220 lines
No EOL
10 KiB
TeX
220 lines
No EOL
10 KiB
TeX
\section{VEM}
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\begin{frame}{Developed formula of variational EM}
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\begin{multline*}
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\ell (\bm{Y};\theta) \geq \color{red}\sum_{m=1}^{M} \bigg( \color{black} \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(Y^{m}_{ij}; \alpha_{qr}) \\
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+ \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} \\
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- \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} \color{red}\bigg) \color{black} \eqcolon
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\mathcal{J}(\tau;\theta),
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\end{multline*}
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\begin{block}{Variational approximation}
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$\tau_{iq}^{1,m} = \mathcal{R}^1_{Y^m,\tau}(Z_{iq}^m = 1)$
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and $\tau_{jr}^{2,m} = \mathcal{R}^2_{Y^m,\tau}(W_{jr}^m = 1)$
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\end{block}
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\end{frame}
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\begin{frame}{\emph{Variational Expectation} Step}
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\[
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\widehat{\tau}^{(t+1)} = \arg \max_{\tau}
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\mathcal{J}(\mathcal{\tau},\bm{\widehat{\theta}}^{(t)})
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\Leftrightarrow \arg\min_{\tau\in\mathcal{T}} \mathbf{KL}[\mathcal{R}_{\mathbf{Y},\tau}, \mathbb{P}(.|\mathbf{Y})]
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\]
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\begin{equation*}
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\begin{cases}
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\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(Y_{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 \\
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\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(Y_{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
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\end{cases}
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\end{equation*}
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\footnotetext[2]{Initialization of $\widehat{\tau}$ with a
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\emph{spectral clustering} on the networks.}
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\end{frame}
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\begin{frame}{\emph{Maximization} Step}
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\[
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\widehat{\theta}^{(t+1)} = \arg \max_{\theta} \mathcal{J}(\mathcal{\bm{\widehat{\tau}}}^{(t+1)},\theta)
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\]
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\begin{block}{Connectivity parameters}
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\begin{align*}
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\widehat{\alpha}_{qr} = \frac{\sum_{m=1}^{M} \sum_{i=1}^{n_1^m} \sum_{j=1}^{n_2^m} \tau_{iq}^{1,m} \tau_{jr}^{2,m} \alert<2>{Y_{ij}^m}}{\sum_{m=1}^{M} \sum_{i=1}^{n_1^m} \sum_{j=1}^{n_2^m} \tau_{iq}^{1,m} \tau_{jr}^{2,m}}
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\end{align*}
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\end{block}
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\only<1>{
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\begin{block}{Proportions for \emph{iid}}
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\begin{align*}
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\widehat{\pi}_q = \frac{\sum_{m=1}^{M} \sum_{i=1}^{n_1^m} \tau_{iq}^{1,m}}{\sum_{m=1}^{M} n_1^m} & &
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\widehat{\rho}_r = \frac{\sum_{m=1}^{M} \sum_{j=1}^{n_2^m} \tau_{jr}^{2,m}}{\sum_{m=1}^{M} n_2^m}
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\end{align*}
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\end{block}
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}
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\only<2>{
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\begin{block}{Proportions for $\pi\rho$}
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\begin{align*}
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\widehat{\pi}^{\color{red}m}_q = \frac{\sum_{i=1}^{n_1^m} \tau_{iq}^{1,m}}{n_1^m} & &
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\widehat{\rho}^{\color{red}m}_r = \frac{\sum_{j=1}^{n_2^m} \tau_{jr}^{2,m}}{n_2^m}
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\end{align*}
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\end{block}
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}
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\end{frame}
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\begin{frame}
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\frametitle{Why does VE minimizes KL ?}
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\begin{align*}
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\ell_c(\bY,\bZ,\bW;\theta) & = \log \Prob(\bZ, \bW|\bY;\theta) + \ell(\bY;\theta) \\
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\Leftrightarrow \ell(\bY;\theta) & = \ell_c(\bY,\bZ,\bW;\theta) - \log \Prob(\bZ, \bW|\bY;\theta) \\
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\Leftrightarrow \Esp_{\Ryt}[\ell(\bY;\theta)] & = \Esp_{\Ryt}[\ell_c(\bY,\bZ,\bW;\theta)] - \Esp_{\Ryt}[\log \Prob(\bZ,\bW|\bY;\theta)] \\
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\Leftrightarrow \ell(\bY;\theta) & = \Esp_{\Ryt}[\ell_c(\bY,\bZ,\bW;\theta)] - \Esp_{\Ryt}[\log \Prob(\bZ,\bW|\bY;\theta)] \\
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\end{align*}
