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Prévision d’un processus à valeurs fonctionnelles.
              Application à la consommation d’électricité.


                               Jaïro Cugliari

                          Groupe select, INRIA Futurs


                            16 décembre 2011




   Directeurs de thèse:   Anestis ANTONIADIS (Univ. Joseph Fourier)
                          Jean-Michel POGGI (Univ. Paris Descartes)
Encadrement industriel:   Xavier BROSSAT (EDF R&D) .
Motivation
                           Kernel-wavelet functional model
                 Clustering functional data using wavelets
              Conditional Autoregression Hilbertian process
                                       Concluding remarks


Outline


  1   Motivation


  2   Kernel-wavelet functional model


  3   Clustering functional data using wavelets


  4   Conditional Autoregression Hilbertian process


  5   Concluding remarks




                      GT Prévision | Déc 2011 | J. Cugliari   Prévision d’un processus à valeurs fonctionnelles.
Motivation
                           Kernel-wavelet functional model
                                                              Functional time series
                 Clustering functional data using wavelets
                                                              Electricity demand data
              Conditional Autoregression Hilbertian process
                                       Concluding remarks


Outline


  1   Motivation


  2   Kernel-wavelet functional model


  3   Clustering functional data using wavelets


  4   Conditional Autoregression Hilbertian process


  5   Concluding remarks




                      GT Prévision | Déc 2011 | J. Cugliari   Prévision d’un processus à valeurs fonctionnelles.
Motivation
                              Kernel-wavelet functional model
                                                                 Functional time series
                    Clustering functional data using wavelets
                                                                 Electricity demand data
                 Conditional Autoregression Hilbertian process
                                          Concluding remarks


FD as slices of a continuous process                                                                                  [Bosq, (1990)]

  The prediction problem
           Suppose one observes a square integrable continuous-time stochastic
           process X = (X (t), t ∈ R) over the interval [0, T ], T > 0;
           We want to predict X all over the segment [T , T + δ], δ > 0
           Divide the interval into n subintervals of equal size δ.
           Consider the functional-valued discrete time stochastic process
           Z = (Zk , k ∈ N), where N = {1, 2, . . .}, defined by

  Xt




                                                                t
       0                                                  T T +δ



                         GT Prévision | Déc 2011 | J. Cugliari   Prévision d’un processus à valeurs fonctionnelles.
Motivation
                              Kernel-wavelet functional model
                                                                 Functional time series
                    Clustering functional data using wavelets
                                                                 Electricity demand data
                 Conditional Autoregression Hilbertian process
                                          Concluding remarks


FD as slices of a continuous process                                                                                  [Bosq, (1990)]

  The prediction problem
           Suppose one observes a square integrable continuous-time stochastic
           process X = (X (t), t ∈ R) over the interval [0, T ], T > 0;
           We want to predict X all over the segment [T , T + δ], δ > 0
           Divide the interval into n subintervals of equal size δ.
           Consider the functional-valued discrete time stochastic process
           Z = (Zk , k ∈ N), where N = {1, 2, . . .}, defined by

  Xt




                                                                t
       0                                                  T T +δ



                         GT Prévision | Déc 2011 | J. Cugliari   Prévision d’un processus à valeurs fonctionnelles.
Motivation
                                 Kernel-wavelet functional model
                                                                    Functional time series
                       Clustering functional data using wavelets
                                                                    Electricity demand data
                    Conditional Autoregression Hilbertian process
                                             Concluding remarks


FD as slices of a continuous process                                                                                     [Bosq, (1990)]

  The prediction problem
            Suppose one observes a square integrable continuous-time stochastic
            process X = (X (t), t ∈ R) over the interval [0, T ], T > 0;
            We want to predict X all over the segment [T , T + δ], δ > 0
            Divide the interval into n subintervals of equal size δ.
            Consider the functional-valued discrete time stochastic process
            Z = (Zk , k ∈ N), where N = {1, 2, . . .}, defined by

  Xt
                           Z3 (t) Z4 (t)               Z6 (t)

                                                                                Zk (t) = X (t + (k − 1)δ)
           Z1 (t) Z2 (t)                     Z5 (t)
                                                                                      k ∈ N ∀t ∈ [0, δ)
                                                                     t
       0       1δ      2δ        3δ       4δ        5δ       6δ T + δ



                            GT Prévision | Déc 2011 | J. Cugliari   Prévision d’un processus à valeurs fonctionnelles.
Motivation
                                 Kernel-wavelet functional model
                                                                    Functional time series
                       Clustering functional data using wavelets
                                                                    Electricity demand data
                    Conditional Autoregression Hilbertian process
                                             Concluding remarks


FD as slices of a continuous process                                                                                     [Bosq, (1990)]

  The prediction problem
            Suppose one observes a square integrable continuous-time stochastic
            process X = (X (t), t ∈ R) over the interval [0, T ], T > 0;
            We want to predict X all over the segment [T , T + δ], δ > 0
            Divide the interval into n subintervals of equal size δ.
            Consider the functional-valued discrete time stochastic process
            Z = (Zk , k ∈ N), where N = {1, 2, . . .}, defined by

  Xt
                           Z3 (t) Z4 (t)               Z6 (t)

                                                                                Zk (t) = X (t + (k − 1)δ)
           Z1 (t) Z2 (t)                     Z5 (t)
                                                                                      k ∈ N ∀t ∈ [0, δ)
                                                                     t
       0       1δ      2δ        3δ       4δ        5δ       6δ T + δ

  If X contents a δ−seasonal component, Z is particularly fruitful.

                            GT Prévision | Déc 2011 | J. Cugliari   Prévision d’un processus à valeurs fonctionnelles.
Motivation
                         Kernel-wavelet functional model
                                                            Functional time series
               Clustering functional data using wavelets
                                                            Electricity demand data
            Conditional Autoregression Hilbertian process
                                     Concluding remarks


Prediction of functional time series
Let (Zk , k ∈ Z) be a stationary sequence of H-valued r.v. Given Z1 , . . . , Zn we
want to predict the future value of Zn+1 .
     A predictor of Zn+1 using Z1 , Z2 , . . . , Zn is

                                     Zn+1 = E[Zn+1 |Zn , Zn−1 , . . . , Z1 ].

Autoregressive Hilbertian process of order 1
The arh(1) centred process states that at each k,

                                            Zk = ρ(Zk−1 ) +           k                                          (1)

where ρ is a compact linear operator and { k }k∈Z is an H−valued strong white
noise.
Under mild conditions, equation (1) has a unique solution which is a strictly
stationary process with innovation { k }k∈Z . [Bosq, (1991)]
When Z is a zero-mean arh(1) process, the best predictor of Zn+1 given
{Z1 , . . . , Zn−1 } is:
                                 Zn+1 = ρ(Zn ).
                    GT Prévision | Déc 2011 | J. Cugliari   Prévision d’un processus à valeurs fonctionnelles.
Motivation
                         Kernel-wavelet functional model
                                                            Functional time series
               Clustering functional data using wavelets
                                                            Electricity demand data
            Conditional Autoregression Hilbertian process
                                     Concluding remarks


Prediction of functional time series
Let (Zk , k ∈ Z) be a stationary sequence of H-valued r.v. Given Z1 , . . . , Zn we
want to predict the future value of Zn+1 .
     A predictor of Zn+1 using Z1 , Z2 , . . . , Zn is

                                     Zn+1 = E[Zn+1 |Zn , Zn−1 , . . . , Z1 ].

