rapport : raccourcissement taille page

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                      \\
-                     \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{ and } \pi\rho
-                 \end{cases} \\
+                 \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                        \\
-        \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{ and } \pi\rho,
     \end{cases}
 \end{align*}
 for the block proportions, and two different structures with the corresponding
@@ -105,5 +105,5 @@ $\pi\rho$ present smaller values and larger variances.
 An explanation for the cases in which our models return lower values than
 expected could be to look for in our simulation parameters. They may, combined
 with the $\rho$ model be a difficult case for the estimation.
-As we currently do not have identifiability results this is just and
+As we currently do not have identifiability results this is just an
 hypothesis.

rapport/chapter4-simulations/network-clustering.tex

@@ -10,13 +10,13 @@ $\bm{\pi}^1 = \left( 0.2, 0.3, 0.5 \right),
     ~\bm{\rho}^1 = \left( 0.2, 0.3, 0.5 \right)$ 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                      \\
-                     \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{ and } \pi\rho
-                 \end{cases} \\
+                 \end{cases},~ &
     \bm{\rho}^m =
     \begin{cases}
-        \bm{\rho}^1             & \text{for } iid\text{-colBiSBM}                                       \\
+        \bm{\rho}^1             & \text{for } iid                       \\
-        \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{ and } \pi\rho
     \end{cases}
 \end{align*}
 where $\sigma_1^m$ and $\sigma_2^m$ are permutations of \{1, 2, 3\} proper to network $m$ and
@@ -35,7 +35,7 @@ parameters as follows:
                                  - \frac{\epsilon}{2} & \epsilon             & \epsilon             \\
                                  \epsilon             & - \frac{\epsilon}{2} & \epsilon             \\
                                  \epsilon             & \epsilon             & - \frac{\epsilon}{2}
-                             \end{pmatrix},                                                                      \\
+                             \end{pmatrix},
                                                                           & \bm{\alpha}^{cp} = .3 + \begin{pmatrix}
                                                                                                         \frac{3 \epsilon}{2} & \epsilon           & \frac{\epsilon}{2}   \\
                                                                                                         \epsilon             & \frac{\epsilon}{2} & 0                    \\
@@ -51,17 +51,19 @@ connections between blocks than within blocks. If $\epsilon = 0$, the three
 matrices are equal and the 9 networks have the same connection structure.
 Increasing $\epsilon$ differentiates the 3 sub-collections of networks.
 
-% ARI boxplot
-\begin{figure}[!ht]
-    \centering
-    \includestandalone{tikz/simulations/clustering/ari-clustering.tex}
-    \caption{ARI obtained for the clustering with the different models in
-        function of $\epsilon$}
-    \label{fig:ari-clustering-boxplot}
-\end{figure}
 
 \paragraph{Results} The evaluation of our method involves a comparison between
 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.
+
+
+% ARI boxplot
+\begin{figure}[!hb]
+    \centering
+    \includestandalone[height=0.25\textheight]{tikz/simulations/clustering/ari-clustering.tex}
+    \caption{ARI obtained for the clustering with the different models in
+        function of $\epsilon$}
+    \label{fig:ari-clustering-boxplot}
+\end{figure}