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 ). Functional time series Conditional Autoregressive Hilbertian Model ( carh ) Outline 1 Motivation Estimation of Autoregressive Hilbertian process ( arh ). 2 Conditional Autoregressive Hilbertian Model ( carh ) 3 Jairo Cugliari @ JSF, Montpellier | June 2012 Conditional Autoregressif Hilbetian process

Motivation Estimation of Autoregressive Hilbertian process ( arh ). Functional time series Conditional Autoregressive Hilbertian Model ( carh ) 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 = ( Z k , k ∈ N ) , where N = { 1 , 2 , . . . } , defined by X t t 0 T T + δ Jairo Cugliari @ JSF, Montpellier | June 2012 Conditional Autoregressif Hilbetian process

Motivation Estimation of Autoregressive Hilbertian process ( arh ). Functional time series Conditional Autoregressive Hilbertian Model ( carh ) 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 = ( Z k , k ∈ N ) , where N = { 1 , 2 , . . . } , defined by X t t 0 T T + δ Jairo Cugliari @ JSF, Montpellier | June 2012 Conditional Autoregressif Hilbetian process

Motivation Estimation of Autoregressive Hilbertian process ( arh ). Functional time series Conditional Autoregressive Hilbertian Model ( carh ) 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 = ( Z k , k ∈ N ) , where N = { 1 , 2 , . . . } , defined by X t Z 3 ( t ) Z 4 ( t ) Z 6 ( t ) Z k ( t ) = X ( t + ( k − 1 ) δ ) Z 1 ( t ) Z 2 ( t ) Z 5 ( t ) k ∈ N ∀ t ∈ [ 0 , δ ) t 0 1 δ 2 δ 3 δ 4 δ 5 δ 6 δ T + δ Jairo Cugliari @ JSF, Montpellier | June 2012 Conditional Autoregressif Hilbetian process

Motivation Estimation of Autoregressive Hilbertian process ( arh ). Functional time series Conditional Autoregressive Hilbertian Model ( carh ) 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 = ( Z k , k ∈ N ) , where N = { 1 , 2 , . . . } , defined by X t Z 3 ( t ) Z 4 ( t ) Z 6 ( t ) Z k ( t ) = X ( t + ( k − 1 ) δ ) Z 1 ( t ) Z 2 ( t ) Z 5 ( t ) k ∈ N ∀ t ∈ [ 0 , δ ) t 0 1 δ 2 δ 3 δ 4 δ 5 δ 6 δ T + 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 ). Functional time series Conditional Autoregressive Hilbertian Model ( carh ) Prediction of functional time series Let ( Z k , k ∈ Z ) be a stationary sequence of H -valued r.v. Given Z 1 , . . . , Z n we want to predict the future value of Z n + 1 . A predictor of Z n + 1 using Z 1 , Z 2 , . . . , Z n is � Z n + 1 = E [ Z n + 1 | Z n , Z n − 1 , . . . , Z 1 ] . Autoregressive Hilbertian process ( arh ) When Z is a zero-mean arh (1) process, then for each k , Z k = ρ ( Z k − 1 ) + ǫ n where ρ is a compact linear operator and { ǫ k } k ∈ Z is an H − valued strong white noise.Then, the best predictor of Z n + 1 given { Z 1 , . . . , Z n − 1 } is: E [ Z n + 1 | Z n , Z n − 1 , . . . , Z 1 ] = ρ ( Z n ) . Two strategies: Estimate ρ with � ρ n , then � ρ n ( Z n ) is the prediction [Bosq (1991), Besse 1 and Cardot (1996), Pumo (1998), Mas (2000)] Directly estimate the functional observation � ρ ( Z n ) [Antoniadis and 2 Sapatinas (2003)] Jairo Cugliari @ JSF, Montpellier | June 2012 Conditional Autoregressif Hilbetian process

Motivation Estimation of Autoregressive Hilbertian process ( arh ). Functional time series Conditional Autoregressive Hilbertian Model ( carh ) Nonparametric approach [Antoniadis et al. (2006), Cugliari(2011)] Estimate � Z n + 1 = E [ Z n + 1 | Z n , Z n − 1 , . . . , Z 1 ] assuming that similar futures correspond to similar pasts. The resulting predictor � Z n + 1 ( t ) is obtained by a kernel regression of Z n over the history { Z n − 1 , . . . , Z 1 } . 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 , � n − 1 � Z n + 1 ( t ) = w n , m Z m + 1 ( t ) . m = 1 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 ). Functional time series Conditional Autoregressive Hilbertian Model ( carh ) 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 of 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 Estimation of Autoregressive Hilbertian process ( arh ). 2 Conditional Autoregressive Hilbertian Model ( carh ) 3 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 E � Z � 4 H < ∞ , three operators can be defined Γ = E [( Z 0 − a ) ⊗ ( Z 0 − a )] covariance operator, ∆ = E [( Z 0 − a ) ⊗ ( Z 1 − 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 = λ j ( e j ⊗ e j ) j ∈ N where λ 1 ≥ λ 2 ≥ . . . ≥ 0 are the eigenvalues and ( e 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 ) Estimation of ρ (1/2) [Mas, 2000] Two relation between the operators ∆ ∗ = Γ ρ ∗ . ∆ = ρ Γ and 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 ( e j ⊗ e j ) within a dense domain D Γ − 1 ⊂ H . j 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 (∆ ∗ ) ⊂ D Γ − 1 allow one to write ρ ∗ = Γ − 1 ∆ ∗ . Then, classical results on linear operators allow us to extend ρ |D Γ − 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 operator. Jairo Cugliari @ JSF, Montpellier | June 2012 Conditional Autoregressif Hilbetian process