rapport : updating rapport

rapport/appendices.tex

@@ -1,6 +1,6 @@
 \clearpage
 \pagenumbering{arabic}% resets `page` counter to 1
-\renewcommand*{\thepage}{A\arabic{page}}
+\renewcommand*{\thepage}{A-\arabic{page}}
 \appendix
 \chapter{Supplementary for~\nameref{chap:simulation-studies}}
 Below are the supplementary material for the~\nameref{chap:simulation-studies}.

rapport/chapter3-structure-detection.tex

@@ -67,7 +67,7 @@ network $m$ is assumed to follow a $BiSBM$ with its own parameters ($\bm{\pi}^m,
     \bm{\rho}^m, \bm{\alpha}^m$).
 % DONE Finish explaining
 
-\section{Definition of the colBiSBM models}\label{sec:definition-of-the-colbisbm-models}
+\section{Definition of the \emph{colBiSBM} models}\label{sec:definition-of-the-colbisbm-models}
 % Here are some common notations and conventions that we will use in the following
 % sections.
 
@@ -77,7 +77,7 @@ all the networks are the independent realizations of the same $Q_1$-$Q_2$-BiSBM
 with identical parameters. The \emph{iid-colBiSBM} is defined as follows:
 
 \begin{align}
-    \tag{\emph{iid-colBiSBM}}
+    \tag{\emph{iid}-colBiSBM}
     X^m \sim \mathcal{F}-BiSBM_{n_1^m,n_2^m} (Q_1, Q_2, \bm{\pi}, \bm{\rho}, \bm{\alpha}), &  & \forall m = 1, \dots M
 \end{align}
 where $\forall (q,r) \in \{1,\dots,Q_1\}\times\{1,\dots,Q_2\}$, $\alpha_{qr} \in \mathcal{A}_{\mathcal{F}}$,
@@ -99,7 +99,7 @@ $\pi$-colBiSBM model still assumes that the networks share a common connectivity
 structure represented by $\bm{\alpha}$ but that each network has its own row
 block proportions. For $m \in \{1,\dots,M\}$, the $X^m$ are independent and
 \begin{align}
-    \tag{\emph{$\pi$-colBiSBM}}
+    \tag{\emph{$\pi$}-colBiSBM}
     X^m \sim \mathcal{F}-BiSBM_{n_1^m,n_2^m} (Q_1, Q_2, \bm{\pi^m}, \bm{\rho}, \bm{\alpha}), &  & \forall m = 1, \dots, M
 \end{align}
 where $\forall (q,r) \in \{1,\dots,Q_1\}\times\{1,\dots,Q_2\}$, $\alpha_{qr} \in \mathcal{A}_{\mathcal{F}}$,
@@ -120,7 +120,7 @@ there is no freedom on the column dimension.
 For a given number of blocks $Q_1$, $Q_2$ and matrix $S^1$ ($S^2$ being in this
 case the matrix full of ones), the number of parameters is:
 \begin{equation*}
-    \text{NP}(\pi\text{-}colBiSBM) = \sum_{m=1}^{M}\Bigg( \sum_{q=1}^{Q_1} S^1_{mq} - 1 \Bigg) + (Q_2 - 1) + \sum_{\substack{q=1,\dots,Q_1 \\ r=1,\dots,Q_2}} \mathbbb{1}_{{(S^{1\prime}S^2)}_{qr}>0}
+    \text{NP}(\pi\text{-colBiSBM}) = \sum_{m=1}^{M}\Bigg( \sum_{q=1}^{Q_1} S^1_{mq} - 1 \Bigg) + (Q_2 - 1) + \sum_{\substack{q=1,\dots,Q_1 \\ r=1,\dots,Q_2}} \mathbbb{1}_{{(S^{1\prime}S^2)}_{qr}>0}
 \end{equation*}
 The first term corresponds to the non-null block proportions in each network.
 The third quantity accounts for the fact that some blocks may never be
@@ -131,7 +131,7 @@ $\rho$-colBiSBM model still assumes that the networks share a common connectivit
 structure represented by $\bm{\alpha}$ but that each network has its own column
 block proportions. For $m \in \{1,\dots,M\}$, the $X^m$ are independent and
 \begin{align}
-    \tag{\emph{$\rho$-colBiSBM}}
+    \tag{\emph{$\rho$}-colBiSBM}
     X^m \sim \mathcal{F}-BiSBM_{n_1^m,n_2^m} (Q_1, Q_2, \bm{\pi}, \bm{\rho^m}, \bm{\alpha}), &  & \forall m = 1, \dots, M
 \end{align}
 where $\forall (q,r) \in \{1,\dots,Q_1\}\times\{1,\dots,Q_2\}$, $\alpha_{qr} \in \mathcal{A}_{\mathcal{F}}$,
@@ -142,13 +142,13 @@ proportions to be
 null in certain networks ($\rho^m_r\in\left[ 0,1 \right]$): if $\rho_r^m = 0$
 then the column block $r$ is not represented in the network $m$.
 
-\enquote{Mirroring} the formulas for the $\pi$-$colBiSBM$ we relax the constraints on
+\enquote{Mirroring} the formulas for the $\pi$-colBiSBM we relax the constraints on
 the column dimension.
 
 For a given number of blocks $Q_1$, $Q_2$ and matrix $S^2$ ($S^1$ being in this
 case the matrix full of ones), the number of parameters is:
 \begin{equation*}
-    \text{NP}(\rho\text{-}colBiSBM) = (Q_1 - 1) + \sum_{m=1}^{M}\Bigg( \sum_{r=1}^{Q_2} S^2_{mr} - 1 \Bigg) + \sum_{\substack{q=1,\dots,Q_1 \\ r=1,\dots,Q_2}} \mathbbb{1}_{{(S^{1\prime}S^2)}_{qr}>0}
+    \text{NP}(\rho\text{-colBiSBM}) = (Q_1 - 1) + \sum_{m=1}^{M}\Bigg( \sum_{r=1}^{Q_2} S^2_{mr} - 1 \Bigg) + \sum_{\substack{q=1,\dots,Q_1 \\ r=1,\dots,Q_2}} \mathbbb{1}_{{(S^{1\prime}S^2)}_{qr}>0}
 \end{equation*}
 
 $\pi\rho$-colBiSBM model still assumes that the networks share a common connectivity
@@ -156,7 +156,7 @@ structure represented by $\bm{\alpha}$ but that each network has its own row and
 column block proportions, it is the less constrained model.
 For $m \in \{1,\dots,M\}$, the $X^m$ are independent and
 \begin{align}
-    \tag{\emph{$\pi\rho$-colBiSBM}}
+    \tag{\emph{$\pi\rho$}-colBiSBM}
     X^m \sim \mathcal{F}-BiSBM_{n_1^m,n_2^m} (Q_1, Q_2, \bm{\pi^m}, \bm{\rho^m}, \bm{\alpha}), &  & \forall m = 1, \dots, M
 \end{align}
 where $\forall (q,r) \in \{1,\dots,Q_1\}\times\{1,\dots,Q_2\}$, $\alpha_{qr} \in \mathcal{A}_{\mathcal{F}}$,
@@ -166,17 +166,18 @@ $\rho^m_r \in \left[ 0,1 \right], \sum_{r=1}^{Q_2} \rho^m_r = 1 $.
 For a given number of blocks $Q_1$, $Q_2$ and matrices $S^1$, $S^2$, the number
 of parameters is:
 \begin{equation*}
-    \text{NP}(\pi\rho\text{-}colBiSBM) = \sum_{m=1}^{M}\Bigg( \sum_{q=1}^{Q_1} S^1_{mq} - 1 \Bigg) + \sum_{m=1}^{M}\Bigg( \sum_{r=1}^{Q_2} S^2_{mr} - 1 \Bigg) + \sum_{\substack{q=1,\dots,Q_1 \\ r=1,\dots,Q_2}} \mathbbb{1}_{{(S^{1\prime}S^2)}_{qr}>0}
+    \text{NP}(\pi\rho\text{-colBiSBM}) = \sum_{m=1}^{M}\Bigg( \sum_{q=1}^{Q_1} S^1_{mq} - 1 \Bigg) + \sum_{m=1}^{M}\Bigg( \sum_{r=1}^{Q_2} S^2_{mr} - 1 \Bigg) + \sum_{\substack{q=1,\dots,Q_1 \\ r=1,\dots,Q_2}} \mathbbb{1}_{{(S^{1\prime}S^2)}_{qr}>0}
 \end{equation*}
 
 \section{Variational estimation of the parameters}\label{sec:variational-estimation-of-the-parameters}
 
 In practice, the estimation of the likelihood is not tractable. Following the
 classical approach defined in~\cite{daudinMixtureModelRandom2008} we use a
-variatonal version of the Expectation Maximization (VEM) algorithm.
+variational version of the Expectation Maximization (VEM) algorithm.
 
