1370 lines
No EOL
60 KiB
TeX
1370 lines
No EOL
60 KiB
TeX
\documentclass{beamer}
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\usetheme{Boadilla}
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\usecolortheme{seahorse}
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% importations
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\usepackage[french]{babel} % pour dire que le texte est en francais
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\usepackage{nth}
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\usepackage{csquotes}
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\usepackage[T1]{fontenc} % pour les font postscript
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\usepackage[cyr]{aeguill} % Police vectorielle TrueType, guillemets francais
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\usepackage{epsfig} % pour gérer les images
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\usepackage{amsmath,amsthm, stmaryrd} % très bon mode mathématique
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\usepackage{amsfonts,amssymb,bm, bbold}% permet la definition des ensembles
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\usepackage{algorithm2e} % pour les algorithmes
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\usepackage{algpseudocode} % pour les algorithmes
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\usepackage{graphicx}
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\usepackage{float} % pour le placement des figure
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\usepackage{url} % pour une gestion efficace des url
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\usepackage{hyperref} % pour les hyperliens dans le document
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\usepackage{tikz} % For graph plots
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\usepackage{adjustbox} % To resize tikzpictures
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\usepackage{fontawesome5}
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\usepackage{makecell}
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\usepackage{appendixnumberbeamer} % Pour cacher la numérotation des backups slides
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\usepackage{chronosys} % Frise chronologique
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\renewcommand\theadfont{\tiny}
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% Beamer
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\setbeamertemplate{headline}{%
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\begin{beamercolorbox}[ht=2.25ex,dp=3.75ex]{section in head/foot}
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\insertnavigation{\paperwidth}
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\end{beamercolorbox}%
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}%
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%\beamertemplatenavigationsymbolsempty % Pas de bar de navigation
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\setbeamerfont{caption}{size=\scriptsize} % Petit titre de figures
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\AtBeginSection[]{
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\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}\secname\par%
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\end{beamercolorbox}
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\vfill
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\end{frame}
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}
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% bibliographie
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\usepackage[style=apa,sorting=none]{biblatex}
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\addbibresource{references.bib}
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% Tikz
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%% Tikz Related
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\usetikzlibrary{calc,shapes,backgrounds,arrows,automata,shadows,positioning}
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\usetikzlibrary{arrows,shapes,positioning,shadows,trees,calc,backgrounds,automata,positioning}
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\usetikzlibrary{decorations.pathreplacing,calligraphy}
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\tikzset{
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basic/.style = {draw, text width=3cm, font=\sffamily, rectangle},
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root/.style = {basic, rounded corners=2pt, thin, align=center,
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fill=green!30},
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level 2/.style = {basic, rounded corners=6pt, thin,align=center,
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fill=green!60,
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text width=8em},
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level 3/.style = {basic, thin, align=left, fill=pink!60, text width=3.5cm}
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}
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% Couleurs
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% pour tickz multilevel
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\definecolor{redorg}{RGB}{215,48,39}
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\definecolor{orangeorg}{RGB}{253,174,97}
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\definecolor{blueind}{RGB}{69,117,233}
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\definecolor{cyanind}{RGB}{116,173,209}
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\definecolor{electricblue}{RGB}{125, 249, 255}
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\definecolor{greenind}{RGB}{112,130,56}
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\definecolor{burntorange}{RGB}{204, 85, 0}
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\definecolor{goldenyellow}{RGB}{255, 192, 0}
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\definecolor{peach}{RGB}{255,255,0}
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\definecolor{gray}{RGB}{128,128,128}
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% Footnote
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\makeatletter
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\newcommand\blfootnote[1]{%
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\begingroup
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\renewcommand{\@makefntext}[1]{\noindent\makebox[1.8em][r]#1}
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\renewcommand\thefootnote{}\footnote{#1}%
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\addtocounter{footnote}{-1}%
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\endgroup
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||
}
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||
\makeatother
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||
|
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\subtitle{Audition candidature de thèse à l'EDMH}
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\title[Comparaison de structures de réseaux]{Comparaison de structures de
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réseaux.
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Applications à des réseaux écologiques}
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\author[L. Lacoste]{Louis \textsc{Lacoste}} % Sous la supervision de Pierre
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\date{23 mai 2024}
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\begin{document}
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% titre
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||
\begin{frame}[noframenumbering,plain]
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||
\maketitle
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||
\end{frame}
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||
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\begin{frame}
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\tableofcontents
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\end{frame}
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\section{Parcours}
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||
\label{sec:parcours}
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\begin{frame}{Formations}
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||
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\begin{itemize}
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||
\item 2023--2024, M2 Mathématiques pour les Sciences du Vivant,
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||
Université Paris-Saclay\\
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{\small UC à choix \nth{2} semestre : modèles à variables
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||
latentes, statistiques spatiales et méthodes de stats en grande dimensions}
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\item 2022--2023, Année de césure
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||
\item 2020--2022, 1ère et 2ème année en formation Ingénieur
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||
AgroParisTech\\
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||
{\small Cours optionnels suivis : statistiques spatiales,
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||
mathématiques pour la santé, ingénierie par la simulation informatique \dots}
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||
\item 2018--2020, Classe Préparatoire BCPST
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||
\end{itemize}
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||
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||
\end{frame}
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||
\begin{frame}{Expériences professionnelles}
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||
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||
\begin{itemize}
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||
\item 2024 Avril--Sept., Détection de structures et clustering de réseaux
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||
écologiques. Stage dans l’UMR MIA Paris-Saclay, supervisé par Pierre Barbillon.
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||
\item 2023 Janv.--Juillet, Détection de structures dans des collections de
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réseaux bipartites et écriture du package implémentant la méthode.
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Stage dans l’UMR MIA Paris-Saclay, supervisé par Pierre Barbillon.
