Functional Methods for Time Series Prediction: A Nonparametric Approach

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1 Journal of Forecating J. Forecat. (200) Publihed online in Wiley InterScience ( DOI: 0.002/for.69 Functional Method for Time Serie Prediction: A Nonparametric Approach GERMÁN ANEIROS-PÉREZ,* RICARDO CAO AND JUAN M. VILAR-FERNÁNDEZ Departamento de Matemática, Facultad de Informática, Univeridade da Coruña, Spain ABSTRACT The problem of prediction in time erie uing nonparametric functional technique i conidered. An extenion of the local linear method to regreion with functional explanatory variable i propoed. Thi forecating method i compared with the functional Nadaraya Waton method and with finitedimenional nonparametric predictor for everal real-time erie. Prediction interval baed on the boottrap and conditional ditribution etimation for thoe nonparametric method are alo compared. Copyright 200 John Wiley & Son, Ltd. key word time erie forecating; functional data; nonparametric regreion; boottrap INTRODUCTION Prediction of future obervation i an important problem in time erie. Given an oberved erie Z, Z 2,..., Z n, the aim i to predict a future value Z n+l, for ome integer l. A ueful approach for the prediction problem i to conider that the erie follow an autoregreive proce of order q: Z = m( Z, Z,..., Z )+ ε t t t 2 t q t where ε t i the error proce, aumed to be independent of the pat of Z t, i.e., Z t, Z t It i clear then that the firt tak i to etimate the function m( ). A claical approach to thi problem conit in auming that m( ) belong to a cla of function, only depending on a finite number of parameter to be etimated. Example of uch clae are the well-known ARIMA model, widely tudied in the literature (ee, among many other, the book by Box and Jenkin, 976; Brockwell and Davi, 987; Makridaki et al., 998). Thi problem can * Correpondence to: Germán Aneiro-Pérez, Departamento de Matemática, Facultad de Informática, Univeridade da Coruña, Campu de Elviña, /n. 507 A Coruña, Spain. ganeiro@udc.e Copyright 200 John Wiley & Son, Ltd.

2 G. Aneiro-Pérez, R. Cao and J. M. Vilar-Fernández alo be addreed via nonparametric method. Thee method do not aume any functional form on m( ), but only impoe regularity condition on it. Nonparametric regreion etimation under dependence i a ueful tool for forecating in time erie. Some relevant work in thi field include Györfi et al. (989), Härdle and Vieu (992), Hart (99, 996), Mary and Tjotheim (995), Härdle et al. (997, 998) and Boq (998). Other paper more pecifically focued on prediction uing nonparametric technique are Carbon and Delecroix (993), Nottingham and Cook (200), Matzner-Lober et al. (998) and Vilar-Fernández and Cao (2007). The literature on method for time erie prediction in the context of functional data i much more limited. The book by Boq (2000) and Ferraty and Vieu (2006) are comprehenive reference for parametric (linear) and nonparametric functional data analyi, repectively. Application of the FAR model (a functional verion of the claical AR model) can be een in Bee et al. (2000). Mary (2005) ha proven aymptotic normality of the kernel regreion functional etimator under dependence, while Antoniadi et al. (2006) propoed a functional wavelet-kernel approach for time erie prediction. Aneiro-Pérez and Vieu (2008) deal with the problem of nonparametric time erie prediction uing a emi-functional partial linear model. In thi paper we adopt a nonparametric view for the problem of time erie prediction uing functional data technique. Specifically, a local-linear regreion etimator for thi problem i been propoed. Thi etimator i compared with the Nadaraya Waton kernel etimator for the regreion functional, a well a with the claical finite-dimenional verion of the Nadaraya Waton and the local-linear regreion etimator for the problem of time erie prediction. Thee four method are applied to three real-time erie concerning electricity conumption, ozone concentration and air temperature. The ret of the paper i organized a follow. The mathematical formulation of the nonparametric prediction problem i preented in the next ection. The third ection contain detail of the local-linear regreion etimator for functional data and how to ue it to contruct point forecat and nonparametric prediction interval. A comparative empirical tudy of the new method and other nonparametric approache i included in the fourth ection, where ome concluion are drawn. FORMULATION OF THE PROBLEM Let u conider a continuou-time tochatic proce, {Z(t)} t, oberved for t [a, b), and uppoe we are intereted in predicting Z(b + r), for ome r 0. Let u aume that Z(t) i (or may be) tational, with eaonal length τ and b = a + ( + )τ. In other word, we aume that the interval [a, b) conit of + eaonal period of length τ of the tochatic proce {Z(t)} t. For implicity we will aume the following Markov property d Zb ( + r) = Zb ( + r) { Zt (), t [ ab, )} { Zt (), t [ b τ, b) } By defining the functional data {(X i, Y i )}, where X i (t) = Z(a + (i )τ + t) with t C = [0, τ), and Y i = Z(a + iτ + r) with r C, we may look at the problem of predicting Z(b + r) by computing nonparametric etimation, mˆ (x + ), of the autoregreion functional: Copyright 200 John Wiley & Son, Ltd. J. Forecat. (200) DOI: 0.002/for

