but the law of $\mathbf{Z,W|Y},\theta^{(t-1)}$ is untractable due to dependence between row and column groups.}
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EM impracticable since $\mathbf{Z,W|Y}$ intractable due to
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conditional dependency.}
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\end{frame}
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\begin{frame}{Parameter estimation}{Solution}
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By \emph{Variational EM}, as proposed
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by~\cite{daudinMixtureModelRandom2008} and adapted for joint simple networks
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by~\cite{chabert-liddellLearningCommonStructures2024}.
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\emph{Variational EM}~\cite{daudinMixtureModelRandom2008,chabert-liddellLearningCommonStructures2024}.
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\begin{block}{Variational approximation of $\bm{Z,W|Y},\theta^{(t-1)}$}
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$\mathcal{R}_{Y^m,\tau}(Z^m, W^m) =
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\mathcal{R}^1_{Y^m,\tau}(Z^m)
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@ -298,26 +296,25 @@
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\end{frame}
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\begin{frame}{Selection criterion for $Q_1, Q_2$}
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\cite{biernackiAssessingMixtureModel2000} introduced the Integrated Classification Likelihood (ICL):
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Integrated Classification Likelihood (ICL)~\cite{biernackiAssessingMixtureModel2000}
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\begin{align*}
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\text{ICL}(\bm{Y}, Q_1, Q_2) & = \mathbb{E} [\ell_c(\bm{Y,Z,W};\hat{\theta})] -\frac{1}{2}\text{pen}(Q_1, Q_2) \\
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& = \ell(\mathbf{Y};\hat{\theta}) - \mathcal{H}(p(\mathbf{Z,W}|\mathbf{Y},\hat{\theta})) - \frac{1}{2}\text{pen}(Q_1, Q_2)
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\end{align*} leads to low entropy clustering. Common in literature for SBM.
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\end{align*} For SBM~\cite{daudinMixtureModelRandom2008}.
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\onslide<2->{
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\begin{align*}
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\text{BIC-L}(\bm{Y}, & Q_1, Q_2) = \mathbb{E}_{\mathcal{R}_{\mathbf{Y},\hat{\tau}}} [\ell_c(\bm{Y,Z,W};\hat{\theta}^{\text{var}})] + \mathcal{H(\mathcal{R}_{\mathbf{Y},\hat{\tau}})} - \frac{1}{2}\text{pen}(Q_1, Q_2) \\
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& = \mathcal{J(\mathcal{R}_{\mathbf{Y},\hat{\tau}}, \hat{\theta}^{\text{var}})} - \frac{1}{2}\text{pen}(Q_1, Q_2) \textcolor{red}{\leq \log p(\mathbf{Y};\hat{\theta}^{\text{MV}})- \frac{1}{2}\text{pen}(Q_1, Q_2)} \\
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\text{BIC-L}(\bm{Y}, Q_1, Q_2) & = \mathbb{E}_{\mathcal{R}_{\mathbf{Y},\hat{\tau}}} [\ell_c(\bm{Y,Z,W};\hat{\theta}^{\text{var}})] + \mathcal{H(\mathcal{R}_{\mathbf{Y},\hat{\tau}})} - \frac{1}{2}\text{pen}(Q_1, Q_2) \\
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& = \mathcal{J(\mathcal{R}_{\mathbf{Y},\hat{\tau}}, \hat{\theta}^{\text{var}})} - \frac{1}{2}\text{pen}(Q_1, Q_2)
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\end{align*}
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because we want fuzzier clustering.