 We maximize a variational lower bound of the log-likelihood of the observed
-data by approximating $p(\bm{Z,W}|\bm{X};\bm{\theta})$ with a distribution on
+data, the so-called Evidence Lower Bound (or ELBO), by approximating
+$p(\bm{Z,W}|\bm{X};\bm{\theta})$ with a distribution on
 $\bm{Z}$ and $\bm{W}$ named $\mathcal{R}$ defined as $\mathcal{R} =
     \otimes_{m=1}^M \mathcal{R}_m$.\
 
@@ -185,7 +186,7 @@ The lower bound is defined as:
     \mathcal{J}(\mathcal{R};\bm{\theta}) \coloneqq \sum_{m=1}^{M} \bigg( \mathbb{E}_{\mathcal{R}_m}[\ell(X^m,Z^m,W^m;\bm{\theta})] + \mathcal{H}(\mathcal{R}_m) \bigg)  \leq \ell(\bm{X};\bm{\theta})
 \end{equation*}
 
-$\bm{Z}$ and $\bm{W}$ are
+$(Z^m_i)_{i=1\dots n_1^m}$ and $(W^m_j)_{j=1\dots n_2^m}$ are
 redefined using the \emph{one-hot encoded} conversion (i.e., $Z_i^m = q
     \rightarrow Z_{iq}^m = 1$ and $W_j^m = r \rightarrow W_{jr}^m = 1$).\\ % W_{jr\prime}^m pour r != r égal 0
 
@@ -201,7 +202,7 @@ we have: $\mathbb{P}_{\mathcal{R}_m} (Z_{iq}^m = 1, W_{jr}^m = 1|X^m) =
 
 The formula for the entropy per network is thus:
 \begin{equation*}
-    \mathcal{H}(\mathcal{R}_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}
+    \mathcal{H}(\mathcal{R}_m) = - \sum_{i=1}^{n_1^m} \tau^{1,m}_{i,q} \log \tau^{1,m}_{i,q} - \sum_{j=1}^{n_2^m} \tau^{2,m}_{j,r} \log \tau^{2,m}_{j,r}
 \end{equation*}
 
 And the expectation of the completed log-likelihood under the $\mathcal{R}_m$
@@ -216,7 +217,7 @@ And thus the lower bound becomes:
 \begin{align*}
     \mathcal{J}(\bm{\tau};\bm{\theta}) \coloneqq \sum_{m=1}^{M} \bigg(\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} \bigg) \color{black}
+    - \sum_{i=1}^{n_1^m} \tau^{1,m}_{i,q} \log \tau^{1,m}_{i,q} - \sum_{j=1}^{n_2^m} \tau^{2,m}_{j,r} \log \tau^{2,m}_{j,r} \bigg) \color{black}
 \end{align*}
 
 where we identify the variational distribution $\mathcal{R}$ with its parameter
@@ -240,11 +241,13 @@ $\bm{\tau}$: $$\widehat{\bm{\tau}}^{(t+1)} = \arg \max_{\bm{\tau}}
     \mathcal{J}(\mathcal{\bm{\tau}},\bm{\widehat{\theta}}^{(t)}).$$
 
 And we obtain the following formulae for the $\bm{\tau^m}$:
+\begin{equation*}
+    \begin{cases}
+        \widehat{\tau}_{iq}^{1,m} \propto \widehat{\pi}_{q}^{m(t)} \prod_{j=1}^{n_2^m}\prod_{r\in\mathcal{Q}_2^m} f(X_{ij}^m;\widehat{\alpha}_{qr}^{(t)})^{\widehat{\tau}_{jr}^{2,m(t+1)}}  & \forall i = 1, \dots , n_1^m, q \in \mathcal{Q}_1^m \\
+        \widehat{\tau}_{jr}^{2,m} \propto \widehat{\rho}_{r}^{m(t)} \prod_{i=1}^{n_1^m}\prod_{q\in\mathcal{Q}_1^m} f(X_{ij}^m;\widehat{\alpha}_{qr}^{(t)})^{\widehat{\tau}_{iq}^{1,m(t+1)}} & \forall j = 1, \dots , n_2^m, r \in \mathcal{Q}_2^m
+    \end{cases}
+\end{equation*}
 
-\begin{align*}
-    \widehat{\tau}_{iq}^{1,m} \propto \widehat{\pi}_{q}^{m(t)} \prod_{j=1}^{n_2^m}\prod_{r\in\mathcal{Q}_2^m} f(X_{ij}^m;\widehat{\alpha}_{qr}^{(t)})^{\widehat{\tau}_{jr}^{2,m(t+1)}}  & \forall i = 1, \dots , n_1^m, q \in \mathcal{Q}_1^m \\
-    \widehat{\tau}_{jr}^{2,m} \propto \widehat{\rho}_{r}^{m(t)} \prod_{i=1}^{n_1^m}\prod_{q\in\mathcal{Q}_1^m} f(X_{ij}^m;\widehat{\alpha}_{qr}^{(t)})^{\widehat{\tau}_{iq}^{1,m(t+1)}} & \forall j = 1, \dots , n_2^m, r \in \mathcal{Q}_2^m
-\end{align*}
 which are used to update iteratively the values by a fixed point algorithm with
 only one step.
 
@@ -269,13 +272,13 @@ The block proportions, in free mixture models,
 $(\pi_q^m)_{q\in\mathcal{Q}_1^m}, (\rho_r^m)_{r\in\mathcal{Q}_2^m}$ are
 estimated as
 \begin{align*}
-    \widehat{\pi}_q^{m}= \frac{n^{1,m}_{q}}{n_1^m}  &  & \text{for } \pi\text{-}colBiSBM \text{ and } \pi\rho\text{-}colBiSBM  \\
+    \widehat{\pi}_q^{m}= \frac{n^{1,m}_{q}}{n_1^m}  &  & \text{for } \pi\text{-colBiSBM} \text{ and } \pi\rho\text{-colBiSBM}  \\
-    \widehat{\rho}_r^{m}= \frac{n^{2,m}_{r}}{n_2^m} &  & \text{for } \rho\text{-}colBiSBM \text{ and } \pi\rho\text{-}colBiSBM
+    \widehat{\rho}_r^{m}= \frac{n^{2,m}_{r}}{n_2^m} &  & \text{for } \rho\text{-colBiSBM} \text{ and } \pi\rho\text{-colBiSBM}
 \end{align*}
 while on the other hand,
 \begin{align*}
-    \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*}
 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
@@ -344,7 +347,7 @@ This leads us to formulate a BIC-like criterion in the following manner:
 We provide below the expression for the penalties for the 4 models that we
 propose.
 \begin{description}
-    \item[\textit{iid-colBiSBM}] For the $\bm\pi$ and $\bm\rho$:
+    \item[\textit{iid}-colBiSBM] For the $\bm\pi$ and $\bm\rho$:
           \begin{align*}
               \text{pen}_{\pi}(Q_1) = (Q_1 - 1)\log(\sum_{m=1}^{M}n_{1}^{m}) & , &
               \text{pen}_{\rho}(Q_2) = (Q_2 - 1)\log(\sum_{m=1}^{M}n_{2}^{m})
@@ -358,15 +361,15 @@ propose.
               \mathcal{J} (\mathcal{\hat{R}}, \bm{\theta})
               - \frac{1}{2} [\text{pen}_{\pi}(Q_1) + \text{pen}_{\rho}(Q_2) +
                   \text{pen}_{\alpha}(Q_1, Q_2)]\]
-    \item[\textit{$\bm{\pi\rho}$-colBiSBM}] The support penalties are
+    \item[$\bm{\pi\rho}$-colBiSBM] The support penalties are
           \begin{align*}
               \text{pen}_{S_1}(Q_1) = -2 \log p_{Q_1} (S_1) & , &
               \text{pen}_{S_2}(Q_2) = -2 \log p_{Q_2} (S_2)
           \end{align*}
           with \begin{align*}
-              \log p_{Q_1}(S_1) = - M \log(Q_1) - \sum_{m=1}^{M} \log {Q_1
+              \textstyle \log p_{Q_1}(S_1) = - M \log(Q_1) - \sum_{m=1}^{M} \log {Q_1
-              \choose Q_1^{(m)}}, &
+              \choose Q_1^{(m)}}, \\
-              \log p_{Q_2}(S_2) = - M \log(Q_2) - \sum_{m=1}^{M} \log {Q_2
+              \textstyle \log p_{Q_2}(S_2) = - M \log(Q_2) - \sum_{m=1}^{M} \log {Q_2
               \choose Q_2^{(m)}}.
           \end{align*}
           And penalties for the $\bm\rho$ and $\bm\pi$ are
@@ -689,12 +692,12 @@ partition $\mathcal{G}$.
 \label{ssec:dissimilarity-between-two-networks}
 The parameters for the dissimilarity are defined as follow:
 \begin{align*}
-    \widetilde{n}_{qr}^m = \sum_{i=1}^{n_1^m} \sum_{j=1}^{n_2^m} \widehat{\tau}_{iq}^{1,m} \widehat{\tau}_{jr}^{2,m},
+    \widetilde{n}_{qr}^m & = \sum_{i=1}^{n_1^m} \sum_{j=1}^{n_2^m} \widehat{\tau}_{iq}^{1,m} \widehat{\tau}_{jr}^{2,m},
-     &  & \widetilde{\alpha}_{qr}^m = \frac{\sum_{i=1}^{n_1^m} \sum_{j=1}^{n_2^m} \widehat{\tau}_{iq}^{1,m} \widehat{\tau}_{jr}^{2,m} X_{ij}^m}{\widetilde{n}_{qr}^m}, \\
+                         &                                                                                              & \widetilde{\alpha}_{qr}^m = \frac{\sum_{i=1}^{n_1^m} \sum_{j=1}^{n_2^m} \widehat{\tau}_{iq}^{1,m} \widehat{\tau}_{jr}^{2,m} X_{ij}^m}{\widetilde{n}_{qr}^m}, \\
-    \widetilde{\pi}_q^m = \frac{\sum_{i=1}^{n_1^m} \widehat{\tau}_{iq}^{1,m}}{n_1^m},
+    \widetilde{\pi}_q^m  & = \frac{\sum_{i=1}^{n_1^m} \widehat{\tau}_{iq}^{1,m}}{n_1^m},
-     &  & \widetilde{\rho}_r^m = \frac{\sum_{j=1}^{n_2^m} \widehat{\tau_{jr}}^{2,m}}{n_2^m}
+                         &                                                                                              & \widetilde{\rho}_r^m = \frac{\sum_{j=1}^{n_2^m} \widehat{\tau_{jr}}^{2,m}}{n_2^m}.
 \end{align*}
-And the dissimilarity between any pair of networks $(m,m')\in\mathcal{M}^2$ is then:
+And the pairwise dissimilarity for networks $(m,m')\in\mathcal{M}^2$ is then:
 \[
     D_{\mathcal{M}}(m,m') = \sum_{q = 1}^{Q_1} \sum_{r = 1}^{Q_2} \max(\widetilde{\pi}_{q}^{m}, \widetilde{\pi}_{q}^{m'}) \left( \widetilde{\alpha}_{qr}^{m} - \widetilde{\alpha}_{qr}^{m'}\right)^{2} \max(\widetilde{\rho}_{r}^{m}, \widetilde{\rho}_{r}^{m'})
 \]
@@ -710,10 +713,10 @@ And the dissimilarity between any pair of networks $(m,m')\in\mathcal{M}^2$ is t
         \tikzstyle{es}=[font=\small, text justified, rectangle,draw,rounded corners=4pt,fill=cyanind!25]
 