3 Functional Method for Time Serie Prediction m( xn )= E( Y ) () + X= x+ with functional explanatory variable, X, and calar repone, Y. In practice, we typically only oberve a dicrete verion of the functional data in equipaced j intant ( ). More pecifically, we only oberve X i (t) for t = τ, with j = 0,,...,. In uch a cae, defining n = ( + ), we may formulate the prediction problem in term of a dicrete-time proce. Given the oberved ample from the time erie: ( ) Z= Z( a), Z2= Z a+ τ,..., ( ) ( ) = n Zn = Z a+ + τ Z b τ our aim i to predict ( ) n l Zn+ l= Z( b+ r)= Z a+ + + τ l with r = τ, for ome fixed l =, 2,...,. LOCAL-LINEAR FUNCTIONAL PREDICTION In thi ection we preent the Nadaraya Waton regreion etimator for functional data and extend it to the local-linear etimator in the context of the functional explanatory variable. Thee two nonparametric etimator are ueful technique for point forecating baed on the autoregreion functional. Two method are alo introduced to compute prediction interval baed on the two previou nonparametric forecat. One i baed on a boottrap reampling of the reidual and the other ue the conditional prediction ditribution. There are plenty of paper in the tatitical literature that are concerned with the ue of boottrap method for time erie prediction. Among them we mention the work by Thomb and Schucany (990), Breidt et al. (995), García-Jurado et al. (995) and Zagdańki (200). Nadaraya Waton etimator for the regreion functional Given the functional ample {(X i, Y i )}, the Nadaraya Waton (NW) kernel etimator evaluated at a given function u, mˆ h,nw (u), i of the form mˆ ( u)= W ( u, X ) Y h, NW h, NW j j j= Copyright 200 John Wiley & Son, Ltd. J. Forecat. (200) DOI: 0.002/for (2)

