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-%--------------------------------------------------------------------------\r
-\r
-\r
-\title{Non supervised classification of individual electricity curves} \r
-\author{Jairo Cugliari}\r
-\institute{%Laboratoire ERIC, Université Lyon 2\r
-% \begin{center}\r
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-\begin{document}\r
-\r
-%--------------------------------------------------------------------------\r
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-\begin{frame}[plain, noframenumbering, b]\r
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-% \includegraphics[height = 1.5cm]{pics/logo_lyon2.jpg} \r
-% \end{center}\r
-\r
-\maketitle\r
-\r
- \begin{center}{\scriptsize \r
- Joint work with Benjamin Auder (LMO, Université Paris-Sud) }\r
- \end{center}\r
-\r
- % \begin{flushright}\r
-% \includegraphics[width = 0.15\textwidth]{pics/by-nc-sa.png} \r
-% \end{flushright}\r
- \r
-\end{frame}\r
-\r
-\r
-% \maketitle\r
-% \begin{center}{\scriptsize \r
-% Joint work with Benjamin Auder (LMO, Université Paris-Sud) }\r
-% \end{center}\r
-% \end{frame}\r
-\r
-%--------------------------------------------------------------------------\r
-\r
-\frame{\frametitle{Outline}\r
- \tableofcontents\r
-}\r
-\r
-%--------------------------------------------------------------------------\r
-\r
-\section{Motivation}\r
-\r
-\r
-\begin{frame}{Industrial motivation}\r
-\r
-\begin{columns}\r
-\column{0.6\textwidth}\r
-\begin{itemize}\r
- \item Smartgrid \& Smart meters : time real information\r
- \item Lot of data of different nature\r
- \item Many problems : transfer protocol, security, privacy, ...\r
- \item The French touch: 35M Linky smartmeter\r
-\end{itemize}\r
-\r
-\vskip 1cm\r
-\r
-What can we do with all these data ?\r
-\r
-\column{0.4\textwidth} \r
-\includegraphics[width = \textwidth]{./pics/smartgrid.jpg} \r
-\r
-\includegraphics[width = \textwidth]{./pics/linky.jpg} \r
-\end{columns}\r
-\end{frame}\r
-\r
-%--------------------------------------------------------------------------\r
-\r
-\begin{frame}{Electricity demand data}\r
-\framesubtitle{Some salient features}\r
-\r
-\begin{figure}[!ht] \centering\r
- \begin{subfigure}[t]{0.45\textwidth}\r
- \includegraphics[width=\textwidth]{pics/longtermload.png}\r
- \caption{Long term trand} %\label{fig:gull}\r
- \end{subfigure}%\r
- ~ %spacing between images\r
- \begin{subfigure}[t]{0.45\textwidth}\r
- \includegraphics[width=\textwidth]{pics/twoyearsload.png}\r
- \caption{Weekly cycle} % \label{fig:tiger}\r
- \end{subfigure}\r
- \r
- \begin{subfigure}[t]{0.45\textwidth}\r
- \includegraphics[width=\textwidth]{pics/dailyloads.png}\r
- \caption{Daily load curve} % \label{fig:mouse}\r
- \end{subfigure}\r
- ~ %spacing between images\r
- \begin{subfigure}[t]{0.45\textwidth}\r
- \includegraphics[width=\textwidth]{pics/consotemp.png}\r
- \caption{Electricity load vs. temperature}\r
- \end{subfigure}\r
-\end{figure}\r
-\end{frame}\r
-\r
-%--------------------------------------------------------------------------\r
-\r
-\begin{frame}[shrink]{FD as slices of a continuous process \r
- \begin{scriptsize} \hfill [Bosq, (1990)] \end{scriptsize}} \r
-% \r
- The prediction problem\r
-\r
-\begin{itemize}\r
- \item Suppose one observes a square integrable continuous-time stochastic process $X=(X(t), t\in\R)$ over the interval $[0,T]$, $T>0$;\r
