TheAnnalsofStatistics 2016,Vol.44,No.1,1–30 DOI:10.1214/15-AOS1323 (cid:13)c InstituteofMathematicalStatistics,2016 FUNCTIONAL LINEAR REGRESSION WITH POINTS OF IMPACT 6 1 By Alois Kneip1, Dominik Poß2 and Pascal Sarda 0 2 Universität Bonn and Institut de Mathématiques de Toulouse n Thepaperconsidersfunctionallinearregression,wherescalarre- a J sponses Y1,...,Yn are modeled in dependence of i.i.d. random functions X ,...,X . We study a generalization of the classical func- 2 1 n tional linear regression model. It is assumed that there exists an 1 unknown number of “points of impact,” that is, discrete observa- ] tion times where the corresponding functional values possess signif- T icant influences on the response variable. In addition to estimating S a functional slope parameter, the problem then is to determine the h. number and locations of points of impact as well as corresponding t regression coefficients.Identifiabilityofthegeneralized modeliscon- a sidered in detail. It is shown that points of impact are identifiable if m the underlying process generating X ,...,X possesses “specific lo- 1 n [ calvariation.”Examplesarewell-known processesliketheBrownian motion, fractional Brownian motion ortheOrnstein–Uhlenbeckpro- 1 v cess. The paper then proposes an easily implementable method for 8 estimatingthenumberandlocations ofpointsofimpact.Itisshown 9 that this number can be estimated consistently. Furthermore, rates 7 of convergence for location estimates, regression coefficients and the 2 slope parameter are derived.Finally, some simulation results as well 0 as a real data application are presented. . 1 0 6 1. Introduction. Weconsiderlinearregressioninvolvingascalarresponse 1 variable Y and a functional predictor variable X L2([a,b]), where [a,b] is : ∈ v a bounded interval of R. It is assumed that data consist of an i.i.d. sample i X (X ,Y ), i=1,...,n, from (X,Y). The functional variable X is such that i i r E( bX2(t)dt)< + and for simplicity the variables are supposed to be a a ∞ centered in the following: E(Y)=0 and E(X(t))=0 for t [a,b] a.e. R ∈ Received July 2014; revised February 2015. 1SupportedbytheDFGthroughGRK1707andtheHausdorffCenterforMathematics. 2Supportedby theDFGthrough GRK 1707. AMS 2000 subject classifications. Primary 62G08, 62M99; secondary 62J05. Key words and phrases. Functional linear regression, model selection, stochastic processes, nonstandard asymptotics. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Statistics, 2016,Vol. 44, No. 1, 1–30. This reprint differs from the original in pagination and typographic detail. 1 2 A.KNEIP,D. POSS ANDP. SARDA In this paper, we study the following functional linear regression model with points of impact b S (1.1) Y = β(t)X (t)dt+ β X (τ )+ε , i=1,...,n, i i r i r i Za r=1 X where ε , i=1,...,n are i.i.d. centered real random variables with E(ε2)= i i σ2 < , which are independent of X (t) for all t, β L2([a,b]) is an un- i ∞ ∈ b known, bounded slope function and β(t)X (t)dt describes a common ef- a i fect of the whole trajectory X () on Y . In addition, the model incorporates i i an unknown number S N of· “poiRnts of impact,” that is, specific time ∈ points τ ,...,τ with the property that the corresponding functional values 1 S X (τ ),...,X (τ )possesssomesignificantinfluenceontheresponsevariable i 1 i S Y . The function β(t), the number S 0, as well as τ and β , r=1,...,S, i r r ≥ areunknownandhavetobeestimatedfromthedata.Throughoutthepaper, we will assume that all points of impact are in the interior of the interval, τ (a,b), r=1,...,S. Standard functional linear regression with S =0 as r ∈ well as the