Applications of Distance Correlation to Time Series

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1 Applications of Distance Correlation to Time Series Richard A. Davis, Columbia University Muneya Matsui, Nanzan University Thomas Mikosch, University of Copenhagen Phyllis Wan, Columbia University May 2-6, 2016 Workshop on Dependence, Stability, and Extremes The Fields Institute

2 Warm-up example: Amazon-returns ( to ) amazon time

3 Warm-up example: Amazon-returns ( to ) Series amazon Series amazon^2 ACF ACF Lag Lag

4 Warm-up example: Amazon-returns ( to ) Series amazon Series amazon^2 ACF ACF Lag Lag Series abs(amazon) ADCF of Amazon ACF ADCF Lag Lag

5 Example: Kilkenny wind speed time series Bonus (teaser) Kilken: 1/1/61 to 1/17/78 Series kilken kilken ACF time Lag

6 Example: Kilkenny wind speed time series Kilken: 1/1/61 to 1/17/78 Series kilken kilken ACF time Lag Series kilkenres ADCF of Kilken Res ACF ADCF 0e+00 8e Lag Lag Figure: Auto-distance correlation function (ADCF) of residuals from AR(9) model applied to Kilkenny daily wind speed.

7 Distance Covariance: a measure of dependence Distance covariance: random vectors X R p and Y R q, T(X, Y; µ) = ϕx,y (s, t) ϕ X (s) ϕ Y (t) 2 µ(ds, dt), R p+q where ϕ X,Y, ϕ X, ϕ Y denote the respective characteristic functions of (X, Y), X, Y, and µ is a measure.

8 Distance Covariance: a measure of dependence Distance covariance: random vectors X R p and Y R q, T(X, Y; µ) = ϕx,y (s, t) ϕ X (s) ϕ Y (t) 2 µ(ds, dt), R p+q where ϕ X,Y, ϕ X, ϕ Y denote the respective characteristic functions of (X, Y), X, Y, and µ is a measure. Distance correlation: R(X, Y; µ) = T(X,Y;µ) T(X,X;µ)T(Y,Y;µ)

9 Distance Covariance: a measure of dependence Distance covariance: random vectors X R p and Y R q, T(X, Y; µ) = ϕx,y (s, t) ϕ X (s) ϕ Y (t) 2 µ(ds, dt), R p+q where ϕ X,Y, ϕ X, ϕ Y denote the respective characteristic functions of (X, Y), X, Y, and µ is a measure. Distance correlation: R(X, Y; µ) = T(X,Y;µ) T(X,X;µ)T(Y,Y;µ) Sample distance covariance: Observations (X 1, Y 1 ),..., (X n, Y n ) from a statioonary time series (X t, Y t ). Then T X,Y n (0) = ˆϕ X,Y (s, t) ˆϕ X (s)ˆϕ Y (t) 2 µ(ds, dt), R p+q where ˆϕ denotes the empirical characteristic function, e.g., ˆϕ X,Y (s, t) = n 1 j=1 e i s,x j, +i t,y j

10 Background Feuerverger et al. (1981,...), Feuerverger and Mureika (1977); applications of the empirical characteristic function, inference using Fourier methods, etc. Feuerverger (1993), a bivariate test for independence, cor(cos(sx), cos(ty)), etc. S. Csörgő (1981a,b,c) Limit behavior of characteristic functions. Meintannis et al. (2008), 2015, Fourier methods for testing multivariate independence Székely, Rizzo, Bakirov (2007), Székely and Rizzo (2009), (2014), special choice of weight function, Brownian distance covariance Dueck et al. (2014) affinely invariant distance correlation Zhou (2012) application to time series. Fokianos and Pitsillou (2016). Testing pairwise dependence in time series.

11 Distance Covariance: choice of weight function T X,Y (0) = ϕ R p+q X,Y (s, t) ϕ X (s)ϕ Y (t) 2 µ(ds, dt) If µ = µ 1 µ 2, let µ i be the Fourier transform of µ i, i.e., µ i (x) = e i s,x dµ i (s), R p and assuming Fubination is okay, T(X, Y; µ) = E[ˆµ 1 (X X ) ˆµ 2 (Y Y )] + E[ˆµ 1 (X X )]E[ˆµ 2 (Y Y )] 2 E[ˆµ 1 (X X ) ˆµ 2 (Y Y )]. where (X, Y), (X, Y ), (X, Y ) are iid copies.

