rapport : small reworking

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.
 
 The idea is to adjust the colBiSBM model over the full collection of $M$