pointimpact modelof McKeague and Sen(2010),which assumes β(t) 0 and S=1, are special cases of the above model. ≡ b If S =0, then (1.1) reduces to Y = β(t)X (t)dt+ε . This model has i a i i been studied in depth in theoretical and applied statistical literature. The R most frequently used approach for estimating β(t) then is based on func- tionalprincipalcomponentsregression[see,e.g.,FrankandFriedman(1993), Bosq (2000), Cardot, Ferraty and Sarda (1999), Cardot, Mas and Sarda (2007) or Mu¨ller and Stadtmu¨ller (2005) in the context of generalized linear models]. Rates of convergence of the estimates are derived in Hall and Horowitz (2007) and Cai and Hall (2006). Alternative approaches and further theoretical results can, for example, be found in Crambes, Kneip and Sarda (2009), Cardot and Johannes (2010), Comte and Johannes (2012) or Delaigle and Hall (2012). There are many successful applications of the standard linear functional regression model. At the same time, results are often difficult to analyze from the points of view of model building and substantial interpretation. b The underlying problem is that β(t)X (t)dt is a weighted average of the a i whole trajectory X () which makes it difficult to assess specific effects of i · R local characteristics of the process. This lead James, Wang and Zhu (2009) to consider “interpretable functional regression” by assuming that β(t)=0 for most points t [a,b] and identifying subintervals of [a,b] with nonzero ∈ β(t). A different approach based on impact points is proposed by Ferraty, Hall and Vieu (2010). For a pre-specified q N, they aim to identify a function g ∈ as well as those design points τ ,...,τ which are “most influential” in the 1 q POINT IMPACT 3 sense that g(X (τ ),...,X (τ )) provides a best possible prediction of Y . i 1 i q i Nonparametric smoothing methods are used to estimate g, while τ ,...,τ 1 q are selected by a cross-validation procedure. The method is applied to data from spectroscopy, where it is of practical interest to know which values X (t) have greatest influence on Y . i i To our knowledge, McKeague and Sen (2010) are the first to explic- itly study identifiability and estimation of a point of impact in a functional regression model. For centered variables, their model takes the form Y = βX (τ)+ε with a single point of impact τ [a,b]. The underlying i i i ∈ process X is assumed to be a fractional Brownian motion with Hurst parameter H. The approach is motivated by the analysis of gene expression data, whereakey problemis toidentify individualgenes associated withthe clinical outcome. McKeague and Sen (2010) show that consistent estimators are obtained by least squares, and that the estimator of τ has the rate of convergence n 1/(2H). The coefficient β can be estimated with a parametric − rate of convergence n 1/2. − There also exists a link between our approach and the work of Hsing and Ren (2009) who for a given grid t ,...,t of observation points propose a 1 p p procedure for estimating linear combinations m(X )= c X (t ) influ- i j=1 j i j encing Y . Their approach is based on an RKHS formulation of the inverse i P regression dimension-reduction problem which for any k=1,2,3,... allows to determine a suitable element (cˆ ,...,cˆ )T of the eigenspace spanned by 1 p the eigenvectors of the k leading eigenvalues of the empirical covariance ma- trix of (X (t ),...,X (t ))T. They then show consistency of the resulting i 1 i p estimators mˆ(X ) as n,p and then k . Note that (1.1) necessarily i →∞ →∞ implies that Y =m(X )+ε , where as p m(X ) may be written as a i i i i →∞ linear combination as considered by