4 G. Aneiro-Pérez, R. Cao and J. M. Vilar-Fernández where W ( )= h, NW ux, j K ( X u ) h j K ( X u ) h i and K t h K t h h( )= ( ) i the recaled kernel function with bandwidth h > 0 and i a uitable eminorm in the functional pace F = { f : C ; f L } R 2 and u F (ee Ferraty and Vieu, 2006, pp , p. 223, for detail on both the etimator (2) and the crucial role of the eminorm, repectively). The kernel K i a non-negative real-valued function uch that Kt () dt =. 0 Local-linear etimator for the regreion functional In thi ection we propoe a local-linear (LL) functional regreion etimator for () at u. We extend the idea in Fan and Gijbel (996) to the cae of functional data. Firt of all, we ue a linear approximation of m in a neighbourhood of a given function u: m( x) β + β() t ( x() t u() t ) t 0 d C for ome β 0 and β F. The contant β 0 play the role of m(u), while the function β i the gradient of m at the point u. Given the ample {(X i, Y i )}, defined above, we contruct the LL etimator of β 0 and β a the minimizer of Ψ( β0, β) = Yi β0 + β( t) ( Xi( t) u( t) ) d t Khi, (3) ( ) C with K h,i = K h ( X i u ), for a uitable eminorm. In order to minimize (3) with repect to β 0 and β, we impoe that the partial derivative with repect to β 0 i zero: 2 Ψ β 0 = 0 β K + K β( t) ( X( t) u( t) ) dt = K Y 0 hi, hi, i hi, i C (4) Copyright 200 John Wiley & Son, Ltd. J. Forecat. (200) DOI: 0.002/for

5 Functional Method for Time Serie Prediction and that the directional derivative of Ψ in the direction of any v F i alo zero: Ψ( β0, β+ εv) ε = 0 β K v( t) ( X( t) u( t) ) dt 0 ε = 0 = + K hi, hi, C i ( v( t) ( Xi( t) u( t) ) dt) β( t) ( Xi( t) u( t) ) dt C ( i ) Khi, Yi v( t) ( X( t) u( t) ) dt C ( ) C (5) In order to olve in β 0 and β the ytem of functional equation (4) and (5) for all v F, we firt write the unknown function β in term of a bai, {e j ( )} j, of F, β()= t λje j() t, and apply j= equation (5) for v = e k, k. Thu we have the following ytem of infinitely many linear equation: β β K + λ K a = K Y (6) 0 hi, j hi, ij hi, i j= 0 hi, ik j h, i ij ik hi, i ik, j= K a + λ K a a = K Ya k =2,,... (7) where aij = ( Xi() t u() t ) ej( t) dt C The ytem (6) (7) can be written in a impler way after introducing ome new notation: λ = β, b = K a a, for k, j N 0 0 kj h, i ik ij b = K a, for j N, b = K a, for k N 0 j h, i ij k0 h, i ik b = K, d = K Ya, for k N, d = K Y 00 hi, k h, i i ik 0 hi, i Thi give the following linear ytem: bkjλ j = dk, k = 0,,... (8) j= 0 Copyright 200 John Wiley & Son, Ltd. J. Forecat. (200) DOI: 0.002/for

6 G. Aneiro-Pérez, R. Cao and J. M. Vilar-Fernández The olution of the previou ytem would give etimator of both m and the gradient of m at u: mˆ h,ll (u) = βˆ 0 = λˆ 0 and ˆ β ˆ h, LL( t)= λ j je j() t =, repectively. Of coure, in general it i not poible to find an explicit olution of thi infinite linear ytem and truncation idea can be applied to olve a finite ytem approximating (8). Fix ome N and conider jut the firt N + equation in (8). Thi require finding the olution λ k, k = 0,,..., N of N bkjλ j = dk, k = 0,,..., N j= 0 Finally we obtain m h,n,ll (u) = β 0 = λ 0 and ˆ N βhn,, LL( t)= λ j je j() t. Thee are approximated = olution of (8), uing only the firt N function in the bai: e ( ), e 2 ( ),..., e N ( ). Reidual-baed boottrap prediction interval (RBB) In thi ubection and the next one, we follow the line of Vilar-Fernández and Cao (2007). For thi reaon we omit the detail. The firt boottrap method for interval prediction i baed on reampling the reidual. A ketch of the algorithm follow.. Compute the reidual, εˆ j = Y j mˆ (X j ), uing ome global cro-validation bandwidth, where mˆ (u) i either the NW or the LL etimator for functional data Ue ε, the tandard deviation of the εˆ j, to compute the moothing parameter g = 4 3 ε. 3. Draw moothed boottrap reidual: ˆ ε* i = ˆ εi + gξi,,..., B i d where Ii = U( {,..., }), ξ d i = N( 0, ) and B i the number of boottrap replication. 4. Sort the boottrap reidual: {εˆ * (i) : i =,..., B} and compute the α prediction interval: ( mˆ ( Xn ˆ B, mˆ Xn ˆ + ) + ε [*( α 2) ] ( + ) + ε* [ ( ( α 2) ) B ) ] Prediction interval baed on the conditional ditribution (CD) Thi method i baed on etimating the conditional ditribution function of Y X=x. The baic idea can be found in Cao (999). For a given real value y, the conditional cumulative ditribution function can be viewed a a regreion function ( )= ( ) Fy E Y y x { } X= x Conequently, the functional NW or LL etimator can be ued to etimate thi conditional ditribution function. The prediction algorithm proceed a follow. Copyright 200 John Wiley & Son, Ltd. J. Forecat. (200) DOI: 0.002/for