- \item {We want to predict $X$ all over the segment $[T, T+\delta], \delta>0$}\r
- \item {Divide the interval into $n$ subintervals of equal\r
- size $\delta$.}\r
- \item Consider the functional-valued discrete time stochastic process $ Z = (Z_k, k\in\N) $, where $ \mathbb{N} = \set{ 1,2,\ldots } $, defined by \r
-\end{itemize}\r
- \r
-\begin{columns}\r
- \column{5cm} \r
- \input{tikz/axis2}\r
- \column{5cm} \r
- \[ Z_k(t) = X(t + (k-1)\delta) \]\r
- \[ k\in\N \;\;\; \forall t \in [0,\delta) \]\r
-\end{columns}\r
-\r
-\vfill\r
- If $X$ contents a $\delta-$seasonal component, \r
- $Z$ is particularly fruitful.\r
-\r
-\end{frame}\r
-\r
-%--------------------------------------------------------------------------\r
-\r
-\begin{frame}{Long term objective}\r
-\r
-\begin{columns}\r
-\column{.6\textwidth}\r
-%\begin{figure}[!ht]\centering\r
- \includegraphics[width = \textwidth]{pics/schema.png} \r
-%\caption{Hierarchical structure of $N$ individual clients among $K$ groups.}\label{fig:schema-hier}\r
-%\end{figure}\r
- \r
-\column{.4\textwidth}\r
-\begin{tikzpicture}[decoration=penciline, decorate]\r
- \node[block, decorate] at (0, 0){$Z_t$} ;\r
- \node[block, decorate] at (3, 0) {$Z_{t + 1}$} ;\r
-\r
- \node[block, decorate] at (0, -2.5) {$\begin{pmatrix}\r
- Z_{t, 1} \\ Z_{t, 2} \\ \vdots \\ Z_{t, K}\r
- \end{pmatrix}$ };\r
-\r
- \node[block, decorate] at (3, -2.5) {$\begin{pmatrix}\r
- Z_{t+1, 1} \\ Z_{t+1, 2} \\ \vdots \\ Z_{t+1, k}\r
- \end{pmatrix} $};\r
-\r
- \draw[decorate, darkblue, line width = 2mm, ->] (1, 0) -- (2, 0);\r
- \draw[decorate, darkgreen, line width = 2mm, ->] (1, -2.5) -- (2, -2.5);\r
- \draw[decorate, black, line width = 2mm, ->] (3, -1.3) -- (3, -0.4);\r
- \draw[decorate, darkred, line width = 2mm, ->] (1, -1.5) -- (2, -0.75);\r
- \end{tikzpicture}\r
-\end{columns}\r
-\r
-\begin{itemize}\r
- \item Groups can express tariffs, geographical dispersion, client class ...\r
- \item \textbf{IDEA}: Use a clustering algorithm to learn groups of customer structure\r
- \item \textbf{Aim}: Set up a classical clustering algorithm to run in parallel \r
-\end{itemize}\r
-\end{frame}\r
-\r
-%--------------------------------------------------------------------------\r
-\r
-\section{Functional clustering}\r
-\r
-\begin{frame}{Aim}\r
-\r
-\begin{columns}\r
- \column{0.6\textwidth}\r
- \begin{block}{ }\r
- \begin{itemize}\r
- \item Segmentation of $X$ may not suffices to render reasonable \r
- the stationary hypothesis.\r
- \item If a grouping effect exists, we may considered stationary within each group. \r
- \item Conditionally on the grouping, functional time series prediction methods \r
- can be applied.\r
- \item We propose a clustering procedure that discover the groups from a bunch\r
- of curves.\r
- \end{itemize}\r
-\r
- We use wavelet transforms to take into account the fact \r
- that curves may present non stationary patters.\r
- \end{block}\r
-\r
- \column{0.4\textwidth}\r
- \includegraphics[width=0.9\textwidth,\r
- height=2.7cm]{pics/conso-traj.png}\r
-\r
- Two strategies to cluster functional time series:\r
- \begin{enumerate}\r
- \item Feature extraction (summary measures of the curves).\r
- \item Direct similarity between curves.\r
- \end{enumerate} \r
-\r
-\end{columns}\r
-\end{frame}\r
-\r
-%---------------------------\r
-\r
-\begin{frame}[plain]{Wavelets to cope with \textsc{fd}}\r
-\r
-\begin{columns}\r
- \column{.6\textwidth}\r
