From: Benjamin Auder Date: Mon, 2 Feb 2015 12:16:29 +0000 (+0100) Subject: remove source latex file from doc X-Git-Url: https://git.auder.net/assets/doc/html/index.html?a=commitdiff_plain;h=ef3c3e248b53f93329e0482e616f1b92281bdd9e;p=synclust.git remove source latex file from doc --- diff --git a/.gitignore b/.gitignore index b5f5640..671f166 100644 --- a/.gitignore +++ b/.gitignore @@ -1,3 +1,4 @@ +/sources_latex/ .Rhistory .RData *.o diff --git a/inst/doc/convex_optimization.tex b/inst/doc/convex_optimization.tex deleted file mode 100644 index 7cf7a08..0000000 --- a/inst/doc/convex_optimization.tex +++ /dev/null @@ -1,83 +0,0 @@ -\documentclass{article} -\usepackage{a4wide} -\usepackage{graphicx} -\def\pen{\textrm{pen}} -\def\L{\mathcal{L}} -\title{\bf Detecting areas with synchronous temporal dynamics} -\author{Christophe Giraud} - -\begin{document} -\maketitle - -\noindent This document summarizes the algorithm used when function \textit{findSyncVarRegions()} is called with first argument method=``convex''. Reading first the article \emph{Delimiting synchronous populations from monitoring data} by Giraud et al. is recommanded, since we use here the same notations. - -\section{Model and estimation procedure} - -\subsection{Goal} - -We write $Z_{stk}$ for the $k$th observations, year $t$, site $s$ and $z_{st}=\sum_{k}Z_{stk}$. -Our goal is to estimate regions $R$ such that -\begin{equation}\label{model} -Z_{stk}\sim \textrm{Poisson}(\exp(\theta_{s}+f(x_{s},t)))\quad\textrm{with } f(x,t)\approx \sum_{R}\rho_{R}(t){\bf 1}_{x\in R}. -\end{equation} - In other words, we try to estimate $f$ with the a priori that -\begin{itemize} -\item for each year $t$ the map $x \to f(x,t)$ is piecewise constant -\item the boundary of the regions where $x \to f(x,t)$ is constant are the same for all year $t$. -\end{itemize} -The main difficulty is to detect the regions $R$. - -\subsection{Estimation procedure} -Let $G$ be a graph and write $V(s)$ for the set of the neighbors of $s$ in G. -The estimators $\widehat \theta$ and $\widehat f$ are defined as minimizers of -$$\mathcal{L}(\theta,f)+\alpha \pen(f):=\sum_{s,t}[e^{\theta_{s}+f_{st}}-z_{st}(\theta_{s}+f_{st})]+\alpha -\sum_{s\stackrel{G}{\sim}u}\|f_{s.}-f_{u.}\|/D_{su}$$ -with boundary conditions: $f_{s1}=0$ for all $s$. We typically choose $D_{su}=1/|V(s)|+1/|V(u)|$. - -\section{Optimization algorithm} - -The following quantity is to be minimized -$$\mathcal{L}(\theta,f)+\alpha \pen(f):=\sum_{s,t}[e^{\theta_{s}+f_{st}}-z_{st}(\theta_{s}+f_{st})]+\alpha\sum_{s\stackrel{G}{\sim}u}\|f_{s.}-f_{u.}\|/D_{su}$$ -with boundary conditions: $f_{s1}=0$ for all $s$. -This last expression can be rewritten into -$$\mathcal{L}(\theta,f)+\alpha \pen(f)=\sum_{s,t}[e^{\theta_{s}+f_{st}}-z_{st}(\theta_{s}+f_{st})]+\alpha -\sum_{s\stackrel{G}{\sim}u}\max_{\|\phi_{su}\|\leq 1}\langle\phi_{su},f_{s.}-f_{u.}\rangle/D_{su}$$ -with $\phi_{su}\in\mathbf R^T$. - -\newpage -\noindent Let us introduce -$$F(\theta,f,\phi)=\sum_{s,t}[e^{\theta_{s}+f_{st}}-z_{st}(\theta_{s}+f_{st})]+\alpha -\sum_{su$ -%$$\partial_{f_{st}}F=e^{\theta_{s}}e^{f_{st}}-z_{st}+\alpha\sum_{u\in V(s)}\phi_{su}/D_{su}$$ -% -% -%\subsection*{Gradient en $\lambda$:} -%pour $s