7 Functional Method for Time Serie Prediction. Ue the ample {(X j, Y j ) : j } to compute Fˆ (y x ) by mean of the NW or the LL method. Thi function i moothed again in the variable y. 2. The α prediction interval i (L, U), where FL ˆ α 2 FU ˆ ( )= and ( )= x+ x+ α 2 EMPIRICAL STUDY The two functional data nonparametric point forecat and prediction interval are compared with their finite-dimenional counterpart. Specifically, thee prediction method are applied to three realtime erie concerning electricity conumption, ozone concentration and air temperature. A we will how later, thee time erie are eaonal and, for each one, we have 2 obervation taken in equipaced intant within each eaonal period. In thi ene, we can conider that the length of the eaonal period i τ = 2 (in ome unit). For the finite-dimenional nonparametric approach we follow the procedure in Vilar-Fernández and Cao (2007). An important quetion in the finite-dimenional etting i how to elect the autoregreor variable, (Z t i, Z t i2,..., Z t ip ), for predicting Z t+l. We adopt the approach by Tjotheim and Auetad (994). It conit in minimizing a nonparametric etimation of the final prediction error. Selection of the tuning parameter In the functional data etup (a well a in the finite-dimenional cae) there are everal tuning parameter that need to be elected. We briefly mention now ome procedure to do thi. The Epanechnikov kernel i been ued for the NW and the LL etimator. Cro-validation method (ee Rachdi and Vieu, 2007; Benhenni et al., 2007), expreed in term of k-nearet neighbour, are ued for moothing parameter election. Global cro-validation i ued for contructing the reidual correponding to the RBB prediction interval, and local cro-validation for computing the nonparametric point forecat and the etimation of the conditional ditribution function. Following the recommendation of Ferraty and Vieu (2006, p. 223), for chooing the eminorm in practical ituation we bae our choice on the moothne or roughne of the explanatory curve. Specifically, when the curve are mooth we ue the L 2 norm of the qth derivative of the curve, q derivative, while for rough curve the eminorm i baed on principal component analyi, q PCA (q being the number of principal component). For the definition of thi cla of eminorm ee Ferraty and Vieu (2006, p ). The Fourier bai i ued in the LL functional etimator. The parameter q in the eminorm and the number of function in the Fourier bai, N, are alo elected by cro-validation. The parameter N i elected within {3, 5, 7}, while q wa elected in the et {,..., 2}, when the eminorm i baed on principal component, and within {0,, 2} for the eminorm baed on the qth derivative. When the prediction horizon i larger than one, point forecat are carried out in two different way. The firt one i the direct method and conit of the approach mentioned in the previou ection. The econd alternative i the recurive method. It compute a one-ahead forecat and Copyright 200 John Wiley & Son, Ltd. J. Forecat. (200) DOI: 0.002/for

8 G. Aneiro-Pérez, R. Cao and J. M. Vilar-Fernández include it in the ample to perform again a one-lag prediction, a many time a needed. Thee two method are compared in the empirical tudy. Method and error criteria Four nonparametric point forecat are computed: (a) a finite-dimenional NW forecat, (b) a finitedimenional LL forecat, (c) a functional NW forecat and (d) a functional LL forecat. Thee forecat are performed uing either the direct method or the recurive one. Four type of prediction interval (only uing the direct method) are computed. Thee are the four combination for the nonparametric method ued for point forecat (NW or LL) and the baic procedure for contructing prediction interval (reidual-baed boottrap and conditional ditribution). Thee four approache are ued for both finite-dimenional and functional autoregreion etimation. The nominal level for the prediction interval i 95%. The number of boottrap replication i et to B = 000. A maximum horizon of = 2 i conidered. The performance of the point forecat i evaluated by excluding the lat eaonal period (lat 2 obervation) from the data, computing the point forecat for thee value and comparing the predicted value, ẑ n (l), with the real one, z n+l. Several error meaure are conidered. The root mean quared error: RMSE = 2 ( () ) zˆ 2 n l zn+ l l= the mean abolute error: and the relative error: MAE = zˆ n() l zn + l RE = where σˆ l2 i the quai-variance of {z (j )τ+l } j=. l= l= zˆ n() l zn + ˆσ l Electricity conumption data The firt dataet analyed conit of monthly electricity conumption in the USA during the period January 972 January 2005 (397 month). The ource of the data wa the US Government (Department of Energy), and they are available at the webite The data are tranformed uing logarithm and then differentiated to eliminate the trend. The eaonal period for thi time erie i one year. Thi give 33 curve (ee Ferraty and Vieu, 2006, pp. 7 20, for detail about thi dataet). From Figure we can oberve that the functional data are quite rough curve. PCA Thu we ue the cla of eminorm { q } 2 q=. Copyright 200 John Wiley & Son, Ltd. J. Forecat. (200) DOI: 0.002/for

9 Functional Method for Time Serie Prediction Electr. conumpt.: diff. log data Yearly curve Month Finite dimenional model Functional model NW-R LL-D NW-D LL-D Month Month Figure. Time erie and functional data (upper panel) together with the bet forecat (differenced log) electricity conumption for each type of model and etimator (lower panel). Both the direct (D) and the recurive (R) method are combined with the Nadaraya Waton (NW) and local-linear (LL) etimator Table I collect the point forecating error. Figure how ome plot of the time erie along time, the functional data and the bet forecat, for every type of model and etimator. In other word, for each kind of model (finite-dimenional or functional) and each kind of etimator (NW or LL), only the reult correponding to the bet (direct or recurive) forecat are hown. Figure 2 report Copyright 200 John Wiley & Son, Ltd. J. Forecat. (200) DOI: 0.002/for

10 G. Aneiro-Pérez, R. Cao and J. M. Vilar-Fernández Table I. Error criteria for the finite dimenional and functional model uing Nadaraya Waton (NW) and the local-linear (LL) forecat with the direct (D) and recurive (R) approach for electricity conumption data Etimator Error criteria RMSE MAE RE Finite-dimenional NW-D NW-R LL-D LL-R Functional NW-D NW-R LL-D LL-R the prediction interval (only uing the direct method) for the four nonparametric forecat (NW and LL either finite-dimenional or functional) with the two poible method for interval contruction (RBB and CD). From a graphical point of view, Figure ugget that the nonparametric forecat have a good behaviour, in the ene that the prediction follow the trend of the data. To compare quantitatively the different prediction method ued in thi paper, we need to conider the information contained in Table I. On the one hand, thi table how that the LL forecat for functional data (direct verion) beat the other nonparametric method (either finite-dimenional or functional) for the analyed erie. On the other hand, we hould mention the problem in the ue of the recurive method, becaue a poor prediction in a pecific intant caue even wore prediction for future intant. In fact, the bad performance of the recurive LL functional predictor ugget that at any intant a poor one-ahead forecat i obtained. Figure 2 report reult on the prediction interval. From thi figure, and focuing on each cla of model (finite-dimenional or functional model), there are no large difference between the interval contructed by NW etimator and thoe contructed uing LL. Neverthele, prediction interval uing the functional data approach are, generally peaking, more narrow than thoe uing finite-dimenional model. In addition, functional LL RBB prediction interval are mot of the time more accurate and narrow than the other. Ozone concentration data The econd erie collect ozone concentration every econd hour from 8 May 2005 to 29 June 2005 (56 item of data) recorded in Getafe (Madrid, Spain). Thee data, publihed by the Autonomou Community of Madrid (Environmental Department), can be found at the webite getiona.madrid.org/aireinternet. There exit a clear daily eaonality in thi erie, which give 43 Copyright 200 John Wiley & Son, Ltd. J. Forecat. (200) DOI: 0.002/for