- %\begin{figure}\r
- \centering\r
- \includegraphics[width = \textwidth]{./pics/weekly-5.png}\r
- % * * * * * * * * * * * * * * * * * * *\r
- \column{.4\textwidth}\r
-\begin{block}{ } %Wavelet transform}\r
-\begin{footnotesize}\r
-\begin{itemize}\r
- \item domain-transform technique for hierarchical decomposing finite energy signals\r
- \item description in terms of a broad trend (\textcolor{PineGreen}{approximation part}), plus a set of localized changes kept in the \textcolor{red}{details parts}.\r
-\end{itemize}\r
-\end{footnotesize}\r
-\end{block}\r
-\end{columns}\r
-\r
-\begin{block}{Discrete Wavelet Transform }\r
-\r
- If $z \in L_2([0, 1])$ we can write it as\r
-\r
- \begin{equation*}\label{eq:zeta}\r
- z(t) = \sum_{k=0}^{2^{j_0}-1} \textcolor{PineGreen}{c_{j_0, k}} \phi_{j_0,k} (t) + \r
- \sum_{j={j_0}}^{\infty} \r
- \sum_{k=0}^{2^j-1} \textcolor{red}{d_{j,k}} \psi_{j,k} (t) ,\r
- \end{equation*}\r
-\r
-%\r
-where $ c_{j,k} = <g, \phi_{j,k} > $, $ d_{j,k} = <g, \varphi_{j,k}>$ are the \r
-\textcolor{PineGreen}{scale coefficients} and \textcolor{red}{wavelet coefficients} respectively, and the functions $\phi$ et $\varphi$ are associated to a orthogonal \textsc{mra} of $L_2([0, 1])$.\r
-\end{block}\r
-\end{frame}\r
-\r
-%---------------------------------------- SLIDE ---------------------\r
-\r
-\begin{frame}{Energy decomposition of the DWT}\r
-\r
-\begin{block}{ }\r
- \begin{itemize}\r
- \item Energy conservation of the signal\r
-%\r
- \begin{equation*}\label{eq:energy} \r
- \| z \|_H^2 \approx \| \widetilde{z_J} \|_2^2 \r
- = c_{0,0}^2 + \sum_{j=0}^{J-1} \sum_{k=0}^{2^j-1} d_{j,k} ^2 = \r
- c_{0,0}^2 + \sum_{j=0}^{J-1} \| \mathbf{d}_{j} \|_2^2.\r
- \end{equation*}\r
-% \item characterization by the set of channel variances estimated at the output of the corresponding filter bank\r
- \item For each $j=0,1,\ldots,J-1$, we compute the absolute and \r
- relative contribution representations by\r
-% \r
- \[ \underbrace{\hbox{cont}_j = ||\mathbf{d_j}||^2}_{\fbox{AC}} \r
- \qquad \text{and} \qquad\r
- \underbrace{\hbox{rel}_j = \r
- \frac{||\mathbf{d_j}||^2}\r
- {\sum_j ||\mathbf{d_j}||^2 }}_{\fbox{RC}} .\]\r
- \item They quantify the relative importance of the scales to the global dynamic.\r
-% \item Only the wavelet coefficients $\set{d_{j,k}}$ are used.\r
- \item RC normalizes the energy of each signal to 1.\r
-\end{itemize}\r
-\end{block}\r
-\end{frame}\r
-% =======================================\r
-\r
-\begin{frame} \r
- \frametitle{Schema of procedure}\r
- \begin{center}\r
- \includegraphics[width = 7cm, height = 2cm]{./pics/Diagramme1.png}\r
- % Diagramme1.png: 751x260 pixel, 72dpi, 26.49x9.17 cm, bb=0 0 751 260\r
- \end{center}\r
- \r
- \begin{footnotesize}\r
-\begin{description}\r
- \item [0. Data preprocessing.] Approximate sample paths of $z_1(t),\ldots,z_n(t)$ %by the truncated wavelet series at the scale $J$ from sampled data $\mathbf{z}_1, \ldots, \mathbf{z}_n$.\r
- \item [1. Feature extraction.] Compute either of the energetic components using absolute contribution (AC) or relative contribution (RC).\r
- \item [2. Feature selection.] Screen irrelevant variables. \begin{tiny} [Steinley \& Brusco ('06)]\end{tiny}\r
- \item [3. Determine the number of clusters.] Detecting significant jumps in the transformed distortion curve.\r
- \begin{tiny} [Sugar \& James ('03)]\end{tiny}\r
- \item [4. Clustering.] Obtain the $K$ clusters using PAM algorithm.\r
