Maj 14/05/2025
This commit is contained in:
Louis
parent
d898e3b292
commit
e8e8a247e9
5 changed files with 210 additions and 201 deletions
134
annexe.tex
134
annexe.tex
|
|
@ -1,5 +1,65 @@
|
|||
\section{VEM}
|
||||
|
||||
\begin{frame}{Developed formula of variational EM}
|
||||
\begin{multline*}
|
||||
\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}) \\
|
||||
+ \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} \\
|
||||
- \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
|
||||
\mathcal{J}(\tau;\theta),
|
||||
\end{multline*}
|
||||
|
||||
\begin{block}{Variational approximation}
|
||||
$\tau_{iq}^{1,m} = \mathcal{R}^1_{Y^m,\tau}(Z_{iq}^m = 1)$
|
||||
and $\tau_{jr}^{2,m} = \mathcal{R}^2_{Y^m,\tau}(W_{jr}^m = 1)$
|
||||
\end{block}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}{\emph{Variational Expectation} Step}
|
||||
\[
|
||||
\widehat{\tau}^{(t+1)} = \arg \max_{\tau}
|
||||
\mathcal{J}(\mathcal{\tau},\bm{\widehat{\theta}}^{(t)})
|
||||
\Leftrightarrow \arg\min_{\tau\in\mathcal{T}} \mathbf{KL}[\mathcal{R}_{\mathbf{Y},\tau}, \mathbb{P}(.|\mathbf{Y})]
|
||||
\]
|
||||
|
||||
\begin{equation*}
|
||||
\begin{cases}
|
||||
\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 \\
|
||||
\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
|
||||
\end{cases}
|
||||
\end{equation*}
|
||||
\footnotetext[2]{Initialization of $\widehat{\tau}$ with a
|
||||
\emph{spectral clustering} on the networks.}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}{\emph{Maximization} Step}
|
||||
\[
|
||||
\widehat{\theta}^{(t+1)} = \arg \max_{\theta} \mathcal{J}(\mathcal{\bm{\widehat{\tau}}}^{(t+1)},\theta)
|
||||
\]
|
||||
|
||||
\begin{block}{Connectivity parameters}
|
||||
\begin{align*}
|
||||
\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}}
|
||||
\end{align*}
|
||||
\end{block}
|
||||
\only<1>{
|
||||
\begin{block}{Proportions for \emph{iid}}
|
||||
\begin{align*}
|
||||
\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} & &
|
||||
\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}
|
||||
\end{align*}
|
||||
\end{block}
|
||||
}
|
||||
\only<2>{
|
||||
\begin{block}{Proportions for $\pi\rho$}
|
||||
\begin{align*}
|
||||
\widehat{\pi}^{\color{red}m}_q = \frac{\sum_{i=1}^{n_1^m} \tau_{iq}^{1,m}}{n_1^m} & &
|
||||
\widehat{\rho}^{\color{red}m}_r = \frac{\sum_{j=1}^{n_2^m} \tau_{jr}^{2,m}}{n_2^m}
|
||||
\end{align*}
|
||||
\end{block}
|
||||
}
|
||||
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{Why does VE minimizes KL ?}
|
||||
\begin{align*}
|
||||
|
|
@ -16,7 +76,79 @@
|
|||
Thus $\ell(\bY;\theta) - \KL{\Ryt}{\log \Prob(\bZ,\bW|\bY;\theta)} = \mathcal{J}(\tau;\theta) \qed$
|
||||
\end{frame}
|
||||
|
||||
\section{Résultats~\cite{baldockSystemsApproachReveals2019,baldockDailyTemporalStructure2011}}
|
||||
\section{Model selection}
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{Choice of $(Q_1,Q_2)$ - Greedy approach}
|
||||
