171 lines
No EOL
7.5 KiB
TeX
171 lines
No EOL
7.5 KiB
TeX
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\documentclass[12pt,a4paper]{rapport1}
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%====En-tête====
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% Ajout des packages
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\usepackage[french]{babel} % pour dire que le texte est en francais
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\usepackage{a4} % pour la taille
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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} % 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{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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% Nouvelles commandes
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\newcommand{\Tau}{\mathcal{T}}
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% titre et auteur
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\title{Rapport de stage dans l'UMR MIA Paris-Saclay}
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\author{Louis Lacoste}
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\begin{document}
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\maketitle
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\tableofcontents
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\chapter{Présentation de l'UMR}
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\chapter{Adaption au cas bipartite : colBiSBM}
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\section{Etape VE de l'algorithme}
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Formule du point fixe pour la distribution de Bernoulli
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% Repasser à l'exponentielle pour la présentation du point fixe
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\begin{itemize}
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\item \textit{iid} :
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\[ \bm{\tau}^{m,1} = ~^{t}\pi + \exp[(\text{Mask}^{m} \odot A^{m})
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\bm{\tau}^{m,2} ~^{t}(\text{logit}(\alpha)) + \text{Mask}^{m}
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\bm{\tau}^{m,2} ~^{t}\log(\bm{1} - \alpha)] \]
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\[ \log(\bm{\tau}^{m,2}) = ~^{t}\log(\rho) + ~^{t}(\text{Mask}^{m} \odot A^{m})
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\bm{\tau}^{m,1} \text{logit}(\alpha) + ~^{t}\text{Mask}^{m}
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\bm{\tau}^{m,1} \log(\bm{1} - \alpha) \]
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\item $\rho\pi$ :
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\[ \log(\bm{\tau}^{m,1}) = ~^{t}\log(\pi^{m}) + (\text{Mask}^{m} \odot A^{m})
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\bm{\tau}^{m,2} ~^{t}(\text{logit}(\alpha)) + \text{Mask}^{m}
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\bm{\tau}^{m,2} ~^{t}\log(\bm{1} - \alpha) \]
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\[ \log(\bm{\tau}^{m,2}) = ~^{t}\log(\rho^{m}) + ~^{t}(\text{Mask}^{m} \odot A^{m})
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\bm{\tau}^{m,1} \text{logit}(\alpha) + ~^{t}\text{Mask}^{m}
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\bm{\tau}^{m,1} \log(\bm{1} - \alpha) \]
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\end{itemize}
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avec $\text{Mask}^{m}$ la matrice qui contient des $0$ si la valeur est un NA et
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des $1$ sinon.
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\section{M step of the algorithm}
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\section{Computation of the variational bound}
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\section{Penalties}
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\paragraph*{\textit{iid-colBiSBM}}
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For the \textit{iid-colBiSBM} the penalties were modified in the following way :
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\begin{itemize}
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\item For the $\pi$s and $\rho$s:
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\[\text{pen}_{\pi}(Q_1) = (Q_1 - 1)\log(\sum_{m=1}^{M}n_{r}^{(m)})\]
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\[\text{pen}_{\rho}(Q_2) = (Q_2 - 1)\log(\sum_{m=1}^{M}n_{c}^{(m)})\]
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\item For the $\alpha$s :
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\[\text{pen}_{\alpha}(Q_1, Q_2) = Q_1 \times Q_2 \log(N_M)\]
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avec
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\[ N_M = \sum_{m = 1}^{M} n_{r}^{(m)} \times n_{c}^{(m)} \]
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\end{itemize}
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And thus the $BIC-L$ formula is now:
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\[ \text{$BIC-L$}(\bm{X},Q_1, Q_2) = \max_{\theta} \mathcal{J} (\mathcal{\hat{R}}, \bm{\theta})
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- \frac{1}{2} [\text{pen}_{\pi}(Q_1) + \text{pen}_{\rho}(Q_2) + \text{pen}_{\alpha}(Q_1, Q_2)]\]
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\paragraph*{\textit{$\rho\pi$-colBiSBM}}
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For the \textit{$\rho\pi$-colBiSBM} the penalties are the following:
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\begin{itemize}
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\item The support penalties are:
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\[ \text{pen}_{S_1}(Q_1) = -2 \log p_{Q_1} (S_1) \]
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\[ \text{pen}_{S_2}(Q_2) = -2 \log p_{Q_2} (S_2) \]
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with
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\[ \log p_{Q_1}(S_1) = - M \log(Q_1) - \sum_{m=1}^{M} \log {Q_1 \choose Q_1^{(m)}} \]
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\[ \log p_{Q_2}(S_2) = - M \log(Q_2) - \sum_{m=1}^{M} \log {Q_2 \choose Q_2^{(m)}} \]
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\item Penalties for the $\rho$s and $\pi$s:
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\[ \text{pen}_{\pi}(Q_1, S_1) = \sum_{m=1}^{M} (Q_{1}^{(m)} - 1) \log n_{r}^{(m)} \]
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\[ \text{pen}_{\rho}(Q_2, S_2) = \sum_{m=1}^{M} (Q_{2}^{(m)} - 1) \log n_{c}^{(m)} \]
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\item Penalties for the $\alpha$s:
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\[ \text{pen}_{\alpha}(Q_1, Q_2, S_1, S_2) = (\sum_{q=1}^{Q_1} \sum_{r=1}^{Q_2} \mathbb{1}_{(S_1)'S_2 > 0}) \log (N_M) \]
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\end{itemize}
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And the corresponding $BIC-L$ formula:
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\[
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\begin{aligned}
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\text{$BIC-L$}(\bm{X},Q_1, Q_2) =
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\max_{S_1,S_2} [
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& \max_{\theta_{S_1,S_2} \in \Theta_{S_1,S_2}} \mathcal{J}(\mathcal{\hat{R}},\theta_{S_1,S_2})\\
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- \frac{1}{2} & (\text{pen}_{\pi}(Q_1, S_1) + \text{pen}_{\rho}(Q_2, S_2)\\
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&+ \text{pen}_{\alpha}(Q_1, Q_2, S_1, S_2)\\
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&+ \text{pen}_{S_1}(Q_1) + \text{pen}_{S_2}(Q_2))]\\
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\end{aligned}
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\]
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\section{Latent space exploration and model selection}
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In order to explorer the bi-dimensional latent space $(Q_1,Q_2)$
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we use the following strategies.
