An Introduction to Wavelets with Applications in Environmental Science

Transcription

1 An Introduction to Wavelets with Applications in Environmental Science Don Percival Applied Physics Lab, University of Washington Data Analysis Products Division, MathSoft overheads for talk available at joint work with: B. Whitcher, S. Byers & P. Guttorp (UW) S. Sardy (UW, MathSoft, EPFL) P. Craigmile (UW)

2 Overview of Talk overview of discrete wavelet transform (DWT) wavelet coefficients and their interpretation DWT as a time series decorrelator three uses for wavelets 1. testing for variance changes 2. bootstrapping auto/cross-correlation estimates 3. estimating d for stationary/nonstationary fractional difference processes with trend warning: results for 2& 3 are preliminary

3 Overview of DWT let X =[X 0,X 1,...,X N 1 ] T be observed time series (for convenience, assume N integer multiple of 2 J 0) let W be N N orthonormal DWT matrix (more precisely: partial DWT of level J 0 ) W = WX is vector of DWT coefficients can partition W as follows: W = W 1. W J0 V J0 W j contains N j = N/2 j wavelet coefficients related to changes of averages at scale τ j =2 j 1 (τ j is jth dyadic scale) related to times spaced 2 j units apart V J0 contains N J0 = N/2 J 0 scaling coefficients related to averages at scale λ J0 =2 J 0 related to times spaced 2 J 0 units apart

4 Example: DWT of FDP X t called fractional difference process (FDP) if it has a spectral density function (SDF) given by S X (f) = where σ 2 > 0 and 1 2 d<1 2 σ 2 2 sin(πf) 2d, note: for small f, have S X (f) C/ f 2d ; i.e., 1/f type process if d = 0, FDP is white noise if 0 <d< 1 2, FDP stationary with long memory can extend definition to d 1 2 nonstationary 1/f type process also called FARIMA(0,d,0) process example: DWT of FDP, d =0.4

5 Two Consequences of Orthonormality multiresolution analysis (MRA) X = W T W = J 0 j=1 W T j W j + V T J 0 V J0 J 0 j=1 (W j partitions W commensurate with W j ) scale-based additive decomposition D j s & S J0 called details & smooth example: Nile River minimum flood levels analysis of variance: ˆσ 2 X 1 N N 1 t=0 (X t X) 2 = 1 N J 0 j=1 D j + S J0 W j 2 + V J0 2 X 2 scale-based decomposition (cf. frequency-based) can define wavelet variance νx(τ 2 j ) for FDP, can deduce d from log/log plots since ν 2 X(τ j ) Cτ 2d 1 j example: Nile River minimum flood levels

6 DWT in Terms of Filters filter X 0,X 1,...,X N 1 to obtain 2 j/2 W j,t L j 1 l=0 h j,l X t l mod N, t =0, 1,...,N 1 where h j,l is jth level wavelet filter note: circular filtering subsample to obtain wavelet coefficients: W j,t =2 j/2 W j,2 j (t+1) 1, t =0, 1,...,N j 1, where W j,t is tth element of W j examples: Haar, D(4), C(6) & LA(8) wavelet filters 1 jth wavelet filter is band-pass with pass-band [, 1 ] 2 j+1 2 j note: jth scale related to interval of frequencies similarly, scaling filters yield V J0 examples: Haar, D(4), C(6) & LA(8) scaling filters J 0 th scaling filter is low-pass with pass-band [0, 1 2 J 0 +1 ]

7 Wavelets as Whitening Filters recall DWT of FDP for d =0.4 since FDP is stationary process, W j is also (ignoring terms influenced by circularity) can compute SDFs for each W j see figure DWT acts as whitening filter requires SDF of X to be flat over pass-band 1 [, 1 ] 2 j+1 2 j if not true, can use wavelet packet transform (DWPT) used by Flandrin, Tewfik & Kim, Wornell, McCoy & Walden three examples built on whitening property 1. testing for variance changes 2. bootstrapping auto/cross-correlation estimates 3. estimating d for stationary/nonstationary fractional difference processes with trend whitening property should help with other problems

8 Homogeneity of Variance: I claim: DWT approximately decorrelates FDPs implication: W j should resemble white noise (ignoring coefficients influenced by circularity) cov {W j,t,w j,t } 0 when t t var {W j,t } should not vary with t (homogeneity of variance) can test for homogeneity of variance using W j suppose Y 0,...,Y N 1 independent normal RVs with E{Y t } = 0 and var {Y t } = σ 2 t want to test null hypothesis H 0 : σ 2 0 = σ 2 1 = = σ 2 N 1 can test H 0 versus a variety of alternatives, e.g., H 1 : σ 2 0 = = σ 2 k σ 2 k+1 = = σ 2 N 1 using normalized cumulative sum of squares