\begin{columns}
|
||||
\begin{column}{0.5\linewidth}
|
||||
\begin{tikzpicture}
|
||||
\input{tikz/greedy-exploration.tex}
|
||||
\end{tikzpicture}
|
||||
\end{column}
|
||||
\begin{column}{0.35\linewidth}
|
||||
\begin{itemize}
|
||||
\item Initial model~:\\
|
||||
\begin{tikzpicture}
|
||||
\draw[fill=gray, draw=gray] circle [radius=0.225cm];
|
||||
\end{tikzpicture}
|
||||
\onslide<2->{
|
||||
\item Model after \emph{split}~:
|
||||
\begin{tikzpicture}
|
||||
\draw[fill=blueind, draw=blueind] circle [radius=0.225cm];
|
||||
\end{tikzpicture}
|
||||
\item Model maximizing the criterion~:\\
|
||||
\begin{tikzpicture}
|
||||
\draw[fill=white, draw=green, very thick] circle [radius=0.225cm];
|
||||
\end{tikzpicture}
|
||||
}
|
||||
\onslide<3->{
|
||||
\item Model after \emph{merge}~:
|
||||
\begin{tikzpicture}
|
||||
\draw[fill=red, draw=red] circle [radius=0.225cm];
|
||||
\end{tikzpicture}
|
||||
}
|
||||
\end{itemize}
|
||||
\end{column}
|
||||
\end{columns}
|
||||
\end{frame}
|
||||
\begin{frame}
|
||||
\frametitle{Choice of $(Q_1,Q_2)$ - Sliding window}
|
||||
\begin{columns}
|
||||
\begin{column}{0.6\textwidth}
|
||||
\begin{figure}
|
||||
\input{tikz/moving-window}
|
||||
\caption{Sliding window}
|
||||
\end{figure}
|
||||
\end{column}
|
||||
\begin{column}{0.4\textwidth}
|
||||
\only<3>{\begin{block}{}
|
||||
Initialization of the model if necessary
|
||||
\end{block}}
|
||||
\only<9>{\begin{block}{}
|
||||
Localization of the new mode
|
||||
\end{block}}
|
||||
\only<10>{\begin{block}{}
|
||||
Move to the new mode then iterate
|
||||
\end{block}}
|
||||
\end{column}
|
||||
\end{columns}
|
||||
\end{frame}
|
||||
|
||||
\section{Clustering}
|
||||
|
||||
\begin{frame}{Clustering algorithm}
|
||||
\centering
|
||||
\vspace{0.25\baselineskip}
|
||||
\begin{tikzpicture}[scale=0.85]
|
||||
\input{tikz/clustering.tex}
|
||||
\end{tikzpicture}
|
||||
\[
|
||||
D_{\mathcal{M}}(m,m') = \sum_{q = 1}^{Q_1} \sum_{r = 1}^{Q_2} \max(\widetilde{\pi}_{q}^{m}, \widetilde{\pi}_{q}^{m'}) \left( \widetilde{\alpha}_{qr}^{m} - \widetilde{\alpha}_{qr}^{m'}\right)^{2} \max(\widetilde{\rho}_{r}^{m}, \widetilde{\rho}_{r}^{m'})
|
||||
\]
|
||||
\end{frame}
|
||||
|
||||
\section{Results~\cite{baldockSystemsApproachReveals2019,baldockDailyTemporalStructure2011}}
|
||||
\begin{frame}[allowframebreaks]
|
||||
\begin{figure}[ht]
|
||||
\centering
|
||||
|
|
|
|||
|
|
@ -111,11 +111,11 @@
|
|||
|
||||
|
||||
|
||||
\subtitle{Présentation LSD}
|
||||
\subtitle{JdS 2025}
|
||||
\title[Bipartite networks collection]{Joint analysis of bipartite networks collection}
|
||||
\author[L. Lacoste]{Louis \textsc{Lacoste}, under the supervision of Pierre Barbillon and
|
||||
Sophie Donnet\newline Laboratoire MIA Paris-Saclay} % Sous la supervision de Pierre
|
||||
\date{\ccbysa}
|
||||
\author[L. Lacoste]{\underline{Louis Lacoste}, Pierre Barbillon and
|
||||
Sophie Donnet\newline Laboratoire MIA Paris-Saclay\newline\ccbysa} % Sous la supervision de Pierre