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}
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\end{frame}
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\begin{frame}{Practical problems of choosing $Q_1, Q_2$}
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\begin{alertblock}{Exploration problems}
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\begin{itemize}
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\item Exploration of a 2D grid is costly. \uncover<2->{$\rightarrow$ \textbf{Greedy
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approach} and \textbf{sliding window}}
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\item Sensitivity to initializations. \uncover<3->{$\rightarrow$ \textbf{Spectral
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\item Sensitivity to initializations. \uncover<2->{$\rightarrow$ \textbf{Spectral
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clustering} and \textbf{split \& merge} approach}
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\item Exploration of a 2D grid is costly. \uncover<3->{$\rightarrow$ \textbf{Greedy
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approach} and \textbf{sliding window}}
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\end{itemize}
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\end{alertblock}
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\end{frame}
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\begin{subfigure}[t]{0.5\textwidth}
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\centering
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\includegraphics[width=0.45\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Bristol.pdf}
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\caption{Bristol, $Q_1 = 3, Q_2 = 5$}
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\caption{Bristol}
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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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\includegraphics[width=0.45\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Edinburgh.pdf}
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\caption{Edinburgh, $Q_1 = 3, Q_2 = 5$}
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\caption{Edinburgh}
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\end{subfigure}
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\newline
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\begin{subfigure}[ht]{0.5\textwidth}
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\centering
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\includegraphics[width=0.5\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Leeds.pdf}
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\caption{Leeds, $Q_1 = 3, Q_2 = 5$}
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\caption{Leeds}
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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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\includegraphics[width=0.5\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Reading.pdf}
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\caption{Reading, $Q_1 = 3, Q_2 = 5$}
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\caption{Reading}
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\end{subfigure}
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\caption{Reordered adjacency matrices by \emph{iid}-colBiSBM,~\cite{baldockSystemsApproachReveals2019}}
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\caption{\emph{iid}-colBiSBM fit, $Q_1 = 3, Q_2 = 5$}
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\end{figure}}
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\end{frame}
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\begin{column}{0.2\textwidth}
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\begin{figure}
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\onslide<3>{
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\onslide<2>{
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\begin{subfigure}[t]{0.7\textwidth}
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\centering
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\includegraphics[width=1\textwidth]{img/baldock/bombus-hortorum.jpeg}
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\caption{\emph{Bombus Hortorum} or garden bumblebee}
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\end{subfigure}
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}
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\onslide<4>{
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\onslide<3>{
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\begin{subfigure}[t]{0.7\textwidth}
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\centering
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\includegraphics[width=1\textwidth]{img/baldock/bombus-lapidarius.jpeg}
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bottom color=blue!1!white,
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anchor=right corner, minimum height=42mm, label={[label distance = 2mm]207:Generalists}, label={[label distance = 12mm]357:Specialists}] (T) at ($(struct.north east)+(-1,-2.5)$) {};
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\only<3>{
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\only<2>{
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\node[left = 3mm of gen] (towns_gen_garden) {B, L};
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\node[left = 3mm of spe] (towns_spe_garden) {\phantom{B, }E, R};
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\path (towns_gen_garden) edge[->,thick] (gen);
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\path (towns_spe_garden) edge[->,thick] (spe);
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}
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\only<4>{
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\only<3>{
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\node[left = 3mm of interm] (towns_interm_red) {L};
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\node[left = 3mm of spe] (towns_spe_red) {B, E, R};
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\path (towns_interm_red) edge[->,thick] (interm);
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\section{Conclusion}
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\begin{frame}
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\frametitle{Conclusion and perspectives}
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\begin{block}{Capabilities}
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\begin{block}{Summary}
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\begin{itemize}
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\item 4 models including 3 with flexibility on at least one of
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the dimensions (adaptability to data).
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\item Detect classic and less classic structures in an agnostic way.
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\item Partition a set of networks according to their structures.
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\item Jointly detect classic and less classic structures agnostically.
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\item Partition a collection in sub-collections with homogeneous structures.
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\item \texttt{R} package \texttt{colSBM} at \url{https://github.com/GrossSBM/colSBM}
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\end{itemize}
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\end{block}
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\begin{block}{Package and applications}
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\begin{block}{Future work}
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\begin{itemize}
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\item Article in redaction
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\item \texttt{R} package \texttt{colSBM} on
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Github\footnote{\url{https://github.com/GrossSBM/colSBM}}
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\item Apply clustering to data from
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\cite{pichonTellingMutualisticAntagonistic2024,doreRelativeEffectsAnthropogenic2021}
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to tell if interaction drives the structure of the network.
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to tell if interaction types drives the structure of the network.
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\end{itemize}
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\end{block}
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\end{frame}
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