Bootstrap condence intervals in functional regression under dependence

Transcription

1 condence intervals in functional regression under dependence Juan Manuel Vilar Fernández Paula Raña, Germán Aneiros and Philippe Vieu Departamento de Matemáticas, Universidade da Coruña Galician Seminar of Nonparametric Statistical Inference 8-9 June 2016

2 1 Introduction 2 3 4

3 Spanish Electricity Market OMIE: `Operador del Mercado Ibérico de Energía' Demand (MWh) Price (Cents/kWh) Hour Hour Figure : Electricity demand and price daily curves in 2012.

4 Electricity demand Introduction 2012 Weekdays Demand (MWh) Demand (MWh) Time Saturday Hour Sunday Demand (MWh) Demand (MWh) Hour Hour

5 Electricity price Introduction 2012 Weekdays Price (Cents/kWh) Price (Cents/kWh) Time Saturday Hour Sunday Price (Cents/kWh) Price (Cents/kWh) Hour Hour

6 Functional time series Daily curves of electricity demand or price along 2012: {χ i } 365 i=1 Discretized curves: χ i (t j ), j = 1,..., 24. Functional time series: {χ i } n i=1 Real-valued continuous time stochastic process {χ (t)} t R Seasonal process, with seasonal length τ, observed on the interval (a, b] with b = a + nτ. Functional time series, {χ i } n i=1, is dened in terms of {χ (t)} t R as: χ i (t) = χ (a + (i 1) τ + t) with t [0, τ).

7 1 Introduction 2 3 4

8 Objective Predict nextday electricity demand/price in Spain during {χ i } N i=1 χ N+1 Functional Autoregressive models Functional nonparametric regression. Semi-functional partial linear regression. Covariates Electricity demand: meteorological variables, temperature. Electricity price: demand, wind power production.

9 Funtional Nonparametric Regression Functional explanatory variable and scalar response Autoregressive model G(χ i+1 ) = m(χ i ) + ε i (i = 1,..., n) General model Y i = m(χ i ) + ε i,i = 1,..., n where {(χ i, Y i )} is α-mixing Functional kernel estimator n i=1 m h (χ) = K(d(χ i, χ)/h)y i n i=1 K(d(χ i, χ)/h) = n w h (χ i, χ)y i i=1

10 Semi-Funtional Partial Linear Regression Functional nonparametric explanatory variable, scalar linear-eect covariate and scalar response Autoregressive model G(χ i+1 ) = X T i β + m(χ i ) + ε i, i = 1,..., n General model Y i = X T i β + m(χ i ) + ε i, i = 1,..., n, where {(X i, χ i, Y i )} is α-mixing

11 Estimators Introduction Denote X = (X 1,..., X n) T, Y = (Y 1,..., Y n ) T, W h = (w h (χ i, χ j )) and, for any (n q) matrix A (q 1), Estimators à h = (I W h )A. β h = ( X T h X h ) 1 X T h Ỹh m h (χ) = Nadaraya-Watson type weights w h (χ i, χ) = n w h (χ i, χ)(y i X T i β h ) i=1 K(d(χ i, χ)/h) n i=1 K(d(χ i, χ)/h), where K( ) is a real function (the kernel) and h > 0 is a smoothing parameter.

12 In practice: predict electricity demand Functional nonparametric regression Functional explanatory variable: previous daily demand curves. Scalar response: electricity demand for the next day, xed hour. Semi-Functional Partial Linear Regression Scalar explanatory variable: daily temperature linear eect. Demand (MWh) HDD CDD Temperature (ºC) Temperature (ºC) Temperature (ºC)

13 In practice: predict electricity price Functional nonparametric regression Functional explanatory variable: previous daily price curves. Scalar response: electricity price for the next day, xed hour. Semi-Functional Partial Linear Regression Scalar explanatory variable: daily demand, wind power. Price (Cents/KWh) Price (Cents/KWh) Demand (MWh) Wind Power

14 1 Introduction 2 3 4

15 Naive bootstrap Introduction From a general functional nonparametric regression model, Y i = m(χ i ) + ε i, built from the sample S = {(χ i, Y i )} n i=1 : Homoscedastic model Naive bootstrap 1 Construct the residuals ε i,b = Y i m b (χ i ), i = 1,..., n. 2 Draw n i.i.d random variables ε 1,..., ε n from the empirical distribution function of ( ε 1,b ε b,..., ε n,b ε b ), where ε b = n 1 n i=1 ε i,b. 3 Obtain Y i = m b (χ i ) + ε i, i = 1,..., n S = {(χ i, Y i )} n i=1 n 4 Dene m hb (χ) = i=1 K(d(χ i, χ)/h)y i n i=1 K(d(χ i, χ)/h).

