141 lines
4.9 KiB
TeX
141 lines
4.9 KiB
TeX
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pdftitle={Netclustering analysis with the CoOPLBM completion},
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{\par}
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\title{Netclustering analysis with the CoOPLBM completion}
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\author{}
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\date{\vspace{-2.5em}}
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\begin{document}
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\maketitle
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\hypertarget{context-of-this-analysis}{%
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\section{Context of this analysis}\label{context-of-this-analysis}}
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After performing a netclustering on the raw data, we will see if the
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detect structure resulting in the clustering comes from the sampling
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effort. To test this we will use the CoOPLBM model by Anakok et al.
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(2022) to complete the data.
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The CoOPLBM model assumes that the observed incidence matrix \(R\) is an
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element-wise product of an \(M\) matrix following an LBM and an \(N\)
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matrix which elements follow Poisson distributions independent on \(M\).
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The model gives us the \(\hat{M}\) matrix, which elements are:
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\begin{itemize}
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\tightlist
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\item
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1 if the interaction was observed
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\item
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a probability, that there should be an interaction but it wasn't
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observed
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\end{itemize}
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This \emph{completed matrix} can be used in different manners to be fed
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to the colSBM model.
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\hypertarget{threshold-based-completions}{%
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\section{Threshold based
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completions}\label{threshold-based-completions}}
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With the thresholds, the infered incidence matrix obtained by CoOPLBM is
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used to generate a completed incidence matrix by the following procedure
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: \[X_{ij} = \begin{cases}
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1 & \text{if the value is over the threshold} \\
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0 & \text{else} \\
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\end{cases}\]
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\hypertarget{completed-threshold}{%
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\subsection{0.5 completed threshold}\label{completed-threshold}}
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Here, the completion threshold is set to \(0.5\).
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\hypertarget{ari-of-networks-clustering-0.5-threshold-vs-raw-data}{%
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\subsubsection{ARI of networks clustering: 0.5 threshold vs raw
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data}\label{ari-of-networks-clustering-0.5-threshold-vs-raw-data}}
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\hypertarget{sample-based-completions}{%
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\section{Sample based completions}\label{sample-based-completions}}
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The \(M\) matrix is used to sample a new \(X\) matrix which elements are
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the realisation of Bernoulli distributions of probability \(M_{i,j}\).
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\[\mathbb{P}(X_{i,j} = 1) = M_{i,j} \]
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\hypertarget{references}{%
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\section*{References}\label{references}}
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\addcontentsline{toc}{section}{References}
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\hypertarget{refs}{}
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\begin{cslreferences}
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\leavevmode\hypertarget{ref-anakokDisentanglingStructureEcological2022}{}%
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Anakok, Emre, Pierre Barbillon, Colin Fontaine, and Elisa Thebault.
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2022. ``Disentangling the Structure of Ecological Bipartite Networks
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from Observation Processes.'' arXiv.
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\url{http://arxiv.org/abs/2211.16364}.
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\end{cslreferences}
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\end{document}
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