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||
\item 2022 Mai--Déc., Stage assistant ingénieur en Qualité chez
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||
Eurofins Food France
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||
\end{itemize}
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||
|
||
\end{frame}
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||
|
||
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\section[Axes de recherche]{Axes de recherche}
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||
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||
\begin{frame}
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\frametitle{Contexte écologique}
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\begin{itemize}
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\item Faire de la détection de structure sur un réseau (SBM, LBM) mais
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intérêt à le faire sur plusieurs
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\item De nombreux réseaux disponibles \parencite{WebLifeEcological} et
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décrivant des interactions similaires. Par exemple des
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interactions proies-prédateurs, plantes-pollinisateurs \dots
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\item Re-grouper les réseaux selon leur similarité (\emph{clustering}
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de réseaux)
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\item Transférer de l'information grâce à la collection (par exemple
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reconstitution de données manquantes)
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\item Déterminer des structures d'interactions fines de manière
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agnostique % Pas d'idee preco
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%\item Vérifier si le regroupement est lié à des co-variables
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||
\end{itemize}
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||
\end{frame}
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||
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||
\begin{frame}{Contexte mathématiques}
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||
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\end{frame}
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||
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\subsection[Axe 1]{Axe 1 : Modèles à variables
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latentes pour une collection de réseaux bipartites}
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\label{sec:axe-1}
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||
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% TODO Demander si suppression car plus techniques qu'important
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% \begin{frame}
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% \frametitle{Réseaux bipartites}
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% \begin{columns}[c]
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% \begin{column}{0.48\textwidth}
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% \centering
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% Réseau bipartite\\
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% \begin{tikzpicture}[scale=.6]
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% \tikzstyle{every edge}=[-,>=stealth',shorten
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||
% >=1pt,auto,draw,line width=1.5pt]
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||
% \tikzstyle{every state}=[draw, text=black,scale=0.95, transform
|
||
% shape]
|
||
% \tikzstyle{every state}=[draw=none,text=black,scale=0.75,
|
||
% transform shape]
|
||
% \tikzstyle{every node}=[fill=blueind]
|
||
|
||
% \node[state, draw=black!50] (A1) at (0,5) {\textbf{R1}};
|
||
% \node[state, draw=black!50] (A2) at (2.5,5) {\textbf{R2}};
|
||
% \node[state, draw=black!50] (A3) at (5,5) {\textbf{R3}};
|
||
|
||
% \tikzstyle{every node}=[fill=greenind, shape=rectangle]
|
||
% \tikzstyle{every state}=[draw=none,text=black,scale=0.75,
|
||
% transform shape, shape=rectangle]
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||
% \node[state, draw=black!50] (B1) at (0,0) {\textbf{C1}};
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||
% \node[state, draw=black!50] (B2) at (1.25,0) {\textbf{C2}};
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||
% \node[state, draw=black!50] (B3) at (2.5,0) {\textbf{C3}};
|
||
% \node[state, draw=black!50] (B4) at (3.75,0) {\textbf{C4}};
|
||
% \node[state, draw=black!50] (B5) at (5,0) {\textbf{C5}};
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||
% \path (A1) edge [] (B1);
|
||
% \path (A1) edge (B2);
|
||
% \path (A1) edge (B3);
|
||
% \path (A1) edge (B4);
|
||
% \path (A2) edge (B3);
|
||
% \path (A2) edge (B4);
|
||
% \path (A3) edge (B5);
|
||
% \path (A2) edge (B5);
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||
% \end{tikzpicture}
|
||
% \end{column}
|
||
% \hfill
|
||
% \begin{column}{0.48\linewidth}
|
||
% Matrice d'incidence
|
||
% \smallskip
|
||
% $X=\left(
|
||
% \begin{array}{rrrrr}
|
||
% 1 & 1 & 1 & 1 & 0 \\
|
||
% 0 & 0 & 1 & 1 & 1 \\
|
||
% 0 & 0 & 0 & 0 & 1 \\
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||
% \end{array}\right)
|
||
% $\\
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||
% \end{column}
|
||
% \end{columns}
|
||
% \smallskip
|
||
% Permet de décrire des interactions impliquant deux agents dont les rôles
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||
% sont de natures différentes.\\
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||
% Par exemple : hôtes-parasites, plantes-pollinisateurs, graines-disperseurs
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||
% \dots
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||
% \end{frame}
|
||
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||
\begin{frame}
|
||
\frametitle{Collections bipartites}
|
||
\begin{center}
|
||
\begin{adjustbox}{trim=0 0 1 1.5cm}
|
||
\begin{tikzpicture}[scale=.33]
|
||
\begin{scope}[xshift=18cm, yshift=2cm]
|
||
\tikzstyle{every state}=[draw=none, text=black,scale=0.75,
|
||
transform shape]
|
||
\tikzset{edge_proba/.style={draw=white, fill=none,
|
||
text=black}}
|
||
|
||
\tikzstyle{every node}=[fill=blueind]
|
||
\node[edge_proba] (pi1) at (1,5.7)
|
||
{\textbf{$\pi_{{\color{blueind}\bullet}}$}};
|
||
\node[state, draw=black!50] (R11) at (0,5) {\textbf{R11}};
|
||
\node[state, draw=black!50] (R12) at (1,5) {\textbf{R12}};
|
||
\node[state, draw=black!50] (R13) at (2,5) {\textbf{R13}};
|
||
|
||
\tikzstyle{every node}=[fill=cyanind]
|
||
\node[edge_proba] (pi2) at (6.75,5.7)
|
||
{\textbf{$\pi_{{\color{cyanind}\bullet}}$}};
|
||
\node[state, draw=black!50] (R21) at (6.25,5)
|
||
{\textbf{R21}};
|
||
\node[state, draw=black!50] (R22) at (7.25,5)
|
||
{\textbf{R22}};
|
||
|
||
\tikzstyle{every node}=[fill=electricblue]
|
||
\node[edge_proba] (pi3) at (10,5.7)
|
||
{\textbf{$\pi_{{\color{electricblue}\bullet}}$}};
|
||
\node[state, draw=black!50] (R31) at (10,5) {\textbf{R31}};
|
||
|
||