|
||||
\date{\today}
|
||||
|
||||
\begin{document}
|
||||
|
||||
|
|
|
|||
234
principal.tex
234
principal.tex
|
|
@ -33,7 +33,7 @@
|
|||
\begin{figure}[ht]
|
||||
\centering
|
||||
\includegraphics[width=0.7\textwidth]{tikz/applications/baldock/graph-Baldock2019_Bristol.pdf}
|
||||
\caption{Plant-pollinator network of
|
||||
\caption{Plant-pollinator network from
|
||||
Bristol\newline\cite{baldockSystemsApproachReveals2019}}
|
||||
\label{fig:label}
|
||||
\end{figure}
|
||||
|
|
@ -41,8 +41,7 @@
|
|||
\begin{column}{0.5\textwidth}
|
||||
\begin{itemize}
|
||||
\item Modeling of various interactions, here ecosystems
|
||||
\item Structure necessary for: biodiversity monitoring, robustness, risk
|
||||
of collapse
|
||||
\item Structure necessary for: biodiversity monitoring, robustness, risk of collapse
|
||||
\item Increasingly available
|
||||
\end{itemize}
|
||||
\end{column}
|
||||
|
|
@ -50,7 +49,7 @@
|
|||
\end{frame}
|
||||
|
||||
\begin{frame}{Analysis methods for a network}
|
||||
Several methods~:
|
||||
TODO (Supprimable) Several methods~:
|
||||
\begin{itemize}
|
||||
\item Metrics~: degree, centrality, nesting \dots
|
||||
\item Network embedding with GNN
|
||||
|
|
@ -132,7 +131,7 @@
|
|||
\section[Bipartite collection models]{Bipartite network collection models}
|
||||
\label{sec:extension-of-colsbm-to-bipartite-networks}
|
||||
\begin{frame}
|
||||
\frametitle{Bipartite collections}
|
||||
\frametitle{Bipartite collections different BiSBM}
|
||||
\[
|
||||
\forall m \in \{1\dots M\}, Y^m \overset{ind}{\sim} \mathcal{F}\text{-BiSBM}_{n_1^m,n_2^m}(Q_1^m, Q_2^m, \pi^m, \rho^m, \alpha^m)
|
||||
\]
|
||||
|
|
@ -158,7 +157,7 @@
|
|||
\end{frame}
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{Different models}
|
||||
\frametitle{Joint models}
|
||||
\onslide<1->{ \begin{block}{\emph{iid}-colBiSBM}
|
||||
\[
|
||||
\forall m \in \{1\dots M\}, Y^m \overset{iid}{\sim}
|
||||
|
|
@ -177,41 +176,41 @@
|
|||
\end{block}
|
||||
}
|
||||
\end{frame}
|
||||
\begin{frame}
|
||||
\frametitle{Parameter estimation}
|
||||
% DONE say that tau i q m c' is the probability that Zim = q, approximation of the variational probability. Because we impose independence
|
||||
% By maximizing a variational lower bound of the
|
||||
% log-likelihood of the observed data.
|
||||
Maximizing the log-likelihood?
|
||||
\begin{block}{log-likelihood and complete log-likelihood}
|
||||
\[
|
||||
\ell(\bm{Y};\theta) = \sum_{\bm{Z,W}\in \bm{\mathcal{Z}\times\mathcal{W}}} \ell_c(\bm{Y}, \bm{Z}, \bm{W};\theta)
|
||||
\]
|
||||
% \begin{frame}
|
||||
% \frametitle{Parameter estimation}
|
||||
% % DONE say that tau i q m c' is the probability that Zim = q, approximation of the variational probability. Because we impose independence
|
||||
% % By maximizing a variational lower bound of the
|
||||
% % log-likelihood of the observed data.
|
||||
% Maximizing the log-likelihood?