         \node[es] (liste) at (0,4) {Supply a collection to partition};
-        \node[instruct, text width=5cm, below = 0.45cm of liste] (1-collection) {Fit \emph{colBiSBM}};
+        \node[instruct, text width=5cm, below = 0.45cm of liste] (1-collection) {Fit 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) {Compute a dissimilarity matrix over the collection};
-        \node[instruct, text width=5cm, below = 0.45cm of dissimi] (2-sous-collection) {Split the \emph{collection in 2 sub-collections} and fit the \emph{colBiSBM}};
+        \node[instruct, text width=5cm, below = 0.45cm of dissimi] (2-sous-collection) {Split the \emph{collection in 2 sub-collections} and fit the 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=0.7] (BICL-test) {$\sum_{i=1}^{2} (\text{BIC-L}(\tikz[baseline=-0.25cm]{\node[second_col] {i};} )) > \text{BIC-L}(\tikz[baseline=-0.25cm]{\node[first_col] {};})$?};
@@ -736,7 +739,7 @@ And the dissimilarity between any pair of networks $(m,m')\in\mathcal{M}^2$ is t
 The above figure (\ref{fig:netclustering-procedure}) shows a condensed
 explanation of the network clustering algorithm.
 
-The idea is to adjust the \emph{colBiSBM} model over the full collection of $M$
+The idea is to adjust the colBiSBM model over the full collection of $M$
 networks and then compute the dissimilarity matrix between all networks of the
 collection. We obtain the collection $\mathcal{G} = \{\mathcal{M}\}$ the
 trivial partition in a unique group.

rapport/chapter4-simulations/inference.tex

@@ -48,7 +48,7 @@ community and dis-assortative community structures, depending on which 3 of the
 blocks are selected for each network. $\eps[\alpha]$ represents the strength of
 these structures, the larger, the easier it is to tell apart one block from
 another.
-The true model of all the simulation is a $\pi\rho\text{-}colBiSBM$.
+The true model of all the simulation is a $\pi\rho$-colBiSBM.
 
 \paragraph{Inference} We want to measure the quality of the
 inference procedure, for this we use the inference described in the section
@@ -57,15 +57,15 @@ inference procedure, for this we use the inference described in the section
 \paragraph{Quality indicators} To assess the quality of the inference, we will
 use the following indicators:
 \begin{itemize}
-    \item First, for each dataset, we put in competition $\pi\text{-}colBiSBM$ with
+    \item First, for each dataset, we put in competition $\pi$-colBiSBM with
-          $sep\text{-}BiSBM$, $iid\text{-}colBiSBM$, $\rho\text{-}colBiSBM$,
+          $sep\text{-}BiSBM$, $iid$-colBiSBM, $\rho$-colBiSBM,
-          $\pi\rho\text{-}colBiSBM$
+          $\pi\rho$-colBiSBM
           respectively. To do so, for each dataset, we compute the
-          BIC-L of each model $\pi\text{-}colBiSBM$ is preferred to $sep\text{-}BiSBM$
+          BIC-L of each model $\pi$-colBiSBM is preferred to $sep\text{-}BiSBM$
-          (resp. $iid\text{-}colBiSBM$, $\rho\text{-}colBiSBM$,
+          (resp. $iid$-colBiSBM, $\rho$-colBiSBM,
-          $\pi\rho\text{-}colBiSBM$) if
+          $\pi\rho$-colBiSBM) if
           its BIC-L is greater.
-    \item When considering our \emph{colBiSBM} models we compare
+    \item When considering our colBiSBM models we compare
           $\widehat{Q_1}$, $\widehat{Q_2}$ to
           their true values. ($Q_1 = 4$ and $Q_2 = 4$)
     \item Finally, we assess the quality of the node grouping by computing the
@@ -76,8 +76,8 @@ use the following indicators:
               negative values if the RI is less than the expected value. This
               indicates a structure in grouping discordance.}.
           For each network, for the
-          $\pi\text{-}colBiSBM$, $\rho\text{-}colBiSBM$,
+          $\pi$-colBiSBM, $\rho$-colBiSBM,
-          $\pi\rho\text{-}colBiSBM$ we compare the inferred block memberships to
+          $\pi\rho$-colBiSBM we compare the inferred block memberships to
           the real ones by computing the mean of the ARI per axis over the two
           networks
           \begin{equation*}
@@ -122,7 +122,7 @@ of a single block on each dimension.
 
 On the figure \ref{fig:inference-prop-modele-pref} one can see that from
 $\eps[\alpha] = 0.06$ around $70\%$ of the time the
-$\pi\rho\text{-}colBiSBM$ model (i.e., the correct one) is selected.
+$\pi\rho$-colBiSBM model (i.e., the correct one) is selected.
 
 An interesting result we can read in the tables is that our models outperform
 the $sep\text{-}BiSBM$ when considering the ARI on the whole set of nodes

rapport/chapter4-simulations/information-transfer.tex

@@ -3,7 +3,7 @@ One of the motivation for collections of networks is \emph{information transfer}
 between the networks, allowing robustness to missing data and enabling the
 finding of finer structures in small networks with the help of bigger ones.
 
-\subsection{Missing edges robustness}
+% \subsection{Missing edges robustness}
 \input{chapter4-simulations/na-robustness}
 
-\subsection{Finer structure detection on small networks}
+% \subsection{Finer structure detection on small networks}

rapport/chapter4-simulations/model-selection.tex

@@ -1,9 +1,9 @@
 \section[Capacity to distinguish models]{Capacity to distinguish
-  $\pi\rho\text{-}colBiSBM$~from\newline
+  $\pi\rho$-colBiSBM~from\newline
-  $iid\text{-}colBiSBM$ and other
+  $iid$-colBiSBM and other
   models}\label{sec:capacity-to-distinguish-pirhotext-colbisbm-from-iidtext-colbisbm-and-other-variants}
 The idea of this model selection simulations is to assess how the model
-select the correct \emph{colBiSBM} model among the possible ones:
+select the correct colBiSBM model among the possible ones:
 \textit{$iid, \pi, \rho, \pi\rho$}. This difference being based on the row and
 col block proportions.\\
 \paragraph{Simulation settings} For this task we choose the same simulation settings as
@@ -40,20 +40,20 @@ $\left[ 0, .28\right]$.\newline
 We simulate 324 different collections for each
 value of $\eps[\pi]$ and $\eps[\rho]$.
 
-$\pi\rho\text{-}colBiSBM$, $\pi\text{-}colBiSBM$,
+$\pi\rho$-colBiSBM, $\pi$-colBiSBM,
-$\rho\text{-}colBiSBM$, $iid\text{-}colBiSBM$ and
+$\rho$-colBiSBM, $iid$-colBiSBM and
 $sep\text{-}BiSBM$ are put in competition and the model with the
 greater BIC-L is selected as the \emph{preferred model}.
 