16 Wild bootstrap Introduction Heteroscedastic model Wild bootstrap 1 Construct the residuals ε i,b = Y i m b (χ i ), i = 1,..., n. 2 Dene ε i = ε i,b V i, i = 1,..., n, where V 1,..., V n are i.i.d. random variables that are independent of the data S and that satisfy E(V 1 ) = 0 and E(V 2 1 ) = 1. 3 Obtain Y i = m b (χ i ) + ε i, i = 1,..., n S = {(χ i, Y i )} n i=1 n 4 Dene m hb (χ) = i=1 K(d(χ i, χ)/h)y i n i=1 K(d(χ i, χ)/h).

17 1 Introduction 2 3 4

18 Notation Introduction For a given xed element χ 0 of the space H, we denote: B(χ 0, l) = {χ 1 H such that d(χ 1, χ 0 ) l}, F χ0 (l) = P(χ B(χ 0, l)) for l > 0, ϕ χ0 (s) = E(m(χ) m(χ 0 ) d(χ, χ 0 ) = s) τ hχ0 (s) = F χ0 (hs)/f χ0 (h) for s (0, 1] and τ 0χ0 (s) = lim τ hχ0 (s). h 0

19 Notation Introduction M 0χ0 = K(1) M 1χ0 = K(1) M 2χ0 = K 2 (1) (sk(s)) τ 0χ0 (s)ds, K (s)τ 0χ0 (s)ds, (K 2 (s)) τ 0χ0 (s)ds and Θ(s) = max{max P(d(χ i, χ 0 ) s, d(χ j, χ 0 ) s), Fχ 2 i j 0 (s)}.

20 Assumptions for the convergence of m h (χ) Distribution m( ) and σ 2 ε( ) are continuous on a neighbourhood of χ 0 ; σ 2 ε(χ 0 ) > 0. F χ0 (0) = 0 and ϕ χ0 (0) = 0 and ϕ χ 0 (0) exists. s [0, 1], lim τ hχ n 0 (s) = τ 0χ0 (s) with τ 0χ0 (s) 1 [0,1] (s). Moments p > 2, M > 0 such that E( ε p χ) M a.s. max{e( Y i Y j p χ i, χ j ), E( Y i p χ i, χ j )} M a.s. i, j Z. Small ball probabilities h(nf χ0 (h)) 1/2 = O(1) and lim nf χ n 0 (h) =.

21 Assumptions for the convergence of m h (χ 0 ) Kernel K( ) is supported on [0, 1], has a continuous derivative on [0, 1), K (s) 0 for s [0, 1) and K(1) > 0. Dependence structure {(χ i, Y i )} n i=1 comes from a α-mixing process with α-mixing coecients α(n) Cn a, where a is given by: v > 0 such that Θ(h) = O(F χ0 (h) 1+v (1 + v)p 2 ) with a > v(p 2) { } 4 γ > 0/ nf χ0 (h) 1+γ and a > max γ, p 2(p 1) + p 2 γ(p 2) 0 Delsol (2009)

22 Assumptions for the convergence of m hb (χ 0) Moments Function E( Y χ = ) is continuous on a neighbourhood of χ 0, and sup d(χ1,χ 0 )<δ E( Y q χ = χ 1 ) < for some δ > 0; q 1. Distribution (χ 1, s) in neighbourhood of (χ 0, 0), ϕ χ1 (0) = 0, ϕ χ 1 (s), ϕ χ 1 (0) 0 and ϕ χ 1 (s) uniformly Lipschitz continuous, order 0 < α 1 in (χ 1, s). χ 1 H, F χ1 (0) = 0 and F χ1 (t)/f χ0 (t) Lipschitz continuous, order α in χ 1, uniformly in t in neighbourhood of 0. 0 Ferraty, van Keilegom and Vieu (2010)

