121 lines
5.5 KiB
TeX
121 lines
5.5 KiB
TeX
\hypertarget{capacity-to-distinguish-pirhotext-colbisbm-from-iidtext-colbisbm-and-other-variants}{%
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\section{\texorpdfstring{Capacity to distinguish
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\(\pi\rho\text{-}colBiSBM\) from \(iid\text{-}colBiSBM\) and other
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variants}{Capacity to distinguish \textbackslash pi\textbackslash rho\textbackslash text\{-\}colBiSBM from iid\textbackslash text\{-\}colBiSBM and other variants}}\label{capacity-to-distinguish-pirhotext-colbisbm-from-iidtext-colbisbm-and-other-variants}}
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The idea of this model selection simulations is to assess how the model
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select the correct \emph{colBiSBM} model among the possible ones:
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\textit{iid, pi, rho, pirho}. This difference being based on the row and
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col block proportions.
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For this task we choose the same simulation context as
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\cite{chabert-liddellLearningCommonStructures2023}.
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Namely \(n_{1}^{m} = 90, n_{2}^{m} = 90, Q_1 = Q_2 = 3\),
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\(\bm{\alpha}, \bm{\pi}\) and \(\bm{\rho}\) are set as follows:
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\begin{align*}
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\bm{\alpha} =.25 + \begin{pmatrix}
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3 \eps[\alpha] & 2 \eps[\alpha] & \eps[\alpha] \\
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2 \eps[\alpha] & 2 \eps[\alpha] & - \eps[\alpha] \\
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\eps[\alpha] & - \eps[\alpha] & \eps[\alpha]
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\end{pmatrix}, & & \bm{\pi}^1 = \begin{pmatrix}
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\frac{1}{3}, & \frac{1}{3}, & \frac{1}{3}
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\end{pmatrix}, & & \bm{\pi}^2 = \sigma\begin{pmatrix}
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\frac{1}{3} - \eps[\pi], & \frac{1}{3}, & \frac{1}{3} + \eps[\pi]
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\end{pmatrix},\\
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& & \bm{\rho}^1 = \begin{pmatrix}
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\frac{1}{3}, & \frac{1}{3}, & \frac{1}{3}
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\end{pmatrix}, & & \bm{\rho}^2 = \sigma\begin{pmatrix}
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\frac{1}{3} - \eps[\rho], & \frac{1}{3}, & \frac{1}{3} + \eps[\rho]
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\end{pmatrix},
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\end{align*} with \(\eps[\alpha] = 0.16\), \(\eps[\pi]\) and
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\(\eps[\rho]\) taking 9 values equally spaced in
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\(\left[ 0, .28\right]\). We simulate 324 different collections for each
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value of \(\eps[\pi]\) and \(\eps[\rho]\).
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\(\pi\rho\text{-}colBiSBM\), \(\pi\text{-}colBiSBM\),
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\(\rho\text{-}colBiSBM\), \(iid\text{-}colBiSBM\) and
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\(sep\text{-}colBiSBM\) are put in competition and the model with the
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greater BIC-L is selected as the \emph{preferred model}.
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When \(\eps[\pi] = 0\), \(\bm{\pi}^1 = \bm{\pi}^2\), \(\eps[\rho] = 0\)
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and \(\bm{\rho}^1 = \bm{\rho}^2\), the generated collection is an
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\(iid\text{-}colBiSBM\). When \(\eps[\pi] > 0\) or
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\(\bm{\pi}^1 \neq \bm{\pi}^2\), the model is a \(\pi\text{-}colBiSBM\).
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When \(\eps[\rho] > 0\) or \(\bm{\rho}^1 \neq \bm{\rho}^2\), the model
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is a \(\rho\text{-}colBiSBM\). Finally, when \(\eps[\pi] > 0\) or
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\(\bm{\pi}^1 \neq \bm{\pi}^2\) and \(\eps[\rho] > 0\) or
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\(\bm{\rho}^1 \neq \bm{\rho}^2\), the model is a
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\(\pi\rho\text{-}colBiSBM\).