Autoregressive Hilbertian process of order 1
The arh(1) centred process states that at each k,

                                            Zk = ρ(Zk−1 ) +           k                                          (1)

where ρ is a compact linear operator and { k }k∈Z is an H−valued strong white
noise.
Under mild conditions, equation (1) has a unique solution which is a strictly
stationary process with innovation { k }k∈Z . [Bosq, (1991)]
When Z is a zero-mean arh(1) process, the best predictor of Zn+1 given
{Z1 , . . . , Zn−1 } is:
                                 Zn+1 = ρ(Zn ).
                    GT Prévision | Déc 2011 | J. Cugliari   Prévision d’un processus à valeurs fonctionnelles.
Motivation
                             Kernel-wavelet functional model
                                                                 Functional time series
                   Clustering functional data using wavelets
                                                                 Electricity demand data
                Conditional Autoregression Hilbertian process
                                         Concluding remarks


Electricity demand data
Some salient features




               (a) Long term trend.                                        (b) Annual and week cycles.




                 (c) Daily pattern.                             (d) Demand (in Gw/h) as a function of
                                                                temperature (in ◦ C)

                        GT Prévision | Déc 2011 | J. Cugliari    Prévision d’un processus à valeurs fonctionnelles.
Motivation
                           Kernel-wavelet functional model
                                                                Functional time series
                 Clustering functional data using wavelets
                                                                Electricity demand data
              Conditional Autoregression Hilbertian process
                                       Concluding remarks


Electricity demand forecast

  Short-term electricity demand forecast in literature
       Time series analysis: sarima(x), Kalman filter                                 [Dordonnat et al. (2009)]

       Machine learning.            [Devaine et al. (2010)]

       Similarity search based methods.                       [Poggi (1994), Antoniadis et al. (2006)]

       Regression: edf modelisation scheme                         [Bruhns et al. (2005)]           , gam        [Pierrot and
       Goude (2011)]


  New challenges
       Market liberalization: may produce variations on clients’ perimeter that
       risk to induce nonstationarities on the signal.
       Development of smart grids and smart meters.
  But, almost all the models rely on a monoscale representation of the data


                      GT Prévision | Déc 2011 | J. Cugliari     Prévision d’un processus à valeurs fonctionnelles.
Motivation
                           Kernel-wavelet functional model
                                                                Functional time series
                 Clustering functional data using wavelets
                                                                Electricity demand data
              Conditional Autoregression Hilbertian process
                                       Concluding remarks


Electricity demand forecast

  Short-term electricity demand forecast in literature
       Time series analysis: sarima(x), Kalman filter                                 [Dordonnat et al. (2009)]

       Machine learning.            [Devaine et al. (2010)]

       Similarity search based methods.                       [Poggi (1994), Antoniadis et al. (2006)]

       Regression: edf modelisation scheme                         [Bruhns et al. (2005)]           , gam        [Pierrot and
       Goude (2011)]


  New challenges
       Market liberalization: may produce variations on clients’ perimeter that
       risk to induce nonstationarities on the signal.
       Development of smart grids and smart meters.
  But, almost all the models rely on a monoscale representation of the data


                      GT Prévision | Déc 2011 | J. Cugliari     Prévision d’un processus à valeurs fonctionnelles.
Motivation
                           Kernel-wavelet functional model    Wavelets
                 Clustering functional data using wavelets    Prediction algorithm
              Conditional Autoregression Hilbertian process   Corrections to handle nonstationarity
                                       Concluding remarks


Outline


  1   Motivation


  2   Kernel-wavelet functional model


  3   Clustering functional data using wavelets


  4   Conditional Autoregression Hilbertian process


  5   Concluding remarks




                      GT Prévision | Déc 2011 | J. Cugliari   Prévision d’un processus à valeurs fonctionnelles.
Motivation
                          Kernel-wavelet functional model      Wavelets
                Clustering functional data using wavelets      Prediction algorithm
             Conditional Autoregression Hilbertian process     Corrections to handle nonstationarity
                                      Concluding remarks


Kernel regression for functional time series                                                       [Antoniadis et al. (2008)]




  For a more general class of process
       The regression function E[Zn+1 |Zn , Zn−1 , . . . , Z1 ] can be estimated by a
       nonparametric approach.
       Key idea: similar futures correspond to similar pasts.


  The resulting predictor Zn+1 (t) of Zn+1
       is obtained by a kernel regression of Zn over the history {Zn−1 , . . . , Z1 }.
       is a weighted mean of futures of past segments. Weights increase with
       similarity between last observed segment n and past segments
       m = 1, . . . , n,
                                                             n−1

                                          Zn+1 (t) =               wn,m Zm+1 (t)
                                                             m=1




                     GT Prévision | Déc 2011 | J. Cugliari     Prévision d’un processus à valeurs fonctionnelles.
Motivation
                          Kernel-wavelet functional model      Wavelets
                Clustering functional data using wavelets      Prediction algorithm
             Conditional Autoregression Hilbertian process     Corrections to handle nonstationarity
                                      Concluding remarks


Kernel regression for functional time series                                                       [Antoniadis et al. (2008)]




  For a more general class of process
       The regression function E[Zn+1 |Zn , Zn−1 , . . . , Z1 ] can be estimated by a
       nonparametric approach.
       Key idea: similar futures correspond to similar pasts.


  The resulting predictor Zn+1 (t) of Zn+1
       is obtained by a kernel regression of Zn over the history {Zn−1 , . . . , Z1 }.
       is a weighted mean of futures of past segments. Weights increase with
       similarity between last observed segment n and past segments
       m = 1, . . . , n,
                                                             n−1

                                          Zn+1 (t) =               wn,m Zm+1 (t)
                                                             m=1




                     GT Prévision | Déc 2011 | J. Cugliari     Prévision d’un processus à valeurs fonctionnelles.
Wavelets to cope with fd


                                                                           domain-transform technique
                                                                           for hierarchical decomposing
                                                                           finite energy signals
                                                                           description in terms of a
                                                                           broad trend (approximation
                                                                           part), plus a set of localized
                                                                           changes kept in the details
                                                                           parts.