23 Assumptions for the convergence of m hb (χ 0) Distribution χ 1 H and s [0, 1], τ 0χ1 (s) exists, sup χ1 H,s [0,1] τ hχ1 (s) τ 0χ1 (s) = o(1), M 0χ0 > 0, M 2χ0 > 0, inf d(χ1,χ 0 )<ε M 1χ0 > 0 for some ε > 0, and M kχ1 is Lipschitz continuous of order α for k = 0, 1, 2. n r n 1, l n > 0 and curves χ 1n,..., χ rnn such that B(χ 0, h) rn k=1 B(χ kn, l n), r n = O(n b/h ) and l n = o(b(nf χ0 (h)) 1/2 ), inf d(χ1,χ 0 )<ε M 1χ0 > 0 for some ε > 0, M kχ1 is Lipschitz continuous of order α for k = 0, 1, 2. Small ball probabilities max{b, h/b, b 1+α (nf χ0 (h)) 1/2, (F χ0 (h)/f χ0 (b)) log n, n 1/p F χ0 (h) 1/2 log n} = o(1) max{bh α 1, F χ0 (b) 1 h/b} = O(1) and lim F χ 0 (b + h)/f χ0 (b) = 1. n 0 Ferraty, van Keilegom and Vieu (2010)

24 Validity of the bootstrap Theorem Under previous assumptions, for the wild bootstrap procedure, we have that sup y R P S ( nf χ (h)( m hb (χ) m b(χ)) y ) P ( nf χ (h)( m h (χ) m(χ)) y ) 0 a.s. In addition, if the model is homoscedastic (i.e. σε( ) 2 = σε 2 ), then the same result holds for the naive bootstrap.

25 Background Introduction Result for independent data Ferraty, Van Keilegom and Vieu (2010) On the Validity of the in Non-Parametric Functional Regression. Asymptotic distribution of m h (χ) m(χ) for independent data Ferraty, Mas and Vieu (2007) Nonparametric Regression on Functional data: Inference and Practical Aspects. Asymptotic distribution of m h (χ) m(χ) for dependent data Delsol (2009) Advances on asymptotic normality in non-parametric functional time series analysis.

26 Proof outline Introduction P S ( nf χ (h)( m hb (χ) m b(χ)) y ) P ( nf χ (h)( m h (χ) m(χ)) y ) = T 1 (y) + T 2 (y) + T 3 (y) where ) T 1 (y) = P ( nf S χ (h)( m hb (χ) m b(χ)) y Φ y nf χ (h) ( E S ( m hb (χ)) m b (χ) ) nf χ (h)var ( S m hb (χ)) 0 P S : probability conditionally on S = {(χ i, Y i)} n i=1

27 Proof outline Introduction T 2 (y) = Φ y nf χ (h) ( E S ( m hb (χ)) m b (χ) ) nf χ (h)var ( S m hb (χ)) ( y ) nf χ (h) (E ( m h (χ)) m(χ)) Φ nfχ (h)var ( m h (χ)) and T 3 (y) = Φ ( y ) nf χ (h) (E ( m h (χ)) m(χ)) nfχ (h)var ( m h (χ)) ( ) P nf χ (h)( m h (χ) m(χ)) y

28 Proof outline Introduction T 3 (y) = Φ ( y ) nf χ (h) (E ( m h (χ)) m(χ)) nfχ (h)var ( m h (χ)) ( ) P nf χ (h)( m h (χ) m(χ)) y Delsol (2009) m h (χ) E ( m h (χ)) Var ( mh (χ)) d N(0, 1), a.s. T 3 (y) 0 a.s. for any xed value of y.

29 Proof outline Introduction ) T 1 (y) = P ( nf S χ (h)( m hb (χ) m b(χ)) y Φ y nf χ (h) ( E S ( m hb (χ)) m b (χ) ) nf χ (h)var ( S m hb (χ)) Lemma: adapted from Ferraty, van Keilegom and Vieu (2010) m hb (χ) ES ( m hb (χ)) Var S ( m hb (χ)) d N(0, 1), a.s.(p S ) T 1 (y) 0 a.s. for any xed value of y.