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\begin{table}[!h]
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\caption{\label{tab:tables}\label{tab:pi-model-sel}Model selection for varying $\pi$ mixture parameters}
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\centering
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\begin{tabular}[t]{lccccl}
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\toprule
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\multicolumn{1}{c}{ } & \multicolumn{4}{c}{Proportions of model selection} & \multicolumn{1}{c}{Blocks} \\
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\cmidrule(l{3pt}r{3pt}){2-5} \cmidrule(l{3pt}r{3pt}){6-6}
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$\eps[\pi]$ & $iid\text{-}colBiSBM$ & $\pi\text{-}colBiSBM$ & $\rho\text{-}colBiSBM$ & $\pi\rho\text{-}colBiSBM$ & Recovered $Q_1$\\
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\midrule
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0.00 & 0.65 & 0.00 & 0.35 & 0.00 & 3.00\\
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0.04 & 0.66 & 0.00 & 0.34 & 0.00 & 3.00\\
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0.07 & 0.64 & 0.01 & 0.34 & 0.01 & 3.01\\
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0.11 & 0.63 & 0.03 & 0.31 & 0.03 & 3.01\\
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0.14 & 0.55 & 0.12 & 0.28 & 0.05 & 3.00\\
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\addlinespace
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0.18 & 0.39 & 0.26 & 0.21 & 0.13 & 3.01\\
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0.21 & 0.23 & 0.42 & 0.13 & 0.23 & 3.01\\
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0.25 & 0.10 & 0.56 & 0.05 & 0.29 & 3.02\\
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0.28 & 0.01 & 0.65 & 0.01 & 0.33 & 3.01\\
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\bottomrule
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\end{tabular}
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\end{table}
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\begin{table}[!h]
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\caption{\label{tab:tables}\label{tab:rho-model-sel}Model selection for varying $\rho$ mixture parameters}
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\centering
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\begin{tabular}[t]{lccccl}
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\toprule
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\multicolumn{1}{c}{ } & \multicolumn{4}{c}{Proportions of model selection} & \multicolumn{1}{c}{Blocks} \\
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\cmidrule(l{3pt}r{3pt}){2-5} \cmidrule(l{3pt}r{3pt}){6-6}
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$\eps[\rho]$ & $iid\text{-}colBiSBM$ & $\pi\text{-}colBiSBM$ & $\rho\text{-}colBiSBM$ & $\pi\rho\text{-}colBiSBM$ & Recovered $Q_2$\\
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\midrule
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0.00 & 0.63 & 0.37 & 0.00 & 0.00 & 3.00\\
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0.04 & 0.65 & 0.34 & 0.00 & 0.01 & 3.00\\
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0.07 & 0.64 & 0.33 & 0.01 & 0.01 & 3.00\\
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0.11 & 0.64 & 0.31 & 0.03 & 0.02 & 3.00\\
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0.14 & 0.53 & 0.29 & 0.11 & 0.06 & 3.00\\
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\addlinespace
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0.18 & 0.42 & 0.20 & 0.24 & 0.14 & 3.01\\
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0.21 & 0.25 & 0.12 & 0.40 & 0.22 & 3.01\\
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0.25 & 0.08 & 0.06 & 0.58 & 0.29 & 3.01\\
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0.28 & 0.01 & 0.01 & 0.65 & 0.32 & 3.00\\
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\bottomrule
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\end{tabular}
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\end{table}
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\begin{figure}[H]
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\includegraphics{./Rcodes/simulation/img/plot_model_function_eps.png}
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\caption{Plot of preferred model in function of $\eps[\pi]$ and $\eps[\rho]$}
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\label{fig:pref_model_func_eps}
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\end{figure}
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\paragraph{Results:}
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On the figure \ref{fig:pref_model_func_eps} and tables
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\ref{tab:pi-model-sel} and \ref{tab:rho-model-sel}, one can see that
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there is a turning point around \(\eps[\pi] = 0.2\) (resp.
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\(\eps[\rho] = 0.2\)), before which \(iid\text{-}colBiSBM\) and
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\(\rho\text{-}colBiSBM\) (resp. \(\pi\text{-}colBiSBM\)) are selected
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most of the times and after \(0.2\) the \(\pi\text{-}colBiSBM\) (resp.
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\(\rho\text{-}colBiSBM\)) and \(\pi\rho\text{-}colBiSBM\) gets more and
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more selected, highlighting our capacity to recover the simulated
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structure.
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\paragraph*{Remark:}
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Please note that when ``Recovered \(Q_1\)(or \(Q_2\))'' is not an
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integer it's because some procedures returned a value other than 3.
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