Conditional Autoregressif Hilbetian process Application to the - - PowerPoint PPT Presentation

Conditional Autoregressif Hilbetian process

Application to the electricity demand Jairo Cugliari

SELECT Research Team

JSF2012, Montpellier 29th June 2012

Motivation Estimation of Autoregressive Hilbertian process (arh). Conditional Autoregressive Hilbertian Model (carh) Functional time series

Outline

1

Motivation

2

Estimation of Autoregressive Hilbertian process (arh).

3

Conditional Autoregressive Hilbertian Model (carh)

Jairo Cugliari @ JSF, Montpellier | June 2012 Conditional Autoregressif Hilbetian process

Motivation Estimation of Autoregressive Hilbertian process (arh). Conditional Autoregressive Hilbertian Model (carh) Functional time series

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 T T + δ

Jairo Cugliari @ JSF, Montpellier | June 2012 Conditional Autoregressif Hilbetian process

Motivation Estimation of Autoregressive Hilbertian process (arh). Conditional Autoregressive Hilbertian Model (carh) Functional time series

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 T T + δ

Jairo Cugliari @ JSF, Montpellier | June 2012 Conditional Autoregressif Hilbetian process

Motivation Estimation of Autoregressive Hilbertian process (arh). Conditional Autoregressive Hilbertian Model (carh) Functional time series

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 1δ 2δ 3δ 4δ 5δ 6δ T + δ

Z1(t) Z2(t) Z5(t) Z3(t) Z4(t) Z6(t)

Zk(t) = X(t + (k − 1)δ) k ∈ N ∀t ∈ [0, δ)

Jairo Cugliari @ JSF, Montpellier | June 2012 Conditional Autoregressif Hilbetian process

Motivation Estimation of Autoregressive Hilbertian process (arh). Conditional Autoregressive Hilbertian Model (carh) Functional time series

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 1δ 2δ 3δ 4δ 5δ 6δ T + δ

Z1(t) Z2(t) Z5(t) Z3(t) Z4(t) Z6(t)

Zk(t) = X(t + (k − 1)δ) k ∈ N ∀t ∈ [0, δ) If X contents a δ−seasonal component, Z is particularly fruitful.

Jairo Cugliari @ JSF, Montpellier | June 2012 Conditional Autoregressif Hilbetian process

Motivation Estimation of Autoregressive Hilbertian process (arh). Conditional Autoregressive Hilbertian Model (carh) Functional time series

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 (arh) When Z is a zero-mean arh(1) process, then for each k, Zk = ρ(Zk−1) + ǫn where ρ is a compact linear operator and {ǫk}k∈Z is an H−valued strong white noise.Then, the best predictor of Zn+1 given {Z1, . . . , Zn−1} is: E[Zn+1|Zn, Zn−1, . . . , Z1] = ρ(Zn). Two strategies:

1

Estimate ρ with ρn, then ρn(Zn) is the prediction [Bosq (1991), Besse

and Cardot (1996), Pumo (1998), Mas (2000)]

2

Directly estimate the functional observation ρ(Zn) [Antoniadis and

Sapatinas (2003)]

Jairo Cugliari @ JSF, Montpellier | June 2012 Conditional Autoregressif Hilbetian process

Motivation Estimation of Autoregressive Hilbertian process (arh). Conditional Autoregressive Hilbertian Model (carh) Functional time series

Nonparametric approach [Antoniadis et al. (2006), Cugliari(2011)] Estimate Zn+1 = E[Zn+1|Zn, Zn−1, . . . , Z1] assuming that similar futures correspond to similar pasts. The resulting predictor Zn+1(t) is obtained by a kernel regression of Zn

• ver the history {Zn−1, . . . , Z1}.

It is a weighted mean of futures of past segments where weights increase with the similarity between last observed segment n and past segments m = 1, . . . , n,

• Zn+1(t) =

n−1

• m=1

wn,mZm+1(t). In the context of a practical application (e.g. french electrical power demand), data may fail to verify the hypothesis of stationarity. However, using corrections that take into account the presence of the nonstationarity the prediction performance is enhance.