|
||||
% \begin{block}{log-likelihood and complete log-likelihood}
|
||||
% \[
|
||||
% \ell(\bm{Y};\theta) = \sum_{\bm{Z,W}\in \bm{\mathcal{Z}\times\mathcal{W}}} \ell_c(\bm{Y}, \bm{Z}, \bm{W};\theta)
|
||||
% \]
|
||||
|
||||
with $\bm{\mathcal{Z}} = \{1,\dots,\alert<2>{Q_1}\}^{\alert<2>{n}},
|
||||
\bm{\mathcal{W}} = \{1,\dots,\alert<2>{Q_2}\}^{\alert<2>{n}}$
|
||||
\end{block}
|
||||
\uncover<3>{So, classic algorithm $\Rightarrow$
|
||||
\emph{Expectation-Maximization} (EM).}
|
||||
\end{frame}
|
||||
% with $\bm{\mathcal{Z}} = \{1,\dots,\alert<2>{Q_1}\}^{\alert<2>{n}},
|
||||
% \bm{\mathcal{W}} = \{1,\dots,\alert<2>{Q_2}\}^{\alert<2>{n}}$
|
||||
% \end{block}
|
||||
% \uncover<3>{So, classic algorithm $\Rightarrow$
|
||||
% \emph{Expectation-Maximization} (EM).}
|
||||
% \end{frame}
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{By classic EM}
|
||||
At iteration $(t)$:
|
||||
\begin{itemize}
|
||||
\item[$\bullet$]\textbf{E Step}: calculate
|
||||
$$ \mathcal{Q}(\theta | \theta^{(t-1)}) = \mathbb E_{\alert<2>{\bm Z, \bm W | \bm Y, \theta^{(t-1)}} } \left[\ell_c(\bm Y, \bm W, \bm Z; \theta) \right] $$
|
||||
\item[$\bullet$]\textbf{M Step}:
|
||||
$$ \theta^{(t)} = \arg \max_{\theta} \mathcal{Q}(\theta | \theta^{(t-1)})$$
|
||||
\end{itemize}
|
||||
\uncover<2>{
|
||||
\begin{alertblock}{Problem for classic EM}
|
||||
Law of $\bm{Z,W|Y},\theta^{(t-1)}$ inaccessible
|
||||
\end{alertblock}}
|
||||
% \begin{frame}
|
||||
% \frametitle{By classic EM}
|
||||
% At iteration $(t)$:
|
||||
% \begin{itemize}
|
||||
% \item[$\bullet$]\textbf{E Step}: calculate
|
||||
% $$ \mathcal{Q}(\theta | \theta^{(t-1)}) = \mathbb E_{\alert<2>{\bm Z, \bm W | \bm Y, \theta^{(t-1)}} } \left[\ell_c(\bm Y, \bm W, \bm Z; \theta) \right] $$
|
||||
% \item[$\bullet$]\textbf{M Step}:
|
||||
% $$ \theta^{(t)} = \arg \max_{\theta} \mathcal{Q}(\theta | \theta^{(t-1)})$$
|
||||
% \end{itemize}
|
||||
% \uncover<2>{
|
||||
% \begin{alertblock}{Problem for classic EM}
|
||||
% Law of $\bm{Z,W|Y},\theta^{(t-1)}$ inaccessible
|
||||
% \end{alertblock}}
|
||||
|
||||
\end{frame}
|
||||
% \end{frame}
|
||||
|
||||
\begin{frame}
|
||||
\begin{frame}{Parameter estimation}
|
||||
By \emph{Variational EM}, as proposed
|
||||
by~\cite{daudinMixtureModelRandom2008,
|
||||
chabert-liddellLearningCommonStructures2024}.