12 Distance Covariance: choice of weight function T X,Y (0) = ϕ R p+q X,Y (s, t) ϕ X (s)ϕ Y (t) 2 µ(ds, dt) If µ = µ 1 µ 2, let µ i be the Fourier transform of µ i, i.e., µ i (x) = e i s,x dµ i (s), R p and assuming Fubination is okay, T(X, Y; µ) = E[ˆµ 1 (X X ) ˆµ 2 (Y Y )] + E[ˆµ 1 (X X )]E[ˆµ 2 (Y Y )] 2 E[ˆµ 1 (X X ) ˆµ 2 (Y Y )]. where (X, Y), (X, Y ), (X, Y ) are iid copies. Choose µ i so that µ i have an explicit and easy to compute form.

13 Distance Covariance: computing T n T X,Y (0) = R p+q ϕ X,Y (s, t) ϕ X (s)ϕ Y (t) 2 µ(ds, dt) Fourier transform: µ i (x) = e i s,x dµ i (s), R p Computing T n. T X,Y n (0) = n 2 s=1 t=1 µ 1 (X s X t ) µ 2 (Y s Y t ) + n 2 µ 1 (X s X t )n 2 µ 2 (Y s Y t ) s,t=1 s,t=1 2n 3 µ 1 (X s X t ) µ 2 (Y s Y u ) s,t,u=1

14 Distance Covariance: choice of weight function finite (probability) measures: normal density: µ(x) = exp{ 1/2x Σx} sub-gaussian α/2-stable: µ(x) = exp{ (x, y) Σ(x, y) α/2 }, α (0, 2).

15 Distance Covariance: choice of weight function finite (probability) measures: normal density: µ(x) = exp{ 1/2x Σx} sub-gaussian α/2-stable: µ(x) = exp{ (x, y) Σ(x, y) α/2 }, α (0, 2). infinite measures: Lévy measure corresponding to an infinitely divisible random vector. Székely and Rizzo (2009): w(s, t) = with α (0, 2) and µ(x) = x α 1 c p c q t α+p p s α+q. q

16 Distance Covariance: choice of weight function finite (probability) measures: normal density: µ(x) = exp{ 1/2x Σx} sub-gaussian α/2-stable: µ(x) = exp{ (x, y) Σ(x, y) α/2 }, α (0, 2). infinite measures: Lévy measure corresponding to an infinitely divisible random vector. Székely and Rizzo (2009): with α (0, 2) and µ(x) = x α w(s, t) = 1 c p c q t α+p p s α+q. q Distance correlation is scale and rotational invariant relative to w(s, t).

17 Results consistency Existence of T(X, Y; µ) = R p+q ϕ X,Y (s, t) ϕ X (s)ϕ Y (t) 2 w(s, t)dsdt,. 1 µ a finite measure. 2 µ is infinite in a neighborhood of the origin and for some α (0, 2], E[ X α ] + E[ Y α ] < and R p+q 1 (s, t) α µ(ds, dt) <.

18 Results consistency Existence of T(X, Y; µ) = R p+q ϕ X,Y (s, t) ϕ X (s)ϕ Y (t) 2 w(s, t)dsdt,. 1 µ a finite measure. 2 µ is infinite in a neighborhood of the origin and for some α (0, 2], E[ X α ] + E[ Y α ] < and R p+q 1 (s, t) α µ(ds, dt) <. Consistency: If (X t, Y t ) is a stationary ergodic sequence satisfying 1 or 2 above, then T X,Y n (h) a.s. T X,Y (h).

19 Results weak convergence Assume that X 0 Y 0, α-mixing ( h=1 α 1/r <, r > 1) + moment condition, h and E[ X α + Y α ] <, [ p E X (l) α] <, [ q E Y (l) α] <, (1) l=1 R p+q (1 s α (1+ɛ)/u )(1 t α (1+ɛ)/u ) µ(ds, dt) < (2) where u = 2r/(r 1), α min(2, α). Then n T n (X, Y; µ) d G 2 µ = G(s, t) 2 µ(ds, dt), (3) R p+q where G is a complex-valued mean-zero Gaussian process. l=1

20 Results weak convergence Assume that X 0 and Y 0 are dependent and for some α (u/2, u] and for α min(2, α) the following hold: E[ X 2α + Y 2α ] <, [ E (1 p q X (l) α )(1 l=1 k=1 ] Y (k) α ) <, (4) and Then R p+q (1 s α /u )(1 t α /u ) µ(ds, dt) <. (5) n (Tn (X, Y; µ) T(X, Y; µ)) d G µ = G (s, t) µ(ds, dt), (6) R p+q where G (s, t) = 2Re{G(s, t)c(s, t)} is a mean-zero Gaussian process.

