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Louis 2025-05-29 11:06:30 +02:00
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presentation.tex
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@ -8,7 +8,7 @@
\usepackage[cyr]{aeguill} % Police vectorielle TrueType, guillemets francais \usepackage[cyr]{aeguill} % Police vectorielle TrueType, guillemets francais
\usepackage{epsfig} % pour gérer les images \usepackage{epsfig} % pour gérer les images
\usepackage{amsmath,amsthm, stmaryrd, mathtools} % très bon mode mathématique \usepackage{amsmath,amsthm, stmaryrd, mathtools} % très bon mode mathématique
\usepackage{amsfonts,amssymb,bm, bbold}% permet la definition des ensembles \usepackage{amsfonts,amssymb,bm}% permet la definition des ensembles
\usepackage{algorithm2e} % pour les algorithmes \usepackage{algorithm2e} % pour les algorithmes
\usepackage{algpseudocode} % pour les algorithmes \usepackage{algpseudocode} % pour les algorithmes
\usepackage{graphicx} \usepackage{graphicx}

37
principal.tex
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@ -1,7 +1,7 @@
\section{Model Context} \section{Model Context}
\label{sec:context-of-the-model} \label{sec:context-of-the-model}
\begin{frame} \begin{frame}
\frametitle{Why a network?} \frametitle{(Why) A network?}
\begin{columns} \begin{columns}
\begin{column}{0.55\textwidth} \begin{column}{0.55\textwidth}
\begin{columns} \begin{columns}
@ -234,11 +234,11 @@
% Maximizing the log-likelihood? % Maximizing the log-likelihood?
% \begin{block}{log-likelihood and complete log-likelihood} % \begin{block}{log-likelihood and complete log-likelihood}
% \[ % \[
% \ell(\bm{Y};\theta) = \sum_{\bm{Z,W}\in \bm{\mathcal{Z}\times\mathcal{W}}} \ell_c(\bm{Y}, \bm{Z}, \bm{W};\theta) % \ell(\mathbf{Y};\theta) = \sum_{\mathbf{Z,W}\in \mathbf{\mathcal{Z}\times\mathcal{W}}} \ell_c(\mathbf{Y}, \mathbf{Z}, \mathbf{W};\theta)
% \] % \]
% with $\bm{\mathcal{Z}} = \{1,\dots,\alert<2>{Q_1}\}^{\alert<2>{n}}, % with $\mathbf{\mathcal{Z}} = \{1,\dots,\alert<2>{Q_1}\}^{\alert<2>{n}},
% \bm{\mathcal{W}} = \{1,\dots,\alert<2>{Q_2}\}^{\alert<2>{n}}$ % \mathbf{\mathcal{W}} = \{1,\dots,\alert<2>{Q_2}\}^{\alert<2>{n}}$
% \end{block} % \end{block}
% \uncover<3>{So, classic algorithm $\Rightarrow$ % \uncover<3>{So, classic algorithm $\Rightarrow$
% \emph{Expectation-Maximization} (EM).} % \emph{Expectation-Maximization} (EM).}
@ -249,13 +249,13 @@
% At iteration $(t)$: % At iteration $(t)$:
% \begin{itemize} % \begin{itemize}
% \item[$\bullet$]\textbf{E Step}: calculate % \item[$\bullet$]\textbf{E Step}: calculate
% $$ \mathcal{Q}(\theta | \theta^{(t-1)}) = \mathbb E_{\alert<2>{\bm Z, \bm W | \bm Y, \theta^{(t-1)}} } \left[\ell_c(\bm Y, \bm W, \bm Z; \theta) \right] $$ % $$ \mathcal{Q}(\theta | \theta^{(t-1)}) = \mathbb E_{\alert<2>{\mathbf Z, \mathbf W | \mathbf Y, \theta^{(t-1)}} } \left[\ell_c(\mathbf Y, \mathbf W, \mathbf Z; \theta) \right] $$
% \item[$\bullet$]\textbf{M Step}: % \item[$\bullet$]\textbf{M Step}:
% $$ \theta^{(t)} = \arg \max_{\theta} \mathcal{Q}(\theta | \theta^{(t-1)})$$ % $$ \theta^{(t)} = \arg \max_{\theta} \mathcal{Q}(\theta | \theta^{(t-1)})$$
% \end{itemize} % \end{itemize}
% \uncover<2>{ % \uncover<2>{
% \begin{alertblock}{Problem for classic EM} % \begin{alertblock}{Problem for classic EM}
% Law of $\bm{Z,W|Y},\theta^{(t-1)}$ inaccessible % Law of $\mathbf{Z,W|Y},\theta^{(t-1)}$ inaccessible
% \end{alertblock}} % \end{alertblock}}
% \end{frame} % \end{frame}
@ -277,34 +277,34 @@
\end{frame} \end{frame}
\begin{frame}{Parameter estimation}{Solution} \begin{frame}{Parameter estimation}{Solution}
\emph{Variational EM}~\cite{daudinMixtureModelRandom2008,chabert-liddellLearningCommonStructures2024}. \emph{Variational EM}~\cite{daudinMixtureModelRandom2008,chabert-liddellLearningCommonStructures2024}.
\begin{block}{Variational approximation of $\bm{Z,W|Y},\theta^{(t-1)}$} \begin{block}{Variational approximation of $\mathbf{Z,W|Y},\theta^{(t-1)}$}
$\mathcal{R}_{Y^m,\tau}(Z^m, W^m) = $\mathcal{R}_{Y^m,\tau}(Z^m, W^m) =
\mathcal{R}^1_{Y^m,\tau}(Z^m) \mathcal{R}^1_{Y^m,\tau}(Z^m)
{\color{red}\times} {\color{red}\times}
\mathcal{R}^2_{Y^m,\tau}(W^m) \Rightarrow$ independence between rows and columns, mean field approximation. \mathcal{R}^2_{Y^m,\tau}(W^m) \Rightarrow$ independence between rows and columns, mean field approximation.