Jairo Cugliari @ JSF, Montpellier | June 2012 Conditional Autoregressif Hilbetian process

Motivation Estimation of Autoregressive Hilbertian process (arh). Conditional Autoregressive Hilbertian Model (carh) Functional time series

Adding exogenous information to the predictors

If additional exogenous information where available, we may want to use it. We aim here at introducing an exogenous covariate in the arh process in such a way that conditionally on the covariate the process becomes an arh. Damon and Guillas (2002) introduce function-valued regressors (that are assumed to follow arh processes). Mas and Pumo (2007) use the derivative of the (function-valued) variable as covariate in the arh framework. Guillas (2002) proposes to model an arh process that randomly chooses between two possible regimes. The resulting process admits to have one

• f the regimes to be explosive if it is not visited too often.

Jairo Cugliari @ JSF, Montpellier | June 2012 Conditional Autoregressif Hilbetian process

Motivation Estimation of Autoregressive Hilbertian process (arh). Conditional Autoregressive Hilbertian Model (carh)

Outline

1

Motivation

2

Estimation of Autoregressive Hilbertian process (arh).

3

Conditional Autoregressive Hilbertian Model (carh)

Jairo Cugliari @ JSF, Montpellier | June 2012 Conditional Autoregressif Hilbetian process

Motivation Estimation of Autoregressive Hilbertian process (arh). Conditional Autoregressive Hilbertian Model (carh)

Covariance operators

If EZ4

H < ∞, three operators can be defined

Γ = E[(Z0 − a) ⊗ (Z0 − a)] covariance operator, ∆ = E[(Z0 − a) ⊗ (Z1 − a)] cross-covariance operator, where x ∈ H → (u ⊗ v)(x) =< u, x > v. Properties: trace-class operators, Hilbert-Schimdt (additionally Γ is positive definite and selfadjoint) Spectral decomposition of Γ Γ =

• j∈N

λj(ej ⊗ ej) where λ1 ≥ λ2 ≥ . . . ≥ 0 are the eigenvalues and (ej)j∈N the associated eigenfunctions.

Jairo Cugliari @ JSF, Montpellier | June 2012 Conditional Autoregressif Hilbetian process

Motivation Estimation of Autoregressive Hilbertian process (arh). Conditional Autoregressive Hilbertian Model (carh)

Estimation of ρ

(1/2)

[Mas, 2000]

Two relation between the operators ∆ = ρΓ and ∆∗ = Γρ∗. If dim(H) < ∞, the inversion of the operator Γ gives us a way to estimate ρ. Problem In the general case, the inverse of Γ is a problem: the operator is not bounded and may not be defined over the whole space H. However, for a well identify ρ we can define a linear measurable mapping Γ−1 =

j∈N λ−1 j

(ej ⊗ ej) within a dense domain DΓ−1 ⊂ H.

Jairo Cugliari @ JSF, Montpellier | June 2012 Conditional Autoregressif Hilbetian process

Motivation Estimation of Autoregressive Hilbertian process (arh). Conditional Autoregressive Hilbertian Model (carh)

Estimation of ρ

(2/2)

[Mas, 2000]

Two important facts the adjoint of a linear operator in H with a dense domain is closed (closed graph theorem) the range(∆∗) ⊂ DΓ−1 allow one to write ρ∗ = Γ−1∆∗. Then, classical results on linear operators allow us to extend ρ|DΓ−1 by continuity to H. This allow us to focus on the estimation of ρ∗, the theoretical properties of a such estimator are applicable to ρ through the composition of ρ∗ by the adjoint

• perator.

