rapport : na robustness interp

rapport/chapter4-simulations/na-robustness.tex

@@ -23,9 +23,9 @@ $\bm{\alpha}$,
                                 0.05 & 0.2  & 0.05 \\
                                 0.05 & 0.05 & 0.8
                             \end{pmatrix}, &
-    \bm{\alpha}^{nested} = \begin{pmatrix}
+    ~\bm{\alpha}^{nested} = \begin{pmatrix}
-                               0.9 & 0.25 & 0.1  \\
+                                0.9  & 0.65 & 0.1  \\
-                               0.3 & 0.15 & 0.05 \\
+                                0.35 & 0.15 & 0.05 \\
                                 0.1  & 0.05 & 0.05
                             \end{pmatrix},
 \end{align*}
@@ -37,7 +37,7 @@ structure detected in ecology with generalist and specialist species and a
 \enquote{nested} structure.
 
 The collections contain two networks ($M=2$) of size $n^{m=1}_1 =
-    n^{m=1}_2 = 40$ and
+    n^{m=1}_2 = 20$ and
 $n^{m=2}_1 = n^{m=2}_2 = 120$. 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.
@@ -85,22 +85,25 @@ corresponding \emph{model}. We will compare the results for one model box plot
 to the corresponding sep-model box plot, serving as a baseline.
 
 % TODO the ARI interpretation
-For the figure~\ref{fig:ari-dim-plot-na},  in almost all cases
+For the figure~\ref{fig:ari-dim-plot-na}, our models almost always do at least
+as good as the sep counterpart. The $iid$ model is the only one for which the
+sep performs better on the columns block memberships.
+The nested structure seems to complexify the block membership attribution with
+only ARI less than 0.75
 
-For the figure~\ref{fig:auc-plot}, overall we observe results similar to
+For the figure~\ref{fig:auc-plot}, in almost all cases and for almost
-the ARIs, namely  our models tend to have a slightly better AUC than their
+all models the differences are not significant but our models seems to perform
-corresponding LBM.
+marginally better and are only a few times under their LBM counterpart.
-This indicates that link prediction benefits from the collection model
+This indicates that information is transferred from the bigger network when estimating the parameters and predicting link values.
-in almost all cases.
-We may even be able to improve the results by using larger collections.
 
-For the cases where our models do not perform better, we observe that
+On the differences between nested and modular structures, the latter shows
-$\pi\rho$-colBiSBM in
+a smaller variance in the AUC with our models predictions contained between
-the modular case seems to do slightly worse than its LBM counterpart for the
+0.7 and 0.9. Whereas for the nested structure, $iid$ and $\pi$ models are
-first values of $p_{\texttt{NA}}$. The $\rho$-colBiSBM seem to suffer from the
+in quite similar value ranges with small variances but $\rho$ and
-same problem, and encounters it too for some values with the nested structure.
+$\pi\rho$ present smaller values and larger variances.
-This may have to do with our simulation parameters giving for this models
-cases that are harder.
-Or this may be due to mis-attribution of the block memberships resulting in
-wrong predictions.
 
+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
+hypothesis.