|
||||
|
|
@ -235,66 +234,6 @@
|
|||
\left[ \ell_c(Y^m,\mathbf{Z}^m,\mathbf{W}^m | \theta) \right] \,$
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}{Developed formula of variational EM}
|
||||
\begin{multline*}
|
||||
\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}) \\
|
||||
+ \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} \\
|
||||
- \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
|
||||
\mathcal{J}(\tau;\theta),
|
||||
\end{multline*}
|
||||
|
||||
\begin{block}{Variational approximation}
|
||||
$\tau_{iq}^{1,m} = \mathcal{R}^1_{Y^m,\tau}(Z_{iq}^m = 1)$
|
||||
and $\tau_{jr}^{2,m} = \mathcal{R}^2_{Y^m,\tau}(W_{jr}^m = 1)$
|
||||
\end{block}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}{\emph{Variational Expectation} Step}
|
||||
\[
|
||||
\widehat{\tau}^{(t+1)} = \arg \max_{\tau}
|
||||
\mathcal{J}(\mathcal{\tau},\bm{\widehat{\theta}}^{(t)})
|
||||
\Leftrightarrow \arg\min_{\tau\in\mathcal{T}} \mathbf{KL}[\mathcal{R}_{\mathbf{Y},\tau}, \mathbb{P}(.|\mathbf{Y})]
|
||||
\]
|
||||
|
||||
\begin{equation*}
|
||||
\begin{cases}
|
||||
\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 \\
|
||||
\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
|
||||
\end{cases}
|
||||
\end{equation*}
|
||||
\footnotetext[2]{Initialization of $\widehat{\tau}$ with a
|
||||
\emph{spectral clustering} on the networks.}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}{\emph{Maximization} Step}
|
||||
\[
|
||||
\widehat{\theta}^{(t+1)} = \arg \max_{\theta} \mathcal{J}(\mathcal{\bm{\widehat{\tau}}}^{(t+1)},\theta)
|
||||
\]
|
||||
|
||||
\begin{block}{Connectivity parameters}
|
||||
\begin{align*}
|
||||
\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}}
|
||||
\end{align*}
|
||||
\end{block}
|
||||
\only<1>{
|
||||
\begin{block}{Proportions for \emph{iid}}
|
||||
\begin{align*}
|
||||
\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} & &
|
||||
\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}
|
||||
\end{align*}
|
||||
\end{block}
|
||||
}
|
||||
\only<2>{
|
||||
\begin{block}{Proportions for $\pi\rho$}
|
||||
\begin{align*}
|
||||
\widehat{\pi}^{\color{red}m}_q = \frac{\sum_{i=1}^{n_1^m} \tau_{iq}^{1,m}}{n_1^m} & &
|
||||
\widehat{\rho}^{\color{red}m}_r = \frac{\sum_{j=1}^{n_2^m} \tau_{jr}^{2,m}}{n_2^m}
|
||||
\end{align*}
|
||||
\end{block}
|
||||
}
|
||||
|
||||
\end{frame}
|
||||
|
||||
\section{Model selection}
|
||||
\begin{frame}
|
||||
\frametitle{Problem of choosing $(Q_1, Q_2)$}
|
||||
|
|
@ -307,69 +246,14 @@
|
|||
|
||||
\begin{alertblock}{Exploration problems}
|
||||
\begin{itemize}
|
||||
\item Exploration of $\mathbb{N}^2$ costly.
|
||||
\item Sensitivity to initializations.
|
||||
\item Exploration of $\mathbb{N}^2$ costly. \uncover<2->{$\rightarrow$ \textbf{Greedy
|
||||
approach} and \textbf{sliding window}}
|
||||
\item Sensitivity to initializations. \uncover<3->{$\rightarrow$ \textbf{Spectral
|
||||
clustering} and \textbf{reuse of previous inits}}
|
||||
\end{itemize}
|
||||
\end{alertblock}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{Choice of $(Q_1,Q_2)$ - Greedy approach}
|
||||
\begin{columns}
|
||||
\begin{column}{0.5\linewidth}
|
||||
\begin{tikzpicture}
|
||||
\input{tikz/greedy-exploration.tex}
|
||||
\end{tikzpicture}
|
||||
\end{column}
|
||||
\begin{column}{0.35\linewidth}
|
||||
\begin{itemize}
|
||||
\item Initial model~:\\
|
||||
\begin{tikzpicture}