\tikzstyle{every node}=[fill=burntorange, shape=rectangle]
|
||
\node[edge_proba] (rho1) at (0.5,-1)
|
||
{\textbf{$\rho_{{\color{burntorange}\bullet}}$}};
|
||
\tikzstyle{every state}=[draw=none,text=black,scale=0.75,
|
||
transform shape, shape=rectangle]
|
||
\node[state, draw=black!50] (B1) at (0,0) {\textbf{C11}};
|
||
\node[state, draw=black!50] (B2) at (1,0) {\textbf{C12}};
|
||
\tikzstyle{every node}=[fill=goldenyellow, shape=rectangle]
|
||
\node[edge_proba] (rho2) at (4,-1)
|
||
{\textbf{$\rho_{{\color{goldenyellow}\bullet}}$}};
|
||
\node[state, draw=black!50] (B3) at (3.5,0) {\textbf{C21}};
|
||
\node[state, draw=black!50] (B4) at (4.5,0) {\textbf{C22}};
|
||
\tikzstyle{every node}=[fill=peach, shape=rectangle]
|
||
\node[edge_proba] (rho3) at (10,-1)
|
||
{\textbf{$\rho_{{\color{peach}\bullet}}$}};
|
||
\node[state, draw=black!50] (B5) at (10,0) {\textbf{C31}};
|
||
|
||
\node[font=\small, text justified,draw=none, fill=none,
|
||
below = 0.05cm of rho2] {LBM};
|
||
|
||
\tikzstyle{every edge}=[-,>=stealth',shorten
|
||
>=1pt,auto,draw,line width=1.5pt,draw opacity=0.2]
|
||
|
||
\path (R11) edge (B2);
|
||
\path (R11) edge (B3);
|
||
\path (R11) edge (B4);
|
||
|
||
\path (R12) edge [] (B1);
|
||
\path (R12) edge (B2);
|
||
\path (R12) edge (B3);
|
||
\path (R12) edge (B4);
|
||
|
||
\path (R13) edge [] (B1);
|
||
\path (R13) edge (B2);
|
||
\path (R13) edge (B3);
|
||
|
||
\path (R21) edge (B4);
|
||
\path (R21) edge (B5);
|
||
|
||
\path (R22) edge (B3);
|
||
\path (R22) edge (B4);
|
||
|
||
\path (R11) edge[-,>=stealth',shorten
|
||
>=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[left,
|
||
fill=none] {$\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}}$}
|
||
(B1);
|
||
\path (R13) edge[-,>=stealth',shorten
|
||
>=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[midway, left,
|
||
fill=none] {$\alpha_{{\color{blueind}\bullet}{\color{goldenyellow}\bullet}}$}
|
||
(B4);
|
||
\path (R21) edge[-,>=stealth',shorten
|
||
>=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[midway,
|
||
anchor=center, fill=none]
|
||
{$\alpha_{{\color{cyanind}\bullet}{\color{goldenyellow}\bullet}}$} (B3);
|
||
\path (R22) edge[-,>=stealth',shorten
|
||
>=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[midway, left,
|
||
fill=none] {$\alpha_{{\color{cyanind}\bullet}{\color{peach}\bullet}}$} (B5);
|
||
\path (R31) edge[-,>=stealth',shorten
|
||
>=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[midway,
|
||
right, fill=none]
|
||
{$\alpha_{{\color{electricblue}\bullet}{\color{peach}\bullet}}$} (B5);
|
||
\end{scope}
|
||
|
||
\begin{scope}[xshift=3cm, yshift = 1cm]
|
||
\node[text justified, fill=none] at (10, 3.5)
|
||
{$\overset{iid}{\sim}$};
|
||
\begin{scope}[yshift = 6cm]
|
||
\tikzstyle{every state}=[draw, text=black,scale=0.75,
|
||
transform shape]
|
||
|
||
\tikzstyle{every node}=[fill=gray]
|
||
\node[state, draw=black!50] (R11) at (0,1.25)
|
||
{\textbf{1}};
|
||
\node[state, draw=black!50] (R12) at (1,1.25)
|
||
{\textbf{2}};
|
||
\node[state, draw=black!50] (R13) at (2,1.25)
|
||
{\textbf{3}};
|
||
\node[state, draw=black!50] (R21) at (3,1.25)
|
||
{\textbf{4}};
|
||
\node[state, draw=black!50] (R31) at (5,1.25)
|
||
{\textbf{6}};
|
||
|
||
\tikzstyle{every
|
||
state}=[draw=none,text=black,scale=0.75, transform shape, shape=rectangle]
|
||
\node[state, draw=black!50] (B1) at (0.5,-1)
|
||
{\textbf{1}};
|
||
|
||
\node[state, draw=black!50] (B31) at (2.5,-1)
|
||
{\textbf{3}};
|
||
\node[state, draw=black!50] (B4) at (3.5,-1)
|
||
{\textbf{4}};
|
||
|
||
\node[state, draw=black!50] (B5) at (4.5,-1)
|
||
{\textbf{5}};
|
||
|
||
\tikzstyle{every edge}=[-,>=stealth',shorten
|
||
>=1pt,auto,draw,line width=1pt, draw=gray, fill=gray]
|
||
\path (R11) edge (B1);
|
||
\path (R11) edge (B31);
|
||
\path (R11) edge (B4);
|
||
|
||
\path (R12) edge [] (B1);
|
||
\path (R12) edge (B31);
|
||
\path (R12) edge (B4);
|
||
|
||
\path (R13) edge [] (B1);
|
||
\path (R13) edge (B31);
|
||
\path (R13) edge (B4);
|
||
|
||
\path (R21) edge (B31);
|
||
\path (R21) edge (B4);
|
||
\path (R21) edge (B5);
|
||
|
||
\path (R31) edge (B5);
|
||
\end{scope}
|
||
\node[text width=3cm,font=\small, text justified,
|
||
rotate=90, fill=none] (dots) at (2.5, 7.5){\dots};
|
||
|
||
\begin{scope}[yshift = 0cm]
|
||
\tikzstyle{every state}=[draw, text=black,scale=0.75,
|
||
transform shape]
|
||
|
||
\tikzstyle{every node}=[fill=gray]
|
||
\node[state, draw=black!50] (R11) at (0,2.25)
|
||
{\textbf{4}};
|
||
\node[state, draw=black!50] (R13) at (2,2.25)
|
||
{\textbf{6}};
|
||
\node[state, draw=black!50] (R21) at (3,2.25)
|
||
{\textbf{3}};
|
||
\node[state, draw=black!50] (R22) at (4,2.25)
|
||
{\textbf{5}};
|
||
\node[state, draw=black!50] (R31) at (5,2.25)
|
||
{\textbf{2}};
|
||
|
||
\tikzstyle{every
|
||
state}=[draw=none,text=black,scale=0.75, transform shape, shape=rectangle]
|
||
\node[state, draw=black!50] (B1) at (0.5,0)
|
||
{\textbf{5}};
|
||
\node[state, draw=black!50] (B2) at (1.5,0)
|
||
{\textbf{1}};
|
||
|
||
\node[state, draw=black!50] (B4) at (3.5,0)
|
||
{\textbf{2}};
|
||
|
||
\node[state, draw=black!50] (B5) at (4.5,0)
|
||
{\textbf{4}};
|
||
|
||
\tikzstyle{every edge}=[-,>=stealth',shorten
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||
>=1pt,auto,draw,line width=1pt, draw=gray, fill=gray]
|
||
\path (R11) edge (B1);
|
||
\path (R11) edge (B2);
|
||
\path (R11) edge (B4);
|
||
|
||
\path (R13) edge [] (B1);
|
||
\path (R13) edge (B2);
|
||
\path (R13) edge (B4);
|
||
|
||
\path (R21) edge (B4);
|
||
\path (R21) edge (B5);
|
||
|
||
\path (R22) edge (B4);
|
||
\path (R22) edge (B5);
|
||
|
||
\path (R31) edge (B5);
|
||
\end{scope}
|
||
\end{scope}
|
||
\end{tikzpicture}
|
||
\end{adjustbox}
|
||
\end{center}
|
||
|
||
Pour
|
||
\begin{itemize}
|
||
\item $Q_1 =
|
||
|\{{\color{blueind}\bullet},{\color{cyanind}\bullet},{\color{electricblue}\bullet}\}|$
|
||
blocs fixés en ligne
|
||
\item $Q_2 =
|
||
|\{{\color{burntorange}\bullet},{\color{goldenyellow}\bullet},{\color{peach}\bullet}\}|$
|
||
blocs fixés en colonne
|
||
\end{itemize}
|
||
\begin{block}{Paramètres}
|
||
\begin{itemize}
|
||
\item $\pi_{\bullet} = \mathbb{P}(Z_i =\bullet)$ en ligne et
|
||
$\rho_{\bullet} = \mathbb{P}(W_j = \bullet)$ en colonne
|
||
\item
|
||
$\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}} =
|
||
\mathbb{P}(X_{ij} = 1 | Z_i = {\color{blueind}\bullet}, W_j =
|
||
{\color{burntorange}\bullet})$
|
||
\end{itemize}
|
||
\end{block}
|
||
\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=0.30\textwidth]{img/iid-meso-1.png}
|
||
\includegraphics[width=0.30\textwidth]{img/iid-meso-2.png}
|
||
\includegraphics[width=0.30\textwidth]{img/iid-meso-3.png}
|
||
\includegraphics[width=0.30\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}
|
||
|
||
\subsection[Axe 2]{Axe 2 : Embedding de n\oe uds par
|
||
apprentissage profond pour comparaison des topologies de réseaux}
|
||
\label{sec:axe-2}
|
||
|
||
\subsection[Axe 3]{Axe 3 : Inférence jointe de réseaux}
|
||
\label{sec:axe-3}
|
||
|
||
\section{Organisation de la thèse}
|
||
\label{sec:organisation-these}
|
||
|
||
\begin{frame}
|
||
\begin{block}{Planning prévisionnel de la thèse}
|
||
TODO Ici une timeline
|
||
\end{block}
|
||
\begin{block}{Financement}
|
||
L'INRAE, par le département MathNum donne 50\% des financements de la
|
||
thèse.
