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\author{B. Auder \and
	    J. Cugliari \and
	    Y. Goude   \and
    	J.-M. Poggi
}
\title{Disaggregated Electricity Forecasting using Clustering of 
	   Individual Consumers}
\subtitle{Réunion mi parcours}
%\logo{}
\institute{IRSDI - RESEARCH INITIATIVE IN INDUSTRIAL DATA SCIENCE}
\date{19 janvier 2017}
%\subject{tito}

\begin{document}
	
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%\maketitle


\section{IRSDI follow up meeting}	

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\begin{frame}{The project in a nutshell}
\begin{block}{Context}
\begin{itemize}
\item 
Industrial : Electricity load forecasting \& smart grids infrastructure 
\item 
Academic : curve's shape \& nonparametric function-valued forecast 
\item 
Past work : clustering with wavelets (RC, Wer), KWF, Enercon 
\end{itemize}
\end{block}

\begin{block}{Aims}
\begin{itemize}
\item evaluate the upscaling capacity of the Energycon strategy 
\item adapt KWF to an exogenous variable (e.g. meteorological)
\end{itemize}
\end{block}
\end{frame}

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\begin{frame}{Clients hierarchical structure and prediction}

\begin{columns}
\column{.6\textwidth}
\begin{figure}[!ht]\centering
 \includegraphics[width = \textwidth]{pics/schema.png} 
\caption{Hierarchical structure of $N$ individual clients among $K$ 
groups.}\label{fig:schema-hier}
\end{figure}
 
\column{.4\textwidth}
\begin{tikzpicture}[decoration=penciline, decorate]
  \node[block, decorate] at (0, 0){$Z_t$} ;
  \node[block, decorate] at (3, 0) {$Z_{t + 1}$} ;

  \node[block, decorate] at (0, -2.5) {$\begin{pmatrix}
                              Z_{t, 1} \\ Z_{t, 2} \\ \vdots \\ Z_{t, K}
                               \end{pmatrix}$ };

  \node[block, decorate] at (3, -2.5) {$\begin{pmatrix}
                           Z_{t+1, 1} \\ Z_{t+1, 2} \\ \vdots \\ Z_{t+1, k}
                               \end{pmatrix} $};

  \draw[decorate, darkblue,  line width = 2mm, ->] (1, 0) -- (2, 0);
  \draw[decorate, darkgreen, line width = 2mm, ->] (1, -2.5) -- (2, -2.5);
  \draw[decorate, black,     line width = 2mm, ->] (3, -1.3) -- (3, -0.4);
  \draw[decorate, darkred,   line width = 2mm, ->] (1, -1.5) -- (2, -0.75);
 \end{tikzpicture}
\end{columns}

\begin{itemize}
 \item $Z_t$: aggregate demand at $t$
 \hfill $Z_{t, k}$:demand of group $k$ at moment $t$
 \item Groups can express tariffs, geographical dispersion, client class ...
 \item Profiling vs Prediction 
 \item We follow Misiti \textit{et al}. (2010) to construct classes of customers to better predict the aggregate.
\end{itemize}
\end{frame}


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\begin{frame}{Expected result}

\includegraphics[width = \textwidth]{pics/perf.pdf}
\end{frame}

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\begin{frame}{Energy decomposition of the DWT}

%\begin{block}{ }
\begin{itemize}
\item Energy conservation of the signal
%
  \begin{equation*}\label{eq:energy}  
     \| z\|^2_H    \approx     \| \widetilde{z_J} \|_2^2 
        = c_{0,0}^2 + \sum_{j=0}^{J-1} \sum_{k=0}^{2^j-1} d_{j,k} ^2  = 
                     c_{0,0}^2 + \sum_{j=0}^{J-1} \| \mathbf{d}_{j} \|_2^2.
  \end{equation*}
%  \item characterization by the set of channel variances estimated at the output of the corresponding filter bank
 \item For each $j=0,1,\ldots,J-1$, we compute the \textcolor{blue}{absolute} and 
 \textcolor{orange}{relative} contribution representations by
%      
   \[ \underbrace{\hbox{cont}_j = ||\mathbf{d_j}||^2}_{\fbox{\textcolor{blue}{AC}}}  
      \qquad  \text{and}  \qquad
       \underbrace{\hbox{rel}_j  = 
     \frac{||\mathbf{d_j}||^2}
          {\sum_j ||\mathbf{d_j}||^2 }}_{\fbox{\textcolor{orange}{RC}}} .\]
 %\item They quantify the relative importance of the scales to the global dynamic.
% \item Only the wavelet coefficients $\set{d_{j,k}}$ are used.
% \item RC normalizes the energy of each signal to 1.
\end{itemize}
%\end{block}
%\end{frame}