 When $\eps[\pi] = 0$, $\bm{\pi}^1 = \bm{\pi}^2$, $\eps[\rho] = 0$
 and $\bm{\rho}^1 = \bm{\rho}^2$, the generated collection is an
-$iid\text{-}colBiSBM$. When $\eps[\pi] > 0$ or
+$iid$-colBiSBM. When $\eps[\pi] > 0$ or
-$\bm{\pi}^1 \neq \bm{\pi}^2$, the model is a $\pi\text{-}colBiSBM$.
+$\bm{\pi}^1 \neq \bm{\pi}^2$, the model is a $\pi$-colBiSBM.
 When $\eps[\rho] > 0$ or $\bm{\rho}^1 \neq \bm{\rho}^2$, the model
-is a $\rho\text{-}colBiSBM$. Finally, when $\eps[\pi] > 0$ or
+is a $\rho$-colBiSBM. Finally, when $\eps[\pi] > 0$ or
 $\bm{\pi}^1 \neq \bm{\pi}^2$ and $\eps[\rho] > 0$ or
 $\bm{\rho}^1 \neq \bm{\rho}^2$, the model is a
-$\pi\rho\text{-}colBiSBM$.
+$\pi\rho$-colBiSBM.
 
 
 \begin{figure}[!ht]
@@ -68,10 +68,10 @@ $\pi\rho\text{-}colBiSBM$.
 
 On the figure \ref{fig:pref_model_func_eps} and table \ref{tab:model-selection}, one can see that
 there is a turning point around $\eps[\pi] = 0.2$ (resp.
-$\eps[\rho] = 0.2$), before which $iid\text{-}colBiSBM$ and
+$\eps[\rho] = 0.2$), before which $iid$-colBiSBM and
-$\rho\text{-}colBiSBM$ (resp. $\pi\text{-}colBiSBM$) are selected
+$\rho$-colBiSBM (resp. $\pi$-colBiSBM) are selected
-very often and after $0.2$ the $\pi\text{-}colBiSBM$ (resp.
+very often and after $0.2$ the $\pi$-colBiSBM (resp.
-$\rho\text{-}colBiSBM$) and $\pi\rho\text{-}colBiSBM$ gets more and
+$\rho$-colBiSBM) and $\pi\rho$-colBiSBM gets more and
 more selected. Moreover, the number of blocks are correctly detected in most
 of the case.
 These two results highlight our capacity to recover the simulated

rapport/chapter4-simulations/na-robustness.tex

@@ -6,13 +6,13 @@ For this purpose we generate collections of networks with the following
 parameters:
 \begin{align*}
     \bm{\pi}^m = \begin{cases}
-                     \bm{\pi} = \left( 0.5, 0.3, 0.2 \right) & \text{for } iid\text{-}colBiSBM                                      \\
+                     \bm{\pi} = \left( 0.5, 0.3, 0.2 \right) & \text{for } iid\text{-colBiSBM}                                      \\
-                     \sigma_1^m(\bm{\pi})                    & \text{for } \pi\text{-}colBiSBM \text{ and } \pi\rho\text{-}colBiSBM
+                     \sigma_1^m(\bm{\pi})                    & \text{for } \pi\text{-colBiSBM} \text{ and } \pi\rho\text{-colBiSBM}
                  \end{cases} \\
     \bm{\rho}^m =
     \begin{cases}
-        \bm{\rho}  = \left( 0.5, 0.3, 0.2 \right) & \text{for } iid\text{-}colBiSBM                                        \\
+        \bm{\rho}  = \left( 0.5, 0.3, 0.2 \right) & \text{for } iid\text{-colBiSBM}                                        \\
-        \sigma_2^m(\bm{\rho})                     & \text{for } \rho\text{-}colBiSBM \text{ and } \pi\rho\text{-}colBiSBM,
+        \sigma_2^m(\bm{\rho})                     & \text{for } \rho\text{-colBiSBM} \text{ and } \pi\rho\text{-colBiSBM},
     \end{cases}
 \end{align*}
 for the block proportions, and two different structures with the corresponding
@@ -38,11 +38,11 @@ structure detected in ecology with generalist and specialist species and a
 
 The collections contain two networks ($M=2$) of size $n^{m=1}_1 =
     n^{m=1}_2 = 40$ and
-$n^{m=2}_1 = n^{m=2}_2 = 120$. One collection is generated for each $colBiSBM$
+$n^{m=2}_1 = n^{m=2}_2 = 120$. One collection is generated for each colBiSBM
 model. And the nodes block memberships (i.e., the row and column blocks they
 belong to) are saved.
 
-Per $colBiSBM$ model, 10 collections are generated and their results are
+Per colBiSBM model, 10 collections are generated and their results are
 averaged.
 
 In the network $m=1$ (i.e., the smaller one) a proportion of the edges
@@ -54,7 +54,7 @@ predicted block memberships are saved, along with the predicted links using the
 inferred parameters. This will serve as a baseline to see if the use of the
 collection benefits the predictions.
 
-A $colBiSBM$ model is then fitted (with a model matching the dataset considered)
+A colBiSBM model is then fitted (with a model matching the dataset considered)
 and we store the same predictions.
 
 \paragraph{Quality metrics} To benchmark the performance we use the
@@ -62,7 +62,7 @@ and we store the same predictions.
 ARI for predicted versus real block memberships.
 
 For the comparison we subtract the metric given by the LBM to the one
-given by $colBiSBM$ and denote it $\Delta\mbox{metric}$.
+given by colBiSBM and denote it $\Delta\mbox{metric}$.
 
 \begin{figure}[ht]
     \centering

rapport/chapter4-simulations/network-clustering.tex

@@ -10,13 +10,13 @@ For the simulations the proportions are the following:
 \end{align*} and for all $m = 2,\dots,9$
 \begin{align*}
     \bm{\pi}^m = \begin{cases}
-                     \bm{\pi}^1             & \text{for } iid\text{-}colBiSBM                                      \\
+                     \bm{\pi}^1             & \text{for } iid\text{-colBiSBM}                                      \\
-                     \sigma_1^m(\bm{\pi}^1) & \text{for } \pi\text{-}colBiSBM \text{ and } \pi\rho\text{-}colBiSBM
+                     \sigma_1^m(\bm{\pi}^1) & \text{for } \pi\text{-colBiSBM} \text{ and } \pi\rho\text{-colBiSBM}
                  \end{cases} \\
     \bm{\rho}^m =
     \begin{cases}
-        \bm{\rho}^1             & \text{for } iid\text{-}colBiSBM                                       \\
+        \bm{\rho}^1             & \text{for } iid\text{-colBiSBM}                                       \\
-        \sigma_2^m(\bm{\rho}^1) & \text{for } \rho\text{-}colBiSBM \text{ and } \pi\rho\text{-}colBiSBM
+        \sigma_2^m(\bm{\rho}^1) & \text{for } \rho\text{-colBiSBM} \text{ and } \pi\rho\text{-colBiSBM}
     \end{cases}
 \end{align*}
 where $\sigma_1^m$ and $\sigma_2^m$ are permutations of \{1, 2, 3\} proper to network $m$ and
@@ -64,4 +64,4 @@ Increasing $\epsilon$ differentiates the 3 sub-collections of networks.
 the resulting partition of the network collection and the simulated partition
 using the ARI index. As the value of $\epsilon$ increases, our ability to
 distinguish between the networks improves, and this distinction becomes nearly
-perfect in all setups of the $colBiSBM$.
+perfect in all setups of the colBiSBM.

rapport/chapter5-applications.tex

@@ -0,0 +1,3 @@
+\addtocounter{customchapter}{1}
+\chapter{Applications on ecological networks}
+\label{chap:applications-ecological-networks}

rapport/conclusions.tex

@@ -1,3 +1,35 @@
 \addtocounter{customchapter}{1}
 \chapter{Conclusions and future work}
-\label{chap:conclusions-and-future-work}
+\label{chap:conclusions-and-future-work}
+\section{Conclusion}
+\label{sec:conclusion}
+
+\section{Future work}
+\label{sec:future-work}
+
+\paragraph{Identifiability}
+As stated in section~\ref{sec:model-identifiability}, we only have
+identifiability for the \emph{iid}-colBiSBM and we will work on establishing
+identifiability for $\pi$, $\rho$ and $\pi\rho$ models.
+
+\paragraph{Finding a trade-off between \emph{iid} and $\pi\rho$}
+We observed while testing clustering with the different models that
+the $\pi$, $\rho$ and $\pi\rho$ model, with their increased number of parameters
+for block memberships parameters tends to give smaller BIC-L criterion values
+while having a higher Evidence Lower Bound than the \emph{iid}.
+This arises because of the penalties on the block memberships and support that
+increase significantly and exceeds the gain on the ELBO and the diminution of
+the connectivity parameters.
+An idea to tackle this problem could be to suppose that the block memberships
+for network $m$ are themselves the realizations of random variables and
+thus introduce sort of a mixed effect model.
+
+\paragraph{Comparison to other graphs clustering methods}
+Recent work have been comparing
+colSBM~\parencite{chabert-liddellLearningCommonStructures2024a} and
+graphclust~\parencite{rebafkaModelbasedClusteringMultiple2023} assessing various
+capabilities of the models and particularly focusing on networks clustering.
+We will reproduce and adapt the analysis to test other simulation settings that
+were not considered in this work.
+
+\section*{Thank you for reading this work}