Hsing and Ren (2009). Their method therefore offers a way to determine consistent estimators mˆ(X ) of m(X ), i i although the structure of the estimator will not allow a straightforward identification of model components. Assuming a linear relationship between Y and X, (1.1) constitutes a unified approach which incorporates the standard linear regression model as well as specific effects of possible point of impacts. The latter may be of substantial interest in many applications. Although in this paper we concentrate on the case of unknown points of impact,wewanttoemphasizethatinpracticealsomodelswithpre-specified points of impact may be of potential importance. This in particular applies tosituations withafunctionalresponsevariable (t),definedover thesame i Y time period t [a,b] as X . For a specified time point τ [a,b], the standard i ∈ ∈ approach [see, e.g., He, Mu¨ller and Wang (2000)] will then assume that Y := (τ)= bβ (t)X (t)dt+ε ,whereβ L2([a,b])mayvarywithτ.But i Yi a τ i i τ ∈ thevalue X (τ) of X atthepointτ ofinterest may have aspecificinfluence, i i R 4 A.KNEIP,D. POSS ANDP. SARDA b and the alternative model Y := (τ)= β (t)X (t)dt+β X (τ)+ε with i Yi a τ i 1 i i S =1 and a fixed point of impact may be seen as a promising alternative. R The estimation procedure proposed in Section 5 can also be applied in this situation, and theoretical results imply that under mild conditions β as 1 well as β (t) can be consistently estimated with nonparametric rates of τ convergence. Asimilar modification may beapplied intherelated context of functional autoregression, where X ,...,X denote a stationary time series 1 n of random function, and (τ) X (τ) is to be predicted from X [see, i i 1 Y ≡ − e.g., Bosq (2000)]. Thefocusofourworkliesondevelopingconditionsensuringidentifiability of the components of model (1.1) as well as on determining procedures for estimating number and locations of points of impact, regression coefficients and slope parameter. Theproblem of identifiability is studiedin detail in Section 2.Thekey assumption is that the process possesses “specific local variation.” Intuitively, this means that at least some part of the local variation of X(t) in a small neighborhood [τ ǫ,τ +ǫ] of a point τ [a,b] is essentially uncorrelated − ∈ with the remainder of the trajectories outside the interval [τ ǫ,τ + ǫ]. − Model (1.1) is uniquely identified for all processes exhibiting specific local variation. It is also shown that the condition of specific local variation is surprisingly weak and only requires some suitable approximation properties of the corresponding Karhunen–Loève basis. Identifiability of (1.1) does not impose any restriction on the degree of smoothness of the random functions X or of the underlying covariance i function.Thesame istruefor thetheoretical resultsof Section 5which yield rates of convergence of coefficient estimates, provided that points of impact are known or that locations can be estimated with sufficient accuracy. But nonsmooth trajectories are advantageous when trying to identify points of impact. In order to define a procedure for estimating number and locations of points of impact, we therefore restrict attention to processes whose covariance function is nonsmooth at the diagonal. It is proved in Sec- tion 3 that any such process has specific local variation. Prominent exam- ples are the fractional Brownian motion or the Ornstein–Uhlenbeck process. From a practical point of view, the setting of processes with nonsmooth trajectories covers awiderangeofapplications.Examplesaregiven inSection 7 and in the supplementary material [Kneip, Poss and