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%\begin{frame} 
%  \frametitle{Schema of procedure}
  \begin{center}
   \includegraphics[width = 7cm, height = 2cm]{./pics/Diagramme1.png}
   % Diagramme1.png: 751x260 pixel, 72dpi, 26.49x9.17 cm, bb=0 0 751 260
  \end{center}
      
        \begin{footnotesize}
       \begin{description}
 \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 $\mathbf{z}_1, \ldots, \mathbf{z}_n$.
 \item [1. Feature extraction.] Compute either of the energetic components using absolute contribution (AC) or relative contribution (RC).
 \item [2. Feature selection.] Screen irrelevant variables. \begin{tiny} [Steinley \& Brusco ('06)]\end{tiny}
 %\item [3. Determine the number of clusters.] Detecting significant jumps %in the transformed distortion curve.
 %\begin{tiny} [Sugar \& James ('03)]\end{tiny}
 %\item [4. Clustering.] Obtain the $K$ clusters using PAM algorithm.
      \end{description}       \end{footnotesize}
    
 \end{frame}

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\begin{frame} 
  \frametitle{A function-based distance}

\begin{columns}
\column{0.6\textwidth}
  \begin{itemize}
    \item Distance based on wavelet-correlation between two time series
    \item Can be used to measure relationship between two functions
    %variables, i.e. temperature and load.
    \item The strength of the relation is hierarchically decomposed across
          scales without losing of time location
   \end{itemize}    

   Drawback: needs more computation time and storage (complex values) 
\column{0.4\textwidth}
  \includegraphics[width  = \textwidth]{pics/conso-week.png} 
                   
  \includegraphics[width  = .96\textwidth]{pics/wsp-week.png} 

\end{columns} 
 \end{frame}

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\begin{frame}{A 2-stages strategy (Energycon)}

\includegraphics[width = \textwidth]{pics/2-stage_strategy.png}

\footnotetext[1]{
	J. Cugliari, Y. Goude and J. M. Poggi, "Disaggregated electricity forecasting using wavelet-based clustering of individual consumers," 2016 IEEE International Energy Conference (ENERGYCON), Leuven, 2016, pp. 1-6.
	}

\end{frame}

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\begin{frame}{Data description}

\begin{columns}
\column{0.45\textwidth}
\begin{block}{Available}
\begin{itemize}
\item 
EDF : 25K professional clients, sampled @ 30min, 5 semesters 
\item 
external open data 
\end{itemize}
\end{block}

\begin{block}{Accesible}
	\begin{itemize}
		\item simulated (very large) data
	\end{itemize}
\end{block}
\column{0.55\textwidth}
\includegraphics[width = \columnwidth]{pics/indiv.jpg}
%\textcolor{red}{A picture here?}
\end{columns}

\end{frame}


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\begin{frame}{Computing resources}
\begin{block}{2 academic testing architectures}
\begin{itemize}
\item Orsay's cluster (500Gb RAM, 80 cores)
\item \texttt{pulpito} : Lyon 2's box with 2 quadricores (HT x 2), 72Gb RAM
\end{itemize}
\end{block}

\begin{block}{1 industrial real-scale architecture}
\begin{itemize}
\item mini cluster @ EDF labs
\end{itemize}
\end{block}

\end{frame}

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\begin{frame}{Strategies for upscaling}

\begin{itemize}
\item From 25K to 25M: in 1000 chunks of 25K
\item Reference values: 
\begin{itemize}
\item $K'=200$ super consumers (SC)
\item  $K\ast=15$ final clusters
\end{itemize}
\end{itemize}



\begin{block}{1st strategy}
\begin{itemize}
\item Do 1000 times ONLY Energycon's 1st-step strategy on 25K clients 

\item With the $1000 \times K'$ SC perform a 2-step run 
      leading to $K^\ast$ clusters
\end{itemize}
\end{block}

\begin{block}{2nd strategy}
\begin{itemize}
\item Do 1000 times Energycon's 2-step strategy on 25K clients
      leading to $1000\times K^\ast$ intermediate clusters
\item Treat the intermediate clusters as individual curves and perform
      a single 2-step run to get $K^\ast$ final clusters
\end{itemize}
\end{block}