|
||||
\draw[fill=gray, draw=gray] circle [radius=0.225cm];
|
||||
\end{tikzpicture}
|
||||
\onslide<2->{
|
||||
\item Model after \emph{split}~:
|
||||
\begin{tikzpicture}
|
||||
\draw[fill=blueind, draw=blueind] circle [radius=0.225cm];
|
||||
\end{tikzpicture}
|
||||
\item Model maximizing the criterion~:\\
|
||||
\begin{tikzpicture}
|
||||
\draw[fill=white, draw=green, very thick] circle [radius=0.225cm];
|
||||
\end{tikzpicture}
|
||||
}
|
||||
\onslide<3->{
|
||||
\item Model after \emph{merge}~:
|
||||
\begin{tikzpicture}
|
||||
\draw[fill=red, draw=red] circle [radius=0.225cm];
|
||||
\end{tikzpicture}
|
||||
}
|
||||
\end{itemize}
|
||||
\end{column}
|
||||
\end{columns}
|
||||
\end{frame}
|
||||
\begin{frame}
|
||||
\frametitle{Choice of $(Q_1,Q_2)$ - Sliding window}
|
||||
\begin{columns}
|
||||
\begin{column}{0.6\textwidth}
|
||||
\begin{figure}
|
||||
\input{tikz/moving-window}
|
||||
\caption{Sliding window}
|
||||
\end{figure}
|
||||
\end{column}
|
||||
\begin{column}{0.4\textwidth}
|
||||
\only<3>{\begin{block}{}
|
||||
Initialization of the model if necessary
|
||||
\end{block}}
|
||||
\only<9>{\begin{block}{}
|
||||
Localization of the new mode
|
||||
\end{block}}
|
||||
\only<10>{\begin{block}{}
|
||||
Move to the new mode then iterate
|
||||
\end{block}}
|
||||
\end{column}
|
||||
\end{columns}
|
||||
\end{frame}
|
||||
|
||||
\section{Application}
|
||||
\label{sec:application}
|
||||
|
||||
|
|
@ -432,6 +316,9 @@
|
|||
}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}
|
||||
TODO Interesting structures detected, functional roles in the british networks.
|
||||
\end{frame}
|
||||
\begin{frame}
|
||||
\frametitle{Network clustering}
|
||||
\begin{figure}[ht]
|
||||
|
|
@ -443,24 +330,6 @@
|
|||
\begin{frame}[allowframebreaks]
|
||||
\frametitle{Application to~\cite{baldockDailyTemporalStructure2011,
|
||||
baldockSystemsApproachReveals2019}}
|
||||
\begin{figure}[t]
|
||||
\centering
|
||||
\begin{subfigure}{0.5\textwidth}
|
||||
\centering
|
||||
\includegraphics[scale=0.2,angle=-90]{backup-app-iid.png}
|
||||
\caption{Model $iid$}
|
||||
\end{subfigure}%
|
||||
~
|
||||
\begin{subfigure}{0.5\textwidth}
|
||||
\centering
|
||||
\includegraphics[scale=0.2,angle=-90]{backup-app-pirho.png}
|
||||
\caption{Model $\pi\rho$}
|
||||
\end{subfigure}%
|
||||
\caption{Partitioning of networks
|
||||
of~\cite{baldockDailyTemporalStructure2011,
|
||||
baldockSystemsApproachReveals2019}}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[t]
|
||||
\centering
|
||||
\begin{subfigure}{0.5\textwidth}
|
||||
|
|
@ -483,17 +352,6 @@
|
|||
\end{figure}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}{Clustering algorithm}
|
||||
\centering
|
||||
\vspace{0.25\baselineskip}
|
||||
\begin{tikzpicture}[scale=0.85]
|
||||
\input{tikz/clustering.tex}
|
||||
\end{tikzpicture}
|
||||
\[
|
||||
D_{\mathcal{M}}(m,m') = \sum_{q = 1}^{Q_1} \sum_{r = 1}^{Q_2} \max(\widetilde{\pi}_{q}^{m}, \widetilde{\pi}_{q}^{m'}) \left( \widetilde{\alpha}_{qr}^{m} - \widetilde{\alpha}_{qr}^{m'}\right)^{2} \max(\widetilde{\rho}_{r}^{m}, \widetilde{\rho}_{r}^{m'})
|
||||
\]
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}{Results}
|
||||
\begin{figure}[ht]
|
||||
\centering
|
||||
|
|
@ -534,15 +392,15 @@
|
|||
\begin{frame}{Perspectives}
|
||||
\begin{itemize}
|
||||
\item Investigate stability against randomness and local \emph{optima}.