30 Proof outline Introduction T 2 (y) = Φ y nf χ (h) ( E S ( m hb (χ)) m b (χ) ) nf χ (h)var ( S m hb (χ)) ( y ) nf χ (h) (E ( m h (χ)) m(χ)) Φ nfχ (h)var ( m h (χ)) Lemma: adapted from Ferraty, van Keilegom and Vieu (2010) nf χ (h) ( E ( m h (χ)) m(χ) E S ( m hb (χ)) + m b(χ) ) 0 a.s. sup T 2 (y) 0 a.s., y R

31 1 Introduction 2 3 4

32 Simulation procedure: building condence intervals Given a curve χ and the FNP regression model Y i = m(χ i ) + ε i (i = 1,..., n), where the process {(χ i, Y i )} is α-mixing and identically distributed as (χ, Y ), and χ is observed from χ, the true, bootstrap and asymptotic (1 α)-condence intervals for m(χ) were constructed: I true χ,1 α = ( m h (χ) + q true α/2 (χ), m h(χ) + q true 1 α/2 (χ)) I χ,1 α = ( m h (χ) + q α/2 (χ), m h(χ) + q 1 α/2 (χ)) I asymp χ,1 α = ( m h(χ) + q asymp α/2 (χ), m h (χ) + q asymp 1 α/2 (χ))

33 Theoretical quantiles Theoretical quantiles (qp true (χ)) 1 Generate n MC samples {(χ s i, Y i s ), i = 1,..., n} n MC s=1 Model. from FNP 2 Carry out n MC = 2000 estimates { m s h (χ)} n MC s=1, where ms h ( ) is the functional kernel estimator derived from the s th sample {(χ s i, Y i s )} n i=1. 3 Compute the set of approximation errors ERRORS.MC = { m s h (χ) m(χ)} n MC s=1. 4 Compute the theoretical quantile, qp true order p of ERROR.MC. (χ), from the quantile of

34 quantiles quantiles (q p(χ)) 1 Generate the sample S = {(χ 1, Y 1 ),..., (χ n, Y n )} from FNP Model. 2 Compute m b (χ) over the dataset S. 3 Repeat B = 500 times the bootstrap algorithm over S by using i.i.d. random variables V i drawn from the two Dirac distributions 0.1(5 + 5)δ (1 5)/ (5 5)δ (1+ 5)/2, giving the B estimates { m,r hb (χ)} B r=1. 4 Compute set of bootstrap errors ERRORS.BOOT { m,r hb (χ) m b(χ) } B r=1. 5 Compute the bootstrap quantile, q p(χ), from the quantile of order p of ERRORS.BOOT.

35 Asymptotic quantiles Asymptotic quantiles (qp asymp (χ)) 1 Generate the sample S = {(χ 1, Y 1 ),..., (χ n, Y n )} from FNP Model. 2 Use the sample S to estimate the constants F χ (h), M 1χ, M 2χ and σ ε as suggested in Delsol (2009). 3 Compute the asymptotic quantile, qp asymp (χ), from the quantile of order p of the corresponding normal distribution.

36 Simulation procedure m h (χ) in each of the three intervals was obtained from S Test sample C = {χ 1,..., χ nc }, consisting in n C = 100 independent curves Empirical coverages: repeat the procedure M = 500 times and computing the proportion of times that each interval contains the value m(χ)

37 Model 1: Smooth curves FNP regression model Y i = m(χ i ) + ε i Functional covariate χ i (t j ) = cos(a i + π(2t j 1)) Regression operator m(χ) = 1 2π 3/4 1/2 (χ (t)) 2 dt {a i } AR(1) gaussian process with correlation coecient ρ a = 0.7 and variance σ 2 a = = t 1 < < t 100 = 1 {ε i } N(0, σ 2 ), σ 2 = 0.1Var(m(χ 1 ),..., m(χ n )) semi-metric d deriv 1 (, ) d deriv 1 (χ i, χ j ) = 1 (χ (t) i χ j (t))2 dt, 0