rapport/rapport.tex

@@ -16,7 +16,16 @@
 \RestyleAlgo{ruled}
 
 \usepackage{url} % pour une gestion efficace des url
-\usepackage[citecolor=blueind,urlcolor=blueps,bookmarks=false,hypertexnames=true]{hyperref}  % pour les hyperliens dans le document
+\usepackage{hyperref}  % pour les hyperliens dans le document
+\hypersetup{
+	colorlinks=true,
+	linkcolor=red!20!black,
+	citecolor=blueps,
+	urlcolor=blueps,
+	bookmarks=false,
+	hypertexnames=true
+}
+
 \usepackage{tocbibind} % Pour avoir des index pour table des matières, biblio
 \usepackage{geometry}
 \geometry{bmargin=25mm}
@@ -212,31 +221,32 @@ automata,positioning}
 % Pour activer les onglets
 \ActivateBG
 \begin{selectlanguage}{french}
-	% \maketitle
+    % \maketitle
-	\pagenumbering{roman}
+    \pagenumbering{roman}
-	\tableofcontents
+    \tableofcontents
-	\include{remerciements}
+    \include{remerciements}
-	% \include{chapter1-presentation_UMR}
+    % \include{chapter1-presentation_UMR}
 \end{selectlanguage}
 
 \begin{selectlanguage}{english}
-	\pagenumbering{arabic}
+    \pagenumbering{arabic}
-	\include{chapter2-context}
+    \include{chapter2-context}
-	\include{chapter3-structure-detection}
+    \include{chapter3-structure-detection}
-	\include{chapter4-simulation-studies}
+    \include{chapter4-simulation-studies}
 
-	% \chapter{Applications}
+    % \chapter{Applications}
-	% \include{Rcodes/real_data/application_dore}
+    % \include{Rcodes/real_data/application_dore}
-	% \include{Rcodes/real_data/CoOPLBM_completion_analyze}
+    % \include{Rcodes/real_data/CoOPLBM_completion_analyze}
+    \include{chapter5-applications}
 
-	\include{conclusions}
+    \include{conclusions}
 
-	\addtocounter{maincontentend}{1}
+    \addtocounter{maincontentend}{1}
-	\addtocounter{customchapter}{1}
+    \addtocounter{customchapter}{1}
-	\printbibliography
+    \printbibliography
-	\input{appendices.tex}
+    \input{appendices.tex}
-	% \listoffigures
+    % \listoffigures
-	% \listoftables
+    % \listoftables
 \end{selectlanguage}
 
 \end{document}

references.bib

@@ -379,6 +379,22 @@
   file = {/home/polarolouis/Zotero/storage/ATY3ZP2X/Ramos-Jiliberto et al. - 2010 - Topological change of Andean plant–pollinator netw.pdf;/home/polarolouis/Zotero/storage/HPBGUP65/ramos-jiliberto2010.pdf.pdf;/home/polarolouis/Zotero/storage/I33MZQQ7/ramos-jiliberto2010.pdf.pdf;/home/polarolouis/Zotero/storage/YJX8XBNW/S1476945X09000622.html}
 }
 
+@online{rebafkaModelbasedClusteringMultiple2023,
+  title = {Model-Based Clustering of Multiple Networks with a Hierarchical Algorithm},
+  author = {Rebafka, Tabea},
+  date = {2023-11-06},
+  eprint = {2211.02314},
+  eprinttype = {arXiv},
+  eprintclass = {math, stat},
+  doi = {10.48550/arXiv.2211.02314},
+  url = {http://arxiv.org/abs/2211.02314},
+  urldate = {2024-07-22},
+  abstract = {The paper tackles the problem of clustering multiple networks, directed or not, that do not share the same set of vertices, into groups of networks with similar topology. A statistical model-based approach based on a finite mixture of stochastic block models is proposed. A clustering is obtained by maximizing the integrated classification likelihood criterion. This is done by a hierarchical agglomerative algorithm, that starts from singleton clusters and successively merges clusters of networks. As such, a sequence of nested clusterings is computed that can be represented by a dendrogram providing valuable insights on the collection of networks. Using a Bayesian framework, model selection is performed in an automated way since the algorithm stops when the best number of clusters is attained. The algorithm is computationally efficient, when carefully implemented. The aggregation of clusters requires a means to overcome the label-switching problem of the stochastic block model and to match the block labels of the networks. To address this problem, a new tool is proposed based on a comparison of the graphons of the associated stochastic block models. The clustering approach is assessed on synthetic data. An application to a set of ecological networks illustrates the interpretability of the obtained results.},
+  pubstate = {prepublished},
+  keywords = {Mathematics - Statistics Theory},
+  file = {/home/polarolouis/Zotero/storage/B9C8S8WQ/Rebafka - 2023 - Model-based clustering of multiple networks with a.pdf;/home/polarolouis/Zotero/storage/GG7C6CNM/2211.html}
+}
+
 @article{snijdersEstimationPredictionStochastic1997,
   title = {Estimation and {{Prediction}} for {{Stochastic Blockmodels}} for {{Graphs}} with {{Latent Block Structure}}},
   author = {Snijders, Tom A.B. and Nowicki, Krzysztof},

tables/simulations/inference/iid.tex

@@ -1,10 +1,10 @@
 
-\begin{table}[!htb]
+\begin{table}[H]
 \centering
 \caption{\label{tab:inference_results_iid}Inference results for $iid$}
 \begin{subtable}{\textwidth}
 \centering
-\caption{\label{subtab:ari_per_model_iid}Quality metrics for $iid$$\text{-}colBiSBM$}
+\caption{\label{subtab:ari_per_model_iid}Quality metrics for $iid$$\text{-colBiSBM}$}
 \centering
 \resizebox{\ifdim\width>\linewidth\linewidth\else\width\fi}{!}{
 \begin{tabular}[t]{rllll}
@@ -27,7 +27,7 @@ $\eps[\alpha]$ & $\overline{\text{ARI}}_{1}$ & $\overline{\text{ARI}}_{2}$ & $\t
 
 \begin{subtable}{\textwidth}
 \centering
-\caption{\label{subtab:blocrecov_per_model_iid}Bloc recovery for $iid$$\text{-}colBiSBM$}
+\caption{\label{subtab:blocrecov_per_model_iid}Bloc recovery for $iid$$\text{-colBiSBM}$}
 \centering
 \resizebox{\ifdim\width>\linewidth\linewidth\else\width\fi}{!}{
 \begin{tabular}[t]{rllllll}

tables/simulations/inference/pi.tex

@@ -1,10 +1,10 @@
 
-\begin{table}[!htb]
+\begin{table}[H]
 \centering
 \caption{\label{tab:inference_results_pi}Inference results for $\pi$}
 \begin{subtable}{\textwidth}
 \centering
-\caption{\label{subtab:ari_per_model_pi}Quality metrics for $\pi$$\text{-}colBiSBM$}
+\caption{\label{subtab:ari_per_model_pi}Quality metrics for $\pi$$\text{-colBiSBM}$}
 \centering
 \resizebox{\ifdim\width>\linewidth\linewidth\else\width\fi}{!}{
 \begin{tabular}[t]{rllll}
@@ -27,7 +27,7 @@ $\eps[\alpha]$ & $\overline{\text{ARI}}_{1}$ & $\overline{\text{ARI}}_{2}$ & $\t
 
 \begin{subtable}{\textwidth}
 \centering
-\caption{\label{subtab:blocrecov_per_model_pi}Bloc recovery for $\pi$$\text{-}colBiSBM$}
+\caption{\label{subtab:blocrecov_per_model_pi}Bloc recovery for $\pi$$\text{-colBiSBM}$}
 \centering
 \resizebox{\ifdim\width>\linewidth\linewidth\else\width\fi}{!}{
 \begin{tabular}[t]{rllllll}

tables/simulations/inference/pirho.tex

@@ -1,10 +1,10 @@
 
-\begin{table}[!htb]
+\begin{table}[H]
 \centering
 \caption{\label{tab:inference_results_pirho}Inference results for $\pi\rho$}
 \begin{subtable}{\textwidth}
 \centering
-\caption{\label{subtab:ari_per_model_pirho}Quality metrics for $\pi\rho$$\text{-}colBiSBM$}
+\caption{\label{subtab:ari_per_model_pirho}Quality metrics for $\pi\rho$$\text{-colBiSBM}$}
 \centering
 \resizebox{\ifdim\width>\linewidth\linewidth\else\width\fi}{!}{
 \begin{tabular}[t]{rllll}
@@ -27,7 +27,7 @@ $\eps[\alpha]$ & $\overline{\text{ARI}}_{1}$ & $\overline{\text{ARI}}_{2}$ & $\t
 
