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rapport : small reworking

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Louis Lacoste 2024-08-11 11:39:57 +02:00
parent 813792184f
commit 2e7bfa2dcf
2 changed files with 14 additions and 11 deletions
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rapport/chapter3-structure-detection.tex
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@ -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{\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} \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*} \end{align*}
the parameters takes into account all the networks at the same time. The the parameters take into account all the networks at the same time.
connectivity parameters $\alpha_{qr}$ for all models are estimated as the ratio The connectivity parameters $\alpha_{qr}$ for all models are estimated as the
of the number of interactions between row block $q$ and column block $r$ among ratio of the number of interactions between row block $q$ and column block $r$
all networks over the number of number of possible interactions: among all networks over the number of number of possible interactions:
\begin{align*} \begin{align*}
\widehat{\alpha}_{qr} = \frac{\sum_{m=1}^{M} e^{m}_{qr}}{\sum_{m=1}^{M} n^{m}_{qr}} \widehat{\alpha}_{qr} = \frac{\sum_{m=1}^{M} e^{m}_{qr}}{\sum_{m=1}^{M} n^{m}_{qr}}
\end{align*} \end{align*}
Please note that those formulae can vary with the emission distribution used.
\section{Model selection}\label{sec:model-selection} \section{Model selection}\label{sec:model-selection}
% DONE % DONE
% Adapt bicl, methode explo car defi % 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. repeat the procedure until the BIC-L stops increasing $2$ times in a row.
\begin{algorithm}[H] \begin{algorithm}[H]
\small
\caption{Greedy Exploration for Mode Estimation} \caption{Greedy Exploration for Mode Estimation}
\SetAlgoLined \SetAlgoLined
\SetKwInOut{Input}{Input} \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} \Output{Estimation of the mode using greedy exploration}
\BlankLine \BlankLine
Initialize $Q = (1,2)$ as the starting point Initialize $Q = (1,2)$ as the starting point\\
Initialize $\text{BIC-L}_{\text{max}}$ as the maximum achieved BIC-L value Initialize $\text{BIC-L}_{\text{max}}$ as the maximum achieved BIC-L value\\
Initialize $consecutive\_count$ as 0 Initialize $consecutive\_count$ as 0
\BlankLine \BlankLine
@ -451,13 +454,12 @@ repeat the procedure until the BIC-L stops increasing $2$ times in a row.
\BlankLine \BlankLine
\If{$\text{BIC-L} > \text{BIC-L}_{\text{max}}$}{ \If{$\text{BIC-L} > \text{BIC-L}_{\text{max}}$}{
$\text{BIC-L}_{\text{max}} \leftarrow \text{BIC-L}$ $\text{BIC-L}_{\text{max}} \leftarrow \text{BIC-L}$\\
$consecutive\_count \leftarrow 0$ $consecutive\_count \leftarrow 0$
} }
\Else{ \Else{
$consecutive\_count \leftarrow consecutive\_count + 1$ $consecutive\_count \leftarrow consecutive\_count + 1$
} }
\BlankLine \BlankLine
$Q \leftarrow$ Next selected point $Q \leftarrow$ Next selected point
} }
@ -489,6 +491,7 @@ consists of two alternating steps:
\end{itemize} \end{itemize}
\begin{algorithm}[t] \begin{algorithm}[t]
\small
\caption{Moving Window Procedure} \caption{Moving Window Procedure}
\SetAlgoLined \SetAlgoLined
\SetKwInOut{Input}{Input} \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{$Q_1 \in \left[ Q_{1,\text{center}} - \text{depth} ; Q_{1,\text{center}} + \text{depth} \right]$}{
\For{$Q_2 \in \left[ Q_{2,\text{center}} - \text{depth}; Q_{2,\text{center}} + \text{depth} \right] $}{ \For{$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 Fit models with the block membership changes
Compare and keep the best model based on BIC-L 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{$Q_1 \in \left[ Q_{1,\text{center}} + \text{depth} ; Q_{1,\text{center}} - \text{depth} \right]$}{
\For{$Q_2 \in \left[ Q_{2,\text{center}} + \text{depth}; Q_{2,\text{center}} - \text{depth} \right] $}{ \For{$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 Fit models with the block membership changes
Compare and keep the best model based on BIC-L 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} \label{fig:netclustering-procedure}
\end{figure} \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. explanation of the network clustering algorithm.
The idea is to adjust the colBiSBM model over the full collection of $M$ The idea is to adjust the colBiSBM model over the full collection of $M$

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rapport/rapport.pdf
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