From 50524dddf67d9e5040451ed8428695595bb1d808 Mon Sep 17 00:00:00 2001 From: Louis Lacoste Date: Thu, 11 Jul 2024 23:54:31 +0200 Subject: [PATCH] =?UTF-8?q?rapport=20:=20ajout=20des=20simus=20d'inf=C3=A9?= =?UTF-8?q?rences=20et=20de=20s=C3=A9lection=20de=20mod=C3=A8les?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- rapport/chapter4-simulation-studies.tex | 5 +- rapport/chapter4-simulations/inference.tex | 122 ++++++ .../chapter4-simulations/model-selection.tex | 128 ++++++ rapport/rapport.pdf | Bin 887375 -> 701834 bytes rapport/rapport.tex | 28 +- tables/simulations/inference/iid.tex | 21 + tables/simulations/inference/pi.tex | 21 + tables/simulations/inference/pirho.tex | 21 + tables/simulations/inference/preferred.tex | 21 + tables/simulations/inference/rho.tex | 21 + tables/simulations/inference/sep.tex | 21 + .../inference/model-proportions.tex | 407 ++++++++++++++++++ 12 files changed, 800 insertions(+), 16 deletions(-) create mode 100644 rapport/chapter4-simulations/inference.tex create mode 100644 rapport/chapter4-simulations/model-selection.tex create mode 100644 tables/simulations/inference/iid.tex create mode 100644 tables/simulations/inference/pi.tex create mode 100644 tables/simulations/inference/pirho.tex create mode 100644 tables/simulations/inference/preferred.tex create mode 100644 tables/simulations/inference/rho.tex create mode 100644 tables/simulations/inference/sep.tex create mode 100644 tikz/simulations/inference/model-proportions.tex diff --git a/rapport/chapter4-simulation-studies.tex b/rapport/chapter4-simulation-studies.tex index 7878aaf..fcf5a4f 100644 --- a/rapport/chapter4-simulation-studies.tex +++ b/rapport/chapter4-simulation-studies.tex @@ -9,6 +9,7 @@ performances and the clustering capacities. the report repository at \url{https://gitea.polarolouis.fr/polarolouis/rapport-CEI-MIA-2023}. -% \include{Rcodes/simulation/inference_analyze} -% \include{Rcodes/simulation/model_selection_analyze} + +\input{chapter4-simulations/inference.tex} +\input{chapter4-simulations/model-selection} % \include{Rcodes/simulation/netclustering_analyze} diff --git a/rapport/chapter4-simulations/inference.tex b/rapport/chapter4-simulations/inference.tex new file mode 100644 index 0000000..1754cfc --- /dev/null +++ b/rapport/chapter4-simulations/inference.tex @@ -0,0 +1,122 @@ +\section{Efficiency of the inference} + +The goal here is to assess the quality of the inference procedure. + +\paragraph{Simulation settings} For this simulation the data is simulated with +$M = 2, n_{1}^{m} = 120, n_{2}^{m} = 120, Q_1 = Q_2 = 4$, $\bm{\alpha}, \bm{\pi}$ +and $\bm{\rho}$ are set as follows: +\begin{align*} + & \bm{\alpha} = .25 + + \begin{pmatrix} + 3 \eps[\alpha] & 2 \eps[\alpha] & \eps[\alpha] & - \eps[\alpha] \\ + 2 \eps[\alpha] & 2 \eps[\alpha] & - \eps[\alpha] & \eps[\alpha] \\ + \eps[\alpha] & - \eps[\alpha] & \eps[\alpha] & 2 \eps[\alpha] \\ + - \eps[\alpha] & \eps[\alpha] & 2 \eps[\alpha] & 0 + \end{pmatrix}, +\end{align*} +\begin{align*} + \bm{\pi}^1 = \sigma_1 + \begin{pmatrix} + 0.2 & 0.4 & 0.4 & 0 + \end{pmatrix}, + & & \bm{\pi}^2 = + \begin{pmatrix} + 0.25 & 0.25 & 0.25 & 0.25 + \end{pmatrix}, \\ + \bm{\rho}^1 = + \begin{pmatrix} + 0.25 & 0.25 & 0.25 & 0.25 + \end{pmatrix}, & & + \bm{\rho}^2 = \sigma_2 + \begin{pmatrix} + 0 & 0.33 & 0.33 & 0.33 + \end{pmatrix}, & & +\end{align*} +with $\eps[\alpha]$ taking nine equally spaced values ranging from 0 to 0.24. +For each value of $\eps[\alpha]$, 108 datasets ($X_1, X_2$) are simulated, +resulting in $9 \times 108 = 972$ datasets. More precisely, for each dataset, +we pick uniformly at random two permutations of $\{ 1, \dots , 4 \}$ +($\sigma_1, \sigma_2$) with the constraint that $\sigma_1(4) \neq \sigma_2(1)$. +This ensures that each of the two networks have a non-empty block that is empty +in the other one. Then the networks are simulated with +$\mathcal{B}$ern-$BiSBM_{120,120}(Q_1 = 4, Q_2 = 4, + \bm{\alpha}, \bm{\pi}^m, \bm{\rho}^m)$ +with the previous parameters. Each network has 2 blocks in common and their +connectivity structures encompass a mix of core-periphery, assortative +community and disassortative community structures, depending on which 3 of the 4 +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 $\pi\rho\text{-}colBiSBM$. + +\paragraph{Inference} We want to measure the quality of the +inference procedure, for this we use the inference described in the section +\ref{sec:variational-estimation-of-the-parameters}. + +\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 + $sep\text{-}BiSBM$, $iid\text{-}colBiSBM$, $\rho\text{-}colBiSBM$, + $\pi\rho\text{-}colBiSBM$ + respectively. To do so, for each dataset, we compute the + BIC-L of each model $\pi\text{-}colBiSBM$ is preferred to $sep\text{-}BiSBM$ + (resp. $iid\text{-}colBiSBM$, $\rho\text{-}colBiSBM$, + $\pi\rho\text{-}colBiSBM$) if + its BIC-L is greater. + \item When considering our \emph{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 + Adjusted Rand Index \parencite{hubertComparingPartitions1985}, ARI = 0 + for a random grouping, ARI = 1 for a perfect recovery. For each + network, for the + $\pi\text{-}colBiSBM$, $\rho\text{-}colBiSBM$, + $\pi\rho\text{-}colBiSBM$ 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*} + \overline{\text{ARI}}_d = \frac{1}{2} \text{ARI}\big( \text{ARI}(\widehat{\bm{Z}^1_d},\bm{Z}^1_d) + \text{ARI}(\widehat{\bm{Z}^2_d},\bm{Z}^2_d) \big), + \end{equation*} + where $d$ is the dimension or axis (i.e., rows, $d=1$, or columns, $d=2$) of + the block memberships. + And we compute the ARI of the whole set of nodes to account for block + pairing between networks + \begin{equation*} + \text{ARI}_d = \text{ARI}\big((\widehat{\bm{Z}^1_d},\widehat{\bm{Z}^2_d}),(\bm{Z}^1_d,\bm{Z}^2_d) \big). + \end{equation*} +\end{itemize} + +All these quality indicators are averaged over the 108 datasets. The results are +provided in the tables \ref{tab:per_model_sep} to \ref{tab:per_model_pirho}. Each line corresponds to the +108 datasets for a given value of value of $\eps[\alpha]$. + +\begin{figure}[ht] + \centering + \input{../tikz/simulations/inference/model-proportions.tex} + \caption{Preferred model proportions over all datasets in function of + $\eps[\alpha]$} + \label{fig:prop-modele-pref} +\end{figure} + +\foreach \modelname in {sep,iid,pi,rho,pirho}{ + \input{../tables/simulations/inference/\modelname.tex} + } + +\paragraph{Results} +For the model comparison, when $\eps[\alpha]$ is small +($\eps[\alpha]\in[0, .04]$), the simulation model is close to the +Erd\H{o}s-Reńyi network and it is very hard to find any structure beyond the one +of a single block on each dimension. + +On the figure \ref{fig:inference-proportion-preferred} and table +\ref{tab:proportion-preferred-table} we 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. + +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 +($\text{ARI}_d$). This means that our models are able to recover the block +pairing \emph{between the networks} in addition to recovering the blocks and +their parameters. +\clearpage \ No newline at end of file diff --git a/rapport/chapter4-simulations/model-selection.tex b/rapport/chapter4-simulations/model-selection.tex new file mode 100644 index 0000000..abb70ae --- /dev/null +++ b/rapport/chapter4-simulations/model-selection.tex @@ -0,0 +1,128 @@ +\section[Capacity to distinguish models]{Capacity to distinguish + $\pi\rho\text{-}colBiSBM$~from\newline$iid\text{-}colBiSBM$ and other + models}\label{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: +\textit{iid, pi, rho, pirho}. This difference being based on the row and +col block proportions. + +For this task we choose the same simulation settings as +\cite{chabert-liddellLearningCommonStructures2024a}. + +Namely $n_{1}^{m} = 90, n_{2}^{m} = 90, Q_1 = Q_2 = 3$, +$\bm{\alpha}, \bm{\pi}$ and $\bm{\rho}$ are set as follows: +\begin{minipage}[l]{0.4\linewidth} + \begin{align*} + \bm{\alpha} =.25 + \begin{pmatrix} + 3 \eps[\alpha] & 2 \eps[\alpha] & \eps[\alpha] \\ + 2 \eps[\alpha] & 2 \eps[\alpha] & - \eps[\alpha] \\ + \eps[\alpha] & - \eps[\alpha] & \eps[\alpha] + \end{pmatrix}, + \end{align*} +\end{minipage} +\hfill +\begin{minipage}[r]{0.4\linewidth} + \begin{align*} + \bm{\pi}^1 = \begin{pmatrix} + \frac{1}{3}, & \frac{1}{3}, & \frac{1}{3} + \end{pmatrix}, & & \bm{\pi}^2 = \sigma\begin{pmatrix} + \frac{1}{3} - \eps[\pi], & \frac{1}{3}, & \frac{1}{3} + \eps[\pi] + \end{pmatrix}, \\ + \bm{\rho}^1 = \begin{pmatrix} + \frac{1}{3}, & \frac{1}{3}, & \frac{1}{3} + \end{pmatrix}, & & \bm{\rho}^2 = \sigma\begin{pmatrix} + \frac{1}{3} - \eps[\rho], & \frac{1}{3}, & \frac{1}{3} + \eps[\rho] + \end{pmatrix}, + \end{align*} +\end{minipage} +with $\eps[\alpha] = 0.16$, $\eps[\pi]$ and +$\eps[\rho]$ taking 9 values equally spaced in +$\left[ 0, .28\right]$. We simulate 324 different collections for each +value of $\eps[\pi]$ and $\eps[\rho]$. + +$\pi\rho\text{-}colBiSBM$, $\pi\text{-}colBiSBM$, +$\rho\text{-}colBiSBM$, $iid\text{-}colBiSBM$ 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 +$\bm{\pi}^1 \neq \bm{\pi}^2$, the