  Discrete Wavelet Transform
  If z ∈ L2 ([0, 1]) we can write it as

                             2j0 −1                         ∞ 2j −1

                    z(t) =            cj0 ,k φj0 ,k (t) +              dj,k ψj,k (t),
                             k=0                            j=j0 k=0

  where cj,k =< g, φj,k >, dj,k =< g, ϕj,k > are the scale coefficients and
  wavelet coefficients respectively, and the functions φ et ϕ are associated to a
  orthogonal mra of L2 ([0, 1]).
Motivation
                        Kernel-wavelet functional model     Wavelets
              Clustering functional data using wavelets     Prediction algorithm
           Conditional Autoregression Hilbertian process    Corrections to handle nonstationarity
                                    Concluding remarks


Approximation and details

     In practice, we don’t dispose of the whole trajectory but only with a
     (possibly noisy) sampling at 2J points, for some integer J.
     Each approximated segment Zi,J (t) is broken up into two terms:
          a smooth approximation Si (t) (lower freqs)
          a set of details Di (t) (higher freqs)
                                        2j0 −1                         J−1 2j −1
                                                  (i)                                 (i)
                       Zi,J (t) =                cj0 ,k φj0 ,k (t) +                dj,k ψj,k (t)
                                         k=0                           j=j0 k=0

                                                   Si (t)                         Di (t)

     The parameter j0 controls the separation. We set j0 = 0.

                                                              J−1 2j −1

                              zJ (t) = c0 φ0,0 (t) +                        dj,k ψj,k (t).
                                                              j=0 k=0



                   GT Prévision | Déc 2011 | J. Cugliari    Prévision d’un processus à valeurs fonctionnelles.
Motivation
                          Kernel-wavelet functional model          Wavelets
                Clustering functional data using wavelets          Prediction algorithm
             Conditional Autoregression Hilbertian process         Corrections to handle nonstationarity
                                      Concluding remarks


A two step prediction algorithm


  Step I: Dissimilarity between segments
  Search the past for segments that are similar to the last one.
  For two observed series of length 2J say Zm and Zl we set for each scale j ≥ j0 :

                                                         2j −1                               1/2
                                                                     (m)         (l)
                           distj (Zm , Zl ) =                      (dj,k   −    dj,k )2
                                                             k=0

  Then, we aggregate over the scales taking into account the number of
  coefficients at each scale
                                                      J−1

                               D(Zm , Zl ) =                  2−j/2 distj (Zm , Zl )
                                                      j=j0




                     GT Prévision | Déc 2011 | J. Cugliari         Prévision d’un processus à valeurs fonctionnelles.
Motivation
                          Kernel-wavelet functional model       Wavelets
                Clustering functional data using wavelets       Prediction algorithm
             Conditional Autoregression Hilbertian process      Corrections to handle nonstationarity
                                      Concluding remarks


A two step prediction algorithm

  Step 2: Kernel regression
  Obtain the prediction of the scale coefficients at the finest resolution
            (n+1)
  Ξn+1 = {cJ,k : k = 0, 1, . . . , 2J − 1} for Zn+1
                                                         n−1

                                           Ξn+1 =               wm,n Ξm+1
                                                        m=1
                                                            D(Zn ,Zm )
                                                         K     hn
                                     wm,n =             n−1
                                                        m=1
                                                            K D(Zhn m )
                                                                    n ,Z


  Finally, the prediction of Zn+1 can be written

                                                        2J −1
                                                                 (n+1)
                                     Zn+1 (t) =                 cJ,k φJ,k (t)
                                                         k=0




                     GT Prévision | Déc 2011 | J. Cugliari      Prévision d’un processus à valeurs fonctionnelles.
Motivation
                         Kernel-wavelet functional model          Wavelets
               Clustering functional data using wavelets          Prediction algorithm
            Conditional Autoregression Hilbertian process         Corrections to handle nonstationarity
                                     Concluding remarks


Corrections proposed to handle nonstationarity

  On mean level
                                          n−1
          base Sn+1 (t) =                 m=1
                                                 wm,n Sm+1 (t)

          prst Sn+1 (t) = Sn (t)
                                                            n−1
           diff Sn+1 (t) = Sn (t) +                         m=2
                                                                  wm,n ∆(Sm )(t)




                     Figure: Daily prediction error (in mapex100).




                    GT Prévision | Déc 2011 | J. Cugliari         Prévision d’un processus à valeurs fonctionnelles.
Motivation
                         Kernel-wavelet functional model          Wavelets
               Clustering functional data using wavelets          Prediction algorithm
            Conditional Autoregression Hilbertian process         Corrections to handle nonstationarity
                                     Concluding remarks


Corrections proposed to handle nonstationarity

  On mean level
                                          n−1
          base Sn+1 (t) =                 m=1
                                                 wm,n Sm+1 (t)

          prst Sn+1 (t) = Sn (t)
                                                            n−1
           diff Sn+1 (t) = Sn (t) +                         m=2
                                                                  wm,n ∆(Sm )(t)




                     Figure: Daily prediction error (in mapex100).




                    GT Prévision | Déc 2011 | J. Cugliari         Prévision d’un processus à valeurs fonctionnelles.
Motivation
                         Kernel-wavelet functional model          Wavelets
               Clustering functional data using wavelets          Prediction algorithm
            Conditional Autoregression Hilbertian process         Corrections to handle nonstationarity
                                     Concluding remarks


Corrections proposed to handle nonstationarity

  On mean level
                                          n−1
          base Sn+1 (t) =                 m=1
                                                 wm,n Sm+1 (t)

          prst Sn+1 (t) = Sn (t)
                                                            n−1
           diff Sn+1 (t) = Sn (t) +                         m=2
                                                                  wm,n ∆(Sm )(t)




                     Figure: Daily prediction error (in mapex100).




                    GT Prévision | Déc 2011 | J. Cugliari         Prévision d’un processus à valeurs fonctionnelles.
Motivation
                            Kernel-wavelet functional model     Wavelets
                  Clustering functional data using wavelets     Prediction algorithm
               Conditional Autoregression Hilbertian process    Corrections to handle nonstationarity
                                        Concluding remarks


Corrections proposed to handle nonstationarity
  On groups by post-treatment
        Define new weights and renormalize.
                                                                                   gr (n) is the group of
                   ww ,m             if               gr (m) = gr (n)              the n-th segment.
        wm,n =
        ˜
                    0            otherwise
    1   Deterministic groups: Calendar or Calendar transitions.
    2   Groups learned from data via clustering analysis. (e.g. temperature curves)
    3   Cross deterministic with clustering groups (e.g. calendar-temperature
        transitions).




                         Figure: Daily prediction error (in mapex100).

                        GT Prévision | Déc 2011 | J. Cugliari   Prévision d’un processus à valeurs fonctionnelles.
Motivation
                            Kernel-wavelet functional model     Wavelets
                  Clustering functional data using wavelets     Prediction algorithm
               Conditional Autoregression Hilbertian process    Corrections to handle nonstationarity
                                        Concluding remarks


Corrections proposed to handle nonstationarity
  On groups by post-treatment
        Define new weights and renormalize.
                                                                                   gr (n) is the group of
                   ww ,m             if               gr (m) = gr (n)              the n-th segment.
        wm,n =
        ˜
                    0            otherwise
    1   Deterministic groups: Calendar or Calendar transitions.
    2   Groups learned from data via clustering analysis. (e.g. temperature curves)
    3   Cross deterministic with clustering groups (e.g. calendar-temperature
        transitions).