Jairo Cugliari @ JSF, Montpellier | June 2012 Conditional Autoregressif Hilbetian process

Motivation Estimation of Autoregressive Hilbertian process (arh). Conditional Autoregressive Hilbertian Model (carh)

Two classes of estimators of ρ∗ [Mas, (2000)]

1

Projection estimator class. Inverts the projection of Γn over Hkn of finite dimension kn, ρ∗

n = (PknΓnPkn)−1∆∗ nPkn

where Pkn is the projector operator over Hkn.

2

Resolvent estimator class. A whole class of estimators can be obtained using the resolvent of Γ ρ∗

n,p,α = bn,p,α(Γn)∆∗ n

where we write bn,p,α(Γn) = (Γn + αnI)−(p+1) with p ≥ 0, αn ≥ 0, n ≥ 0.

Jairo Cugliari @ JSF, Montpellier | June 2012 Conditional Autoregressif Hilbetian process

Motivation Estimation of Autoregressive Hilbertian process (arh). Conditional Autoregressive Hilbertian Model (carh) Some results

Outline

1

Motivation

2

Estimation of Autoregressive Hilbertian process (arh).

3

Conditional Autoregressive Hilbertian Model (carh)

Jairo Cugliari @ JSF, Montpellier | June 2012 Conditional Autoregressif Hilbetian process

Motivation Estimation of Autoregressive Hilbertian process (arh). Conditional Autoregressive Hilbertian Model (carh) Some results

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 . Assume that a transition probability exists that associates to each v ∈ Rd a probability measure Pv on (H, B) such that Pv

Z|V (A) = Pv(A),

for every A ∈ B, v ∈ Rd. The probability measure Pv induces the expectation Ev. Let H = {z(t), t ∈ [0, 1] : z(t) is continuous and Ez4

H < ∞}

Jairo Cugliari @ JSF, Montpellier | June 2012 Conditional Autoregressif Hilbetian process

Motivation Estimation of Autoregressive Hilbertian process (arh). Conditional Autoregressive Hilbertian Model (carh) Some results

(Z, V ) is a carh(1) if it is stationary and and such that, Zk = a + ρVk (Zk−1 − a) + ǫk, k ∈ Z, (1) where for each v ∈ Rd, av = Ev[Z0|V ], {ǫk}k∈Z is an H−white noise independent of V , and {ρVk }k∈Z is a sequence of linear compact operators. Theorem (Existence and uniqueness) If supn ρVnL < 1 a.s., then (1) defines a carh process with an unique stationary solution given by Zk = a +

∞

• j=0

j−1

• p=0

ρVk−p

• (ǫk−j),

with the convention j−1

p=0 ρVk−p is the identity operator for j = 0.

Jairo Cugliari @ JSF, Montpellier | June 2012 Conditional Autoregressif Hilbetian process

Motivation Estimation of Autoregressive Hilbertian process (arh). Conditional Autoregressive Hilbertian Model (carh) Some results

Conditional covariance operators

Conditional covariance and cross covariance operators (on V at the point v ∈ Rd) are respectively defined by z ∈ H → Γvz = Ev[(Z0 − a) ⊗ (Z0 − a)(z)|V ] and z ∈ H → ∆vz = Ev[(Z0 − a) ⊗ (Z1 − a)(z)|V ], where x ∈ H → (u ⊗ v)(x) =< u, x > v. For each v ∈ Rd: these are trace-class operators, thus Hilbert-Schimdt (additionally Γv is positive definite and selfadjoint) Spectral decomposition of Γv, Γv =

j∈N λv,j(ev,j ⊗ ev,j), where

λv,1 ≥ λv,2 ≥ . . . ≥ 0 are the eigenvalues and (ev,j)j∈N the associated eigenfunctions.