38 Simulated data: Model 1 t(curves.c)

39 Model 2: Rough curves FNP regression model Y i = m(χ i ) + ε i Functional covariate j χ i (t j ) = b 2i cos(b 1i t j ) + B ik /b Regression operator m(χ) = π k=1 (χ(t)) 2 dt 0 {b 1i } MA(1), {b 2i } AR(1) with θ b1 = 0.5 and ρ b2 = 0.9 and σ 2 b 1 = σ 2 b 2 = 0.1 b = 5, B ik N(0, 0.1) 0 = t 1 < < t 100 = π {ε i } N(0, σ 2 ), σ 2 = 0.1Var(m(χ 1 ),..., m(χ n )) semi-metric d proj 4 (, ) d proj 4 (χ i, χ j ) = 4 π ( (χ i (t) χ j (t))v k (t)dt) 2. k=1 0

40 Simulated data: Model 2 t(curves.c)

41 Average over C of the empirical coverage of the true, bootstrap and asymptotic condence intervals. Model 1: smooth curves 1 α n I true 0.95 (0.12) 0.95 (0.01) 0.90 (0.02) 0.90 (0.02) I 0.89 (0.12) 0.92 (0.08) 0.85 (0.12) 0.88 (0.08) I asymp 0.85 (0.14) 0.90 (0.11) 0.79 (0.14) 0.84 (0.12) Model 2: rough curves 1 α n I true 0.95 (0.01) 0.95 (0.01) 0.90 (0.02) 0.90 (0.02) I 0.80 (0.18) 0.89 (0.07) 0.77 (0.18) 0.86 (0.07) I asymp 0.76 (0.17) 0.82 (0.06) 0.69 (0.16) 0.75 (0.06)

42 Model 1: CI coverage. True Boot Asymp

43 Model 1: Condence interval for each χ in C Segment: bootstrap CI, points: true CI.

44 Model 2: CI coverage. True Boot Asymp

45 Model 2: Condence interval for each χ in C Segment: bootstrap CI, points: true CI.

46 1 Introduction 2 3 4

47 Electricity demand Introduction Dataset: workdays of the second quarter of the year Predict one day (24 hours) χ i+1 (t) = m t (χ i ) + ε i,t (t = 1,..., 24, i = 1,..., n); Predict one hour for 21 days χ i+1,d (9) = m d (χ i,d ) + ε i,d (d = 1,..., 21, i = 1,..., n);

48 Condence intervals for electricity demand Predicted demand Predicted demand Observed demand Observed demand Figure : Left: CI for the 24 hours of Friday, June 29, Right: CI the workdays in June, 2012 (xed hour: 09:00 a.m.).

49 Condence intervals for electricity price Predicted price Predicted price Observed price Observed price Figure : Left: CI for the 24 hours of Friday, June 29, Right: CI the workdays in June, 2012 (xed hour: 09:00 a.m.).

50 1 Introduction 2 3 4

51 Semi-Funtional Partial Linear Regression Functional nonparametric explanatory variable, scalar linear-eect covariate and scalar response Autoregressive model G(χ i+1 ) = X T i β + m(χ i ) + ε i, i = 1,..., n General model Y i = X T i β + m(χ i ) + ε i, i = 1,..., n, where {(X i, χ i, Y i )} is α-mixing Estimators β h = ( X T h X h ) 1 X T h Ỹh m h (χ) = n w h (χ i, χ)(y i X T i i=1 β h )

52 1 Introduction 2 3 4

53 Naive bootstrap Introduction Homoscedastic model 1 Construct the residuals ε i,b = Y i X T i β b m b (χ i ), i = 1,..., n. 2 Draw n i.i.d. random variables ε 1,..., ε n from the empirical distribution function of ( ε 1,b ε b,..., ε n,b ε b ), where ε b = n 1 n i=1 ε i,b. 3 Obtain Y i = X T i β b + m b (χ i ) + ε i, i = 1,..., n. 4 Dene β b = ( X T X b b ) 1 X T b Ỹ b and m hb (χ) = n i=1 w h (χ i, χ)(y i X T i β b),

54 Wild bootstrap Introduction Heteroscedastic model 1 Construct the residuals ε i,b = Y i X T i β b m b (χ i ), i = 1,..., n. 2 Dene ε i = ε i,b V i, i = 1,..., n, where V 1,..., V n are i.i.d. random variables that are independent of the data S and that satisfy E(V 1 ) = 0 and E(V1 2) = 1. 3 Obtain Y i = X T i β b + m b (χ i ) + ε i, i = 1,..., n. 4 Dene β b = ( X T X b b ) 1 X T b Ỹ b and m hb (χ) = n i=1 w h (χ i, χ)(y i X T i β b),