                         Figure: Daily prediction error (in mapex100).

                        GT Prévision | Déc 2011 | J. Cugliari   Prévision d’un processus à valeurs fonctionnelles.
Predict of 10 September 2005 (Saturday)

                 SimilIndex
                                    date      SimilIndex
                                 2004-09-10     0.455
                                 2003-09-05     0.141
                                 2002-09-06     0.083
                                 2004-09-03     0.070
                                 2003-09-19     0.068
                                 2000-09-08     0.058
                                 2000-09-15     0.019
                                 1999-09-10     0.017




  similar past                                 similar future
Motivation
                           Kernel-wavelet functional model
                 Clustering functional data using wavelets    Clustering via feature extraction
              Conditional Autoregression Hilbertian process
                                       Concluding remarks


Outline


  1   Motivation


  2   Kernel-wavelet functional model


  3   Clustering functional data using wavelets


  4   Conditional Autoregression Hilbertian process


  5   Concluding remarks




                      GT Prévision | Déc 2011 | J. Cugliari   Prévision d’un processus à valeurs fonctionnelles.
Motivation
                           Kernel-wavelet functional model
                 Clustering functional data using wavelets    Clustering via feature extraction
              Conditional Autoregression Hilbertian process
                                       Concluding remarks


Aim

      Segmentation of X may not suffices to
      render reasonable the stationary
      hypothesis.
      If a grouping effect exists, we may
      considered stationary within each group.
      Conditionally on the grouping, functional
      time series prediction methods can be
                                                                            Two strategies to cluster
      applied.
                                                                            functional time series:
      We propose a clustering procedure that                                     1   Feature extraction
      discover the groups from a bunch of
                                                                                     (summary measures of the
      curves.
                                                                                     curves).
We use wavelet transforms to take into account                                   2   Direct similarity between
the fact that curves may present non stationary                                      curves.
patters.

                      GT Prévision | Déc 2011 | J. Cugliari   Prévision d’un processus à valeurs fonctionnelles.
Motivation
                           Kernel-wavelet functional model
                 Clustering functional data using wavelets    Clustering via feature extraction
              Conditional Autoregression Hilbertian process
                                       Concluding remarks


Aim

      Segmentation of X may not suffices to
      render reasonable the stationary
      hypothesis.
      If a grouping effect exists, we may
      considered stationary within each group.
      Conditionally on the grouping, functional
      time series prediction methods can be
                                                                            Two strategies to cluster
      applied.
                                                                            functional time series:
      We propose a clustering procedure that                                     1   Feature extraction
      discover the groups from a bunch of
                                                                                     (summary measures of the
      curves.
                                                                                     curves).
We use wavelet transforms to take into account                                   2   Direct similarity between
the fact that curves may present non stationary                                      curves.
patters.

                      GT Prévision | Déc 2011 | J. Cugliari   Prévision d’un processus à valeurs fonctionnelles.
Motivation
                        Kernel-wavelet functional model
              Clustering functional data using wavelets        Clustering via feature extraction
           Conditional Autoregression Hilbertian process
                                    Concluding remarks


Energy decomposition of the DWT

     Energy conservation of the signal

                                                           J−1 2j −1                          J−1
                       2
                   z       ≈    zJ 2
                                   2     =    2
                                             c0,0   +                   2      2
                                                                       dj,k = c0,0 +                   dj    2
                                                                                                             2.
                                                           j=0 k=0                            j=0


     For each j = 0, 1, . . . , J − 1, we compute the absolute and relative
     contribution representations by

                                                                                        ||dj ||2
                           contj = ||dj ||2                and          relj =                      .
                                                                                         j
                                                                                           ||dj ||2
                                  AC
                                                                                    RC

     They quantify the relative importance of the scales to the global dynamic.
     RC normalizes the energy of each signal to 1.

                   GT Prévision | Déc 2011 | J. Cugliari       Prévision d’un processus à valeurs fonctionnelles.
Motivation
                            Kernel-wavelet functional model
                  Clustering functional data using wavelets    Clustering via feature extraction
               Conditional Autoregression Hilbertian process
                                        Concluding remarks


Schema of procedure




  0. Data preprocessing. Approximate sample paths of z1 (t), . . . , zn (t)
  1. Feature extraction. Compute either of the energetic components using absolute
                   contribution (AC) or relative contribution (RC).
  2. Feature selection. Screen irrelevant variables.                  [Steinley & Brusco (’06)]

  3. Determine the number of clusters. Detecting significant jumps in the transformed
                  distortion curve. [Sugar & James (’03)]
    4. Clustering. Obtain the K clusters using PAM algorithm.




                       GT Prévision | Déc 2011 | J. Cugliari   Prévision d’un processus à valeurs fonctionnelles.
Motivation
                        Kernel-wavelet functional model
              Clustering functional data using wavelets    Clustering via feature extraction
           Conditional Autoregression Hilbertian process
                                    Concluding remarks


Electricity demand data

     Feature extraction:
        - Significant scales are associated to mid-frequencies.
        - The retained scales parametrize the represented cycles of 1.5, 3 and
          6 hours (AC) and to the cycles of 30 minutes, 1.5 and 3 hours (RC).
     Number of clusters:
          8 for AC and 5 for RC.
     For instance, we found a class of segments that can be recognized as
     summer Monday




                   GT Prévision | Déc 2011 | J. Cugliari   Prévision d’un processus à valeurs fonctionnelles.
Motivation
                   Kernel-wavelet functional model
         Clustering functional data using wavelets    Clustering via feature extraction
      Conditional Autoregression Hilbertian process
                               Concluding remarks




Towards a dissimilarity based on
      wavelet coherence
Distance based on wavelet-correlation between two time series.
Can be used to measure relationship between two (possibly non
stationary) time series, i.e. temperature and load.
The strength of the relationship is hierarchically decomposed on scales
(≈frequencies), without loose of time location.




              GT Prévision | Déc 2011 | J. Cugliari   Prévision d’un processus à valeurs fonctionnelles.
(a) Curves                            (b) Calendar
Figure: Curves membership of the clustering using wer based dissimilarity (a)
and the corresponding calendar positioning (b).
Motivation
                           Kernel-wavelet functional model
                                                              carh: Conditional Autoregressive Hilbertian Model
                 Clustering functional data using wavelets
                                                              Some results
              Conditional Autoregression Hilbertian process
                                       Concluding remarks


Outline


  1   Motivation


  2   Kernel-wavelet functional model


  3   Clustering functional data using wavelets


  4   Conditional Autoregression Hilbertian process


  5   Concluding remarks




                      GT Prévision | Déc 2011 | J. Cugliari   Prévision d’un processus à valeurs fonctionnelles.
Motivation
                           Kernel-wavelet functional model
                                                              carh: Conditional Autoregressive Hilbertian Model
                 Clustering functional data using wavelets
                                                              Some results
              Conditional Autoregression Hilbertian process
                                       Concluding remarks


carh process


  Let (Z , V ) = {(Zk , Vk ), k ∈ Z} be stationary sequences of H × Rd − valued r.v.
  defined over (Ω, F, P).