|
||
\end{block}
|
||
\end{frame}
|
||
|
||
\section*{Remerciements}
|
||
\begin{frame}[noframenumbering]
|
||
\vfill
|
||
\centering
|
||
\huge Merci pour votre attention.
|
||
\vfill
|
||
\end{frame}
|
||
|
||
\renewcommand{\pgfuseimage}[1]{\scalebox{.75}{\includegraphics{#1}}}
|
||
\begin{frame}[noframenumbering,plain,allowframebreaks]
|
||
\frametitle{Bibliographie}
|
||
\printbibliography
|
||
\end{frame}
|
||
|
||
\appendix
|
||
|
||
\section{Modèles à variables latentes pour collection de réseaux bipartites}
|
||
\begin{frame}
|
||
\frametitle{Latent Block Model (LBM\footnotemark[2])}
|
||
%DONE remplacer i \in bullet par Zi = \bullet
|
||
Proposé par~\cite{govaertEMAlgorithmBlock2005}.
|
||
\begin{columns}
|
||
\begin{column}{0.40\linewidth}
|
||
\begin{figure}[H]
|
||
\center
|
||
\begin{tikzpicture}[scale=0.35]
|
||
\tikzstyle{every state}=[draw, text=black,scale=0.95,
|
||
transform shape]
|
||
\tikzstyle{every state}=[draw=none,text=black,scale=0.75,
|
||
transform shape]
|
||
\tikzset{edge_proba/.style={draw=white, fill=none,
|
||
text=black}}
|
||
|
||
\tikzstyle{every node}=[fill=blueind]
|
||
\node[edge_proba] (pi1) at (1,5.7)
|
||
{\textbf{$\pi_{{\color{blueind}\bullet}}$}};
|
||
\node[state, draw=black!50] (R11) at (0,5) {\textbf{R11}};
|
||
\node[state, draw=black!50] (R12) at (1,5) {\textbf{R12}};
|
||
\node[state, draw=black!50] (R13) at (2,5) {\textbf{R13}};
|
||
|
||
\tikzstyle{every node}=[fill=cyanind]
|
||
\node[edge_proba] (pi2) at (6.75,5.7)
|
||
{\textbf{$\pi_{{\color{cyanind}\bullet}}$}};
|
||
\node[state, draw=black!50] (R21) at (6.25,5)
|
||
{\textbf{R21}};
|
||
\node[state, draw=black!50] (R22) at (7.25,5)
|
||
{\textbf{R22}};
|
||
|
||
\tikzstyle{every node}=[fill=electricblue]
|
||
\node[edge_proba] (pi3) at (10,5.7)
|
||
{\textbf{$\pi_{{\color{electricblue}\bullet}}$}};
|
||
\node[state, draw=black!50] (R31) at (10,5) {\textbf{R31}};
|
||
|
||
\tikzstyle{every node}=[fill=burntorange, shape=rectangle]
|
||
\node[edge_proba] (pi3) at (0.5,-0.7)
|
||
{\textbf{$\rho_{{\color{burntorange}\bullet}}$}};
|
||
\tikzstyle{every state}=[draw=none,text=black,scale=0.75,
|
||
transform shape, shape=rectangle]
|
||
\node[state, draw=black!50] (B1) at (0,0) {\textbf{C11}};
|
||
\node[state, draw=black!50] (B2) at (1,0) {\textbf{C12}};
|
||
\tikzstyle{every node}=[fill=goldenyellow, shape=rectangle]
|
||
\node[edge_proba] (pi3) at (4,-0.7)
|
||
{\textbf{$\rho_{{\color{goldenyellow}\bullet}}$}};
|
||
\node[state, draw=black!50] (B3) at (3.5,0) {\textbf{C21}};
|
||
\node[state, draw=black!50] (B4) at (4.5,0) {\textbf{C22}};
|
||
\tikzstyle{every node}=[fill=peach, shape=rectangle]
|
||
\node[edge_proba] (pi3) at (10,-0.7)
|
||
{\textbf{$\rho_{{\color{peach}\bullet}}$}};
|
||
\node[state, draw=black!50] (B5) at (10,0) {\textbf{C31}};
|
||
|
||
\tikzstyle{every edge}=[-,>=stealth',shorten
|
||
>=1pt,auto,draw,line width=1.5pt,draw opacity=0.2]
|
||
|
||
\path (R11) edge[-,>=stealth',shorten
|
||
>=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[left,
|
||
fill=none] {$\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}}$}
|
||
(B1);
|
||
\path (R11) edge (B2);
|
||
\path (R11) edge (B3);
|
||
\path (R11) edge (B4);
|
||
|
||
\path (R12) edge [] (B1);
|
||
\path (R12) edge (B2);
|
||
\path (R12) edge (B3);
|
||
\path (R12) edge (B4);
|
||
|
||
\path (R13) edge [] (B1);
|
||
\path (R13) edge (B2);
|
||
\path (R13) edge (B3);
|
||
\path (R13) edge[-,>=stealth',shorten
|
||
>=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[midway, left,
|
||
fill=none] {$\alpha_{{\color{blueind}\bullet}{\color{goldenyellow}\bullet}}$}
|
||
(B4);
|
||
|
||
\path (R21) edge[-,>=stealth',shorten
|
||
>=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[midway,
|
||
right, fill=none]
|
||
{$\alpha_{{\color{cyanind}\bullet}{\color{goldenyellow}\bullet}}$} (B3);
|
||
\path (R21) edge (B4);
|
||
\path (R21) edge (B5);
|
||
|
||
\path (R22) edge (B3);
|
||
\path (R22) edge (B4);
|
||
\path (R22) edge[-,>=stealth',shorten
|
||
>=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[midway, left,
|
||
fill=none] {$\alpha_{{\color{cyanind}\bullet}{\color{peach}\bullet}}$} (B5);
|
||
|
||
\path (R31) edge[-,>=stealth',shorten
|
||
>=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[midway,
|
||
right, fill=none]
|
||
{$\alpha_{{\color{electricblue}\bullet}{\color{peach}\bullet}}$} (B5);
|
||
|
||
