From 2e7bfa2dcf768ec2e8dc31768cc29f817f44f86f Mon Sep 17 00:00:00 2001 From: Louis Lacoste Date: Sun, 11 Aug 2024 11:39:57 +0200 Subject: [PATCH] rapport : small reworking --- rapport/chapter3-structure-detection.tex | 25 +++++++++++++---------- rapport/rapport.pdf | Bin 2444268 -> 2452622 bytes 2 files changed, 14 insertions(+), 11 deletions(-) diff --git a/rapport/chapter3-structure-detection.tex b/rapport/chapter3-structure-detection.tex index f2062b3..835966e 100644 --- a/rapport/chapter3-structure-detection.tex +++ b/rapport/chapter3-structure-detection.tex @@ -281,14 +281,16 @@ while on the other hand, \widehat{\pi}_q = \frac{\sum_{m=1}^{M} n^{1,m}_{q}}{\sum_{m=1}^{M} n_1^m} & & \text{for } iid\text{-colBiSBM} \text{ and } \rho\text{-colBiSBM} \\ \widehat{\rho}_r = \frac{\sum_{m=1}^{M} n^{2,m}_{r}}{\sum_{m=1}^{M} n_2^m} & & \text{for } iid\text{-colBiSBM} \text{ and } \pi\text{-colBiSBM} \end{align*} -the parameters takes into account all the networks at the same time. The -connectivity parameters $\alpha_{qr}$ for all models are estimated as the ratio -of the number of interactions between row block $q$ and column block $r$ among -all networks over the number of number of possible interactions: +the parameters take into account all the networks at the same time. +The connectivity parameters $\alpha_{qr}$ for all models are estimated as the +ratio of the number of interactions between row block $q$ and column block $r$ +among all networks over the number of number of possible interactions: \begin{align*} \widehat{\alpha}_{qr} = \frac{\sum_{m=1}^{M} e^{m}_{qr}}{\sum_{m=1}^{M} n^{m}_{qr}} \end{align*} +Please note that those formulae can vary with the emission distribution used. + \section{Model selection}\label{sec:model-selection} % DONE % Adapt bicl, methode explo car defi @@ -430,6 +432,7 @@ maximizes the BIC-L as the next point from which to repeat the procedure. We repeat the procedure until the BIC-L stops increasing $2$ times in a row. \begin{algorithm}[H] + \small \caption{Greedy Exploration for Mode Estimation} \SetAlgoLined \SetKwInOut{Input}{Input} @@ -439,8 +442,8 @@ repeat the procedure until the BIC-L stops increasing $2$ times in a row. \Output{Estimation of the mode using greedy exploration} \BlankLine - Initialize $Q = (1,2)$ as the starting point - Initialize $\text{BIC-L}_{\text{max}}$ as the maximum achieved BIC-L value + Initialize $Q = (1,2)$ as the starting point\\ + Initialize $\text{BIC-L}_{\text{max}}$ as the maximum achieved BIC-L value\\ Initialize $consecutive\_count$ as 0 \BlankLine @@ -451,13 +454,12 @@ repeat the procedure until the BIC-L stops increasing $2$ times in a row. \BlankLine \If{$\text{BIC-L} > \text{BIC-L}_{\text{max}}$}{ - $\text{BIC-L}_{\text{max}} \leftarrow \text{BIC-L}$ + $\text{BIC-L}_{\text{max}} \leftarrow \text{BIC-L}$\\ $consecutive\_count \leftarrow 0$ } \Else{ $consecutive\_count \leftarrow consecutive\_count + 1$ } - \BlankLine $Q \leftarrow$ Next selected point } @@ -489,6 +491,7 @@ consists of two alternating steps: \end{itemize} \begin{algorithm}[t] + \small \caption{Moving Window Procedure} \SetAlgoLined \SetKwInOut{Input}{Input} @@ -507,7 +510,7 @@ consists of two alternating steps: \For{$Q_1 \in \left[ Q_{1,\text{center}} - \text{depth} ; Q_{1,\text{center}} + \text{depth} \right]$}{ \For{$Q_2 \in \left[ Q_{2,\text{center}} - \text{depth}; Q_{2,\text{center}} + \text{depth} \right] $}{ - Compute possible splits from predecessors $(Q_1 - 1, Q_2)$ and $(Q_1, Q_2 - 1)$ + Compute possible splits from predecessors $(Q_1 - 1, Q_2)$ and $(Q_1, Q_2 - 1)$\\ Fit models with the block membership changes Compare and keep the best model based on BIC-L } @@ -518,7 +521,7 @@ consists of two alternating steps: \For{$Q_1 \in \left[ Q_{1,\text{center}} + \text{depth} ; Q_{1,\text{center}} - \text{depth} \right]$}{ \For{$Q_2 \in \left[ Q_{2,\text{center}} + \text{depth}; Q_{2,\text{center}} - \text{depth} \right] $}{ - Compute possible merges from predecessors $(Q_1 + 1, Q_2)$ and $(Q_1, Q_2 + 1)$ + Compute possible merges from predecessors $(Q_1 + 1, Q_2)$ and $(Q_1, Q_2 + 1)$\\ Fit models with the block membership changes Compare and keep the best model based on BIC-L } @@ -737,7 +740,7 @@ And the pairwise dissimilarity for networks $(m,m')\in\mathcal{M}^2$ is then: \label{fig:netclustering-procedure} \end{figure} -The above figure (\ref{fig:netclustering-procedure}) shows a condensed +The above figure,~\ref{fig:netclustering-procedure}, shows a condensed explanation of the network clustering algorithm. 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