-\end{description} \end{footnotesize}\r
- \r
-\footnotetext[1]{Antoniadis, X. Brossat, J. Cugliari et J.-M. Poggi (2013), Clustering Functional Data Using Wavelets, {\it IJWMIP}, 11(1), 35--64}\r
- \r
-\end{frame}\r
-\r
-% ===========================================\r
-\r
-\section{Parallel $k$-medoids}\r
-\r
-\begin{frame}{Partitioning Around Medoids (PAM)\r
- \begin{scriptsize} \hfill [Kaufman et Rousseeuw~(1987)] \end{scriptsize}}\r
-\r
-\begin{itemize}\r
- \item Partition the $n$ points $R^d$-scatter into $K$ clusters\r
- \item Optimization problem :\r
- \[ D(x) = \min_{m_1,\dots,m_k \in \mathbb{R}^d} \sum_{i=1}^{n} \min_{j=1,\dots,k} \| x_i - m_j \| \, ,\]\r
-with $x = (x_1,\dots,x_n)$, $\|\,.\,\|$ can be any norm. Here we choose to use the euclidean norm. \r
- \item Robust version of $k$-means\r
- \item Computational burden : medians instead of means\r
- \item Several heuristics allow to reduce the computation time.\r
-\end{itemize}\r
-\end{frame}\r
-\r
-% ===========================================\r
-\r
-\begin{frame}{Parallelization with MPI}\r
-\r
-\begin{columns}\r
-\column{.8\textwidth}\r
-\begin{itemize}\r
- \item Easy to use library routines allowing to write algorithms in parallel\r
- \item Available on several languages \r
- \item We use the master-slave mode\r
-\end{itemize}\r
-\r
-\column{.2\textwidth}\r
-\includegraphics[width=\textwidth]{./pics/open-mpi-logo.png} \r
-\end{columns}\r
-\r
-\vfill\r
-\r
-\begin{block}{The outline of code:}\r
-\begin{enumerate}\r
- \item The master process splits the problem in tasks over the data set and sends it to the workers;\r
- \item Each worker reduces the functional nature of the data using the DWT, applies the clustering and returns the centers;\r
- \item The master recuperates and clusters the centers into $K$ meta centers. \r
-\end{enumerate}\r
-\end{block}\r
-\r
-The source code is open and will be available to download from \r
-\href{https://github.com/}{github}.\r
-\r
-\footnotetext[1]{B. Auder \& J. Cugliari. Parallélisation de l'algorithme des $k$-médoïdes. Application au clustering de courbes. (2014, submitted)}\r
-\end{frame}\r
-\r
-\section{Numerical experiences}\r
-\r
-% ===========================================\r
-\r
-\begin{frame}{Application I: Starlight curves}\r
-\r
-\begin{itemize}\r
- \item Data from UCR Time Series Classification/Clustering\r
- \item 1000 curves learning set + 8236 validation set ($d= 1024$)% discretization points\r
-\end{itemize}\r
-\r
-\begin{figure}[H]\r
-\begin{minipage}[c]{.32\linewidth}\r
- \includegraphics[width=\linewidth,height=3.5cm]{pics/slgr1.png}\r
- %\vspace*{-0.3cm}\r
- \caption{Groupe 1}\r
-\end{minipage}\r
-\begin{minipage}[c]{.32\linewidth}\r
- \includegraphics[width=\linewidth,height=3.5cm]{pics/slgr2.png}\r
- %\vspace*{-0.3cm}\r
- \caption{Groupe 2}\r
-\end{minipage}\r
-\begin{minipage}[c]{.32\linewidth}\r
- \includegraphics[width=\linewidth,height=3.5cm]{pics/slgr3.png}\r
- %\vspace*{-0.3cm}\r
- \caption{Groupe 3}\r
-\end{minipage}\r
-\label{figsltr3clusts}\r
-\end{figure}\r
-\r
-\begin{table}[H]\r
-\centering\r
-\begin{tabular}{lccc} \toprule\r
- & & \multicolumn{2}{c}{Adequacy} \\\r
- & Distortion & Internal & External \\ \midrule\r
-Training (sequential) & 1.31e4 & 0.79 & 0.77 \\\r
-Training (parallel) & 1.40e4 & 0.79 & 0.68 \\\r
-Test (sequential) & 1.09e5 & 0.78 & 0.76 \\\r