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\begin{align*}
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\text{But }\KL{\Ryt}{\log \Prob(\bZ,\bW|\bY;\theta)} & = - \Esp_{\Ryt} [\log \frac{\Prob(\bZ,\bW|\bY;\theta)}{\Ryt}] \\
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= - \Esp_{\Ryt} [\log \Prob(\bZ,\bW|\bY;\theta)] + & \underbrace{\Esp_{\Ryt[\log \Ryt]}}_{-\Hshannon(\Ryt)} \\
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\Leftrightarrow \KL{\Ryt}{\log \Prob(\bZ,\bW|\bY;\theta)} + \Hshannon(\Ryt) & = - \Esp_{\Ryt} [\log \Prob(\bZ,\bW|\bY;\theta)]
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\end{align*}
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Thus $\ell(\bY;\theta) - \KL{\Ryt}{\log \Prob(\bZ,\bW|\bY;\theta)} = \mathcal{J}(\tau;\theta) \qed$
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\end{frame}
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\begin{frame}
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\frametitle{On the BIC-L}
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\begin{align*}
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& \text{ICL}(\hat{\theta}) = \Esp_{\mathbf{Z}, \mathbf{W}|\mathbf{Y}} [\log p(\mathbf{Y}|\mathbf{Z},\mathbf{W};\hat{\theta})] - \frac{1}{2} \text{pen}(\dots) \\
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& \text{BIC}(\hat{\theta}) = \Esp_{\mathbf{Z}, \mathbf{W}|\mathbf{Y}} [\log p(\mathbf{Y}|\mathbf{Z},\mathbf{W};\hat{\theta})] + \mathcal{H}(p(\mathbf{Z},\mathbf{W}|\mathbf{Y})) - \frac{1}{2} \text{pen}(\dots) \\
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& \text{BIC-L}(\hat{\theta}, \hat{\tau}) = \Esp_{\mathcal{R}_{\mathbf{Y}, \hat{\tau}}}[\log \ell_c(\mathbf{Y},\mathbf{Z},\mathbf{W};\hat{\theta}^{\text{var}})] + \mathcal{H}(\mathcal{R}_{\mathbf{Y}, \hat{\tau}}) - \frac{1}{2} \text{pen}(\dots) \\
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\end{align*}
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\end{frame}
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\section{Model selection}
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\begin{frame}
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\frametitle{Choice of $(Q_1,Q_2)$ - Greedy approach}
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\begin{columns}
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\begin{column}{0.5\linewidth}
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\begin{tikzpicture}
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\input{tikz/greedy-exploration.tex}
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\end{tikzpicture}
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\end{column}
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\begin{column}{0.35\linewidth}
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\begin{itemize}
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\item Initial model~:\\
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\begin{tikzpicture}
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\draw[fill=gray, draw=gray] circle [radius=0.225cm];
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\end{tikzpicture}
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\onslide<2->{
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\item Model after \emph{split}~:
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\begin{tikzpicture}
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\draw[fill=blueind, draw=blueind] circle [radius=0.225cm];
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\end{tikzpicture}
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\item Model maximizing the criterion~:\\
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\begin{tikzpicture}
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\draw[fill=white, draw=green, very thick] circle [radius=0.225cm];
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\end{tikzpicture}
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}
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\onslide<3->{
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\item Model after \emph{merge}~:
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\begin{tikzpicture}
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\draw[fill=red, draw=red] circle [radius=0.225cm];
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\end{tikzpicture}
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}
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\end{itemize}
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\end{column}
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\end{columns}
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\end{frame}
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\begin{frame}
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\frametitle{Choice of $(Q_1,Q_2)$ - Sliding window}
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\begin{columns}
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\begin{column}{0.6\textwidth}
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\begin{figure}
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\input{tikz/moving-window}
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\caption{Sliding window}
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\end{figure}
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\end{column}
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\begin{column}{0.4\textwidth}
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\only<3>{\begin{block}{}
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Initialization of the model if necessary
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\end{block}}
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\only<9>{\begin{block}{}
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Localization of the new mode
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\end{block}}
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\only<10>{\begin{block}{}
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Move to the new mode then iterate