|
||||
\item Proof of identifiability of the $\pi\rho$ model.
|
||||
\end{itemize}
|
||||
|
||||
\begin{block}{Package and applications}
|
||||
\begin{itemize}
|
||||
\item Integration into the \texttt{colSBM} package, improvement of user interface and
|
||||
addition of ecologists' feedback
|
||||
\item CRAN publication
|
||||
\item Integrate the possibility of an additional criterion for clustering
|
||||
\item CRAN submission
|
||||
\item Integrate the possibility of an additional criterion for clustering (e.g.
|
||||
urbanization gradient~\cite{fisogniSeasonalTrajectoriesPlantpollinator2022})
|
||||
\item Apply clustering to data from
|
||||
\cite{pichonTellingMutualisticAntagonistic2024,doreRelativeEffectsAnthropogenic2021}
|
||||
\end{itemize}
|
||||
|
|
|
|||
|
|
@ -384,6 +384,25 @@
|
|||
file = {/home/louis/snap/zotero-snap/common/Zotero/storage/WRSY3FZV/Erdős et Rényi - 2022 - On random graphs. I..pdf}
|
||||
}
|
||||
|
||||
@article{fisogniSeasonalTrajectoriesPlantpollinator2022,
|
||||
title = {Seasonal Trajectories of Plant-Pollinator Interaction Networks Differ Following Phenological Mismatches along an Urbanization Gradient},
|
||||
author = {Fisogni, Alessandro and Hautekèete, Nina and Piquot, Yves and Brun, Marion and Vanappelghem, Cédric and Ohlmann, Marc and Franchomme, Magalie and Hinnewinkel, Christelle and Massol, François},
|
||||
date = {2022-10-01},
|
||||
journaltitle = {Landscape and Urban Planning},
|
||||
shortjournal = {Landscape and Urban Planning},
|
||||
volume = {226},
|
||||
pages = {104512},
|
||||
issn = {0169-2046},
|
||||
doi = {10.1016/j.landurbplan.2022.104512},
|
||||
url = {https://www.sciencedirect.com/science/article/pii/S016920462200161X},
|
||||
urldate = {2025-05-14},
|
||||
abstract = {Urbanization may significantly alter the abundance, composition and phenology of natural communities of plants and pollinators. However, how such alterations eventually affect the structure of plant-pollinator interaction networks is still poorly known. Here, we investigate how the structure of plant-pollinator networks changes along an urbanization gradient, which coincides with a phenological mismatch between plants and pollinators. We examined changes in plant-pollinator network structure in 12 sites sown with standardized native flower mixes along an urbanization gradient in a metropolis in Northern France. We used network-level metrics in combination with more detailed methodologies to identify changes in network structure, species clustering, and species roles through urban classes and time. We also evaluated the temporal trajectories of α- and β-diversity of species and interactions along the gradient. Network-level metrics showed limited spatial–temporal variability in the connectance, distribution of interactions and network-level specialization. Finer-scale analyses showed that generalist plant and pollinator species with long phenology were the most central and played key roles in defining the composition of cohesive groups of interacting species in all networks. Network motifs and species positions showed higher temporal variability in less urbanized areas, and interactions were more dissimilar between urbanization classes earlier in the season. We showed evidence of alterations in plant-pollinator network structure across space and time along an urbanization gradient, likely driven by the significant advancement in flowering phenology observed in the more urbanized areas. Our results emphasize the importance of targeted measures to maintain functional plant-pollinator communities, especially early in the season in highly urbanized areas.},