\end{frame}


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\begin{frame}
\frametitle{Course + Workshop}
	
\begin{block}{1-day IRSDI-ECAS Course}
\begin{itemize}
\item 
GAM : from classical to distributed environments
\item
October 19, 2017 @ EDF Labs, Paris-Saclay, France
\item 
Simon Wood \& Matteo Fasiolo (University Walk, Bristol, UK)
\end{itemize}
\end{block}


\begin{block}{1-day Worshop}
\begin{itemize}
\item 
Individual Electricity Consumers, Data, Packages and Methods
\item
October 20, 2017 @ EDF Labs, Paris-Saclay, France
\item 
5 keynote speakers 
\begin{itemize}
\item
Souhaib Ben Taieb, Monash University, Melbourne, Australia
\item
Ram Rajagopal, Stanford Univ., USA
\item
Gavin Shaddick, University of Bath, UK
\item
Bei Chen, IBM Research, Ireland
\item
Jack Kelly, University of London, Imperial College of Science, UK
\end{itemize}
\end{itemize}
\end{block}
\end{frame}


\section{Point sur les codes}

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\begin{frame}{Résumé point sur le code (BA, JC @ Lyon déc 2016)}
	
Nous avons réussi à 
\begin{itemize}
\item
faire une fresh installation du code de BA sur une nouvelle machine
(problèmes divers liés à la compilation, configuration, libraries exotiques)
\item 
conduire des expériences sur les données pour mesurer le temps de calcul (le code est blazing fast: 30sec pour obtenir 500 groupes sur 4 procs)
\item
identifier de problèmes : manque un installateur et une interface de pretraitement indépendant du calcul
\end{itemize}
	
A faire:
\begin{itemize}
\item
finir les experiences (sur nb de classes, nb de curves / chunk, nb de procs) et sur d'autres architectures
\item
interface matrice -> binaire
\item 
obtenir les courbes synchrones
\end{itemize}
	
Piste à explorer pour les comparaisons: \texttt{h2o} 
\end{frame}

\section{Expériences numériques}

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\begin{frame}
\frametitle{Code C/MPI}
	
\begin{enumerate}
\item [0]] Sérialisation des données : on écrit d'abord la longueur de la série	puis les puissances sont codées sur 3 octets, permettant une excellente compression et une lecture facile (accès en O(1) à n'importe quelle série)
		
\item [1]] Algorithme PAM appliqué en parallèle via la librairie MPI.
		
\item [2]] Agrégation des médoïdes obtenus, (re-)sérialisation, puis on ré-applique l'algorithme PAM.

\end{enumerate}

\begin{itemize}
\item 
		%Plusieurs astuces : sérialisation des données, calcul en parallèle
\item 
		Très rapide : environ 5 minutes from raw to 1st stage clustering
		\item 
		Divergences par rapport à Energycon (moyennes au lieu d'aggrégation)
	\end{itemize}
	
	%\begin{verbatim}
	%> time ./ppam.exe serialize 2009.csv 2009.bin 1 0
	%real	7m34.182s
	%\end{verbatim}
\end{frame}


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\begin{frame}
\frametitle{R Code}

\begin{itemize}
\item Enercon's code update 
\item \texttt{data.table} is used for readings and writtings \footnote{\texttt{tidyverse} toolbox is much slower}
\item Disk spaces of the plain text associated object
\item Timmings on \texttt{pulpito}  (16 cores, 64Gb RAM, SSD) 
\end{itemize}
\begin{center}
\begin{tabular}{lccc}\toprule
Task & Time & Memory & Disk \\ \midrule
Raw (15Gb) to matrix  &   7 min  &
  30 Gb\footnote{\texttt{ff} is a promising alternative if needed} &
  2.7 Gb \\
Compute contributions  &   7 min  & <1Gb & 7 Mb \\
1st stage clustering   &   3 min  & <1Gb & -- \\
Aggregation            &   1 min  &  6Gb & 30 Mb \\
Wer distance matrix    &  40 min  & 64Gb\footnote{Embarransgly parallel but still too slow} & 150 Kb \\
Forecasts              &  10 min  & <1Gb & --\\ 
\bottomrule
\end{tabular}
\end{center}
\end{frame}