9 Homogeneity of Variance: II D + to define test statistic D, start with P k and then compute max k +1 0 k N 2 k j=0 Yj 2 N 1 j=0 Yj 2, k =0,...,N 2 N 1 P k & D max 0 k N 2 from which we form D max (D +,D ) can reject H 0 if observed D is too large can quantify too large by considering distribution of D under H 0 need to find critical value x α such that P[D x α ]=α for, e.g., α =0.01, 0.05 or 0.1 P k once determined, can perform α level test of H 0 : compute D statistic from data Y 0,...,Y N 1 reject H 0 at level α if D x α k N 1

10 Homogeneity of Variance: III can determine critical values x α in two ways Monte Carlo simulations large sample approximation to distribution of D: P[(N/2) 1/2 D x] 1+2 ( 1) l e 2l2 x 2 l=1 (reasonable approximation for N 128) idea: given time series X, compute D using W j,t =2 j/2 W j,2 j (t+1) 1, (L 2) j t N 2 1, j where L is length of j = 1 level wavelet filter and 2 j/2 W j,t L j 1 l=0 h j,l X t l mod N results in level by level tests above formulation allows for general N (i.e., N need not be multiple of 2 J 0) Q: is DWT decorrelation of FDPs good enough? Fig. 9 says yes!

11 Example: Nile River Minima recall MRA & wavelet variance plots application of homogeneity of variance test: scale D x 0.1 x 0.05 x year years years years if H 0 rejected, use nondecimated DWT to detect change point: compute rotated cumulative variance curve look for time of largest excursion from 0 Fig. 10: change point detection 720 AD for level j =1 722 AD for level j =2 agrees well with mosque construction in 715 AD interpretation differs from Beran & Terrin (1996)

12 Wavelet-Based Bootstrapping: I Davison & Hinkley, 1998, Chapter 8, discusses bootstrapping in context of time series analysis whitening allows wavelet-based bootstrapping for certain statistics (but not all) first example: lag 1 autocorrelation estimate ˆr 1 N 2 t=0 X t X t+1 N 1 t=0 X 2 t idea: to get standard error of ˆr 1, compute DWT of X sample with replacement from W j to form W (b) j (do same with V J0 ) synthesize X (b) using W (b) j s & V (b) J 0 compute ˆr (b) 1 for X (b) repeat until computer gets tired use standard error of ˆr (b) 1 s for X (b) s to assess standard error of ˆr 1 for X

13 Wavelet-Based Bootstrapping: II to test scheme, did Monte Carlo study involving AR(1) process: X t =0.9X t 1 + ɛ t MA(1) process: X t = ɛ t + ɛ t 1 FDP with d =0.45 average ˆr (b) 1 s have negligible bias comparison of standard errors, N = 128: LA(8) DWT LA(8) DWPT true AR(1) ± ± MA(1) ± ± FDP ± ± comparison of standard errors, N = 1024: LA(8) DWT LA(8) DWPT true AR(1) ± ± MA(1) ± ± FDP ± ± handles both short & long memory models

14 Wavelet-Based Bootstrapping: III second example: cross-correlation estimate ˆr (XY ) 0 N 1 ( N 1 t=0 X 2 t t=0 X t Y t N 1 t=0 Yt 2 ) 1/2 to assess null hypothesis r (XY ) 0 =0, separately generate X (b) & Y (b) bootstrapped ˆr (XY ) 0 should reflect variability in ˆr (XY ) 0 under null example: cross-correlation ˆr (XY ) 0 = 0.26 between maximum annual snow-pack level at Mt. Rainier Pacific decadel oscillation index can reject null at critical level p =0.01 lots of questions to be addressed! what is class of applicable statistics?

15 Estimation for FDPs: I extension of work by Wornell, McCoy & Walden problem: estimate d from time series U t such that where U t = T t + X t T t r j=0 a j t j is polynomial trend X t is FDP process, but can have d 1 2 DWT wavelet filter of width L has embedded differencing operation of order L/2 if L 2 r + 1, reduces polynomial trend to 0 can partition DWT coefficients as W = W s + W b + W w where W s has scaling coefficients and 0s elsewhere W s has boundary-dependent wavelet coefficients W w has boundary-independent wavelet coefficients

16 Estimation for FDPs: II since U = W T W, can write U = W T (W s + W b )+W T W w T + X example: microbolometric camera data can use values in W w to form likelihood: L(d, σ 2 ɛ ) J 0 N j j=1 t=1 where σ 2 ɛ is innovations variance; 1 ( 2πσ 2 j ) 1/2 e W 2 j,t+l j 1/(2σ2 j ) σj 2 1/2 H 1/2 j(f) df ; 2 sin(πf) 2d and H j ( ) is squared gain for {h j,l } σ 2 ɛ works well in Monte Carlo simulations for Hansen Lebedeff series, get ˆd. =0.40 ± 0.08 lots of questions to be addressed! what are properties of T? how to assess significance of T? (extension of Brillinger, 1994)