 \begin{subtable}{\textwidth}
 \centering
-\caption{\label{subtab:blocrecov_per_model_pirho}Bloc recovery for $\pi\rho$$\text{-}colBiSBM$}
+\caption{\label{subtab:blocrecov_per_model_pirho}Bloc recovery for $\pi\rho$$\text{-colBiSBM}$}
 \centering
 \resizebox{\ifdim\width>\linewidth\linewidth\else\width\fi}{!}{
 \begin{tabular}[t]{rllllll}

tables/simulations/inference/rho.tex

@@ -1,10 +1,10 @@
 
-\begin{table}[!htb]
+\begin{table}[H]
 \centering
 \caption{\label{tab:inference_results_rho}Inference results for $\rho$}
 \begin{subtable}{\textwidth}
 \centering
-\caption{\label{subtab:ari_per_model_rho}Quality metrics for $\rho$$\text{-}colBiSBM$}
+\caption{\label{subtab:ari_per_model_rho}Quality metrics for $\rho$$\text{-colBiSBM}$}
 \centering
 \resizebox{\ifdim\width>\linewidth\linewidth\else\width\fi}{!}{
 \begin{tabular}[t]{rllll}
@@ -27,7 +27,7 @@ $\eps[\alpha]$ & $\overline{\text{ARI}}_{1}$ & $\overline{\text{ARI}}_{2}$ & $\t
 
 \begin{subtable}{\textwidth}
 \centering
-\caption{\label{subtab:blocrecov_per_model_rho}Bloc recovery for $\rho$$\text{-}colBiSBM$}
+\caption{\label{subtab:blocrecov_per_model_rho}Bloc recovery for $\rho$$\text{-colBiSBM}$}
 \centering
 \resizebox{\ifdim\width>\linewidth\linewidth\else\width\fi}{!}{
 \begin{tabular}[t]{rllllll}

tables/simulations/inference/sep.tex

@@ -1,10 +1,10 @@
 
-\begin{table}[!htb]
+\begin{table}[H]
 \centering
 \caption{\label{tab:inference_results_sep}Inference results for $sep$}
 \begin{subtable}{\textwidth}
 \centering
-\caption{\label{subtab:ari_per_model_sep}Quality metrics for $sep\text{-}BiSBM$}
+\caption{\label{subtab:ari_per_model_sep}Quality metrics for $sep\text{-BiSBM}$}
 \centering
 \resizebox{\ifdim\width>\linewidth\linewidth\else\width\fi}{!}{
 \begin{tabular}[t]{rllll}