\end{tikzpicture}
|
||
\caption{Exemple de LBM\footnotemark}
|
||
\label{fig:LBMvisu}
|
||
\end{figure}
|
||
\end{column}
|
||
\begin{column}{0.51\linewidth}
|
||
Pour \begin{itemize}
|
||
\item $Q_1 =
|
||
|\{{\color{blueind}\bullet},{\color{cyanind}\bullet},{\color{electricblue}\bullet}\}|$
|
||
blocs fixés en ligne
|
||
\item $Q_2 =
|
||
|\{{\color{burntorange}\bullet},{\color{goldenyellow}\bullet},{\color{peach}\bullet}\}|$
|
||
blocs fixés en colonne
|
||
\end{itemize}
|
||
\begin{block}{Paramètres}
|
||
\begin{itemize}
|
||
\item $\pi_{\bullet} = \mathbb{P}(Z_i = \bullet)$ en ligne
|
||
et $\rho_{\bullet} = \mathbb{P}(W_j = \bullet)$ en colonne
|
||
\item
|
||
$\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}} =
|
||
\mathbb{P}(X_{ij} = 1 | Z_i = {\color{blueind}\bullet}, W_j =
|
||
{\color{burntorange}\bullet})$
|
||
\end{itemize}
|
||
\end{block}
|
||
\end{column}
|
||
\end{columns}
|
||
|
||
\footnotetext{Que j'appellerai par la suite BiSBM}
|
||
|
||
\end{frame}
|
||
\begin{frame}
|
||
\frametitle{\emph{colSBM}}
|
||
Le modèle \emph{colSBM}
|
||
\parencite{chabert-liddellLearningCommonStructures2023}.\\
|
||
% Difficulté estimer les parametres
|
||
|
||
% DONE Modifier les realisations pour variabilite, mettre iid au dessus du sim et inverser modele et realisations
|
||
\smallskip
|
||
\definecolor{yellow}{RGB}{255,190,60}
|
||
\begin{center}
|
||
\begin{adjustbox}{trim=0 0 0 1cm}
|
||
\begin{tikzpicture}[scale=.32]
|
||
\tikzstyle{every edge}=[-,>=stealth',shorten
|
||
>=1pt,auto,draw,line width=.5pt, bend left]
|
||
\tikzstyle{every state}=[draw, text=black,scale=0.95, transform
|
||
shape]
|
||
\tikzset{edge_proba/.style={draw=white, fill=none, text=black}}
|
||
|
||
\tikzstyle{every node}=[fill=yellow]
|
||
\node[state, draw=black!50] (A1) at (0,2) {\textbf{A1}};
|
||
\node[state, draw=black!50] (A2) at (1.5, 2) {\textbf{A2}};
|
||
\node[state, draw=black!50] (A3) at (0.75,3.25) {\textbf{A3}};
|
||
|
||
\tikzstyle{every node}=[fill=blueind]
|
||
\node[state, draw=black!50] (B1) at (4.5,3) {\textbf{B1}};
|
||
\node[state, draw=black!50] (B2) at (4,4.75) {\textbf{B2}};
|
||
\node[state, draw=black!50] (B3) at (5.5,6) {\textbf{B3}};
|
||
\node[state, draw=black!50] (B4) at (7,4.75) {\textbf{B4}};
|
||
\node[state, draw=black!50] (B5) at (6.5,3) {\textbf{B5}};
|
||
|
||
\tikzstyle{every node}=[fill=greenind]
|
||
\node[state, draw=black!50] (C1) at (5,0) {\textbf{C1}};
|
||
\node[state, draw=black!50] (C2) at (7,1) {\textbf{C2}};
|
||
|
||
\path (A1) edge[bend right] (A2);
|
||
\path (A1) edge node[midway, left, fill=none]
|
||
{$\alpha_{{\color{yellow}\bullet}{\color{yellow}\bullet}}$} (A3);
|
||
\path (A3) edge (A2);
|
||
|
||
\path (A3) edge node[midway, above, fill=none]
|
||
{$\alpha_{{\color{yellow}\bullet}{\color{blueind}\bullet}}$} (B3);
|
||
|
||
\path (B1) edge (B2);
|
||
\path (B2) edge (B3);
|
||
\path (B3) edge (B4);
|
||
\path (B4) edge (B5);
|
||
\path (B5) edge (B1);
|
||
|
||
\path (B1) edge[bend left=0] (B4);
|
||
\path (B5) edge[bend left=0] (B2);
|
||
|
||
\path (A2) edge[bend right] node[midway, below, fill=none]
|
||
{$\alpha_{{\color{yellow}\bullet}{\color{greenind}\bullet}}$} (C1);
|
||
\path (C1) edge[bend right] node[midway, below, fill=none]
|
||
{$\alpha_{{\color{greenind}\bullet}{\color{greenind}\bullet}}$} (C2);
|
||
\path (C2) edge[bend right] node[midway, right, fill=none]
|
||
{$\alpha_{{\color{greenind}\bullet}{\color{blueind}\bullet}}$} (B4);
|
||
|
||
\node[font=\small, text justified,draw=none, fill=none] at
|
||
(4.5,-1.5) {SBM};
|
||
|
||
% Sampled network
|
||
\begin{scope}[xshift=-16cm,yshift=4cm]
|
||
\node[font=\small, text justified, fill=none] at (10, -2.5)
|
||
{$\overset{iid}{\sim}$};
|
||
\tikzstyle{every node}=[fill=gray, scale=0.95]
|
||
\tikzstyle{every edge}=[-,>=stealth',shorten
|
||
>=1pt,auto,draw,line width=.5pt, bend left]
|
||
\tikzstyle{every state}=[draw, text=black,scale=0.95,
|
||
transform shape]
|
||
|
||
\node[state, draw=black!50] (A1) at (0,0) {\textbf{10}};
|
||
\node[state, draw=black!50] (A2) at (1, 0) {\textbf{2}};
|
||
\node[state, draw=black!50] (A3) at (0.5,1) {\textbf{5}};
|
||
|
||
\node[state, draw=black!50] (B2) at (2,2.75) {\textbf{9}};
|
||
\node[state, draw=black!50] (B3) at (3.5,4) {\textbf{6}};
|
||
\node[state, draw=black!50] (B4) at (5,2.75) {\textbf{3}};
|
||
\node[state, draw=black!50] (B5) at (4.5,1) {\textbf{7}};
