diff --git a/presentation.tex b/presentation.tex index 1c5c1e7..1d33452 100644 --- a/presentation.tex +++ b/presentation.tex @@ -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} diff --git a/principal.tex b/principal.tex index 3123a9b..d9655b5 100644 --- a/principal.tex +++ b/principal.tex @@ -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} @@ -277,34 +277,34 @@ \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) \\ - & = \mathcal{J(\mathcal{R}_{\mathbf{Y},\hat{\tau}}, \hat{\theta}^{\text{var}})} - \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*} } \end{frame} @@ -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} \ No newline at end of file