-Test (parallel) & 1.15e5 & 0.78 & 0.69 \\ \bottomrule\r
-\end{tabular}\r
-%\caption{Distorsions et indices d'adéquation des partitions}\r
-\label{tabDistorSl}\r
-\end{table}\r
-\end{frame}\r
-\r
-% ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++\r
-\r
-\begin{frame}{Application II: EDF data}\r
- \begin{figure}\r
- \centering\r
- \includegraphics[width= 0.9\textwidth]{pics/conso-shapes.png}\r
- % conso-traj.eps: 0x0 pixel, 300dpi, 0.00x0.00 cm, bb=18 18 577 824\r
- \caption{ \begin{footnotesize}\r
-French electricity power demand on autumn (top left), winter (bottom left), spring (top right) and summer (bottom right). \end{footnotesize} }\r
- \label{fig:conso-shapes}\r
- \end{figure}\r
- \r
- \begin{footnotesize}\r
- Feature extraction:\r
- \begin{itemize}\r
- \item The significant scales for revealing the cluster structure are independent of the possible number of clusters.\r
- \item Significant scales are associated to mid-frequencies. \r
- \item The retained scales parametrize the represented cycles of 1.5, 3 and 6 hours (AC). \r
- \end{itemize} \end{footnotesize}\r
-\end{frame}\r
-\r
-\r
-% ===========================================\r
-\r
-\begin{frame}\r
-\begin{figure}\r
- \centering\r
- \includegraphics[width= 0.9\textwidth]{./pics/conso_jump_AC.png} \\\r
- \caption{ \begin{footnotesize}\r
-Number of clusters by feature extraction of the AC (top). From left to right: distortion curve, transformed distortion curve and first difference on the transformed distortion curve. \end{footnotesize} }\r
- \label{fig:conso-jumps}\r
-\end{figure}\r
- \end{frame}\r
-\r
-% ===========================================\r
-\r
-\begin{frame}\r
-\begin{figure} \centering\r
- \begin{subfigure}[t]{0.45\textwidth}\r
- \includegraphics[width=\textwidth]{./pics/conso_AC-curves.png}\r
- \caption{Cluster}\r
- \end{subfigure}\r
- ~ \r
- \begin{subfigure}[t]{0.45\textwidth}\r
- \includegraphics[width=\textwidth]{./pics/conso_AC-calendar.png}\r
- \caption{Calendar}\r
- \end{subfigure}\r
-% \subfloat[Calendar]{\label{fig:conso_clust_AC_cal}\r
-% \includegraphics[width = 0.45\textwidth]{./pics/conso_AC-calendar.png}} \r
-\caption{Curves membership of the clustering using AC based dissimilarity (a) and the corresponding calendar positioning (b).}\r
- \end{figure}\r
-\end{frame}\r
-\r
-\r
-% ===========================================\r
-\r
-\r
-\begin{frame}{Application III: Electricity Smart Meter CBT (ISSDA)} \small\r
-\r
-\footnotetext[1]{\textit{Irish Social Science Data Archive}, \url{http://www.ucd.ie/issda/data/}}\r
-\r
-\begin{itemize}\r
- \item 4621 Irish households smart meter data % eséries de consommation électrique de foyers irlandais\r
- \item About 25K discretization points \r
- \item We test with $K=$ 3 or 5 classes\r
- \item We compare sequential and parallel versions \r
-\end{itemize}\r
-\r
-\r
-\begin{table}[H]\r
-\centering\r
-\begin{tabular}{lcc} \toprule\r
-% & & \\\r
- & Distortion & Internal adequacy \\ \midrule\r
-3 clusters sequential & 1.90e7 & 0.90 \\\r
-3 clusters parallel & 2.15e7 & 0.90 \\\r
-5 clusters sequential & 1.61e7 & 0.89 \\\r
-5 clusters parallel & 1.84e7 & 0.89 \\ \bottomrule\r
-\end{tabular}\r
-% \caption{Distorsions et indices d'adéquation des partitions}\r
-\label{tabDistorIr}\r
-\end{table}\r
-\r
-\end{frame}\r
-\r
-%--------------------------------------------------------------------------\r
-\r
-\section{Conclusion}\r
-\r