Jairo Cugliari @ JSF, Montpellier | June 2012 Conditional Autoregressif Hilbetian process

Motivation Estimation of Autoregressive Hilbertian process (arh). Conditional Autoregressive Hilbertian Model (carh) Some results

Estimation of conditional covariance operators

Now, for (s, t) ∈ [0, 1]2, v ∈ Rd, Γvz(t) =

1

γ(v, s, t)z(t)ds, ∆vz(t) =

1

δ(v, s, t)z(t)ds. with γ(v, ., .) and δ(v, ., .) continuous kernels (additionally, γ is symmetric and positive definite.) The kernels turn to be the conditional covariance function and the one-ahead conditional cross-covariance function valued at the point v ∈ Rd, γ(v, s, t) = Ev[(Z0(s) − a(s))(Z0(t) − a(t))|V ], δ(v, s, t) = Ev[(Z0(s) − a(s))(Z1(t) − a(t))|V ], (s, t) ∈ [0, 1]2. Finally, we have for each v ∈ Rd ∆v = ρvΓv.

Jairo Cugliari @ JSF, Montpellier | June 2012 Conditional Autoregressif Hilbetian process

Motivation Estimation of Autoregressive Hilbertian process (arh). Conditional Autoregressive Hilbertian Model (carh) Some results

Estimation of the conditional covariance operators

Nonparametric Nadaraya-Watson like estimators. Context of dependent data (α−mixing framework) ˆ an(v) =

n

• i=1

wn,i(v, ha)Zi ˆ Γn,v =

n

• i=1

wn,i(v, hγ)(Zi − ˆ an(v)) ⊗ (Zi − ˆ an(v)) ˆ ∆n,v =

n

• i=2

wn,i(v, hδ)(Zi−1 − ˆ an(v)) ⊗ (Zi − ˆ an(v)) where the weights wn,i are defined by wn,i(v, h) = K(h−1(Vi − v))

n

i=1 K(h−1(Vi − v)).

(2)

Jairo Cugliari @ JSF, Montpellier | June 2012 Conditional Autoregressif Hilbetian process

Motivation Estimation of Autoregressive Hilbertian process (arh). Conditional Autoregressive Hilbertian Model (carh) Some results

Estimation of ρv

Let us call Pkn

v

the projection operator from H to Hkn

v . Then, we define the

projection estimator of ρ∗

v by

• ρ∗

v,n = (Pkn v

Γv,nPkn

v )−1

∆∗

v,nPkn v .

(3) As before, be can also define a class of resolvent estimators by

• ρ∗

v,n,p = bn,p(

Γv,n) ∆∗

v,n,

(4) which is parametrized by p ≥ 0 and b > 0. The same kind of remarks done for the arh case stands here. Finally, we obtain the convergence on probability of both predictors ρ∗

v,n(Zn+1)

and ρ∗

v,n,p(Zn+1).

Jairo Cugliari @ JSF, Montpellier | June 2012 Conditional Autoregressif Hilbetian process

Motivation Estimation of Autoregressive Hilbertian process (arh). Conditional Autoregressive Hilbertian Model (carh) Some results

Simulation and prediction

We extend the simulation strategies for arh processes [Guillas & Damon

(2000)] to the simple case of an carh process with d = 1 and V is a i.i.d.

sequence of Beta(β1, β2) rv. Numerical experience: prediction of the electricity demand using the temperature as exogenous information Figure: Prediction of one simulated curve of an carh process (full line).

Jairo Cugliari @ JSF, Montpellier | June 2012 Conditional Autoregressif Hilbetian process

Motivation Estimation of Autoregressive Hilbertian process (arh). Conditional Autoregressive Hilbertian Model (carh) Some results

Some references

• 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.

• J. Cugliari.

Prévision non paramétrique de processus à valeurs fonctionnelles. Application à la consommation d’électricité. PhD thesis, Université Paris Sud, 2011.

• A. Mas.

Estimation d’opérateurs de corrélation de processus fonctionnels: lois limites, tests, déviations modérées. PhD thesis, Université Paris 6, 2000. ☞Jairo.Cugliari@math.u-psud.fr http://www.math.u-psud.fr/~cugliari

Jairo Cugliari @ JSF, Montpellier | June 2012 Conditional Autoregressif Hilbetian process