21 Distance correlation and AR(p) models Let (X t ) be the causal AR(p) process given by p X t = φ k X t k + Z t, (Z t ) IID(0, σ 2 ). k=1 Least squares estimate: observations X 1,..., X n where X t 1 = (X t 1,..., X t p ) ˆφ φ = Γ 1 1 n,p n ˆΓ n,p = 1 n t=p+1 X t 1 Z t, X T t 1 X t 1. t=p+1

22 Distance correlation and AR(p) models Let (X t ) be the causal AR(p) process given by p X t = φ k X t k + Z t, (Z t ) IID(0, σ 2 ). k=1 Least squares estimate: observations X 1,..., X n where X t 1 = (X t 1,..., X t p ) ˆφ φ = Γ 1 1 n,p n ˆΓ n,p = 1 n t=p+1 t=p+1 X t 1 Z t, X T t 1 X t 1. n 1/2 (ˆφ φ) d Q N(0, σ 2 Σ 1 p ).

23 Distance correlation and AR(p) residuals Fitted residuals Ẑ k = X k ˆφ X k 1, k = p , n. = Z k + (φ ˆφ) X k 1

24 Distance correlation and AR(p) residuals Fitted residuals Ẑ k = X k ˆφ X k 1, k = p , n. = Z k + (φ ˆφ) X k 1 Difference in characteristic functions between noise and residuals: n 1/2 e isẑ k +itẑ k+h n 1/2 k=p+1 k=p+1 e isz k +itz k+h ( = n 1/2 e isz k +itz k+h e is(φ ˆφ) X k 1 +it(φ ˆφ) X k+h 1 ) 1 k=p+1

25 Distance correlation and AR(p) residuals Fitted residuals Ẑ k = X k ˆφ X k 1, k = p , n. = Z k + (φ ˆφ) X k 1 Difference in characteristic functions between noise and residuals: n 1/2 e isẑ k +itẑ k+h n 1/2 k=p+1 k=p+1 e isz k +itz k+h ( = n 1/2 e isz k +itz k+h e is(φ ˆφ) X k 1 +it(φ ˆφ) X k+h 1 ) 1 k=p+1 ( ) n 1/2 e isz k +itz k+h is(φ ˆφ) X k 1 + it(φ ˆφ) X k+h 1 k=p+1

26 Distance correlation and AR(p) residuals Fitted residuals Ẑ k = X k ˆφ X k 1, k = p , n. = Z k + (φ ˆφ) X k 1 Difference in characteristic functions between noise and residuals: n 1/2 e isẑ k +itẑ k+h n 1/2 k=p+1 k=p+1 e isz k +itz k+h ( = n 1/2 e isz k +itz k+h e is(φ ˆφ) X k 1 +it(φ ˆφ) X k+h 1 ) 1 k=p+1 ( ) n 1/2 e isz k +itz k+h is(φ ˆφ) X k 1 + it(φ ˆφ) X k+h 1 k=p+1 = n 1/2 (φ ˆφ) n 1 e isz k +itz k+h (isx k 1 + itx k+h 1 ) k=p+1

27 Distance correlation and AR(p) residuals Fitted residuals Ẑ k = X k ˆφ X k 1, k = p , n. = Z k + (φ ˆφ) X k 1 Difference in characteristic functions between noise and residuals: n 1/2 k=p+1 e isẑ k +itẑ k+h n 1/2 = n 1/2 k=p+1 n 1/2 k=p+1 = n 1/2 (φ ˆφ) n 1 k=p+1 e isz k +itz k+h e isz k +itz k+h ( e is(φ ˆφ) X k 1 +it(φ ˆφ) X k+h 1 1 ) e isz k +itz k+h ( is(φ ˆφ) X k 1 + it(φ ˆφ) X k+h 1 ) k=p+1 d ( QE e isz 1+itZ h+1 (isx 0 + itx h ) ) e isz k +itz k+h (isx k 1 + itx k+h 1 )

28 ADCF of AR(p) residuals Z t finite and infinite variance where Ẑ t = X t p k=1 ˆφ k X t k. Theorem R n (h) := R n (Ẑ 1, Ẑ h+1 ; µ), Assume [(1 s 2 ) (1 t 2 ) + (s 2 + t 2 ) 1 { s t >1} ]µ(ds, dt) < 1 If EZ 2 t <, then n Rn (h) d GZ (s, t) + ξ h (s, t) 2 µ(ds, dt). T(0)