11 Functional Method for Time Serie Prediction Finite dimenional model: NW-D Finite dimenional model: LL-D RBB CD RBB CD Month Month Functional model: NW-D Functional model: LL-D RBB CD RBB CD Month Month Figure 2. Prediction interval correponding to the (differenced log) electricity conumption data. They are baed on both finite-dimenional (upper panel) and functional (lower panel) model. The Nadaraya Waton (NW) and local-linear (LL) etimator are ued for contructing the reidual-baed boottrap (RBB) prediction interval and prediction interval baed on the conditional ditribution (CD). Only the direct (D) method i conidered Copyright 200 John Wiley & Son, Ltd. J. Forecat. (200) DOI: 0.002/for

12 G. Aneiro-Pérez, R. Cao and J. M. Vilar-Fernández Ozone concentration Daily curve Hour Finite dimenional model Functional model NW-R LL-D NW-D LL-R Hour Hour Figure 3. Time erie and functional data (upper panel) together with the bet forecat ozone concentration for each type of model and etimator (lower panel). Both the direct (D) and recurive (R) method are combined with the Nadaraya Waton (NW) and local-linear (LL) etimator curve (ee Aneiro-Pérez and Vieu, 2008, for more information about thi erie). The mooth hape of the curve (ee Figure 3) ugget ue of the cla of eminorm { q derivative } 2 q=0. The point forecating error can be een in Table II. Figure 3 collect ome plot of the time erie along time, the functional data and the bet forecat, for every type of model and etimator. Copyright 200 John Wiley & Son, Ltd. J. Forecat. (200) DOI: 0.002/for

13 Functional Method for Time Serie Prediction Table II. Error criteria for the finite-dimenional and functional model uing the Nadaraya Waton (NW) and local-linear (LL) forecat with the direct (D) and recurive (R) approach for ozone concentration data Etimator Error criteria RMSE MAE RE Finite-dimenional NW-D NW-R LL-D LL-R Functional NW-D NW-R LL-D LL-R The dicuion given in the previou example on the point forecat applie eentially for the ozone concentration data, a can be een in Table II and Figure 3. The only difference i baed on the fact that the recurive verion of the LL functional predictor how now a better behaviour than the direct verion. Thi ugget that all the one-ahead forecat involved in the recurive approach have a good performance. For the ake of brevity, we omit the reult for the prediction interval. The concluion are the ame a thoe obtained from Figure 2. Air temperature data The Mabegondo data comprie the third time erie we analye. It i available at the webite (ource: Xunta of Galicia). Air temperature wa recorded every 2 hour at Mabegondo meteorological tation (Mabegondo, Galicia, Spain) over the period January 30 March The eaonal period i one day (092 item of data and 9 curve). A in the cae of the ozone concentration data, we are in a ituation in which the curve are mooth (ee Figure 4). Thu we ue the cla of eminorm { q derivative } 2 q=0. Table III report the point forecating error. Figure 4 how ome plot of the time erie along time, the functional data and the bet forecat, for every type of model and etimator. Both Table III and Figure 4 how poor behaviour of the finite-dimenional predictor. The good performance of the functional forecat remain here (epecially in the cae of the LL forecat). In addition, we oberve in Table III imilar value for the error criteria when the direct method or the recurive one i ued in the LL functional forecat. A in the previou ubection, we do not report reult on the prediction interval. In fact, the concluion for the prediction interval are imilar to thoe preented for the electricity conumption data. In ummary, it i worth mentioning that the curve correponding to the electricity data are rough, while thoe correponding to both ozone concentration and air temperature data are mooth. On the other hand, we note that the curve correponding to the lat two dataet are more pare than thoe correponding to the electricity data. Thu the empirical tudy cover different ituation that are common in practice. Copyright 200 John Wiley & Son, Ltd. J. Forecat. (200) DOI: 0.002/for

