fix accent (TODO: fix gitweb...)

[ppam-mpi.git] / communication / slides / 201402-JClust.tex

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%--------------------------------------------------------------------------\r
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\title{Non supervised classification of individual electricity curves} \r
\author{Jairo Cugliari}\r
\institute{%Laboratoire ERIC, Université Lyon 2\r
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  \begin{center}{\scriptsize \r
    Joint work with Benjamin Auder (LMO, Université Paris-Sud) }\r
  \end{center}\r
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%     Joint work with Benjamin Auder (LMO, Université Paris-Sud) }\r
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%--------------------------------------------------------------------------\r
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\frame{\frametitle{Outline}\r
       \tableofcontents\r
}\r
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%--------------------------------------------------------------------------\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
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  \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
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% \begin{frame}{Motivation académique: Big Data} \r
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%  \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
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% \r
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%  \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
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