tables/simulations/model_selection/model-selection.tex

@@ -22,165 +22,165 @@ $\epsilon_{\pi}$ & $\epsilon_{\rho}$ & $\mathbbb{1}_{\widehat{Q_1}_{iid}=3}$ & $
 \endfoot
 \bottomrule
 \endlastfoot
- & 0.000 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.991} & 0.009 &  & \\
+ & 0.000 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{1} &  &  & \\
 \cmidrule{2-14}\nopagebreak
  & 0.035 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{1} &  &  & \\
 \cmidrule{2-14}\nopagebreak
- & 0.070 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.981} &  & 0.019 & \\
+ & 0.070 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.981} & 0.019 &  & \\
 \cmidrule{2-14}\nopagebreak
- & 0.105 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.917} &  & 0.083 & \\
+ & 0.105 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.944} & 0.056 &  & \\
 \cmidrule{2-14}\nopagebreak
- & 0.140 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.815} &  & 0.185 & \\
+ & 0.140 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.833} & 0.157 & 0.009 & \\
 \cmidrule{2-14}\nopagebreak
- & 0.175 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.676} &  & 0.324 & \\
+ & 0.175 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.556} & 0.444 &  & \\
 \cmidrule{2-14}\nopagebreak
- & 0.210 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.361 &  & \textbf{0.630} & 0.009\\
+ & 0.210 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.361 & \textbf{0.639} &  & \\
 \cmidrule{2-14}\nopagebreak
- & 0.245 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.111 &  & \textbf{0.889} & \\
+ & 0.245 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.102 & \textbf{0.898} &  & \\
 \cmidrule{2-14}\nopagebreak
-\multirow{-9}{*}[4\dimexpr\aboverulesep+\belowrulesep+\cmidrulewidth]{\raggedright\arraybackslash 0.000} & 0.280 & 1 & 1 & 1 & \textbf{0.991} & 1 & 1 & 1 & 1 & 0.009 & 0.009 & \textbf{0.963} & 0.019\\
+\multirow{-9}{*}[4\dimexpr\aboverulesep+\belowrulesep+\cmidrulewidth]{\raggedright\arraybackslash 0.000} & 0.280 & 1 & \textbf{0.991} & 1 & 1 & 1 & 1 & 1 & 1 & 0.009 & \textbf{0.991} &  & \\
 \cmidrule{1-14}\pagebreak[0]
- & 0.000 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.981} & 0.009 & 0.009 & \\
+ & 0.000 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{1} &  &  & \\
 \cmidrule{2-14}\nopagebreak
- & 0.035 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.981} & 0.019 &  & \\
+ & 0.035 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.991} & 0.009 &  & \\
 \cmidrule{2-14}\nopagebreak
- & 0.070 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.991} &  & 0.009 & \\
+ & 0.070 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.991} & 0.009 &  & \\
 \cmidrule{2-14}\nopagebreak
- & 0.105 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.935} &  & 0.065 & \\
+ & 0.105 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.917} & 0.083 &  & \\
 \cmidrule{2-14}\nopagebreak
- & 0.140 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.880} &  & 0.111 & 0.009\\
+ & 0.140 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.843} & 0.157 &  & \\
 \cmidrule{2-14}\nopagebreak
- & 0.175 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.593} &  & 0.407 & \\
+ & 0.175 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.657} & 0.333 & 0.009 & \\
 \cmidrule{2-14}\nopagebreak
- & 0.210 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.454 &  & \textbf{0.546} & \\
+ & 0.210 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.315 & \textbf{0.685} &  & \\
 \cmidrule{2-14}\nopagebreak
- & 0.245 & 1 & \textbf{0.991} & 1 & 1 & 1 & 1 & 1 & 1 & 0.157 &  & \textbf{0.843} & \\
+ & 0.245 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.176 & \textbf{0.824} &  & \\
 \cmidrule{2-14}\nopagebreak
-\multirow{-9}{*}[4\dimexpr\aboverulesep+\belowrulesep+\cmidrulewidth]{\raggedright\arraybackslash 0.035} & 0.280 & \textbf{0.991} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.019 &  & \textbf{0.972} & 0.009\\
+\multirow{-9}{*}[4\dimexpr\aboverulesep+\belowrulesep+\cmidrulewidth]{\raggedright\arraybackslash 0.035} & 0.280 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.028 & \textbf{0.972} &  & \\
 \cmidrule{1-14}\pagebreak[0]
- & 0.000 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.991} & 0.009 &  & \\
+ & 0.000 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.972} &  & 0.028 & \\
 \cmidrule{2-14}\nopagebreak
- & 0.035 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.972} & 0.019 & 0.009 & \\
+ & 0.035 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.963} & 0.009 & 0.028 & \\
 \cmidrule{2-14}\nopagebreak
- & 0.070 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.972} & 0.019 & 0.009 & \\
+ & 0.070 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.917} & 0.065 & 0.019 & \\
 \cmidrule{2-14}\nopagebreak
- & 0.105 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.880} & 0.056 & 0.065 & \\
+ & 0.105 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.917} & 0.065 & 0.019 & \\
 \cmidrule{2-14}\nopagebreak
- & 0.140 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.806} & 0.009 & 0.176 & 0.009\\
+ & 0.140 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.824} & 0.157 & 0.009 & 0.009\\
 \cmidrule{2-14}\nopagebreak
- & 0.175 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.602} & 0.009 & 0.389 & \\
+ & 0.175 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.565} & 0.426 &  & 0.009\\
 \cmidrule{2-14}\nopagebreak
- & 0.210 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.324 &  & \textbf{0.648} & 0.028\\
+ & 0.210 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.343 & \textbf{0.639} & 0.009 & 0.009\\
 \cmidrule{2-14}\nopagebreak
- & 0.245 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.139 &  & \textbf{0.843} & 0.019\\
+ & 0.245 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.991} & 1 & 0.139 & \textbf{0.843} &  & 0.019\\
 \cmidrule{2-14}\nopagebreak
-\multirow{-9}{*}[4\dimexpr\aboverulesep+\belowrulesep+\cmidrulewidth]{\raggedright\arraybackslash 0.070} & 0.280 & \textbf{0.991} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.019 &  & \textbf{0.963} & 0.019\\
+\multirow{-9}{*}[4\dimexpr\aboverulesep+\belowrulesep+\cmidrulewidth]{\raggedright\arraybackslash 0.070} & 0.280 & 1 & 1 & 1 & 1 & \textbf{0.991} & 1 & 1 & 1 &  & \textbf{0.963} &  & 0.037\\
 \cmidrule{1-14}\pagebreak[0]
- & 0.000 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.917} & 0.083 &  & \\
+ & 0.000 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.963} &  & 0.037 & \\
 \cmidrule{2-14}\nopagebreak
- & 0.035 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.917} & 0.065 & 0.019 & \\
+ & 0.035 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.898} & 0.019 & 0.083 & \\
 \cmidrule{2-14}\nopagebreak
- & 0.070 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.935} & 0.056 & 0.009 & \\
+ & 0.070 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.926} & 0.009 & 0.065 & \\
 \cmidrule{2-14}\nopagebreak
- & 0.105 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.806} & 0.056 & 0.139 & \\
+ & 0.105 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.935} & 0.056 & 0.009 & \\
 \cmidrule{2-14}\nopagebreak
- & 0.140 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.815} & 0.028 & 0.139 & 0.019\\
+ & 0.140 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.778} & 0.157 & 0.065 & \\
 \cmidrule{2-14}\nopagebreak
- & 0.175 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.657} & 0.009 & 0.315 & 0.019\\
+ & 0.175 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.657} & 0.296 & 0.028 & 0.019\\
 \cmidrule{2-14}\nopagebreak
- & 0.210 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.333 & 0.019 & \textbf{0.593} & 0.056\\
+ & 0.210 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.324 & \textbf{0.611} & 0.037 & 0.028\\
 \cmidrule{2-14}\nopagebreak
- & 0.245 & \textbf{0.991} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.102 &  & \textbf{0.843} & 0.056\\
+ & 0.245 & \textbf{0.991} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.185 & \textbf{0.750} & 0.009 & 0.056\\
 \cmidrule{2-14}\nopagebreak
-\multirow{-9}{*}[4\dimexpr\aboverulesep+\belowrulesep+\cmidrulewidth]{\raggedright\arraybackslash 0.105} & 0.280 & 1 & 1 & 1 & \textbf{0.991} & 1 & 1 & 1 & 1 & 0.037 &  & \textbf{0.935} & 0.028\\
+\multirow{-9}{*}[4\dimexpr\aboverulesep+\belowrulesep+\cmidrulewidth]{\raggedright\arraybackslash 0.105} & 0.280 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.046 & \textbf{0.917} &  & 0.037\\
 \cmidrule{1-14}\pagebreak[0]
- & 0.000 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.769} & 0.231 &  & \\
+ & 0.000 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.815} &  & 0.185 & \\
 \cmidrule{2-14}\nopagebreak
- & 0.035 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.796} & 0.194 & 0.009 & \\
+ & 0.035 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.741} & 0.009 & 0.250 & \\
 \cmidrule{2-14}\nopagebreak
- & 0.070 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.806} & 0.176 & 0.019 & \\
+ & 0.070 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.861} &  & 0.139 & \\
 \cmidrule{2-14}\nopagebreak
- & 0.105 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.769} & 0.111 & 0.111 & 0.009\\
+ & 0.105 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.759} & 0.056 & 0.185 & \\
 \cmidrule{2-14}\nopagebreak
- & 0.140 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.657} & 0.120 & 0.194 & 0.028\\
+ & 0.140 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.667} & 0.157 & 0.157 & 0.019\\
 \cmidrule{2-14}\nopagebreak
- & 0.175 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.435} & 0.111 & 0.370 & 0.083\\
+ & 0.175 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.583} & 0.269 & 0.046 & 0.102\\
 \cmidrule{2-14}\nopagebreak
- & 0.210 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.287 & 0.046 & \textbf{0.556} & 0.111\\
+ & 0.210 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.278 & \textbf{0.546} & 0.065 & 0.111\\
 \cmidrule{2-14}\nopagebreak
- & 0.245 & \textbf{0.991} & \textbf{0.991} & 1 & 1 & 1 & 1 & 1 & 1 & 0.083 & 0.037 & \textbf{0.731} & 0.148\\
+ & 0.245 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.111 & \textbf{0.704} & 0.037 & 0.148\\
 \cmidrule{2-14}\nopagebreak
-\multirow{-9}{*}[4\dimexpr\aboverulesep+\belowrulesep+\cmidrulewidth]{\raggedright\arraybackslash 0.140} & 0.280 & \textbf{0.991} & \textbf{0.991} & 1 & 1 & 1 & 1 & 1 & 1 & 0.019 & 0.009 & \textbf{0.833} & 0.139\\
+\multirow{-9}{*}[4\dimexpr\aboverulesep+\belowrulesep+\cmidrulewidth]{\raggedright\arraybackslash 0.140} & 0.280 & 1 & \textbf{0.991} & 1 & 1 & \textbf{0.991} & 1 & 1 & 1 & 0.028 & \textbf{0.852} & 0.009 & 0.111\\
 \cmidrule{1-14}\pagebreak[0]
- & 0.000 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.528} & 0.472 &  & \\
+ & 0.000 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.630} &  & 0.370 & \\
 \cmidrule{2-14}\nopagebreak
- & 0.035 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.685} & 0.315 &  & \\
+ & 0.035 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.694} &  & 0.306 & \\
 \cmidrule{2-14}\nopagebreak
- & 0.070 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.620} & 0.370 &  & 0.009\\
+ & 0.070 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.528} & 0.009 & 0.444 & 0.019\\
 \cmidrule{2-14}\nopagebreak
- & 0.105 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.546} & 0.370 & 0.065 & 0.019\\
+ & 0.105 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.574} & 0.028 & 0.389 & 0.009\\
 \cmidrule{2-14}\nopagebreak
- & 0.140 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.546} & 0.287 & 0.046 & 0.120\\
+ & 0.140 & 1 & \textbf{0.991} & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.556} & 0.130 & 0.250 & 0.065\\
 \cmidrule{2-14}\nopagebreak
- & 0.175 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.407} & 0.204 & 0.194 & 0.194\\
+ & 0.175 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.991} & 1 & \textbf{0.417} & 0.167 & 0.231 & 0.185\\
 \cmidrule{2-14}\nopagebreak
- & 0.210 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.269 & 0.111 & \textbf{0.352} & 0.269\\
+ & 0.210 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.250 & \textbf{0.380} & 0.157 & 0.213\\
 \cmidrule{2-14}\nopagebreak
- & 0.245 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.083 & 0.037 & \textbf{0.546} & 0.333\\
+ & 0.245 & 1 & 1 & 1 & 1 & \textbf{0.991} & 1 & 1 & 1 & 0.102 & \textbf{0.463} & 0.019 & 0.417\\
 \cmidrule{2-14}\nopagebreak
-\multirow{-9}{*}[4\dimexpr\aboverulesep+\belowrulesep+\cmidrulewidth]{\raggedright\arraybackslash 0.175} & 0.280 & 1 & 0.991 & 1 & 1 & \textbf{0.981} & 1 & 1 & 1 & 0.009 &  & \textbf{0.731} & 0.259\\
+\multirow{-9}{*}[4\dimexpr\aboverulesep+\belowrulesep+\cmidrulewidth]{\raggedright\arraybackslash 0.175} & 0.280 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 &  & \textbf{0.694} & 0.009 & 0.296\\
 \cmidrule{1-14}\pagebreak[0]
- & 0.000 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.296 & \textbf{0.704} &  & \\
+ & 0.000 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.361 &  & \textbf{0.639} & \\
 \cmidrule{2-14}\nopagebreak
- & 0.035 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.370 & \textbf{0.620} &  & 0.009\\
+ & 0.035 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.343 &  & \textbf{0.657} & \\
 \cmidrule{2-14}\nopagebreak
- & 0.070 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.343 & \textbf{0.657} &  & \\
+ & 0.070 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.370 & 0.009 & \textbf{0.611} & 0.009\\
 \cmidrule{2-14}\nopagebreak
- & 0.105 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.343 & \textbf{0.546} & 0.037 & 0.074\\
+ & 0.105 & \textbf{0.991} & \textbf{0.991} & 1 & 1 & 1 & 1 & 1 & 1 & 0.287 & 0.009 & \textbf{0.648} & 0.056\\
 \cmidrule{2-14}\nopagebreak
- & 0.140 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.259 & \textbf{0.565} & 0.028 & 0.148\\
+ & 0.140 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.361 & 0.083 & \textbf{0.481} & 0.074\\
 \cmidrule{2-14}\nopagebreak
- & 0.175 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.167 & \textbf{0.454} & 0.185 & 0.194\\
+ & 0.175 & \textbf{0.991} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.278 & 0.167 & \textbf{0.361} & 0.194\\
 \cmidrule{2-14}\nopagebreak
- & 0.210 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.148 & 0.269 & 0.176 & \textbf{0.407}\\
+ & 0.210 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.157 & 0.231 & 0.176 & \textbf{0.435}\\
 \cmidrule{2-14}\nopagebreak
- & 0.245 & \textbf{0.991} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.037 & 0.065 & 0.324 & \textbf{0.574}\\
+ & 0.245 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.028 & 0.380 & 0.093 & \textbf{0.500}\\
 \cmidrule{2-14}\nopagebreak
-\multirow{-9}{*}[4\dimexpr\aboverulesep+\belowrulesep+\cmidrulewidth]{\raggedright\arraybackslash 0.210} & 0.280 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 &  &  & 0.417 & \textbf{0.583}\\
+\multirow{-9}{*}[4\dimexpr\aboverulesep+\belowrulesep+\cmidrulewidth]{\raggedright\arraybackslash 0.210} & 0.280 & 1 & \textbf{0.991} & 1 & \textbf{0.991} & 1 & 1 & 1 & 1 &  & 0.333 & 0.037 & \textbf{0.630}\\
 \cmidrule{1-14}\pagebreak[0]
- & 0.000 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.148 & \textbf{0.852} &  & \\
+ & 0.000 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.148 &  & \textbf{0.852} & \\
 \cmidrule{2-14}\nopagebreak
- & 0.035 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.157 & \textbf{0.843} &  & \\
+ & 0.035 & \textbf{0.991} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.194 &  & \textbf{0.796} & 0.009\\
 \cmidrule{2-14}\nopagebreak
- & 0.070 & 1 & 1 & 1 & 1 & \textbf{0.991} & 1 & 1 & 1 & 0.148 & \textbf{0.852} &  & \\
+ & 0.070 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.176 &  & \textbf{0.824} & \\
 \cmidrule{2-14}\nopagebreak
- & 0.105 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.139 & \textbf{0.833} &  & 0.028\\
+ & 0.105 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.139 & 0.009 & \textbf{0.778} & 0.074\\
 \cmidrule{2-14}\nopagebreak
- & 0.140 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.148 & \textbf{0.685} & 0.028 & 0.139\\
+ & 0.140 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.120 & 0.037 & \textbf{0.657} & 0.185\\
 \cmidrule{2-14}\nopagebreak
- & 0.175 & 1 & 1 & 1 & 1 & \textbf{0.991} & 1 & 1 & 1 & 0.074 & \textbf{0.509} & 0.037 & 0.380\\
+ & 0.175 & 1 & 1 & 1 & 1 & \textbf{0.991} & 1 & 1 & 1 & 0.139 & 0.093 & \textbf{0.509} & 0.259\\
 \cmidrule{2-14}\nopagebreak
- & 0.210 & 1 & \textbf{0.981} & 1 & 1 & 1 & 1 & 1 & 1 & 0.028 & 0.343 & 0.111 & \textbf{0.519}\\
+ & 0.210 & \textbf{0.991} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.028 & 0.093 & 0.370 & \textbf{0.509}\\
 \cmidrule{2-14}\nopagebreak
- & 0.245 & 1 & 1 & 1 & 1 & \textbf{0.991} & 1 & 1 & 1 &  & 0.130 & 0.083 & \textbf{0.787}\\
+ & 0.245 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.009 & 0.102 & 0.102 & \textbf{0.787}\\
 \cmidrule{2-14}\nopagebreak
-\multirow{-9}{*}[4\dimexpr\aboverulesep+\belowrulesep+\cmidrulewidth]{\raggedright\arraybackslash 0.245} & 0.280 & \textbf{0.991} & \textbf{0.991} & 1 & 1 & \textbf{0.991} & 1 & 1 & 1 & 0.009 & 0.037 & 0.111 & \textbf{0.843}\\
+\multirow{-9}{*}[4\dimexpr\aboverulesep+\belowrulesep+\cmidrulewidth]{\raggedright\arraybackslash 0.245} & 0.280 & \textbf{0.991} & 1 & 1 & 1 & \textbf{0.991} & 1 & 1 & 1 & 0.009 & 0.148 & 0.019 & \textbf{0.824}\\
 \cmidrule{1-14}\pagebreak[0]
- & 0.000 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.991} & 1 & 0.019 & \textbf{0.981} &  & \\
+ & 0.000 & \textbf{0.991} & 1 & 1 & 1 & 1 & 1 & \textbf{0.991} & 1 & 0.009 &  & \textbf{0.991} & \\
 \cmidrule{2-14}\nopagebreak
- & 0.035 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.981} & 1 & 0.046 & \textbf{0.954} &  & \\
+ & 0.035 & \textbf{0.981} & 1 & 1 & 1 & 1 & 1 & 1 & 1 &  &  & \textbf{0.981} & 0.019\\
 \cmidrule{2-14}\nopagebreak
- & 0.070 & 1 & 0.991 & 1 & 0.991 & 0.991 & 1 & 0.991 & \textbf{0.981} & 0.019 & \textbf{0.954} &  & 0.028\\
+ & 0.070 & \textbf{0.991} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.037 &  & \textbf{0.954} & 0.009\\
 \cmidrule{2-14}\nopagebreak
- & 0.105 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.019 & \textbf{0.898} &  & 0.083\\
+ & 0.105 & \textbf{0.991} & \textbf{0.991} & 1 & \textbf{0.991} & 1 & 1 & 1 & \textbf{0.991} & 0.028 & 0.009 & \textbf{0.889} & 0.074\\
 \cmidrule{2-14}\nopagebreak
- & 0.140 & \textbf{0.981} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0.009 & \textbf{0.815} &  & 0.176\\
+ & 0.140 & \textbf{0.991} & 1 & 1 & 1 & 1 & 1 & \textbf{0.991} & 1 & 0.019 &  & \textbf{0.889} & 0.093\\
 \cmidrule{2-14}\nopagebreak
- & 0.175 & \textbf{0.981} & 1 & 1 & 1 & 1 & 1 & 0.991 & 1 & 0.019 & \textbf{0.685} &  & 0.296\\
+ & 0.175 & \textbf{0.991} & 1 & 1 & 1 & 1 & 1 & 1 & 1 &  &  & \textbf{0.602} & 0.398\\
 \cmidrule{2-14}\nopagebreak
- & 0.210 & \textbf{0.981} & 0.991 & 1 & 1 & 1 & 1 & 1 & 1 &  & 0.435 & 0.009 & \textbf{0.556}\\
+ & 0.210 & 0.981 & 1 & 1 & 1 & 1 & 1 & \textbf{0.972} & 1 & 0.009 &  & 0.324 & \textbf{0.667}\\
 \cmidrule{2-14}\nopagebreak
- & 0.245 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 &  & 0.167 & 0.019 & \textbf{0.815}\\
+ & 0.245 & 1 & 1 & 1 & 1 & 1 & 1 & \textbf{0.991} & 1 & 0.009 & 0.009 & 0.194 & \textbf{0.787}\\
 \cmidrule{2-14}\nopagebreak
-\multirow{-9}{*}[4\dimexpr\aboverulesep+\belowrulesep+\cmidrulewidth]{\raggedright\arraybackslash 0.280} & 0.280 & 1 & \textbf{0.991} & 1 & 1 & 1 & 1 & \textbf{0.991} & 1 &  & 0.019 & 0.028 & \textbf{0.954}\\*
+\multirow{-9}{*}[4\dimexpr\aboverulesep+\belowrulesep+\cmidrulewidth]{\raggedright\arraybackslash 0.280} & 0.280 & 1 & \textbf{0.991} & 1 & 1 & 1 & 1 & \textbf{0.991} & 1 &  & 0.019 & 0.046 & \textbf{0.935}\\*
 \end{longtable}