Sarda (2015)], where the methodology is applied to temperature curves and near infrared data. An easily implementable and computationally efficient algorithm for estimating number and locations of points of impact is presented in Sec- tion 4. The basic idea is to perform a decorrelation. Instead of regressing on X (t), we analyze the empirical correlation between Y and a process i i Z (t):=X (t) 1(X (t δ)+X (t+δ)) for some δ >0. For the class of δ,i i − 2 i − i POINT IMPACT 5 processes defined in Section 3, Z (t) is highly correlated with X (t) but δ,i i only possesses extremely weak correlations with X (s) if t s is large. This i | − | implies that under model (1.1) local maxima τ of the empirical correla- r tion between Y and Z (t) should be found at locations close to existing i δ,i points of impact. The number S is then estimated by a cut-off criterion. It b is proved that the resulting estimator S of S is consistent, and we derive rates of convergence for the estimators τ . In the special case of a fractional r Brownian motion and S=1, we retrievebthe basic results of McKeague and Sen (2010). b In Section 5, we introduce least squares estimates of β(t) and β , r = r 1,...,S, based on a Karhunen–Loève decomposition. Rates of convergence for these estimates are then derived. A simulation study is performed in Section 6, while applications to a dataset is presented in Section 7. The Appendixisdevotedtotheproofsofsomeofthemainresults.Theremaining proofs as well as the application of our method to a second dataset are gathered in the supplementary material. 2. Identifiability. Our setup implies that X ,...,X are i.i.d. random 1 n functions with the same distribution as a generic X L2([a,b]). In the fol- ∈ lowing,wewilladditionallyassumethatX possessesacontinuouscovariance function σ(t,s), t,s [a,b]. ∈ In a natural way, the components of model (1.1) possess different inter- b pretations. The linear functional β(t)X (t)dt describes a common effect a i of the whole trajectory X () on Y . The additional terms S β X (τ ) i · R i r=1 r i r quantify specific effects of the functional values X (τ ),...,X (τ ) at the i 1 i S P points of impact τ ,...,τ . Identifiability of an impact point τ quite ob- 1 S r viously requires that at least some part of the local variation of X (t) in i small neighborhoods of τ , is uncorrelated with the remainder of the trajec- r tories. This idea is formalized by introducing the concept of “specific local variation.” Definition1. AprocessX L2([a,b])withcontinuouscovariancefunc- ∈ tion σ(, ) possesses specific local variation if for any t (a,b) and all suffi- · · ∈ ciently small ǫ>0 thereexists a real random variable ζ (X) such that with ǫ,t f (s):= cov(X(s),ζǫ,t(X)) the following conditions are satisfied: ǫ,t var(ζǫ,t(X)) (i) 00, ǫ,t (iii) f (s) (1+ǫ)f (t) for all s [a,b], ǫ,t ǫ,t | |≤ ∈ (iv) f (s) ǫ f (t) for all s [a,b] with s /[t ǫ,t+ǫ]. ǫ,t ǫ,t | |≤ · ∈ ∈ − The definition of course implies that for given t (a,b) and small ǫ>0 ∈ any process X with specific local variation can be decomposed into (2.1) X(s)=X (s)+ζ (X)f (s), s [a,b], ǫ,t ǫ,t ǫ,t ∈ 6 A.KNEIP,D. POSS ANDP. SARDA Fig. 1. The figure illustrates the decomposition of a trajectory from a Brownian motion X (blackline)inX (redline)andζ (X)f (blueline).Thecomponent ζ (X)f can ǫ,t ǫ,t ǫ,t ǫ,t ǫ,t be seen to quantify the local variation of X in an interval around t. where X (s)=X(s) ζ (X)f (s) is a process which is uncorrelated with ǫ,t ǫ,t ǫ,t − ζ (X). If σ (, ) denotes the covariance function of X (s), then obviously ǫ,t ǫ,t ǫ,t · · (2.2) σ(s,u)=σ (s,u)+var(ζ (X))f (s)f (u), s,u [a,b]. ǫ,t ǫ,t ǫ,t