55 1 Introduction 2 3 4

56 Assumptions for the linear part of the SFPLR model Semi-metric space χ is valued in come given compact subset C of H such that C τn B(z k, l n ), where τ n l γ n = C, τ n and l n 0 as n. k=1 Kernel K has support [0, 1], Lipschitz continuous on [0, ). k/ u [0, 1], K (u) > k > 0. Smoothness Denote g j (χ) = E(X ij χ i = χ), 1 i n, 1 j p. All the operators to be estimated are smooth, ie, for some c < and α > 0, (u, v) C C, f m, g 1,..., g p : f (u) f (v) cd(u, v) α.

57 Assumptions for the linear part of the SFPLR model Distributions For the probability distribution of the innite-dimensional process χ, it is assumed that exists F, positive valued function on (0, ) and positive constants α 0, α 1, α 2 such that, t C, h > 0 : 1 0 F (hs)ds > α 0 F (h) and α 1 F (h) P(χ B(t, h)) α 2 F (h). The joint probability distribution of (χ i, χ j ) is assumed that exists a function ψ(h) = cf (h) 1+ε (c > 0, 0 ε 1) and positive constants α 3, α 4 such that t C, h > 0: 0 < α 3 ψ(h) sup P[(χ j, χ j ) B(t, h) B(t, h)] α 4 ψ(h). i j

58 Assumptions for the linear part of the SFPLR model Dependence structure {(X i, χ i, Y i )} n i=1 come from some stationary strong mixing process, with mixing coecients {α(n)} that verify while α(n) cn a, a > 4.5. η i is independent of ε i, (i = 1,..., n), where η i = (η i1,..., η ip ) T, η ij = X ij E(X ij χ i ) = X ij g j (χ), j = 1,..., p.

59 Assumptions for the linear part of the SFPLR model Moments Denote V ε = E(εε T ), ε T = (ε 1,..., ε n ), η T = (η 1,..., η n ). E Y 1 r + E X 11 r E X 1p r < for some r > 4. sup i,j E( Y i Y j (χ i, χ j )) < max sup E( X i1 jx i2 j (χ i1,jχ i2,j)) < 1 j p i 1,i 2 B = E(η 1 η T 1 ), C = lim n n 1 E(η T V ε η). B and C are positive denite matrix.

60 Assumptions for the linear part of the SFPLR model Moments r(a+1) 2(a+r) sn = o(n θ ) for some θ > 2, where s n = sup χ C (s n,1 (χ) + s n,2 (χ) + s n,3 (χ)), with s n,1 (χ) = s n,2 (χ) = s n,3 (χ) = max 1 k p n n i=1 j=1 n n i=1 j=1 n n i=1 j=1 Cov( i (χ), j (χ) with i (χ) = K( d(χ i, χ) ) h Cov(Γ i (χ), Γ j (χ) with Γ i (χ) = Y i K( d(χ i, χ) ) h Cov(Γ ik (χ), Γ jk (χ) with Γ ik (χ) = X ik K( d(χ i, χ) ) h

61 Assumptions for the linear part of the SFPLR model Small ball probabilities In order to manage the convergence rates found in the development of the Theorem, it is necessary to consider the following assumptions: nh 4α 0, F (h) 1 n 1/4+1/r logn 0, nf (h) εa(r 2) r 1 =O(1) F (h) 2( θ(a+r) 1 ) n r(a+1) 2logn= O(1) as n where α > 0, 0 ε 1, a > 4.5, r > 4 and θ > 2.

62 Validity of the bootstrap for the linear part Theorem (Naive) Under previous assumptions, if the model is homoscedastic and a R p, for the naive bootstrap we have: ( sup P S na T ( β b β ) ( na b ) y P T ( β b β) y) P 0 y R Theorem (Wild) Under previous assumptions if, in addition ε i <, i = 1,..., n, F (h) 1 n 1/4+1/r logn(loglogn) 1/4 0, E ηη T <, E η 3 < and a R p, for the wild bootstrap procedure we have that ( sup P S na T ( β b β ) ( na b ) y P T ( β b β) y) P 0 y R

63 Validity of the bootstrap for the nonparametric part Theorem (Naive and Wild bootstrap) Under previous assumptions, if X i C <, we have: ) ( sup P S nf (h)( m y R hb (χ) m b(χ)) y P ( nf (h)( mh (χ) m(χ)) y ) P 0