  We will focus on the behaviour of Z conditioned to V .
  (Z , V ) is a carh(1) if it is stationary and and such that,

                           Zk = a + ρVk (Zk−1 − a) +                  k,           k ∈ Z,                          (2)
                               d              v
  where for each v ∈ R , av = E [Z0 |V ], { k }k∈Z is an H−white noise
  independent of V , and {ρVk }k∈Z is a sequence of linear compact operators.




                      GT Prévision | Déc 2011 | J. Cugliari   Prévision d’un processus à valeurs fonctionnelles.
Motivation
                            Kernel-wavelet functional model
                                                               carh: Conditional Autoregressive Hilbertian Model
                  Clustering functional data using wavelets
                                                               Some results
               Conditional Autoregression Hilbertian process
                                        Concluding remarks


Simulation and prediction
     We extend the simulation strategies for arh processes [Guillas & Damon
           to the simple case of an carh process with d = 1 and V is a i.i.d.
     (2000)]
     sequence of Beta(β1 , β2) rv.
     Numerical experience: prediction of the electricity demand using the
     temperature as exogenous information




                       GT Prévision | Déc 2011 | J. Cugliari   Prévision d’un processus à valeurs fonctionnelles.
Motivation
                           Kernel-wavelet functional model
                 Clustering functional data using wavelets
              Conditional Autoregression Hilbertian process
                                       Concluding remarks


Outline


  1   Motivation


  2   Kernel-wavelet functional model


  3   Clustering functional data using wavelets


  4   Conditional Autoregression Hilbertian process


  5   Concluding remarks




                      GT Prévision | Déc 2011 | J. Cugliari   Prévision d’un processus à valeurs fonctionnelles.
Concluding remarks

       We study the problem of prediction of E[Zn+1 |Zn , Vn+1 ] for H−valued rv.
       Stationary assumptions are held conditionally on an exogenous rv V .
       First, V is multiclass and prediction is done by kwf.
       When the states of V are unknown, clustering is used to discover them.
       Last, carh uses V ∈ Rd and arh ideas.

  The contribution of wavelets
       To exploit information from past data that was observed under a different
       regime from the actual one.
       Appropriate tool to detect useful similarities between nonstationary
       patterns of rough trajectories.
       Allow fast computations: online prediction version of kwf.

  The contribution of the kernel regression on fts
       Accurate alternative for heavy parametric model.
       Interpretation ability through the study of past similar behaviours.
       Allow fast computations: online prediction version of kwf.
       Special attention must be paid to transitions.
Motivation
                            Kernel-wavelet functional model
                  Clustering functional data using wavelets
               Conditional Autoregression Hilbertian process
                                        Concluding remarks


Some references
     A. Antoniadis, X. Brossat, J. Cugliari, and J.-M. Poggi.
     Clustering functional data using wavelets.
     arXiv:1101.4744, 2011.

     A. Antoniadis, E. Paparoditis, and T. Sapatinas.
     A functional wavelet-kernel approach for time series prediction.
     Journal of the Royal Statistical Society, Series B, Methodological, 68(5):837, 2006.

     D. Bosq.
     Linear processes in function spaces: Theory and applications.
     Springer-Verlag, New York, 2000.

     A. Mas.
     Estimation d’opérateurs de corrélation de processus fonctionnels: lois limites, tests,
     déviations modérées.
     PhD thesis, Université Paris 6, 2000.

     J.-M. Poggi.
     Prévision nonprametrique de la consommation électrique.
     Rev. Statistiqué Appliquée, XLII(4):93–98, 1994.
                                                                                 www.math.u-psud.fr/~cugliari

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Prévision consommation électrique par processus à valeurs fonctionnelles