|
||
|
||
\node[state, draw=black!50] (C1) at (3,-0.5) {\textbf{4}};
|
||
|
||
\path (A1) edge[bend right] (A2);
|
||
\path (A1) edge (A3);
|
||
\path (A3) edge (A2);
|
||
|
||
\path (A3) edge (B3);
|
||
|
||
\path (B2) edge (B3);
|
||
\path (B3) edge (B4);
|
||
\path (B4) edge (B5);
|
||
|
||
\path (B5) edge[bend left=0] (B2);
|
||
|
||
\path (A2) edge[bend right] (C1);
|
||
|
||
\node[text width=3cm,font=\small, text justified,
|
||
rotate=90, fill=none, below = -0.8cm of C1] (dots) {\dots};
|
||
|
||
\end{scope}
|
||
\begin{scope}[xshift=-16cm,yshift=-4cm]
|
||
\tikzstyle{every node}=[fill=gray, scale=0.95]
|
||
\tikzstyle{every edge}=[-,>=stealth',shorten
|
||
>=1pt,auto,draw,line width=.5pt, bend left]
|
||
\tikzstyle{every state}=[draw, text=black,scale=0.95,
|
||
transform shape]
|
||
|
||
\node[state, draw=black!50] (A2) at (1, 0) {\textbf{2}};
|
||
\node[state, draw=black!50] (A3) at (0.5,1) {\textbf{1}};
|
||
|
||
\node[state, draw=black!50] (B1) at (2.5,1) {\textbf{5}};
|
||
\node[state, draw=black!50] (B2) at (2,2.75) {\textbf{10}};
|
||
\node[state, draw=black!50] (B4) at (5,2.75) {\textbf{8}};
|
||
\node[state, draw=black!50] (B5) at (4.5,1) {\textbf{7}};
|
||
|
||
\node[state, draw=black!50] (C2) at (5,0) {\textbf{3}};
|
||
|
||
\path (A3) edge (A2);
|
||
|
||
\path (B1) edge (B2);
|
||
\path (B4) edge (B5);
|
||
\path (B5) edge (B1);
|
||
|
||
\path (B1) edge[bend left=0] (B4);
|
||
\path (B5) edge[bend left=0] (B2);
|
||
|
||
\path (C2) edge[bend right] (B4);
|
||
\end{scope}
|
||
\end{tikzpicture}
|
||
\end{adjustbox}
|
||
\end{center}
|
||
Pour $Q =
|
||
|\{{\color{yellow}\bullet},{\color{blueind}\bullet},{\color{greenind}\bullet}\}|$
|
||
blocs fixés :
|
||
\begin{block}{Paramètres}
|
||
\begin{itemize}
|
||
\item $\pi_{\bullet} = \mathbb{P}(Z_i =\bullet)$
|
||
\item $\alpha_{{\color{greenind}\bullet}{\color{blueind}\bullet}} =
|
||
\mathbb{P}(X_{ij} = 1 | Z_i = {\color{greenind}\bullet}, Z_j =
|
||
{\color{blueind}\bullet})$
|
||
\end{itemize}
|
||
\end{block}
|
||
\end{frame}
|
||
|
||
\begin{frame}
|
||
\frametitle{Collections bipartites}
|
||
\begin{center}
|
||
\begin{adjustbox}{trim=0 0 1 1.5cm}
|
||
\begin{tikzpicture}[scale=.33]
|
||
\begin{scope}[xshift=18cm, yshift=2cm]
|
||
\tikzstyle{every state}=[draw=none, text=black,scale=0.75,
|
||
transform shape]
|
||
\tikzset{edge_proba/.style={draw=white, fill=none,
|
||
text=black}}
|
||
|
||
\tikzstyle{every node}=[fill=blueind]
|
||
\node[edge_proba] (pi1) at (1,5.7)
|
||
{\textbf{$\pi_{{\color{blueind}\bullet}}$}};
|
||
\node[state, draw=black!50] (R11) at (0,5) {\textbf{R11}};
|
||
\node[state, draw=black!50] (R12) at (1,5) {\textbf{R12}};
|
||
\node[state, draw=black!50] (R13) at (2,5) {\textbf{R13}};
|
||
|
||
\tikzstyle{every node}=[fill=cyanind]
|
||
\node[edge_proba] (pi2) at (6.75,5.7)
|
||
{\textbf{$\pi_{{\color{cyanind}\bullet}}$}};
|
||
\node[state, draw=black!50] (R21) at (6.25,5)
|
||
{\textbf{R21}};
|
||
\node[state, draw=black!50] (R22) at (7.25,5)
|
||
{\textbf{R22}};
|
||
|
||
\tikzstyle{every node}=[fill=electricblue]
|
||
\node[edge_proba] (pi3) at (10,5.7)
|
||
{\textbf{$\pi_{{\color{electricblue}\bullet}}$}};
|
||
\node[state, draw=black!50] (R31) at (10,5) {\textbf{R31}};
|
||
|
||
\tikzstyle{every node}=[fill=burntorange, shape=rectangle]
|
||
\node[edge_proba] (rho1) at (0.5,-1)
|
||
{\textbf{$\rho_{{\color{burntorange}\bullet}}$}};
|
||
\tikzstyle{every state}=[draw=none,text=black,scale=0.75,
|
||
transform shape, shape=rectangle]
|
||
\node[state, draw=black!50] (B1) at (0,0) {\textbf{C11}};
|
||
\node[state, draw=black!50] (B2) at (1,0) {\textbf{C12}};
|
||
\tikzstyle{every node}=[fill=goldenyellow, shape=rectangle]
|
||
\node[edge_proba] (rho2) at (4,-1)
|
||
{\textbf{$\rho_{{\color{goldenyellow}\bullet}}$}};
|
||
\node[state, draw=black!50] (B3) at (3.5,0) {\textbf{C21}};
|
||
\node[state, draw=black!50] (B4) at (4.5,0) {\textbf{C22}};
|
||
\tikzstyle{every node}=[fill=peach, shape=rectangle]
|
||
\node[edge_proba] (rho3) at (10,-1)
|
||
{\textbf{$\rho_{{\color{peach}\bullet}}$}};
|
||
\node[state, draw=black!50] (B5) at (10,0) {\textbf{C31}};
|
||
|
||
\node[font=\small, text justified,draw=none, fill=none,
|
||
below = 0.05cm of rho2] {BiSBM};
|
||
|
||
\tikzstyle{every edge}=[-,>=stealth',shorten
|
||
>=1pt,auto,draw,line width=1.5pt,draw opacity=0.2]
|
||
|
||
\path (R11) edge (B2);
|
||
\path (R11) edge (B3);