ǫ,t ∈ By condition (iv), we can infer that for small ǫ>0 the component ζ (X) ǫ,t × f (s) essentially quantifies local variation in a small interval around the ǫ,t given point t, since fǫ,t(s)2 ǫ2 for all s /[t ǫ,t+ǫ].When X is a standard fǫ,t(t)2 ≤ ∈ − Brownian motion it is easily verified that conditions (i)–(iv) are satisfied for ζ (X)=X(t) 1(X(t ǫ)+X(t+ǫ)). Then f (s):= cov(X(s),ζǫ,t(X)) =1 ǫ,t − 2 − ǫ,t var(ζǫ,t(X)) for t=s, while f (s)=0 for all s [a,b] with t s ǫ.Figure 1 illustrates ǫ,t ∈ | − |≥ the decomposition of X(s) in X (s) and ζ (X)f (s) for a trajectory of a ǫ,t ǫ,t ǫ,t Brownian motion. The following theorem shows that under our setup all impact points in model (1.1) are uniquely identified for any process possessing specific local variation. Recall that (1.1) implies that b S m(X):=E(Y X)= β(t)X(t)dt+ β X(τ ). r r | Za r=1 X Theorem 1. Under our setup, assume that X possesses specific local variation. Then, for any bounded function β L2([a,b]), all S S, all ∗ ∗ β1∗,...,βS∗∗ ∈R, and all τ1,...,τS∗ ∈ (a,b) with∈τk ∈/ {τ1,...,τS}, ≥k =S + 1,...,S , we obtain ∗ b S∗ 2 (2.3) E m(X) β (t)X(t)dt β X (τ ) >0, − ∗ − r∗ i r Za r=1 ! ! X POINT IMPACT 7 whenever E(( b(β(t) β (t))X(t)dt)2)>0, or sup β β >0, or a − ∗ r=1,...,S| r − r∗| sup β >0. r=S+1,...,S∗R| r∗| Thequestion arises whether it is possibleto find general conditions which ensure that a process possesses specific variation. From a theoretical point of view, the Karhunen–Loève decomposition provides a tool for analyzing this problem. For f,g L2([a,b]) let f,g = bf(t)g(t)dt and f theassociated norm. ∈ h i a k k Wewilluseλ λ todenotethenonzeroeigenvalues of thecovariance 1 2 ≥ ≥··· R operatorΓofX,whileψ ,ψ ,...denoteacorrespondingsystemoforthonor- 1 2 mal eigenfunctions. It is then well known that X can be decomposed in the form ∞ (2.4) X(t)= X,ψ ψ (t), r r h i r=1 X where E( X,ψ 2)=λ , and X,ψ is uncorrelated with X,ψ for l=r. r r r l h i h i h i 6 The existence of specific local variation requires that the structure of the process is not too simple in the sense that the realizations X a.s. i lie in a finite dimensional subspace of L2([a,b]). Indeed, if Γ only possesses a finite number K < of nonzero eigenvalues, then model (1.1) ∞ is not identifiable. This is easily verified: X(t)= K X,ψ ψ (t) implies r=1h ri r that bβ(t)X(t)dt= K α X,ψ with α = ψ ,β . Hence, there are in- a r=1 rh ri r h Pr i finitely many different collections of K points τ ,...,τ and corresponding 1 K R P coefficients β ,...,β such that 1 K b K K K K β(t)X(t)dt= α X,ψ = X,ψ β ψ (τ )= β X(τ ). s s s r s r r r h i h i Za s=1 s=1 r=1 r=1 X X X X Most work in functional data analysis, however, relies on the assumption that Γ possesses infinitely many nonzero eigenvalues. In theoretically ori- ented papers, it is often assumed that ψ ,ψ ,... form a complete orthonor- 1 2 mal system of L2([a,b]) such that k ∞r=1hf,ψriψr−fk=0 for any function f L2([a,b]). ∈ P ThefollowingtheoremshowsthatX possessesspecificlocalvariationiffor a suitable class of functions L2-convergence generalizes to L -convergence. ∞ For t (a,b) and ǫ>0, let (t,ǫ,[a,b]) denote the space of all continuous ∈ C functionsf L2([a,b])withthepropertiesthatf(t)=sup f(s)=1and f(s)=0 for∈s /[t ǫ,t+ǫ]. s∈[a,b] ∈ − Theorem 2. Let ψ ,ψ ,... be a system of orthonormal eigenfunctions 1 2 corresponding to the nonzero eigenvalues of the covariance operator Γ of X. 8 A.KNEIP,D. POSS ANDP. SARDA If for all t (a,b) there exists an ǫ >0 such that t ∈ k lim inf sup f(s) f,ψ ψ (s) =0 r r (2.5) k→∞f∈C(t,ǫ,[a,b])s∈[a,b](cid:12)(cid:12)(cid:12) −Xr=1h i (cid:12)(cid:12)(cid:12) (cid:12) fo(cid:12)r every 0<ǫ<ǫ , (cid:12) (cid:12) t then the process X possesses specific local variation. The message of the theorem is that existence of specific local variation only requires that the underlying basis ψ ,ψ ,... possesses suitable approx- 1 2 imation properties. Somewhat surprisingly, the degree of smoothness of the realized trajectories does not play any role. As an example consider a standard Brownian motion defined on [a,b]= [0,1]. The corresponding Karhunen–Loève decomposition possesses eigenvalues λ = 1 and eigenfunctions ψ (t)=√2sin((r 1/2)πt), r = r (r 0.5)2π2 r − − 1,2,.... In the Supplementary Appendix B [Kneip, Poss and Sarda (2015)], it is verified that this system of orthonormal eigenfunctions satisfies (2.5). Although all eigenfunctions are smooth, it is well known that realized trajectories ofaBrownianmotionarea.s.notdifferentiable.Thiscanbeseenas a consequence of the fact that the eigenvalues λ 1 decrease fairly slowly, r ∼ r2 and, therefore, the sequence E(( k X,ψ ψ (t))2)= k λ (ψ (t))2 di- r=1h ri r′ r=1 r r′ verges as k . At the same time, another process with the same system →∞ P P ofeigenfunctions butexponentially decreasingeigenvalues λ exp( r)will ∗r ∼ − a.s. show sample paths possessing an infinite number of derivatives. Theo- rem 2 states that any process of this type still has specific local variation. 3. Covariance functions which are nonsmooth at the diagonal. In the following, we will concentrate on developing a theoretical framework which allows to define an efficient procedure for estimating number and locations of points of impact. Although specific local variation may well be present for processes possessing very smooth sample paths, it is clear that detection of points of impact will profit from a high local variability which goes along with nons- moothness.AspointedoutintheIntroduction,wealsobelievethatassuming nonsmooth trajectories reflect the situation encountered in a number of important applications. McKeague and Sen (2010) convincingly demonstrate that genomics data lead to sample paths with fractal behavior. All important processes analyzed in economics exhibit strong random fluctuations. Observed temperatures or precipitation rates show wiggly trajectories over time,ascanbeseeninourapplicationinSection7.Furthermore,anygrowth process willtosomeextent beinfluencedby randomchanges inenvironmen- tal conditions. In functional data analysis, it is common practice to smooth POINT IMPACT 9 observed (discrete) sample paths and to interpret nonsmooth components as “errors.” We want to emphasize that, unless observations are inaccurate and there exists some important measurement error, such components are an intrinsic part of the process. For many purposes as, for example, functional principal component analysis, smoothing makes a lot of sense since local variation has to be seen as nuisance. But in the present context local variation actually is a key property for identifying impact points. Therefore, further development will focus on processes with nonsmooth sample paths which will be expressed in terms of a nonsmooth diagonal of the correspondingcovariance function σ(t,s). It will beassumed that σ(t,s) possessesnonsmoothtrajectorieswhenpassingfromσ(t,t ∆)toσ(t,t+∆), − but is twice continuously differentiable for all (t,s), t=s. An example is the 6 standard Brownian motion whose covariance function σ(t,s)=min(t,s) has a kink at the diagonal. Indeed, in view of decomposition (2.2) a nonsmooth transition at diagonal may be seen as a natural consequence of pronounced specific local variation. For a precise analysis, it will be useful to reparametrize the