Jairo Cugliari

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.
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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.