14 G. Aneiro-Pérez, R. Cao and J. M. Vilar-Fernández Air temperature Daily curve Hour Finite dimenional model Functional model NW-D LL-R NW-R LL-D Hour Figure 4. Time erie and functional data (upper panel) together with the bet forecat air temperature for each type of model and etimator (lower panel). Both the direct (D) and recurive (R) method are combined with the Nadaraya Waton (NW) and local-linear (LL) etimator Hour Copyright 200 John Wiley & Son, Ltd. J. Forecat. (200) DOI: 0.002/for

15 Functional Method for Time Serie Prediction Table III. Error criteria for the finite-dimenional and functional model uing the Nadaraya Waton (NW) and local-linear (LL) forecat with the direct (D) and recurive (R) approach for air temperature data Etimator Error criteria RMSE MAE RE Finite-dimenional NW-D NW-R LL-D LL-R Functional NW-D NW-R LL-D LL-R ACKNOWLEDGEMENTS We are grateful to Hervé Cardot, who kindly provided u ome R code for functional data analyi. Thi reearch wa partially upported by Grant PGIDIT07PXIB05259PR and 07SIN0205PR from Xunta de Galicia (Spain), by Grant number MTM from Miniterio de Ciencia e Innovación (Spain), and by the reearch group MODES. REFERENCES Aneiro-Pérez G, Vieu P Nonparametric time erie prediction: a emi-functional partial linear modeling. Journal of Multivariate Analyi 99: Antoniadi A, Paparoditi E, Sapatina T A functional wavelet-kernel approach for time erie prediction. Journal of the Royal Statitical Society, Serie B 68: Benhenni K, Ferraty F, Rachdi M, Vieu P Local moothing regreion with functional data. Computational Statitic 22: Bee PC, Cardot H, Stephenon DB Autoregreive forecating of ome functional climatic variation. Scandinavian Journal of Statitic 27: Boq D Nonparametric Statitic for Stochatic Procee: Etimation and Prediction. Lecture Note in Statitic 0. Springer: Berlin. Boq D Linear Procee in Function Space: Theory and Application. Lecture Note in Statitic 49. Springer: Berlin. Box GEP, Jenkin GM Time Serie Analyi: Forecating and Control. Holden-Day: San Francico, CA. Breidt FJ, Davi RA, Dunmuir WTM Improved boottrap prediction interval for autoregreion. Journal of Time Serie Analyi 6: Copyright 200 John Wiley & Son, Ltd. J. Forecat. (200) DOI: 0.002/for

