Weakly dependent functional data. Piotr Kokoszka. Utah State University. Siegfried Hörmann. University of Utah

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1 Weakly dependent functional data Piotr Kokoszka Utah State University Joint work with Siegfried Hörmann University of Utah

2 Outline Examples of functional time series L 4 m approximability Convergence of Eigenvalues and Eigenfunctions Estimation of the long run variance Change point detection Functional linear model with dependent dependent regressors 1

3 Seven functional time series observations (1440 measurements per day) The horizontal component of the magnetic field measured in one minute resolution at Honolulu magnetic observatory from 1/1/ :00 UT to 1/7/ :00 UT

4 Functional time series can be transformed to stationary series (removal of trend, differencing, ad hoc)

5 Autoregressive dependence: Three weeks of a time series derived from credit card transaction data. The vertical dotted lines separate days

6 Definition The sequence {X n } of functions in L 2 is L 4 m approximable if it admits the representation X n = f(ε n, ε n 1,...), where the ε i are iid elements in a measurable space S, and f is a measurable function f : S L 2. Let {ε i } be an independent copy of {ε i}, set X (m) n = f(ε n, ε n 1,..., ε n m+1, ε n m, ε n m 1,...) Main Condition: ( E X m X m (m) ) 1/4 4 <. m=1 5

7 Discussion The main condition depends only on ν 4 ( X 0 X (m) 0 ) = ( E X 0 X (m) 0 4 ) 1/4 {ε (n) k, k Z} independent copy of {ε k, k Z} Sequences {ε (n), k Z} are independent k X (m) n = f(ε n, ε n 1,..., ε n m+1, ε (n) n m, ε(n) n m 1,...) For each m, the sequence dependent. { X (m) n }, n Z is m X (2) 0 = f(ε 0, ε 1, ε (0) 2, ε(0) 3,...) X (2) 2 = f(ε 2, ε 1, ε (2) 0, ε(2) 1,...) 6

8 An alternative definition X (m) n = f (m) (ε n, ε n 1,..., ε n m+1 ) For each m, the sequence dependent. { X (m) n }, n Z is m does not have the same distribu- (extra line in proofs.) This X n (m) tion as X n Functions f (m) very natural in examples, e.g. f (m) (ε n, ε n 1,..., ε n m+1 ) = f(ε n, ε n 1,..., ε n m+1,0,0,...) This definition produces a (theoretically) narrower class. 7

9 Example: AR(1) X n (t) = ψ(t, s)x n 1 (s)ds + ε n (t), X n = j=0 Ψ j (ε n j ), ( Ψ < 1) or X (m) n = m 1 j=0 X (m) n = Ψ j (ε n j ) + m 1 j=0 j=m Ψ j (ε n j ) Ψ j (ε (n) n j ). m=1 ν 4 (X m X (m) m ) (2) j=m ν 4 (X m X m (m) ) O(1) Ψ j ν 4 (ε 0 ) m=1 Ψ m <, 8

10 Example: bilinear model X n (t) = U n Y n (t) {U n } independent of {Y n }. X (m) n (t) = U (m) n Y n (m) (t) Example: Functional ARCH y k (t) = ε k (t)σ k (t), where σk 2 1 (t) = δ(t) + β(t, 0 s)σ2 k 1 (s)ε2 k 1 (s)ds 9

11 Bounds on Eigenfunctions and Eigenvalues Theorem: where NE Ĉ C 2 S U X, U X = ν 4 4 (X) ν 3 4 (X) r=1 ν 4 (X r X (r) r ). Corollary: NE [ λ j ˆλ j 2] U X. Set ĉ j = sign(< ˆv j, v j >) NE [ ĉ jˆv j v j 2] 8U X α 2 j where α 1 = λ 1 λ 2 and α j = min(λ j 1 λ j, λ j λ j+1 ), 2 j d., 10