  • 1. Prévision d’un processus à valeurs fonctionnelles. Application à la consommation d’électricité. Jaïro Cugliari Groupe select, INRIA Futurs 16 décembre 2011 Directeurs de thèse: Anestis ANTONIADIS (Univ. Joseph Fourier) Jean-Michel POGGI (Univ. Paris Descartes) Encadrement industriel: Xavier BROSSAT (EDF R&D) .
  • 2. Motivation Kernel-wavelet functional model Clustering functional data using wavelets Conditional Autoregression Hilbertian process Concluding remarks Outline 1 Motivation 2 Kernel-wavelet functional model 3 Clustering functional data using wavelets 4 Conditional Autoregression Hilbertian process 5 Concluding remarks GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 3. Motivation Kernel-wavelet functional model Functional time series Clustering functional data using wavelets Electricity demand data Conditional Autoregression Hilbertian process Concluding remarks Outline 1 Motivation 2 Kernel-wavelet functional model 3 Clustering functional data using wavelets 4 Conditional Autoregression Hilbertian process 5 Concluding remarks GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 4. Motivation Kernel-wavelet functional model Functional time series Clustering functional data using wavelets Electricity demand data Conditional Autoregression Hilbertian process Concluding remarks FD as slices of a continuous process [Bosq, (1990)] The prediction problem Suppose one observes a square integrable continuous-time stochastic process X = (X (t), t ∈ R) over the interval [0, T ], T > 0; We want to predict X all over the segment [T , T + δ], δ > 0 Divide the interval into n subintervals of equal size δ. Consider the functional-valued discrete time stochastic process Z = (Zk , k ∈ N), where N = {1, 2, . . .}, defined by Xt t 0 T T +δ GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 5. Motivation Kernel-wavelet functional model Functional time series Clustering functional data using wavelets Electricity demand data Conditional Autoregression Hilbertian process Concluding remarks FD as slices of a continuous process [Bosq, (1990)] The prediction problem Suppose one observes a square integrable continuous-time stochastic process X = (X (t), t ∈ R) over the interval [0, T ], T > 0; We want to predict X all over the segment [T , T + δ], δ > 0 Divide the interval into n subintervals of equal size δ. Consider the functional-valued discrete time stochastic process Z = (Zk , k ∈ N), where N = {1, 2, . . .}, defined by Xt t 0 T T +δ GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 6. Motivation Kernel-wavelet functional model Functional time series Clustering functional data using wavelets Electricity demand data Conditional Autoregression Hilbertian process Concluding remarks FD as slices of a continuous process [Bosq, (1990)] The prediction problem Suppose one observes a square integrable continuous-time stochastic process X = (X (t), t ∈ R) over the interval [0, T ], T > 0; We want to predict X all over the segment [T , T + δ], δ > 0 Divide the interval into n subintervals of equal size δ. Consider the functional-valued discrete time stochastic process Z = (Zk , k ∈ N), where N = {1, 2, . . .}, defined by Xt Z3 (t) Z4 (t) Z6 (t) Zk (t) = X (t + (k − 1)δ) Z1 (t) Z2 (t) Z5 (t) k ∈ N ∀t ∈ [0, δ) t 0 1δ 2δ 3δ 4δ 5δ 6δ T + δ GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 7. Motivation Kernel-wavelet functional model Functional time series Clustering functional data using wavelets Electricity demand data Conditional Autoregression Hilbertian process Concluding remarks FD as slices of a continuous process [Bosq, (1990)] The prediction problem Suppose one observes a square integrable continuous-time stochastic process X = (X (t), t ∈ R) over the interval [0, T ], T > 0; We want to predict X all over the segment [T , T + δ], δ > 0 Divide the interval into n subintervals of equal size δ. Consider the functional-valued discrete time stochastic process Z = (Zk , k ∈ N), where N = {1, 2, . . .}, defined by Xt Z3 (t) Z4 (t) Z6 (t) Zk (t) = X (t + (k − 1)δ) Z1 (t) Z2 (t) Z5 (t) k ∈ N ∀t ∈ [0, δ) t 0 1δ 2δ 3δ 4δ 5δ 6δ T + δ If X contents a δ−seasonal component, Z is particularly fruitful. GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 8. Motivation Kernel-wavelet functional model Functional time series Clustering functional data using wavelets Electricity demand data Conditional Autoregression Hilbertian process Concluding remarks Prediction of functional time series Let (Zk , k ∈ Z) be a stationary sequence of H-valued r.v. Given Z1 , . . . , Zn we want to predict the future value of Zn+1 . A predictor of Zn+1 using Z1 , Z2 , . . . , Zn is Zn+1 = E[Zn+1 |Zn , Zn−1 , . . . , Z1 ]. Autoregressive Hilbertian process of order 1 The arh(1) centred process states that at each k, Zk = ρ(Zk−1 ) + k (1) where ρ is a compact linear operator and { k }k∈Z is an H−valued strong white noise. Under mild conditions, equation (1) has a unique solution which is a strictly stationary process with innovation { k }k∈Z . [Bosq, (1991)] When Z is a zero-mean arh(1) process, the best predictor of Zn+1 given {Z1 , . . . , Zn−1 } is: Zn+1 = ρ(Zn ). GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 9. Motivation Kernel-wavelet functional model Functional time series Clustering functional data using wavelets Electricity demand data Conditional Autoregression Hilbertian process Concluding remarks Prediction of functional time series Let (Zk , k ∈ Z) be a stationary sequence of H-valued r.v. Given Z1 , . . . , Zn we want to predict the future value of Zn+1 . A predictor of Zn+1 using Z1 , Z2 , . . . , Zn is Zn+1 = E[Zn+1 |Zn , Zn−1 , . . . , Z1 ]. Autoregressive Hilbertian process of order 1 The arh(1) centred process states that at each k, Zk = ρ(Zk−1 ) + k (1) where ρ is a compact linear operator and { k }k∈Z is an H−valued strong white noise. Under mild conditions, equation (1) has a unique solution which is a strictly stationary process with innovation { k }k∈Z . [Bosq, (1991)] When Z is a zero-mean arh(1) process, the best predictor of Zn+1 given {Z1 , . . . , Zn−1 } is: Zn+1 = ρ(Zn ). GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 10. Motivation Kernel-wavelet functional model Functional time series Clustering functional data using wavelets Electricity demand data Conditional Autoregression Hilbertian process Concluding remarks Electricity demand data Some salient features (a) Long term trend. (b) Annual and week cycles. (c) Daily pattern. (d) Demand (in Gw/h) as a function of temperature (in ◦ C) GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 11. Motivation Kernel-wavelet functional model Functional time series Clustering functional data using wavelets Electricity demand data Conditional Autoregression Hilbertian process Concluding remarks Electricity demand forecast Short-term electricity demand forecast in literature Time series analysis: sarima(x), Kalman filter [Dordonnat et al. (2009)] Machine learning. [Devaine et al. (2010)] Similarity search based methods. [Poggi (1994), Antoniadis et al. (2006)] Regression: edf modelisation scheme [Bruhns et al. (2005)] , gam [Pierrot and Goude (2011)] New challenges Market liberalization: may produce variations on clients’ perimeter that risk to induce nonstationarities on the signal. Development of smart grids and smart meters. But, almost all the models rely on a monoscale representation of the data GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 12. Motivation Kernel-wavelet functional model Functional time series Clustering functional data using wavelets Electricity demand data Conditional Autoregression Hilbertian process Concluding remarks Electricity demand forecast Short-term electricity demand forecast in literature Time series analysis: sarima(x), Kalman filter [Dordonnat et al. (2009)] Machine learning. [Devaine et al. (2010)] Similarity search based methods. [Poggi (1994), Antoniadis et al. (2006)] Regression: edf modelisation scheme [Bruhns et al. (2005)] , gam [Pierrot and Goude (2011)] New challenges Market liberalization: may produce variations on clients’ perimeter that risk to induce nonstationarities on the signal. Development of smart grids and smart meters. But, almost all the models rely on a monoscale representation of the data GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 13. Motivation Kernel-wavelet functional model Wavelets