16 G. Aneiro-Pérez, R. Cao and J. M. Vilar-Fernández Brockwell PJ, Davi RA Time erie: Theory and method. Springer: Berlin. Cao R An overview of boottrap method for etimating and predicting in time erie. Tet 8: Carbon M, Delecroix M Non-parametric v parametric forecating in time erie: a computational point of view. Applied Stochatic Model and Analyi 9: Fan J, Gijbel I Local Polynomial Modelling and it Application. Chapman & Hall: London. Ferraty F, Vieu P Nonparametric Functional Analyi. Springer Serie in Statitic. Springer: New York. García-Jurado I, González-Manteiga W, Prada-Sánchez JM, Febrero-Bande M, Cao R Predicting uing Box Jenkin, nonparametric, and boottrap technique. Technometric 37: Györfi L, Härdle W, Sarda P, Vieu P Nonparametric Curve Etimation from Time Serie. Lecture Note in Statitic 60. Springer: Berlin. Härdle W, Vieu P Kernel regreion moothing of time erie. Journal of the Time Serie Analyi 3: Härdle W, Lütkepohl H, Chen R A review of nonparametric time erie analyi. International Statitical Review 65: Härdle W, Tybakov A, Yang L Nonparametric vector autoregreion. Journal of Statitical Planning and Inference 68: Hart JD. 99. Kernel regreion etimation with time erie error. Journal of the Royal Statitical Society, Serie B 53: Hart JD Some automated method of moothing time-dependent data. Journal of Nonparametric Statitic 6: Makridaki S, Wheelwright SC, Hyndman RJ Forecating: Method and Application. Wiley: Chicheter. Mary E Nonparametric regreion etimation for dependent functional data: aymptotic normality. Stochatic Procee and their Application 5: Mary E, Tjotheim D Nonparametric etimation and identification of nonlinear ARCH time erie. Econometric Theory : Matzner-Lober E, Gannoun A, De Gooijer JG Nonparametric forecating: a comparion of three kernel baed method. Communication in Statitic: Theory and Method 27: Nottingham QJ, Cook DC Local linear regreion for etimating time erie data. Computational Statitic and Analyi 37: Rachdi M, Vieu P Nonparametric regreion for functional data: automatic moothing parameter election. Journal of Statitical Planning and Inference 37: Thomb LA, Schucany WR Boottrap prediction interval for autoregreion. Journal of the American Statitical Aociation 85: Tjotheim D, Auetad BH Nonparametric identification of nonlinear time erie: electing ignificant lag. Journal of the American Statitical Aociation 89: Vilar-Fernández JM, Cao R Nonparametric forecating in time erie: a comparative tudy. Communication in Statitic: Simulation and Computation 36: Zagdańki A Prediction interval for tationary time erie uing the ieve boottrap method. Demontratio Mathematica 34: Author biographie: Germán Aneiro-Pérez i Aociate Profeor of Statitic at Univerity of A Coruña (Spain) ince He wa awarded a Ph.D. in Statitic from Univerity of Santiago de Compotela (Spain) in 200. He ha been working on everal reearch topic over the pat few year. Thi include nonparametric method for time erie forecating and, more recently, reearch in the etting of nonparametric method for dependent functional data. Ricardo Cao i Profeor of Statitic at Univerity of A Coruña (Spain) ince 999. He i Editor in Chief of the cientific journal Tet, and Aociate Editor of Computational Statitic and Journal of Nonparametric Statitic. He ha publihed over ixty paper in international cientific journal in the topic of nonparametric curve etimation, boottrap method, urvival analyi and time erie, ome of them with application to the environment, material cience and forenic cience. Copyright 200 John Wiley & Son, Ltd. J. Forecat. (200) DOI: 0.002/for

17 Functional Method for Time Serie Prediction Juan M. Vilar-Fernández i Profeor of Statitic at Univerity of A Coruña (Spain) ince 994. Hi recent reearch deal with nonparametric method under dependence condition, including nonparametric forecating in time erie, local polynomial regreion, time erie clutering, comparion tet for regreion curve and conditional variance etimation. Author addree: Germán Aneiro-Pérez, Ricardo Cao and Juan M. Vilar-Fernández, Departamento de Matemática, Facultad de Informática, Univeridade da Coruña, Campu de Elviña, /n. 507 A Coruña, Spain. Copyright 200 John Wiley & Son, Ltd. J. Forecat. (200) DOI: 0.002/for