tikz/clustering.tex

@@ -6,10 +6,10 @@ 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[instruct, text width=5cm, below = 0.45cm of liste] (1-collection) {Ajuster 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[instruct, text width=5cm, below = 0.45cm of dissimi] (2-sous-collection) {Séparer la \emph{collection en 2 sous-collections} et ajuster les 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) {$\sum_{i=1}^{2} (\text{BIC-L}(\tikz[baseline=-0.25cm]{\node[second_col] {i};} )) > \text{BIC-L}(\tikz[baseline=-0.25cm]{\node[first_col] {};})$?};

tikz/simulations/inference/ari-plots.tex

@@ -1,4 +1,4 @@
-% Created by tikzDevice version 0.12.6 on 2024-07-17 09:19:02
+% Created by tikzDevice version 0.12.6 on 2024-07-22 15:42:36
 % !TEX encoding = UTF-8 Unicode
 \begin{tikzpicture}[x=1pt,y=1pt]
 \definecolor{fillColor}{RGB}{255,255,255}
@@ -2229,7 +2229,7 @@
 \path[clip] (  0.00,  0.00) rectangle (433.62,361.35);
 \definecolor{drawColor}{RGB}{0,0,0}
 
-\node[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale=  0.88] at (398.67,193.19) {iid};
+\node[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale=  0.88] at (398.67,193.19) {$iid$};
 \end{scope}
 \begin{scope}
 \path[clip] (  0.00,  0.00) rectangle (433.62,361.35);

tikz/simulations/inference/model-proportions.tex

@@ -1,4 +1,4 @@
-% Created by tikzDevice version 0.12.6 on 2024-07-17 09:18:50
+% Created by tikzDevice version 0.12.6 on 2024-07-22 15:42:35
 % !TEX encoding = UTF-8 Unicode
 \begin{tikzpicture}[x=1pt,y=1pt]
 \definecolor{fillColor}{RGB}{255,255,255}
@@ -286,7 +286,7 @@
 \path[clip] (  0.00,  0.00) rectangle (289.08,216.81);
 \definecolor{drawColor}{RGB}{0,0,0}
 
-\node[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale=  0.88] at (256.33,129.21) {iid};
+\node[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale=  0.88] at (256.33,129.21) {$iid$};
 \end{scope}
 \begin{scope}
 \path[clip] (  0.00,  0.00) rectangle (289.08,216.81);