covariance function. Obviously, the symmetry of σ(t,s) implies that σ(t,s) = σ(1(t+s+ t s ),1(t+s t s )) 2 | − | 2 −| − | =:ω (t+s, t s ) for all t,s [a,b]. ∗ | − | ∈ Instead of σ(t,s), we may thus equivalently consider the function ω (x,y) ∗ with x=t+s and y= t s . When passing from s=t ∆ to s=t+∆, | − | − the degree of smoothness of σ(t,s) at s=t is reflected by the behavior of ω (2t,y) as y 0. ∗ → First, consider the case that σ is twice continuously differentiable and for fixed x and y>0 let ∂ ω (x,y) denote the right (partial) derivative of ∂y+ ∗ |y=0 ω (x,y) as y 0. It is easy to check that in this case for all t (a,b) we ∗ → ∈ obtain ∂ ∂ y y ω (2t,y) = σ t+ ,t ∗ ∂y ∂y 2 − 2 + (cid:12)y=0 (cid:18) (cid:19)(cid:12)y=0 (3.1) (cid:12) (cid:12) (cid:12) 1 ∂ (cid:12)∂ (cid:12) (cid:12) = σ(s,t) σ(t,s) =0. 2 ∂s − ∂s (cid:18) (cid:12)s=t (cid:12)s=t(cid:19) (cid:12) (cid:12) In contrast, any process with ∂ ω (x(cid:12),y) =0 is(cid:12)nonsmooth at the ∂y+ ∗ (cid:12) |y=0 6 (cid:12) diagonal. If this function is smooth for all other points (x,y), y >0, then theprocess,similartotheBrownianmotion,possessesakinkatthediagonal. Now note that, for any process with σ(t,s)=ω (t+s, t s ) continuously ∗ | − | differentiable for t=s but ∂ ω (x,y) <0, it is possible to find a twice 6 ∂y+ ∗ |y=0 continuously differentiable function ω(x,y,z) with σ(t,s) = ω(t,s, t s ) such that ∂ ω (t+t,y) = ∂ ω(t,t,y) . | − | ∂y+ ∗ |y=0 ∂y |y=0 10 A.KNEIP,D. POSS ANDP. SARDA In a still more general setup, the above ideas are formalized by Assump- tion 1 belowwhich,as willbeshowninTheorem 3,provides sufficientcondi- tions in order to guarantee that the underlying process X possesses specific variation. We will also allow for unbounded derivatives as t s 0. | − |→ Assumption1. ForsomeopensubsetΩ R3 with[a,b]2 [0,b a] Ω, there exists a twice continuously differentiabl⊂e function ω:Ω× R a−s we⊂ll as → some 0<κ<2 such that for all t,s [a,b] ∈ (3.2) σ(t,s)=ω(t,s, t s κ). | − | Moreover, ∂ (3.3) 0< inf c(t) where c(t):= ω(t,t,z) . t [a,b] −∂z ∈ (cid:12)z=0 (cid:12) (cid:12) One can infer from (3.1) that for every twice continuo(cid:12)usly differentiable covariance functionσ thereexistssomefunctionω suchthat(3.2)holdswith κ=2. But note that formally introducing t s κ as an extra argument es- | − | tablishes an easy way of capturing nonsmooth behavior as t s 0, since | − |→ σ is not twice differentiable at the diagonal if κ<2. In Assumption 1, the value of κ<2 thus quantifies the degree of smoothness of σ at the diagonal. A very small κ will reflect pronounced local variability and extremely nonsmooth sample paths. There are many well-known processes satisfying this assumption. Fractional Brownian motion with Hurst coefficient 00: The covariance function is then given by σ(t,s)= 1(t2H +s2H t s 2H). 2 −| − | Inthiscase,Assumption1issatisfiedwithκ=2H,ω(t,s,z)= 1(t2H+s2H 2 − z) and c(t)=1/2. Ornstein–Uhlenbeck process with parameters σ2,θ > 0: The covariance u function is then defined by σ2 σ(t,s)= u(exp( θ t s ) exp( θ(t+s))). 2θ − | − | − − σ2 Then Assumption 1 is satisfied with κ = 1, ω(t,s,z) = u(exp( θz) 2θ − − exp( θ(t+s))) and c(t)=σ2/2. − u Theorem 3 below now states that any process respecting Assumption 1 possessesspecificlocalvariation.InSection2,wealreadydiscussedthestruc- tureofanappropriater.v.ζ (X)forthespecialcaseofastandardBrownian ǫ,t motion. The same type of functional may now be used in a more general setting. For δ>0 and [t δ,t+δ] [a,b], define − ⊂ (3.4) Z (X,t)=X(t) 1(X(t δ)+X(t+δ)). δ − 2 −

Functional linear regression with points of impact

Similar Functional linear regression with points of impact