|
||
\path (R11) edge (B4);
|
||
|
||
\path (R12) edge [] (B1);
|
||
\path (R12) edge (B2);
|
||
\path (R12) edge (B3);
|
||
\path (R12) edge (B4);
|
||
|
||
\path (R13) edge [] (B1);
|
||
\path (R13) edge (B2);
|
||
\path (R13) edge (B3);
|
||
|
||
\path (R21) edge (B4);
|
||
\path (R21) edge (B5);
|
||
|
||
\path (R22) edge (B3);
|
||
\path (R22) edge (B4);
|
||
|
||
\path (R11) edge[-,>=stealth',shorten
|
||
>=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[left,
|
||
fill=none] {$\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}}$}
|
||
(B1);
|
||
\path (R13) edge[-,>=stealth',shorten
|
||
>=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[midway, left,
|
||
fill=none] {$\alpha_{{\color{blueind}\bullet}{\color{goldenyellow}\bullet}}$}
|
||
(B4);
|
||
\path (R21) edge[-,>=stealth',shorten
|
||
>=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[midway,
|
||
anchor=center, fill=none]
|
||
{$\alpha_{{\color{cyanind}\bullet}{\color{goldenyellow}\bullet}}$} (B3);
|
||
\path (R22) edge[-,>=stealth',shorten
|
||
>=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[midway, left,
|
||
fill=none] {$\alpha_{{\color{cyanind}\bullet}{\color{peach}\bullet}}$} (B5);
|
||
\path (R31) edge[-,>=stealth',shorten
|
||
>=1pt,auto,draw=gray,line width=1.5pt, fill=gray, opacity=1] node[midway,
|
||
right, fill=none]
|
||
{$\alpha_{{\color{electricblue}\bullet}{\color{peach}\bullet}}$} (B5);
|
||
\end{scope}
|
||
|
||
\begin{scope}[xshift=3cm, yshift = 1cm]
|
||
\node[text justified, fill=none] at (10, 3.5)
|
||
{$\overset{iid}{\sim}$};
|
||
\begin{scope}[yshift = 6cm]
|
||
\tikzstyle{every state}=[draw, text=black,scale=0.75,
|
||
transform shape]
|
||
|
||
\tikzstyle{every node}=[fill=gray]
|
||
\node[state, draw=black!50] (R11) at (0,1.25)
|
||
{\textbf{1}};
|
||
\node[state, draw=black!50] (R12) at (1,1.25)
|
||
{\textbf{2}};
|
||
\node[state, draw=black!50] (R13) at (2,1.25)
|
||
{\textbf{3}};
|
||
\node[state, draw=black!50] (R21) at (3,1.25)
|
||
{\textbf{4}};
|
||
\node[state, draw=black!50] (R31) at (5,1.25)
|
||
{\textbf{6}};
|
||
|
||
\tikzstyle{every
|
||
state}=[draw=none,text=black,scale=0.75, transform shape, shape=rectangle]
|
||
\node[state, draw=black!50] (B1) at (0.5,-1)
|
||
{\textbf{1}};
|
||
|
||
\node[state, draw=black!50] (B31) at (2.5,-1)
|
||
{\textbf{3}};
|
||
\node[state, draw=black!50] (B4) at (3.5,-1)
|
||
{\textbf{4}};
|
||
|
||
\node[state, draw=black!50] (B5) at (4.5,-1)
|
||
{\textbf{5}};
|
||
|
||
\tikzstyle{every edge}=[-,>=stealth',shorten
|
||
>=1pt,auto,draw,line width=1pt, draw=gray, fill=gray]
|
||
\path (R11) edge (B1);
|
||
\path (R11) edge (B31);
|
||
\path (R11) edge (B4);
|
||
|
||
\path (R12) edge [] (B1);
|
||
\path (R12) edge (B31);
|
||
\path (R12) edge (B4);
|
||
|
||
\path (R13) edge [] (B1);
|
||
\path (R13) edge (B31);
|
||
\path (R13) edge (B4);
|
||
|
||
\path (R21) edge (B31);
|
||
\path (R21) edge (B4);
|
||
\path (R21) edge (B5);
|
||
|
||
\path (R31) edge (B5);
|
||
\end{scope}
|
||
\node[text width=3cm,font=\small, text justified,
|
||
rotate=90, fill=none] (dots) at (2.5, 7.5){\dots};
|
||
|
||
\begin{scope}[yshift = 0cm]
|
||
\tikzstyle{every state}=[draw, text=black,scale=0.75,
|
||
transform shape]
|
||
|
||
\tikzstyle{every node}=[fill=gray]
|
||
\node[state, draw=black!50] (R11) at (0,2.25)
|
||
{\textbf{4}};
|
||
\node[state, draw=black!50] (R13) at (2,2.25)
|
||
{\textbf{6}};
|
||
\node[state, draw=black!50] (R21) at (3,2.25)
|
||
{\textbf{3}};
|
||
\node[state, draw=black!50] (R22) at (4,2.25)
|
||
{\textbf{5}};
|
||
\node[state, draw=black!50] (R31) at (5,2.25)
|
||
{\textbf{2}};
|
||
|
||
\tikzstyle{every
|
||
state}=[draw=none,text=black,scale=0.75, transform shape, shape=rectangle]
|
||
\node[state, draw=black!50] (B1) at (0.5,0)
|
||
{\textbf{5}};
|
||
\node[state, draw=black!50] (B2) at (1.5,0)
|
||
{\textbf{1}};
|
||
|
||
\node[state, draw=black!50] (B4) at (3.5,0)
|
||
{\textbf{2}};
|
||
|
||
\node[state, draw=black!50] (B5) at (4.5,0)
|
||
{\textbf{4}};
|
||
|
||
\tikzstyle{every edge}=[-,>=stealth',shorten
|
||
>=1pt,auto,draw,line width=1pt, draw=gray, fill=gray]
|
||
\path (R11) edge (B1);
|
||
\path (R11) edge (B2);
|
||
\path (R11) edge (B4);
|
||
|
||
\path (R13) edge [] (B1);
|
||
\path (R13) edge (B2);
|
||
\path (R13) edge (B4);
|
||
|
||
\path (R21) edge (B4);
|
||
\path (R21) edge (B5);
|
||
|
||