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\begin{frame}
\frametitle{Why wer distance is so slow ?}


\begin{block}{2nd step strategy (Enercon way)}
% We proceed as follows:
 \begin{itemize}
    \item Transform data $z_1(t), \ldots, z_n(t)$ using the CWT and Morlet wavelet to obtain $n$ matrices of size $J\times N$.
    \item Compute the wer-based dissimilarity matrix 
    \item Obtain the PAM-based clustering.
\end{itemize}

\begin{block}{Current choices on the computation}
\begin{itemize}
\item From (\texttt{Rwave} \& \texttt{sowas}) to \texttt{biwavelt}
\item About 1 sec to compute \texttt{werd(x, y)} with current 
      filtering ($J \sim 52$ with 13 octaves, 4 voices )
\item Need to compute $n (n - 1) / 2$ pairwise distances 
     (20K, 130K, 500K entries for $n = 200, 500, 1000$)
\item Need an efficient \texttt{werd} function (maybe in 
      RcppParallel ?)
\end{itemize}

\end{block}

  
\end{block}

\end{frame}



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\begin{frame}{CWT}

\begin{block}{Continuous WT}
Starting with a mother wavelet $\psi$ consider $\psi_{a, \tau} = a^{-1/2} \psi\left(\frac{t-\tau}{a}\right)$.

The CWT of a function $z\in L^2 (\mathbb{R})$ is,
$$ W_z(a, \tau) = \int_{-\infty}^{\infty} z(t) \psi_{a, \tau}^* (t) dt$$

As for Fourier transform, a spectral approach is possible.


\begin{eqnarray*}
S_z(a, \tau)  &=& |W_z(a, \tau)|^2 \qquad\qquad \hbox{wavelet spectrum} \\
\mathcal{W}_{z, x}(a, \tau)  &=& W_z(a, \tau)W_x^*(a, \tau) \qquad \hbox{cross-wavelet transform}
\end{eqnarray*}

%$$ S_z(a, \tau) = |W_z(a, \tau)|^2 \qquad \hbox{wavelet spectrum}$$

%$$ \mathcal{W}_{z, x}(a, \tau) = W_z(a, \tau)W_x(a, \tau)^* \qquad \hbox{cross-wavelet transform}$$
\end{block}

\end{frame}


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\begin{frame}{Wavelet coherence}
\begin{block}{ }
\begin{equation*} \label{coherence}
R_{z,x}^2(a,\tau) = \frac{ |\tilde{\mathcal{W}}_{x,y}(a, \tau)|^2 }{|\tilde{\mathcal{W}}_{x,x}(a, \tau)| |\tilde{\mathcal{W}}_{y,y}(a, \tau) | },
\end{equation*}

Based on the extended $R^2$ coefficient, we can construct an coefficient of determination between two wavelet spectrums
 
\begin{equation*}\label{eq:wer}
  WER_{z, x}^2 = \frac{ 
 \int_0^\infty  \left( \int_{-\infty}^\infty |\tilde{\mathcal{W}}_{z, x}(a, \tau)|  d\tau \right)^2 da} { \int_0^\infty \left( \int_{-\infty}^\infty |\tilde{\mathcal{W}}_{z, z}(a, \tau)| d\tau \int_{-\infty}^\infty |\tilde{\mathcal{W}}_{x, x}(a, \tau)| d\tau\right) da}.
 \end{equation*}

And obtain a dissimilarity based on it
 
\begin{equation*}\label{eq:dist-wer}
    d(z, x) = \sqrt{ JN(1 - \widehat{WER}_{z, x}^2)} 
\end{equation*}
\end{block}
\end{frame}

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%\begin{frame} \frametitle{Wavelet coherence}
%\begin{block}{ }
% We proceed as follows:
% \begin{itemize}
%    \item Transform data $z_1(t), \ldots, z_n(t)$ using the  CWT and Morlet wavelet to obtain $n$ matrices of size $J\times N$.
%    \item Compute a dissimilarity matrix with the coherency based dissimilarity.
%    \item Using PAM obtain clusters $k=8$ clusters.
%   \end{itemize}
%
%  Rand Index (AC, WER) = 0.26
%  
%\end{block}
%
%\end{frame}


\end{document}


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\begin{frame}
\frametitle{Misc jc}
	
\begin{itemize}
\item simulated dataset : howto ?
\item temperature
\item Rcpp
\end{itemize}
	