12 Long run variance introduction: scalar case. NVar[ X N ] j= γ j, γ j = Cov(X 0, X j ) If {X n } is L 2 m approximable, then j= γ j < Proof: Cov(X 0, X j ) = Cov(X 0, X j X (j) j )+Cov(X 0, X (j) j ). X 0 = f(ε 0, ε 1,...), are independent X (j) j = f (j) (ε j, ε j 1,..., ε 1 ), γ j [EX 2 0 ]1/2 [E(X j X (j) j ) 2 ] 1/2. 11

13 Kernel estimator: ˆσ = j q ω q (j)ˆγ j, Consistency requires cumulant conditions: sup h r= s= κ(h, r, s) <. κ(h, r, s) = E[(X 0 µ)(x h µ)(x r µ)(x s µ)] (γ h γ r s + γ r γ h s + γ r γ h r ), For an L 4 m approximable sequence ( Cov X 0 (X k X (k) ), X r (r) sup k,l 0 r=1 X k = ( ) X 0 X k X (k) k c j ε k j ; j=1 k X (k) k = k 1 j=1 depends on {ε j, j 0} c j ε k j X r (r) X (r+l) r+l depends on {ε j,1 j r + l}. Condition holds trivially Approximability: m=1 j=m c j < X (r+l) r+l 12 ) <.

14 Long run variance for functional time series Project functions on principal components The resulting sequence of vectors has a long run covariance matrix It inherits L 4 m approximability kernel estimator is consistent under the condition max 1 i,j d q(n) N 1 k,l=0 N 1 r=1 ( Cov X i0 (X jk X (k) jk ), X(r) ir X(r+l) j,r+l ) 0. X jk projection of kth observation on jth principal component 13

15 Change point detection H 0 : X i (t) = µ(t) + Y i (t), EY i (t) = 0. H A : X i (t) = { µ1 (t) + Y i (t), 1 i k, µ 2 (t) + Y i (t), k < i N, Test statistic is based on 1 i k X i (t) k N 1 i N X i (t) ˆv l(t)dt = 1 i k ˆη li k N 1 i N ˆη li. For functional time series it is different than for iid data (long run variance). Required asymptotic properties can be established under L 4 m approximability. 14

16 Functional linear model with dependent regressors Y n (t) = ψ(t, s)x n (s) + ε n. Idea of Yao, Müller, Wang: X(s) = i=1 ξ i v i (s), Y (t) = j=1 ζ j u j (t), ψ(t, s) = k=1 l=1 E[ξ l ζ k ] E[ξ 2 l ] u k(t)v l (s). ˆψ KL (t, s) = K L k=1 l=1 ˆλ 1 l ˆσ lk û k (t)ˆv l (s), YMW focus on smoothing, assumptions (2+ pages) deal mostly with smoothing parameters, K and L depend on the smoothing bandwidths, integrals of FT of kernels, and the rate of decay of the eigenfunctions. 15

17 Examples of Magnetometer data High latitude High latitude CMO Mid latitude BOU Low latitude HON CMO Mid latitude BOU Low latitude HON Horizontal intensities of the magnetic field measured at a high-, mid- and low-latitude stations during a sub-storm (left column) and a quiet day (right column). Note the different vertical scales for high-latitude records. 16

18 1st observed FPC Score (Y) st observed FPC Score (Y) st observed FPC Score (X) 2nd observed FPC Score (X) 2nd observed FPC Score (Y) nd observed FPC Score (Y) st observed FPC Score (X) 2nd observed FPC Score (X) Functional predictor-response plots of FPC scores of response functions versus FPC of explanatory functions for magnetometer data (CMO vs THY0) 17

19 For functional time series data, the focus is on dependence. We assume that the X n are L 4 m approximable. λ j, γ j eigenvalues corresponding to v j u j. Recall α j = min(λ j 1 λ j, λ j λ j+1 ), 2 j d. Define α j correspondingly for γ j Set g L = min{λ j 1 j L}, h L = min{α j α j 1 j L}. Assumption for consistency: The same for K. L = o(n 1/12 (g L h L ) 1/2 ). 18