64 1 Introduction 2 3 4

65 Electricity demand Introduction Dataset: workdays of the second quarter of the year Predict one day (24 hours) χ i+1 (t) = X T i β + m t (χ i ) + ε i,t (t = 1,..., 24, i = 1,..., n); Temperature covariates: X i = (X i1, X i2 ) T = (HDD i, CDD i ) T Model length: mean (sd) FNP (353.44) SFPLR (250.00)

66 Condence intervals for electricity demand Observed Predicted t Observed Figure : CI for the 24 hours of Friday, June 29, 2012.

67 Electricity price Introduction Dataset: workdays of the second quarter of the year Predict one day (24 hours) χ i+1 (t) = X T i β + m t (χ i ) + ε i,t (t = 1,..., 24, i = 1,..., n); Covariates: X i = (X i1, X i2 ) T = (Demand i, Wind i ) T Model length: mean (sd) FNP 7.44 (1.63) SFPLR (Demand) 6.50 (1.55) SFPLR (Demand+Wind) 8.40 (1.21)

68 Condence intervals for electricity price Observed Predicted t Observed Figure : CI for the 24 hours of Friday, June 29, 2012.

69 Electricity price Introduction Predict one hour for 21 days χ i+1,d (20) = X T i β + m d (χ i,d ) + ε i,d (d = 1,..., 21, i = 1,..., n); Covariates: X i = (X i1, X i2 ) T = (Demand i, Wind i ) T Model length: mean (sd) FNP 6.21 (1.54) SFPLR (Demand) 6.23 (1.57) SFPLR (Demand+Wind) 8.34 (2.64)

70 Condence intervals for electricity price Observed Predicted t Observed Figure : CI the workdays in June, 2012 (xed hour: 20:00 a.m.).

71 References Introduction Ferraty, F., Van Keilegom, I. and Vieu, P. (2010), On the Validity of the in Non-Parametric Functional Regression, Scandinavian Journal of Statistics, 37, Delsol, L. (2009) Advances on asymptotic normality in non-parametric functional time series analysis, Statistics, 43:1, Raña, P., Aneiros, G., Vilar, J. and Vieu, P. condence intervals in functional nonparametric regression under dependence. (Submitted). Aneiros, G. and Vieu, P. (2008) Nonparametric time series prediction: A semi-functional partial linear modeling, Journal of Multivariate Analysis, 99, Raña, P., Aneiros, G., Vilar, J. and Vieu, P. condence intervals in semi-functional partial linear regression. (Preprint).

72 Thanks for your attention!

73 Proofs outline: linear part ( P S na T ( β b β ) ( na ) b ) y P T ( β b β) y = T 1 (y)+t 2 (y) where a is a constant vector in R p, ( T 1 (y) = P S na T ( β b β ) ( ) y b ) y Φ at Aa ( ) y ( na ) T 2 (y) = Φ P T ( β b β) y. at Aa

74 Proofs outline: linear part ( ) y ( na ) T 2 (y) = Φ P T ( β b β) y. at Aa Theorem 1, Aneiros and Vieu (2008) n( βh β) D N(0, A)where A = B 1 CB 1. T 2 (y) 0 for any xed value of y.

75 Proofs outline: linear part ( T 1 (y) = P S na T ( β b β ) ( ) y b ) y Φ at Aa Lemma n( β b β d b ) P N(0, A),conditionally on the sample S. T 1 (y) 0 for any xed value of y.

76 Proofs outline: linear part Proof of the Lemma: For a given function g( ) = m( ) or g( ) = m b ( ), we denote n g b (χ) = g(χ) w b (χ i, χ)g(χ i ). i=1 Then, one can write n( β b β b ) = (n 1 X T b X b ) 1 n 1/2 (S n1 S n2 + S n3). Asymptotic normality is obtained by: n S n1 S n2 + S n3 = η i ε i + o P (n 1/2 ) (P S ), and n 1/2 n η i ε i i=1 i=1 D N(0, C), in P S,