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\end{block}}
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\end{column}
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\end{columns}
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\end{frame}
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\section{Clustering}
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\begin{frame}{Clustering algorithm}
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\centering
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\vspace{0.25\baselineskip}
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\begin{tikzpicture}[scale=0.85]
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\input{tikz/clustering.tex}
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\end{tikzpicture}
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\[
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D_{\mathcal{M}}(m,m') = \sum_{q = 1}^{Q_1} \sum_{r = 1}^{Q_2} \max(\widetilde{\pi}_{q}^{m}, \widetilde{\pi}_{q}^{m'}) \left( \widetilde{\alpha}_{qr}^{m} - \widetilde{\alpha}_{qr}^{m'}\right)^{2} \max(\widetilde{\rho}_{r}^{m}, \widetilde{\rho}_{r}^{m'})
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\]
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\end{frame}
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\section{Results~\cite{baldockSystemsApproachReveals2019,baldockDailyTemporalStructure2011}}
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\begin{frame}[allowframebreaks]
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\begin{figure}[ht]
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\centering
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\begin{subfigure}[t]{0.5\textwidth}
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\centering
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\includegraphics[width=0.45\textwidth]{tikz/applications/baldock/mat-Baldock2019_Bristol.pdf}
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\caption{Donnée}
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\end{subfigure}\hfil
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\begin{subfigure}[t]{0.5\textwidth}
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\centering
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\includegraphics[width=0.45\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Bristol.pdf}
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\caption{Reordered}
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\end{subfigure}
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\caption{Bristol}
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\end{figure}
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\begin{figure}[ht]
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\centering
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\begin{subfigure}[t]{0.5\textwidth}
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\centering
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\includegraphics[width=0.45\textwidth]{tikz/applications/baldock/mat-Baldock2019_Edinburgh.pdf}
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\caption{Donnée}
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\end{subfigure}\hfil
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\begin{subfigure}[t]{0.5\textwidth}
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\centering
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\includegraphics[width=0.5\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Edinburgh.pdf}
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\caption{Reordered}
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\end{subfigure}
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\caption{Edinburgh}
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\end{figure}
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\begin{figure}
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\begin{subfigure}[ht]{0.5\textwidth}
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\centering
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\includegraphics[width=0.5\textwidth]{tikz/applications/baldock/mat-Baldock2019_Leeds.pdf}
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\caption{Donnée}
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\end{subfigure}\hfil
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\begin{subfigure}[ht]{0.5\textwidth}
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\centering
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\includegraphics[width=0.5\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Leeds.pdf}
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\caption{Réordonnée}
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\end{subfigure}
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\caption{Leeds}
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\end{figure}
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\begin{figure}
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\begin{subfigure}[ht]{0.5\textwidth}
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\centering
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\includegraphics[width=0.5\textwidth]{tikz/applications/baldock/mat-Baldock2019_Reading.pdf}
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\caption{Donnée}
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\end{subfigure}\hfil
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\begin{subfigure}[ht]{0.5\textwidth}
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\centering
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\includegraphics[width=0.5\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Reading.pdf}
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\caption{Réordonnée}
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\end{subfigure}
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\caption{Reading}
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\end{figure}
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\end{frame} |