-\begin{frame}{Conclusion}\r
-\r
-\begin{itemize}\r
- \item Identification of customers groups from smartmeter data\r
- \item Wavelets allow to capture the functional nature of the data\r
- \item Clustering algorithm upscale envisaged for millions of curves\r
- \item \textit{Divide-and-Conquer} approach thanks to MPI library %pour l'algorithme des $k$-médoïdes : d'abord sur des groupes de données courbes, puis des groupes de médoïdes jusqu'à obtenir un seul ensemble traité sur un processseur.\r
- %\item %Les résultats obtenus sur les deux jeux de données présentés sont assez encourageants, et permettent d'envisager une utilisation à plus grande échelle.\r
-\end{itemize}\r
-\r
-\begin{block}{Further work}\r
-\begin{itemize}\r
- \item Go back to the prediction task\r
- \item Apply the algorithm over many hundreds of processors \r
- \item Connect the clustering method with a prediction model\r
-\end{itemize}\r
-\end{block}\r
-\end{frame}\r
-\r
-%--------------------------------------------------------------------------\r
-\r
-\begin{frame}[plain]{Bibliographie}\small\r
-\r
-\begin{thebibliography}{10}\r
-\bibitem{1} A. Antoniadis, X. Brossat, J. Cugliari et J.-M. Poggi (2013), Clustering Functional Data Using Wavelets, {\it IJWMIP}, 11(1), 35--64\r
-\r
-\bibitem{2} R. Bekkerman, M. Bilenko et J. Langford - éditeurs (2011), Scaling up Machine Learning: Parallel and Distributed Approaches, {\it Cambridge University Press}\r
-\r
-\bibitem{3} P. Berkhin (2006), A Survey of Clustering Data Mining Techniques, {\it Grouping Multidimensional Data, éditeurs : J. Kogan, C. Nicholas, M. Teboulle}.\r
-\r
-\bibitem{6} J. Dean et S. Ghemawat (2004), MapReduce: Simplified Data Processing on Large Clusters, {\it Sixth Symposium on Operating System Design and Implementation}.\r
-\r
-\bibitem{7} G. De Francisci Morales et A. Bifet (2013), G. De Francisci Morales SAMOA: A Platform for Mining Big Data Streams Keynote Talk at RAMSS ’13: 2nd International Workshop on Real-Time Analysis and Mining of Social Streams WWW, Rio De Janeiro\r
-\r
-\bibitem{10} L. Kaufman et P.J. Rousseeuw (1987), Clustering by means of Medoids, {\it Statistical Data Analysis Based on the L\_1-Norm and Related Methods, éditeur : Y. Dodge}.\r
-\end{thebibliography}\r
-\end{frame}\r
-\r
-\r
-\end{document}\r
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-\r
-% \begin{frame}{Motivation académique: Big Data} \r
-% \begin{itemize}\r
-% \item Besoins spécifiques: très grands volumes de données, grande dimension\r
-% \item Réponses: algorithmes opérant sur de grands graphes (Kang et al.~2009), sur des flux de données haut débit (De Francisci Morales et Bifet~2013)\r
-% \item Bekkerman et al.~(2011): algorithmes de Machine Learning s'exécutant en parallèle \r
-% \end{itemize}\r
-% \r
-% \begin{itemize}\r
-% \item classification non supervisée (\textit{clustering}): regrouper les données en \textit{clusters} homogènes, suffisamment distincts deux à deux\r
-% \item nombreux algorithmes depuis Tyron~(1939) (voir Berkhin~2006 pour une revue) \r
-% \item cependant la notion de cluster varie en fonction des données, du contexte et de l'algorithme utilisé\r
-% \item technique très populaire qui permet \r
-% de réduire la taille des données en les résumant à quelques représentants \r
-% \end{itemize}\r
-% \end{frame}\r
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projects / ppam-mpi.git / blobdiff