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Louis 2025-05-29 11:06:30 +02:00
parent 8e8c87a123
commit e179f05773
2 changed files with 19 additions and 20 deletions
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2
presentation.tex
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@ -8,7 +8,7 @@
\usepackage[cyr]{aeguill} % Police vectorielle TrueType, guillemets francais
\usepackage{epsfig} % pour gérer les images
\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{algpseudocode} % pour les algorithmes
\usepackage{graphicx}

35
principal.tex
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@ -1,7 +1,7 @@
\section{Model Context}
\label{sec:context-of-the-model}
\begin{frame}
\frametitle{Why a network?}
\frametitle{(Why) A network?}
\begin{columns}
\begin{column}{0.55\textwidth}
\begin{columns}
@ -234,11 +234,11 @@
% Maximizing the 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 $\bm{\mathcal{Z}} = \{1,\dots,\alert<2>{Q_1}\}^{\alert<2>{n}},
% \bm{\mathcal{W}} = \{1,\dots,\alert<2>{Q_2}\}^{\alert<2>{n}}$
% with $\mathbf{\mathcal{Z}} = \{1,\dots,\alert<2>{Q_1}\}^{\alert<2>{n}},
% \mathbf{\mathcal{W}} = \{1,\dots,\alert<2>{Q_2}\}^{\alert<2>{n}}$
% \end{block}
% \uncover<3>{So, classic algorithm $\Rightarrow$
% \emph{Expectation-Maximization} (EM).}
@ -249,13 +249,13 @@
% At iteration $(t)$:
% \begin{itemize}
% \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}:
% $$ \theta^{(t)} = \arg \max_{\theta} \mathcal{Q}(\theta | \theta^{(t-1)})$$
% \end{itemize}
% \uncover<2>{
% \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{frame}
@ -277,33 +277,33 @@
\end{frame}
\begin{frame}{Parameter estimation}{Solution}
\emph{Variational EM}~\cite{daudinMixtureModelRandom2008,chabert-liddellLearningCommonStructures2024}.
\begin{block}{Variational approximation of $\bm{Z,W|Y},\theta^{(t-1)}$}
\begin{block}{Variational approximation of $\mathbf{Z,W|Y},\theta^{(t-1)}$}
$\mathcal{R}_{Y^m,\tau}(Z^m, W^m) =
\mathcal{R}^1_{Y^m,\tau}(Z^m)
{\color{red}\times}
\mathcal{R}^2_{Y^m,\tau}(W^m) \Rightarrow$ independence between rows and columns, mean field approximation.
\end{block}
\begin{multline*}
\ell (\bm{Y};\theta) \geq \color{red}\sum_{m=1}^{M} \bigg(
\begin{align*}
\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)}
\left[ \ell_c(Y^m,Z^m,W^m ; \theta^{(t)}) \right] + \\
\mathcal{H}(\mathcal{R}_{Y^m,\theta^{(t)}}
(Z^m, W^m))
\color{red}\bigg) \color{black}
\eqcolon \mathcal{J}(\mathcal{R}_{\mathbf{Y},\tau};\theta)
\end{multline*}
\end{align*}
where $\theta = (\pi, \rho, \alpha)$ for \emph{iid}-colBiSBM
\end{frame}
\begin{frame}{Selection criterion for $Q_1, Q_2$}
Integrated Classification Likelihood (ICL)~\cite{biernackiAssessingMixtureModel2000}
\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) \\
& = \ell(\mathbf{Y};\hat{\theta}) - \mathcal{H}(p(\mathbf{Z,W}|\mathbf{Y},\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)
\end{align*} For SBM~\cite{daudinMixtureModelRandom2008}.
\onslide<2->{
\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)
\end{align*}
}
@ -470,8 +470,7 @@
\frametitle{Conclusion and perspectives}
\begin{block}{Summary}
\begin{itemize}
\item 4 models including 3 with flexibility on at least one of
the dimensions (adaptability to data).
\item 4 models including 3 flexible on at least one dimension (adaptability to data).
\item Jointly detect classic and less classic structures agnostically.
\item Partition a collection in sub-collections with homogeneous structures.
\item \texttt{R} package \texttt{colSBM} at \url{https://github.com/GrossSBM/colSBM}
@ -479,10 +478,10 @@
\end{block}
\begin{block}{Future work}
\begin{itemize}
\item Article in redaction
\item Preprint in redaction
\item Apply clustering to data from
\cite{pichonTellingMutualisticAntagonistic2024,doreRelativeEffectsAnthropogenic2021}
to tell if interaction types drives the structure of the network.
\cite{pichonTellingMutualisticAntagonistic2024,doreRelativeEffectsAnthropogenic2021}.
Do interaction type drives the structure of the network?
\end{itemize}
\end{block}
\end{frame}