\path (R22) edge (B4);
|
||
\path (R22) edge (B5);
|
||
|
||
\path (R31) edge (B5);
|
||
\end{scope}
|
||
\end{scope}
|
||
\end{tikzpicture}
|
||
\end{adjustbox}
|
||
\end{center}
|
||
|
||
Pour
|
||
\begin{itemize}
|
||
\item $Q_1 =
|
||
|\{{\color{blueind}\bullet},{\color{cyanind}\bullet},{\color{electricblue}\bullet}\}|$
|
||
blocs fixés en ligne
|
||
\item $Q_2 =
|
||
|\{{\color{burntorange}\bullet},{\color{goldenyellow}\bullet},{\color{peach}\bullet}\}|$
|
||
blocs fixés en colonne
|
||
\end{itemize}
|
||
\begin{block}{Paramètres}
|
||
\begin{itemize}
|
||
\item $\pi_{\bullet} = \mathbb{P}(Z_i =\bullet)$ en ligne et
|
||
$\rho_{\bullet} = \mathbb{P}(W_j = \bullet)$ en colonne
|
||
\item
|
||
$\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}} =
|
||
\mathbb{P}(X_{ij} = 1 | Z_i = {\color{blueind}\bullet}, W_j =
|
||
{\color{burntorange}\bullet})$
|
||
\end{itemize}
|
||
\end{block}
|
||
\end{frame}
|
||
|
||
\begin{frame}
|
||
\frametitle{Différents modèles}
|
||
\begin{block}{\emph{iid-colBiSBM}}
|
||
$\bm{\pi} = (\pi_1, \dots \pi_{Q_1})$ et $\bm{\rho} = (\rho_1, \dots
|
||
\rho_{Q_2})$ %{$\forall q \in \llbracket 1, Q_1 - 1\rrbracket, \pi_q > 0$ et $\forall r \in \llbracket 1, Q_2 - 1\rrbracket, \rho_r > 0$}
|
||
, tous les réseaux partagent les mêmes paramètres\footnotemark
|
||
\end{block}
|
||
|
||
\begin{block}{\emph{$\pi\rho$-colBiSBM}}
|
||
$\bm{\pi} = ((\pi_{\color{black}1}^{\color{red}m}, \dots
|
||
\pi_{\color{black}Q_1}^{\color{red}m}))_{m=1,\dots M}$ et $\bm{\rho} =
|
||
((\rho_{\color{black}1}^{\color{red}m}, \dots
|
||
\rho_{\color{black}Q_2}^{\color{red}m}))_{m=1,\dots M}$
|
||
%{$\forall q \in \llbracket 1, Q_1 - 1\rrbracket, \pi_q > 0$ et $\forall r \in \llbracket 1, Q_2 - 1\rrbracket, \rho_r > 0$}
|
||
\small \\
|
||
avec $\forall q,m \in \llbracket 1, Q_1 \rrbracket \times \llbracket 1,
|
||
M \rrbracket, \pi_q^m \in \left[ 0,1 \right]$
|
||
et $\forall r,m \in \llbracket 1, Q_2 \rrbracket \times \llbracket 1, M
|
||
\rrbracket, \rho_r^m \in \left[ 0,1 \right]$
|
||
\end{block}
|
||
Et également deux autres modèles ($\pi$-colBiSBM et $\rho$-colBiSBM) où
|
||
seulement une des deux dimensions est libre.
|
||
\footnotetext{Dans tous les modèles la structure de connectivité est
|
||
supposée identique au sein de la collection.}
|
||
\end{frame}
|
||
\begin{frame}
|
||
\frametitle{Estimation des paramètres}
|
||
% DONE dire que tau i q m c' est la proba que Zim = q, approximation de la proba variationnelle. Parce qu on impose lindependance
|
||
Maximisation d'une borne inférieure de la log-vraisemblance des données
|
||
observées.
|
||
\begin{multline*}
|
||
\ell (\bm{X};\bm{\theta}) \geq \color{red}\sum_{m=1}^{M} \bigg(
|
||
\color{black} \sum_{i = 1}^{n_1^m}\sum_{j=1}^{n_2^m}\sum_{q \in
|
||
\mathcal{Q}_{1,m}} \sum_{r \in \mathcal{Q}_{2,m}} \tau^{1,m}_{i,q}
|
||
\tau^{2,m}_{j,r} \log f(X^{m}_{ij}; \alpha_{qr}) \\
|
||
+ \sum_{i=1}^{n_1^m} \sum_{q \in \mathcal{Q}_{1,m}} \tau^{1,m}_{i,q}
|
||
\log \pi_{\color{black}q}^{\color{gray}m} + \sum_{j=1}^{n_2^m} \sum_{r \in
|
||
\mathcal{Q}_{2,m}} \tau^{2,m}_{j,r} \log \rho_{\color{black}r}^{\color{gray}m}
|
||
\\
|
||
- \sum_{i=1}^{n_1} \tau^{1,m}_{i,q} \log \tau^{1,m}_{i,q} -
|
||
\sum_{j=1}^{n_2} \tau^{2,m}_{j,r} \log \tau^{2,m}_{j,r} \color{red}\bigg)
|
||
\color{black} =: J(\bm{\tau};\bm{\theta}) $$
|
||
\end{multline*}
|
||
|
||
\begin{block}{Approximation variationnelle}
|
||
$\tau_{i,q}^{1,m} = P(Z_i = q | X^m_{ij})$ et $\tau_{j,r}^{2,m} = P(W_j
|
||
= r | X^m_{ij})$ tels que $P(Z_i = q, W_j = r | X^m_{ij}) =
|
||
\tau_{i,q}^{1,m}\times\tau_{j,r}^{2,m}$
|
||
\end{block}
|
||
|
||
\end{frame}
|
||
|
||
\begin{frame}
|
||
\frametitle{Sélection de modèle : choix de $(Q_1,Q_2)$ - Approche
|
||
gloutonne}
|
||
% DONE But maximiser un critere le BICL, deplacer voir St Clair dans la note
|
||
% VEM a Q1 Q2 fixer
|
||
% Choix de Q1 Q2 par maximisation du BICL
|
||
% Itemize dans la box : init, explo voisin, arrets
|
||
\underline{Le VEM se fait à $Q_1, Q_2$ fixés}, il faut donc déterminer les
|
||
\enquote*{meilleures} coordonnées.
|
||
Nous maximisons un BIC-L\footnote{\emph{Bayesian Information Criterion -
|
||
Like}, en adaptant les formules
|
||
de~\cite{chabert-liddellLearningCommonStructures2023}}.
|
||
|
||
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}
|
||
\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}
|
||
|
||
\end{document} |