model is a $\pi\text{-}colBiSBM$. +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 +$\bm{\pi}^1 \neq \bm{\pi}^2$ and $\eps[\rho] > 0$ or +$\bm{\rho}^1 \neq \bm{\rho}^2$, the model is a +$\pi\rho\text{-}colBiSBM$. + +\begin{table}[!h] + + \caption{\label{tab:tables}\label{tab:pi-model-sel}Model selection for varying $\pi$ mixture parameters} + \centering + \begin{tabular}[t]{lccccl} + \toprule + \multicolumn{1}{c}{ } & \multicolumn{4}{c}{Models} & \multicolumn{1}{c}{Blocks} \\ + \cmidrule(l{3pt}r{3pt}){2-5} \cmidrule(l{3pt}r{3pt}){6-6} + $\eps[\pi]$ & $iid\text{-}colBiSBM$ & $\pi\text{-}colBiSBM$ & $\rho\text{-}colBiSBM$ & $\pi\rho\text{-}colBiSBM$ & Recovered $Q_1$ \\ + \midrule + 0.00 & 0.65 & 0.00 & 0.35 & 0.00 & 3 \\ + 0.04 & 0.66 & 0.00 & 0.34 & 0.00 & 3 \\ + 0.07 & 0.64 & 0.01 & 0.34 & 0.01 & 3.01 $\pm$ 0.01 \\ + 0.11 & 0.63 & 0.03 & 0.31 & 0.03 & 3.01 $\pm$ 0.01 \\ + 0.14 & 0.55 & 0.12 & 0.28 & 0.05 & 3 \\ + \addlinespace + 0.18 & 0.39 & 0.26 & 0.21 & 0.13 & 3.01 \\ + 0.21 & 0.23 & 0.42 & 0.13 & 0.23 & 3.01 \\ + 0.25 & 0.10 & 0.56 & 0.05 & 0.29 & 3.02 $\pm$ 0.01 \\ + 0.28 & 0.01 & 0.65 & 0.01 & 0.33 & 3.01 $\pm$ 0.01 \\ + \bottomrule + \end{tabular} +\end{table} + +\begin{table}[!h] + + \caption{\label{tab:tables}\label{tab:rho-model-sel}Model selection for varying $\rho$ mixture parameters} + \centering + \begin{tabular}[t]{lccccl} + \toprule + \multicolumn{1}{c}{ } & \multicolumn{4}{c}{Models} & \multicolumn{1}{c}{Blocks} \\ + \cmidrule(l{3pt}r{3pt}){2-5} \cmidrule(l{3pt}r{3pt}){6-6} + $\eps[\rho]$ & $iid\text{-}colBiSBM$ & $\pi\text{-}colBiSBM$ & $\rho\text{-}colBiSBM$ & $\pi\rho\text{-}colBiSBM$ & Recovered $Q_2$ \\ + \midrule + 0.00 & 0.63 & 0.37 & 0.00 & 0.00 & 3 \\ + 0.04 & 0.65 & 0.34 & 0.00 & 0.01 & 3 \\ + 0.07 & 0.64 & 0.33 & 0.01 & 0.01 & 3 \\ + 0.11 & 0.64 & 0.31 & 0.03 & 0.02 & 3 \\ + 0.14 & 0.53 & 0.29 & 0.11 & 0.06 & 3 \\ + \addlinespace + 0.18 & 0.42 & 0.20 & 0.24 & 0.14 & 3.01 \\ + 0.21 & 0.25 & 0.12 & 0.40 & 0.22 & 3.01 \\ + 0.25 & 0.08 & 0.06 & 0.58 & 0.29 & 3.01 \\ + 0.28 & 0.01 & 0.01 & 0.65 & 0.32 & 3 \\ + \bottomrule + \end{tabular} +\end{table} + +\begin{figure}[H] + % \includegraphics{./Rcodes/simulation/img/plot_model_function_eps.png} + \caption{Plot of preferred model in function of $\eps[\pi]$ and $\eps[\rho]$} + \label{fig:pref_model_func_eps} +\end{figure} + +\paragraph{Results:} + +On the figure \ref{fig:pref_model_func_eps} and tables +\ref{tab:pi-model-sel} and \ref{tab:rho-model-sel}, 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 +$\rho\text{-}colBiSBM$ (resp. $\pi\text{-}colBiSBM$) are selected +most of the times and after $0.2$ the $\pi\text{-}colBiSBM$ (resp. +$\rho\text{-}colBiSBM$) and $\pi\rho\text{-}colBiSBM$ gets more and +more selected, highlighting our capacity to recover the simulated +structure. + +\paragraph*{Remark:} + +Please note that when ``Recovered $Q_1$(or $Q_2$)'' is not an +integer it's because some procedures returned a value other than 3. 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