|
||||
keywords = {/unread},
|
||||
annotation = {Read\_Status: New\\
|
||||
Read\_Status\_Date: 2025-05-14T20:18:00.025Z},
|
||||
file = {/home/louis/snap/zotero-snap/common/Zotero/storage/CCJWEIBD/Fisogni et al. - 2022 - Seasonal trajectories of plant-pollinator interaction networks differ following phenological mismatc.pdf;/home/louis/snap/zotero-snap/common/Zotero/storage/HMUM8AMZ/S016920462200161X.html}
|
||||
}
|
||||
|
||||
@incollection{Frontmatter1990,
|
||||
title = {Frontmatter},
|
||||
booktitle = {Finding {{Groups}} in {{Data}}},
|
||||
|
|
|
|||
|
|
@ -5,22 +5,22 @@
|
|||
draw=blue,fill=yellow!50,text=blue]
|
||||
\tikzstyle{es}=[font=\small, text justified, rectangle,draw,rounded corners=4pt,fill=cyanind!25]
|
||||
|
||||
\node[es] (liste) at (0,4) {Donner une collection à partitionner};
|
||||
\node[instruct, text width=5cm, below = 0.45cm of liste] (1-collection) {Ajuster colBiSBM};
|
||||
\node[es] (liste) at (0,4) {Provide a collection to partition};
|
||||
\node[instruct, text width=5cm, below = 0.45cm of liste] (1-collection) {Fit colBiSBM};
|
||||
\node[first_col, right = 0.5cm of 1-collection] (1-col-obj) {};
|
||||
\node[instruct, text width=5cm, below = 0.45cm of 1-collection] (dissimi) {Calculer une matrice de dissimilarité de la collection};
|
||||
\node[instruct, text width=5cm, below = 0.45cm of dissimi] (2-sous-collection) {Séparer la \emph{collection en 2 sous-collections} et ajuster les colBiSBM};
|
||||
\node[instruct, text width=5cm, below = 0.45cm of 1-collection] (dissimi) {Compute a dissimilarity matrix of the collection};
|
||||
\node[instruct, text width=5cm, below = 0.45cm of dissimi] (2-sous-collection) {Split the \emph{collection into 2 sub-collections} and fit colBiSBMs};
|
||||
\node[second_col, right = 0.25cm of 2-sous-collection] (1-sec-col-obj) {1};
|
||||
\node[second_col, right = 0.25cm of 1-sec-col-obj] (1-sec-col-obj) {2};
|
||||
\node[test,below = 0.45cm of 2-sous-collection, scale=0.5] (BICL-test) {$\sum_{i=1}^{2} (\text{BIC-L}(\tikz[baseline=-0.25cm]{\node[second_col] {i};} )) > \text{BIC-L}(\tikz[baseline=-0.25cm]{\node[first_col] {};})$?};
|
||||
\node[es, right = 0.55cm of BICL-test] (sortie) {Renvoyer \tikz{\node[rectangle, draw, fill=gray!50, rounded corners=0pt] {};}};
|
||||
\node[es, left = 0.45cm of dissimi, text width = 2cm] (recursion) {Recommencer sur \tikz{\node[second_col] {1};} et \tikz{\node[second_col] {2};} };
|
||||
\node[es, right = 0.55cm of BICL-test] (sortie) {Return \tikz{\node[rectangle, draw, fill=gray!50, rounded corners=0pt] {};}};
|
||||
\node[es, left = 0.45cm of dissimi, text width = 2cm] (recursion) {Repeat on \tikz{\node[second_col] {1};} and \tikz{\node[second_col] {2};} };
|
||||
|
||||
\tikzstyle{suite}=[->,>=stealth,thick,rounded corners=4pt]
|
||||
\draw[suite] (liste) -- (1-collection);
|
||||
\draw[suite] (1-collection) -- (dissimi);
|
||||
\draw[suite] (dissimi) -- (2-sous-collection);
|
||||
\draw[suite] (2-sous-collection) -- (BICL-test);
|
||||
\draw[suite] (BICL-test) -| node[near start, above, fill=none] {Oui} (recursion);
|
||||
\draw[suite] (BICL-test) -| node[near start, above, fill=none] {Yes} (recursion);
|
||||
\draw[suite] (recursion.east) -- (dissimi.west);
|
||||
\draw[suite] (BICL-test) -- node[near start, above, fill=none] {Non} (sortie);
|
||||
\draw[suite] (BICL-test) -- node[near start, above, fill=none] {No} (sortie);
|
||||
|
|
|
|||
Loading…
Add table
Reference in a new issue