77 Proofs outline: nonparametric part ( sup P S nf (h)( m y R hb (χ) m b(χ)) y P ( nf (h)( mh (χ) m(χ)) y ) ) P 0 (nf (h)) 1/2 ( m h (χ) m(χ)) N(0, σ 2 (χ)) (nf (h)) 1/2 ( m hb (χ) m b(χ)) N(0, σ 2 (χ))

78 Proofs outline: nonparametric part (nf (h)) 1/2 ( m h (χ) m(χ)) = n (nf (h)) 1/2 ( w h (χ i, χ)(y i X T i β h ) m(χ))) = i=1 n (nf (h)) 1/2 ( w h (χ i, χ)(x T i β + m(χ i ) + ε i X T i β h ) m(χ))) = (nf (h)) 1/2 ( (nf (h)) 1/2 i=1 n w h (χ i, χ)(m(χ i ) + ε i ) m(χ)) i=1 S 1 (χ) S 2 (χ) n w h (χ i, χ)x T i ( β h β) = i=1

79 Proofs outline: nonparametric part S 1 (χ) = (nf (h)) 1/2 ( n w h (χ i, χ)(m(χ i ) + ε i ) m(χ)) = i=1 = (nf (h)) 1/2 ( m NP h (χ) mnp (χ)) Delsol (2009) (nf (h)) 1/2 ( m NP h (χ) mnp (χ)) D N(0, σ 2 (χ)) S 1 (χ) D N(0, σ 2 (χ))

80 Proofs outline: nonparametric part n S 2 (χ) = (nf (h)) 1/2 w h (χ i, χ)x T i ( β h β) i=1 Theorem 1, Aneiros and Vieu (2008) n( βh β) D N(0, A)where A = B 1 CB 1. Lemma max w h (χ i, χ) = O((nF (h)) 1 ) S 2 (χ) = o P (1)

81 Proofs outline: nonparametric part (nf (h)) 1/2 ( m hb (χ) m b(χ)) = n (nf (h)) 1/2 ( w h (χ i, χ)(y i i=1 i=1 X T i β b) m b (χ)) = n (nf (h)) 1/2 ( w h (χ i, χ)(x T i β b + m b (χ i ) + ε i X T i (nf (h)) 1/2 ( (nf (h)) 1/2 n w h (χ i, χ)( m b (χ i ) + ε i ) m b (χ)) i=1 n w h (χ i, χ)x T i ( β b β b ) = i=1 S 1 (χ) S 2 (χ) β b) m b (χ))

82 Proofs outline: nonparametric part n S 1 (χ) = (nf (h)) 1/2 ( w h (χ i, χ)( m b (χ i ) + ε i ) m b (χ)) i=1 = S 1,1(χ) + S 1,2(χ) S 1,1 S 1,2 (χ) contains the nonparametric part of the expression. (χ) contains the linear part of the expression.

83 Proofs outline: nonparametric part S 1,1(χ) = (nf (h)) 1/2 ( m NP hb (χ) mnp (χ)) D N(0, σ 2 (χ)) b Raña, Aneiros, Vilar and Vieu Delsol (2009) sup y R P S ( nf χ (h)( m NP hb (χ) mnp b (χ)) y) P ( nf χ (h)( m NP h (χ) mnp (χ)) y ) 0 a.s. (nf (h)) 1/2 ( m NP h (χ) mnp (χ)) D N(0, σ 2 (χ))

84 Proofs outline: nonparametric part n n S1,2(χ) = (nf (h)) 1/2 ( w h (χ i, χ)[ w b (χ j, χ i )X T j (β β b ) + +X T j (β β b ) i=1 j=1 n w b (χ l, χ j )X T l (β β b ) 1 n l=1 n w b (χ l, χ k )X T l (β β b ))] l=1 Aneiros and Vieu (2008) n( βh β) D N(0, A) n (X T k (β β b ) k=1 n w b (χ i, χ)x T i (β β b )) i=1 Assumption S 1,2(χ) = o P (1)(P S ). X i C <

85 Proofs outline: nonparametric part n S 2 (χ) = (nf (h)) 1/2 w h (χ i, χ)x T i ( β b β b ) = o P (1)(P S ) i=1 Raña, Aneiros, Vilar and Vieu ( sup P S na T ( β b β ) ( na b ) y P T ( β b β) y) P 0 y R Aneiros and Vieu (2008) n( βh β) D N(0, A) Assumption X i C <