Prévision consommation électrique par processus à valeurs fonctionnelles
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Jairo Cugliari

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Prévision consommation électrique par processus à valeurs fonctionnelles Presentation Transcript

  • 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 remarksOutline 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 remarksOutline 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 remarksFD 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 remarksFD 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 remarksFD 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 remarksFD 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 remarksPrediction of functional time seriesLet (Zk , k ∈ Z) be a stationary sequence of H-valued r.v. Given Z1 , . . . , Zn wewant 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 1The 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 whitenoise.Under mild conditions, equation (1) has a unique solution which is a strictlystationary 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 remarksPrediction of functional time seriesLet (Zk , k ∈ Z) be a stationary sequence of H-valued r.v. Given Z1 , . . . , Zn wewant 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 1The 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 whitenoise.Under mild conditions, equation (1) has a unique solution which is a strictlystationary 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 remarksElectricity demand dataSome 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 remarksElectricity 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 remarksElectricity 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 remarksOutline 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 remarksKernel 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 remarksKernel 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 remarksApproximation 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 remarksA 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 remarksA 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 remarksCorrections 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 remarksCorrections 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 remarksCorrections 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 remarksCorrections 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 remarksCorrections 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 remarksOutline 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 remarksAim 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 betweenthe 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 remarksAim 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 betweenthe 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 remarksEnergy 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 remarksSchema 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 remarksElectricity 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 remarksTowards a dissimilarity based on wavelet coherenceDistance based on wavelet-correlation between two time series.Can be used to measure relationship between two (possibly nonstationary) 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) CalendarFigure: 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 remarksOutline 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 remarkscarh 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 remarksSimulation 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 remarksOutline 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 remarksSome 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.