Clustering functional data using wavelets Prediction algorithm Conditional Autoregression Hilbertian process Corrections to handle nonstationarity Concluding remarks Outline 1 Motivation 2 Kernel-wavelet functional model 3 Clustering functional data using wavelets 4 Conditional Autoregression Hilbertian process 5 Concluding remarks GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 14. Motivation Kernel-wavelet functional model Wavelets Clustering functional data using wavelets Prediction algorithm Conditional Autoregression Hilbertian process Corrections to handle nonstationarity Concluding remarks Kernel regression for functional time series [Antoniadis et al. (2008)] For a more general class of process The regression function E[Zn+1 |Zn , Zn−1 , . . . , Z1 ] can be estimated by a nonparametric approach. Key idea: similar futures correspond to similar pasts. The resulting predictor Zn+1 (t) of Zn+1 is obtained by a kernel regression of Zn over the history {Zn−1 , . . . , Z1 }. is a weighted mean of futures of past segments. Weights increase with similarity between last observed segment n and past segments m = 1, . . . , n, n−1 Zn+1 (t) = wn,m Zm+1 (t) m=1 GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 15. Motivation Kernel-wavelet functional model Wavelets Clustering functional data using wavelets Prediction algorithm Conditional Autoregression Hilbertian process Corrections to handle nonstationarity Concluding remarks Kernel regression for functional time series [Antoniadis et al. (2008)] For a more general class of process The regression function E[Zn+1 |Zn , Zn−1 , . . . , Z1 ] can be estimated by a nonparametric approach. Key idea: similar futures correspond to similar pasts. The resulting predictor Zn+1 (t) of Zn+1 is obtained by a kernel regression of Zn over the history {Zn−1 , . . . , Z1 }. is a weighted mean of futures of past segments. Weights increase with similarity between last observed segment n and past segments m = 1, . . . , n, n−1 Zn+1 (t) = wn,m Zm+1 (t) m=1 GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 16. Wavelets to cope with fd domain-transform technique for hierarchical decomposing finite energy signals description in terms of a broad trend (approximation part), plus a set of localized changes kept in the details parts. Discrete Wavelet Transform If z ∈ L2 ([0, 1]) we can write it as 2j0 −1 ∞ 2j −1 z(t) = cj0 ,k φj0 ,k (t) + dj,k ψj,k (t), k=0 j=j0 k=0 where cj,k =< g, φj,k >, dj,k =< g, ϕj,k > are the scale coefficients and wavelet coefficients respectively, and the functions φ et ϕ are associated to a orthogonal mra of L2 ([0, 1]).
  • 17. Motivation Kernel-wavelet functional model Wavelets Clustering functional data using wavelets Prediction algorithm Conditional Autoregression Hilbertian process Corrections to handle nonstationarity Concluding remarks Approximation and details In practice, we don’t dispose of the whole trajectory but only with a (possibly noisy) sampling at 2J points, for some integer J. Each approximated segment Zi,J (t) is broken up into two terms: a smooth approximation Si (t) (lower freqs) a set of details Di (t) (higher freqs) 2j0 −1 J−1 2j −1 (i) (i) Zi,J (t) = cj0 ,k φj0 ,k (t) + dj,k ψj,k (t) k=0 j=j0 k=0 Si (t) Di (t) The parameter j0 controls the separation. We set j0 = 0. J−1 2j −1 zJ (t) = c0 φ0,0 (t) + dj,k ψj,k (t). j=0 k=0 GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 18. Motivation Kernel-wavelet functional model Wavelets Clustering functional data using wavelets Prediction algorithm Conditional Autoregression Hilbertian process Corrections to handle nonstationarity Concluding remarks A two step prediction algorithm Step I: Dissimilarity between segments Search the past for segments that are similar to the last one. For two observed series of length 2J say Zm and Zl we set for each scale j ≥ j0 : 2j −1 1/2 (m) (l) distj (Zm , Zl ) = (dj,k − dj,k )2 k=0 Then, we aggregate over the scales taking into account the number of coefficients at each scale J−1 D(Zm , Zl ) = 2−j/2 distj (Zm , Zl ) j=j0 GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 19. Motivation Kernel-wavelet functional model Wavelets Clustering functional data using wavelets Prediction algorithm Conditional Autoregression Hilbertian process Corrections to handle nonstationarity Concluding remarks A two step prediction algorithm Step 2: Kernel regression Obtain the prediction of the scale coefficients at the finest resolution (n+1) Ξn+1 = {cJ,k : k = 0, 1, . . . , 2J − 1} for Zn+1 n−1 Ξn+1 = wm,n Ξm+1 m=1 D(Zn ,Zm ) K hn wm,n = n−1 m=1 K D(Zhn m ) n ,Z Finally, the prediction of Zn+1 can be written 2J −1 (n+1) Zn+1 (t) = cJ,k φJ,k (t) k=0 GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 20. Motivation Kernel-wavelet functional model Wavelets Clustering functional data using wavelets Prediction algorithm Conditional Autoregression Hilbertian process Corrections to handle nonstationarity Concluding remarks Corrections proposed to handle nonstationarity On mean level n−1 base Sn+1 (t) = m=1 wm,n Sm+1 (t) prst Sn+1 (t) = Sn (t) n−1 diff Sn+1 (t) = Sn (t) + m=2 wm,n ∆(Sm )(t) Figure: Daily prediction error (in mapex100). GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 21. Motivation Kernel-wavelet functional model Wavelets Clustering functional data using wavelets Prediction algorithm Conditional Autoregression Hilbertian process Corrections to handle nonstationarity Concluding remarks Corrections proposed to handle nonstationarity On mean level n−1 base Sn+1 (t) = m=1 wm,n Sm+1 (t) prst Sn+1 (t) = Sn (t) n−1 diff Sn+1 (t) = Sn (t) + m=2 wm,n ∆(Sm )(t) Figure: Daily prediction error (in mapex100). GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 22. Motivation Kernel-wavelet functional model Wavelets Clustering functional data using wavelets Prediction algorithm Conditional Autoregression Hilbertian process Corrections to handle nonstationarity Concluding remarks Corrections proposed to handle nonstationarity On mean level n−1 base Sn+1 (t) = m=1 wm,n Sm+1 (t) prst Sn+1 (t) = Sn (t) n−1 diff Sn+1 (t) = Sn (t) + m=2 wm,n ∆(Sm )(t) Figure: Daily prediction error (in mapex100). GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 23. Motivation Kernel-wavelet functional model Wavelets Clustering functional data using wavelets Prediction algorithm Conditional Autoregression Hilbertian process Corrections to handle nonstationarity Concluding remarks Corrections proposed to handle nonstationarity On groups by post-treatment Define new weights and renormalize. gr (n) is the group of ww ,m if gr (m) = gr (n) the n-th segment. wm,n = ˜ 0 otherwise 1 Deterministic groups: Calendar or Calendar transitions. 2 Groups learned from data via clustering analysis. (e.g. temperature curves) 3 Cross deterministic with clustering groups (e.g. calendar-temperature transitions). Figure: Daily prediction error (in mapex100). GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 24. Motivation Kernel-wavelet functional model Wavelets Clustering functional data using wavelets Prediction algorithm Conditional Autoregression Hilbertian process Corrections to handle nonstationarity Concluding remarks Corrections proposed to handle nonstationarity On groups by post-treatment Define new weights and renormalize. gr (n) is the group of ww ,m if gr (m) = gr (n) the n-th segment. wm,n = ˜ 0 otherwise 1 Deterministic groups: Calendar or Calendar transitions. 2 Groups learned from data via clustering analysis. (e.g. temperature curves) 3 Cross deterministic with clustering groups (e.g. calendar-temperature transitions). Figure: Daily prediction error (in mapex100). GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 25. Predict of 10 September 2005 (Saturday) SimilIndex date SimilIndex 2004-09-10 0.455 2003-09-05 0.141 2002-09-06 0.083 2004-09-03 0.070 2003-09-19 0.068 2000-09-08 0.058 2000-09-15 0.019 1999-09-10 0.017 similar past similar future