\end{block} \end{block}
\begin{multline*} \begin{align*}
\ell (\bm{Y};\theta) \geq \color{red}\sum_{m=1}^{M} \bigg( \ell (\mathbf{Y};\theta) \geq \color{red}\sum_{m=1}^{M} \bigg(
\color{black} \mathbb{E}_{\mathcal{R}_{Y^m,\tau}(Z^m,W^m)} \color{black} \mathbb{E}_{\mathcal{R}_{Y^m,\tau}(Z^m,W^m)}
\left[ \ell_c(Y^m,Z^m,W^m ; \theta^{(t)}) \right] + \\ \left[ \ell_c(Y^m,Z^m,W^m ; \theta^{(t)}) \right] + \\
\mathcal{H}(\mathcal{R}_{Y^m,\theta^{(t)}} \mathcal{H}(\mathcal{R}_{Y^m,\theta^{(t)}}
(Z^m, W^m)) (Z^m, W^m))
\color{red}\bigg) \color{black} \color{red}\bigg) \color{black}
\eqcolon \mathcal{J}(\mathcal{R}_{\mathbf{Y},\tau};\theta) \eqcolon \mathcal{J}(\mathcal{R}_{\mathbf{Y},\tau};\theta)
\end{multline*} \end{align*}
where $\theta = (\pi, \rho, \alpha)$ for \emph{iid}-colBiSBM where $\theta = (\pi, \rho, \alpha)$ for \emph{iid}-colBiSBM
\end{frame} \end{frame}
\begin{frame}{Selection criterion for $Q_1, Q_2$} \begin{frame}{Selection criterion for $Q_1, Q_2$}
Integrated Classification Likelihood (ICL)~\cite{biernackiAssessingMixtureModel2000} Integrated Classification Likelihood (ICL)~\cite{biernackiAssessingMixtureModel2000}
\begin{align*} \begin{align*}
\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) \\ \text{ICL}(\mathbf{Y}, Q_1, Q_2) & = \mathbb{E}_{\mathbf{Z,W|Y}} [\ell_c(\mathbf{Y,Z,W};\hat{\theta})] -\frac{1}{2}\text{pen}(Q_1, Q_2) \\
& = \ell(\mathbf{Y};\hat{\theta}) - \mathcal{H}(p(\mathbf{Z,W}|\mathbf{Y},\hat{\theta})) - \frac{1}{2}\text{pen}(Q_1, Q_2) & = \ell(\mathbf{Y};\hat{\theta}) - \mathcal{H}(p(\mathbf{Z,W}|\mathbf{Y};\hat{\theta})) - \frac{1}{2}\text{pen}(Q_1, Q_2)
\end{align*} For SBM~\cite{daudinMixtureModelRandom2008}. \end{align*} For SBM~\cite{daudinMixtureModelRandom2008}.
\onslide<2->{ \onslide<2->{
\begin{align*} \begin{align*}
\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) \\ \text{BIC-L}(\mathbf{Y}, Q_1, Q_2) & = \mathbb{E}_{\mathcal{R}_{\mathbf{Y},\hat{\tau}}} [\ell_c(\mathbf{Y,Z,W};\hat{\theta}^{\text{var}})] + \mathcal{H(\mathcal{R}_{\mathbf{Y},\hat{\tau}})} - \frac{1}{2}\text{pen}(Q_1, Q_2) \\
& = \mathcal{J(\mathcal{R}_{\mathbf{Y},\hat{\tau}}, \hat{\theta}^{\text{var}})} - \frac{1}{2}\text{pen}(Q_1, Q_2) & = \mathcal{J(\mathcal{R}_{\mathbf{Y},\hat{\tau}}, \hat{\theta}^{\text{var}})} - \frac{1}{2}\text{pen}(Q_1, Q_2)
\end{align*} \end{align*}
} }
\end{frame} \end{frame}
@ -470,8 +470,7 @@
\frametitle{Conclusion and perspectives} \frametitle{Conclusion and perspectives}
\begin{block}{Summary} \begin{block}{Summary}
\begin{itemize} \begin{itemize}
\item 4 models including 3 with flexibility on at least one of \item 4 models including 3 flexible on at least one dimension (adaptability to data).
the dimensions (adaptability to data).
\item Jointly detect classic and less classic structures agnostically. \item Jointly detect classic and less classic structures agnostically.
\item Partition a collection in sub-collections with homogeneous structures. \item Partition a collection in sub-collections with homogeneous structures.
\item \texttt{R} package \texttt{colSBM} at \url{https://github.com/GrossSBM/colSBM} \item \texttt{R} package \texttt{colSBM} at \url{https://github.com/GrossSBM/colSBM}
@ -479,10 +478,10 @@
\end{block} \end{block}
\begin{block}{Future work} \begin{block}{Future work}
\begin{itemize} \begin{itemize}
\item Article in redaction \item Preprint in redaction
\item Apply clustering to data from \item Apply clustering to data from
\cite{pichonTellingMutualisticAntagonistic2024,doreRelativeEffectsAnthropogenic2021} \cite{pichonTellingMutualisticAntagonistic2024,doreRelativeEffectsAnthropogenic2021}.
to tell if interaction types drives the structure of the network. Do interaction type drives the structure of the network?
\end{itemize} \end{itemize}
\end{block} \end{block}
\end{frame} \end{frame}