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\subsection{Model selection}
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In the following steps the model selection consists of using the $BIC-L$
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criterion to select the model. We choose among the proposed models the one that
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maximizes the $BIC-L$
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\subsection{Initialization and pairing of the models}
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First to combine the information from the $M$ networks we fit a collection model
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for each network at the two points $Q = (1, 2)$ and $Q = (2, 1)$. Using the
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previously described VEM algorithm we obtain for each network its parameters
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($\rho,\pi,\alpha$).
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We then compute the marginal laws for each dimension, for each network. Then
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we order the network blocks by the probabilities obtained in decreasing order.
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\begin{itemize}
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\item For the memberships on the columns:
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$col~order_m = order\left(\pi_m \times \alpha_m\right)$
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\item For the memberships on the rows:
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$row~order_m = order\left(\rho_m \times ~^{t}(\alpha_m)\right)$
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\end{itemize}
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Using this order we relabel the memberships for the $M$ fitted collection of a
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single network.
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Then we use the $M$ memberships to fit a collection containing the $M$ networks.
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\subsection{Greedy exploration to find an estimation of the mode}
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Using the previously fitted models for $Q = (1,2)$ and $Q = (2,1)$ we choose to
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perform a greedy exploration to find a first mode.
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Meaning that for a given $Q = (Q_1, Q_2)$ we will compute all the possible
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memberships for the points $Q = (Q_1 + 1, Q_2)$ and $Q = (Q_1, Q_2 + 1)$, fit
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the corresponding models and choose the one that maximizes the $BIC-L$ as the
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next point from which to repeat the procedure. We repeat the procedure until the
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$BIC-L$ stops increasing $3$ times in a row.
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% \begin{algorithm}
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% \caption{Greedy exploration of the latent space $Q_1$, $Q_2$}
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% \label{alg:greedy_explotation}
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% \textbf{Commencer} initialisation
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% \begin{itemize}
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% \item Pour chacun des $M$ réseaux, inférer les paramètre avec $Q_1 = q_{1,0}$ et $Q_2 = q_{2,0}$
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% \item Apparier les clusterings obtenus en utilisant les probabilités marginales, afin de faire correspondre les étiquettes des clusters obtenus.
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% \end{itemize}
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% \textbf{tant que} le BICL du meilleur voisin sélectionné à chaque itération n'a pas augmenté durant 3 itérations consécutives, continuer :
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% \begin{itemize}
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% \item Calculer toutes les séparations possible de chacun des
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% \end{itemize}
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% \end{algorithm}
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When this first estimation of the $BIC-L$ mode has been find we apply the moving
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window on it.
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\subsection{Fenêtre glissante pour mettre à jour les clusterings et les $BIC-L$}
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\section{Clustering des réseaux}
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\subsection{Adaptation de la distance entre les paramètres du modèle}
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La distance pondère désormais avec les $\pi$ et les $\rho$.
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\[
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D_{\mathcal{M}}(m,m') = \sum_{q = 1}^{Q_1} \sum_{r = 1}^{Q_2} \max(\widetilde{\pi}_{q}^{m}, \widetilde{\pi}_{q}^{m'}) \left( \frac{\widetilde{\alpha}_{qr}^{m}}{\widehat{\delta}_{m}} - \frac{\widetilde{\alpha}_{qr}^{m'}}{\widehat{\delta}_{m'}}\right)^{2} \max(\widetilde{\rho}_{r}^{m}, \widetilde{\rho}_{r}^{m'})
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\]
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\listoffigures
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\listoftables
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\end{document} |