  • 26. Motivation Kernel-wavelet functional model Clustering functional data using wavelets Clustering via feature extraction Conditional Autoregression Hilbertian process Concluding remarks Outline 1 Motivation 2 Kernel-wavelet functional model 3 Clustering functional data using wavelets 4 Conditional Autoregression Hilbertian process 5 Concluding remarks GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 27. Motivation Kernel-wavelet functional model Clustering functional data using wavelets Clustering via feature extraction Conditional Autoregression Hilbertian process Concluding remarks Aim Segmentation of X may not suffices to render reasonable the stationary hypothesis. If a grouping effect exists, we may considered stationary within each group. Conditionally on the grouping, functional time series prediction methods can be Two strategies to cluster applied. functional time series: We propose a clustering procedure that 1 Feature extraction discover the groups from a bunch of (summary measures of the curves. curves). We use wavelet transforms to take into account 2 Direct similarity between the fact that curves may present non stationary curves. patters. GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 28. Motivation Kernel-wavelet functional model Clustering functional data using wavelets Clustering via feature extraction Conditional Autoregression Hilbertian process Concluding remarks Aim Segmentation of X may not suffices to render reasonable the stationary hypothesis. If a grouping effect exists, we may considered stationary within each group. Conditionally on the grouping, functional time series prediction methods can be Two strategies to cluster applied. functional time series: We propose a clustering procedure that 1 Feature extraction discover the groups from a bunch of (summary measures of the curves. curves). We use wavelet transforms to take into account 2 Direct similarity between the fact that curves may present non stationary curves. patters. GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 29. Motivation Kernel-wavelet functional model Clustering functional data using wavelets Clustering via feature extraction Conditional Autoregression Hilbertian process Concluding remarks Energy decomposition of the DWT Energy conservation of the signal J−1 2j −1 J−1 2 z ≈ zJ 2 2 = 2 c0,0 + 2 2 dj,k = c0,0 + dj 2 2. j=0 k=0 j=0 For each j = 0, 1, . . . , J − 1, we compute the absolute and relative contribution representations by ||dj ||2 contj = ||dj ||2 and relj = . j ||dj ||2 AC RC They quantify the relative importance of the scales to the global dynamic. RC normalizes the energy of each signal to 1. GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 30. Motivation Kernel-wavelet functional model Clustering functional data using wavelets Clustering via feature extraction Conditional Autoregression Hilbertian process Concluding remarks Schema of procedure 0. Data preprocessing. Approximate sample paths of z1 (t), . . . , zn (t) 1. Feature extraction. Compute either of the energetic components using absolute contribution (AC) or relative contribution (RC). 2. Feature selection. Screen irrelevant variables. [Steinley & Brusco (’06)] 3. Determine the number of clusters. Detecting significant jumps in the transformed distortion curve. [Sugar & James (’03)] 4. Clustering. Obtain the K clusters using PAM algorithm. GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 31. Motivation Kernel-wavelet functional model Clustering functional data using wavelets Clustering via feature extraction Conditional Autoregression Hilbertian process Concluding remarks Electricity demand data Feature extraction: - Significant scales are associated to mid-frequencies. - The retained scales parametrize the represented cycles of 1.5, 3 and 6 hours (AC) and to the cycles of 30 minutes, 1.5 and 3 hours (RC). Number of clusters: 8 for AC and 5 for RC. For instance, we found a class of segments that can be recognized as summer Monday GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 32. Motivation Kernel-wavelet functional model Clustering functional data using wavelets Clustering via feature extraction Conditional Autoregression Hilbertian process Concluding remarks Towards a dissimilarity based on wavelet coherence Distance based on wavelet-correlation between two time series. Can be used to measure relationship between two (possibly non stationary) time series, i.e. temperature and load. The strength of the relationship is hierarchically decomposed on scales (≈frequencies), without loose of time location. GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 33. (a) Curves (b) Calendar Figure: Curves membership of the clustering using wer based dissimilarity (a) and the corresponding calendar positioning (b).
  • 34. Motivation Kernel-wavelet functional model carh: Conditional Autoregressive Hilbertian Model Clustering functional data using wavelets Some results Conditional Autoregression Hilbertian process Concluding remarks Outline 1 Motivation 2 Kernel-wavelet functional model 3 Clustering functional data using wavelets 4 Conditional Autoregression Hilbertian process 5 Concluding remarks GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 35. Motivation Kernel-wavelet functional model carh: Conditional Autoregressive Hilbertian Model Clustering functional data using wavelets Some results Conditional Autoregression Hilbertian process Concluding remarks carh process Let (Z , V ) = {(Zk , Vk ), k ∈ Z} be stationary sequences of H × Rd − valued r.v. defined over (Ω, F, P). We will focus on the behaviour of Z conditioned to V . (Z , V ) is a carh(1) if it is stationary and and such that, Zk = a + ρVk (Zk−1 − a) + k, k ∈ Z, (2) d v where for each v ∈ R , av = E [Z0 |V ], { k }k∈Z is an H−white noise independent of V , and {ρVk }k∈Z is a sequence of linear compact operators. GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 36. Motivation Kernel-wavelet functional model carh: Conditional Autoregressive Hilbertian Model Clustering functional data using wavelets Some results Conditional Autoregression Hilbertian process Concluding remarks Simulation and prediction We extend the simulation strategies for arh processes [Guillas & Damon to the simple case of an carh process with d = 1 and V is a i.i.d. (2000)] sequence of Beta(β1 , β2) rv. Numerical experience: prediction of the electricity demand using the temperature as exogenous information GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 37. Motivation Kernel-wavelet functional model Clustering functional data using wavelets Conditional Autoregression Hilbertian process Concluding remarks Outline 1 Motivation 2 Kernel-wavelet functional model 3 Clustering functional data using wavelets 4 Conditional Autoregression Hilbertian process 5 Concluding remarks GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.
  • 38. Concluding remarks We study the problem of prediction of E[Zn+1 |Zn , Vn+1 ] for H−valued rv. Stationary assumptions are held conditionally on an exogenous rv V . First, V is multiclass and prediction is done by kwf. When the states of V are unknown, clustering is used to discover them. Last, carh uses V ∈ Rd and arh ideas. The contribution of wavelets To exploit information from past data that was observed under a different regime from the actual one. Appropriate tool to detect useful similarities between nonstationary patterns of rough trajectories. Allow fast computations: online prediction version of kwf. The contribution of the kernel regression on fts Accurate alternative for heavy parametric model. Interpretation ability through the study of past similar behaviours. Allow fast computations: online prediction version of kwf. Special attention must be paid to transitions.
  • 39. Motivation Kernel-wavelet functional model Clustering functional data using wavelets Conditional Autoregression Hilbertian process Concluding remarks Some references A. Antoniadis, X. Brossat, J. Cugliari, and J.-M. Poggi. Clustering functional data using wavelets. arXiv:1101.4744, 2011. A. Antoniadis, E. Paparoditis, and T. Sapatinas. A functional wavelet-kernel approach for time series prediction. Journal of the Royal Statistical Society, Series B, Methodological, 68(5):837, 2006. D. Bosq. Linear processes in function spaces: Theory and applications. Springer-Verlag, New York, 2000. A. Mas. Estimation d’opérateurs de corrélation de processus fonctionnels: lois limites, tests, déviations modérées. PhD thesis, Université Paris 6, 2000. J.-M. Poggi. Prévision nonprametrique de la consommation électrique. Rev. Statistiqué Appliquée, XLII(4):93–98, 1994. www.math.u-psud.fr/~cugliari
  • 40. Jairo.Cugliari@math.u-psud.fr GT Prévision | Déc 2011 | J. Cugliari Prévision d’un processus à valeurs fonctionnelles.