29 ADCF of AR(p) residuals Z t finite and infinite variance where Ẑ t = X t p k=1 ˆφ k X t k. Theorem R n (h) := R n (Ẑ 1, Ẑ h+1 ; µ), Assume [(1 s 2 ) (1 t 2 ) + (s 2 + t 2 ) 1 { s t >1} ]µ(ds, dt) < 1 If EZ 2 t <, then n Rn (h) d GZ (s, t) + ξ h (s, t) 2 µ(ds, dt). T(0) ξ h (s, t) = tϕ Z (t) ϕ Z (s)ψt h Q

30 ADCF of AR(p) residuals Z t finite and infinite variance where Ẑ t = X t p k=1 ˆφ k X t k. Theorem R n (h) := R n (Ẑ 1, Ẑ h+1 ; µ), Assume [(1 s 2 ) (1 t 2 ) + (s 2 + t 2 ) 1 { s t >1} ]µ(ds, dt) < 1 If EZ 2 t <, then n Rn (h) d GZ (s, t) + ξ h (s, t) 2 µ(ds, dt). T(0) ξ h (s, t) = tϕ Z (t) ϕ Z (s)ψt h Q 2 If Z t DOA(α) with index α (0, 2), then n Rn (h) d GZ (s, t) 2 µ(ds, dt). T(0)

31 Example: ADCF of AR(10) residuals: Z t N(0, 1) vs. Z t t(1.5) N(0,1) ADCF lag iid noise ADCF t(1.5) iid noise lag Figure: Empirical 5%, 50%, 95% quantiles of R n for Z t N(0, 1) (upper panel) and Z t t(1.5) (lower panel)

32 Example: ADCF of AR(10) residuals: Z t N(0, 1) vs. Z t t(1.5) N(0,1) ADCF residuals iid noise lag ADCF t(1.5) iid noise lag Figure: Empirical 5%, 50%, 95% quantiles of R n and Rn for Z t N(0, 1) (upper panel) and Z t t(1.5) (lower panel)

33 Example: ADCF of AR(10) residuals: Z t N(0, 1) vs. Z t t(1.5) N(0,1) ADCF residuals iid noise lag ADCF t(1.5) residuals iid noise lag Figure: Empirical 5%, 50%, 95% quantiles of R n and Rn for Z t N(0, 1) (upper panel) and Z t t(1.5) (lower panel)

34 Example: ADCF of AR(10) residuals: Z t N(0, 1) vs. Z t t(1.5) N(0,1) ADCF residuals bootstrap iid noise lag ADCF t(1.5) residuals iid noise lag Figure: Empirical 5%, 50%, 95% quantiles of R n and Rn for Z t N(0, 1) (upper panel) and Z t t(1.5) (lower panel)

35 Example: ADCF of AR(10) residuals: Z t N(0, 1) vs. Z t t(1.5) N(0,1) ADCF residuals bootstrap iid noise lag ADCF t(1.5) residuals bootstrap iid noise lag Figure: Empirical 5%, 50%, 95% quantiles of R n and Rn for Z t N(0, 1) (upper panel) and Z t t(1.5) (lower panel)

36 Example: ADCF of AR(10) residuals: symmetric Gamma(0.2,0.5) noise Székely and Rizzo weight function, need not work. w(s, t) = 1 c p c q t 1+p p s 1+q q, ADCF ADCF residuals noise lag lag Figure: Left: Box-plots from 500 independent replications. Right panel: empirical 5%, 50%, 95% quantiles from simulated residuals and from iid noise.

37 Example: Kilkenny wind speed time series bs residu iid noise ADCF 0e+00 4e 04 8e Lag Figure: Auto-distance correlation function (ADCF) of residuals from AR(9) model applied to Kilkenny daily wind speed.

38 Example: Kilkenny wind speed time series bs residu iid noise ADCF 0e+00 4e 04 8e Lag Figure: Auto-distance correlation function (ADCF) of residuals from AR(9) model applied to Kilkenny daily wind speed.

39 Example: Kilkenny wind speed time series Series kilkengarchres Series kilkengarchres^2 ACF ACF Lag Lag Series abs(kilkengarchres) ACF ADCF Lag Lag Figure: Auto-distance correlation function (ADCF) of residuals from AR(9) followed by a GARCH(1,1) model applied to Kilkenny daily wind speed.

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