Adaptive wavelet decompositions of stochastic processes and some applications

Transcription

1 Adaptive wavelet decompositions of stochastic processes and some applications Vladas Pipiras University of North Carolina at Chapel Hill SCAM meeting, June 1, 2012 (joint work with G. Didier, P. Abry) Vladas Pipiras University of North Carolina atadaptive Chapel Hill wavelet () decompositions of stochastic processes and some applications 1 / 31

2 Outline Motivation: fractional Brownian motion Wavelet-based decomposition of Meyer, Sellan & Taqqu (1999) Some applications (path regularity and synthesis) Extension I: general stationary Gaussian processes Continuous time (Ornstein-Uhlenbeck process, synthesis) Discrete time (FARIMA(0, s, 0) series, estimation) Extension II: Rosenblatt process Synthesis Some related work Vladas Pipiras University of North Carolina atadaptive Chapel Hill wavelet () decompositions of stochastic processes and some applications 2 / 31

3 FRACTIONAL BROWNIAN MOTION Vladas Pipiras University of North Carolina atadaptive Chapel Hill wavelet () decompositions of stochastic processes and some applications 3 / 31

4 Fractional Brownian motion Fractional Brownian motion (FBM) {B H (t)} t R, H (0, 1), is the only Gaussian H self-similar processes with stationary increments. It has the following representations: B H (t) = c ((t u) H ( u) H )B(du) R = C R e ixt 1 (ix) 1 2 H B(dx). ix Fractional Gaussian noise : db H dt (t) = C R eixt (ix) 1 2 H B(dx) or db H dt (x) = C(ix) 1 H B(dx) 2 dx. Vladas Pipiras University of North Carolina atadaptive Chapel Hill wavelet () decompositions of stochastic processes and some applications 4 / 31

5 Fractional Brownian motion Wavelet-based decomposition (Meyer, Sellan & Taqqu (1999)): For suitable biorthogonal scaling and wavelet functions Φ H and Ψ H, B H (t) = k= Φ H (t k)s (H) k + j=0 k= 2 jh Ψ H (2 j t k)ɛ j,k b 0, where S (H) k is a Gaussian FARIMA(0, H + 1/2, 0) sequence, ɛ j,k are independent Gaussian N (0, 1) variables and b 0 is such that B H (0) = 0. The functions Φ H and Ψ H are defined through their Fourier transforms as Φ H (x) = ( ) ix 1 2 H φ(x), 1 e ix ΨH (x) = (ix) 1 2 H ψ(x), where φ and ψ are scaling function and wavelet of an orthogonal wavelet basis. Vladas Pipiras University of North Carolina atadaptive Chapel Hill wavelet () decompositions of stochastic processes and some applications 5 / 31

6 Fractional Brownian motion Basic idea: With Φ ( ) 1 H (x) = ix 2 +H φ(x), 1 e ΨH (x) = ( ix) 1 ix 2 +H ψ(x), for example, ɛ 0,k = B H (t)ψ H db H (t k)dt = c dt (t)ψh 1 (t k)dt = c and R S (H) k = R db H dt (x) Ψ H 1 (x)e ikx dx = c (ix) 1 2 R = c ψ(x)e ikx B(dx) = ψ(u k)b(du) R R B H (t)φ H (t k)dt = c = m=0 R R R H B(dx) (ix) 1 2 +H ψ(x)e ikx dx dx (1 e ix ) ( 1 2 +H) φ(x)e ikx B(dx) c m ( φ(u (k m))b(du)) R where (1 e ix ) ( 1 2 +H) = m=0 c me imx. Vladas Pipiras University of North Carolina atadaptive Chapel Hill wavelet () decompositions of stochastic processes and some applications 6 / 31

7 Fractional Brownian motion Application 1 (e.g. Jaffard, Lashermes and Abry (2006)): Finer sample properties of FBM: In the adaptive wavelet basis, the wavelet coefficients d j,k are independent Gaussian and one can manipulate easily the wavelet leaders L j,k = sup (j,k ) 3(j,k) d j,k. In particular, it can be shown that, with probability 1, for any q R, 1 log(2 j ) log ( 1 2 j L j,k q) Hq. 2 j k=1 This allows concluding that the Hölder exponent for FBM is exactly H at every point. Vladas Pipiras University of North Carolina atadaptive Chapel Hill wavelet () decompositions of stochastic processes and some applications 7 / 31

8 Fractional Brownian motion Application 2 (Abry & Sellan (1996), Pipiras (2005)): Synthesis of FBM: FWT at the reconstruction: S (H) j. = u (s) ( 2 S (H) j 1, ) + v (s) ( 2 ɛ j 1, ) where s = H + 1/2, the fractional low- and high-pass filters are û (s) (w) = f (s) (w)û(w), v (s) (w) = ĝ (s) (w) v(w) with CMFs u and v associated with an orthogonal MRA and f (s) (w) = (1 + e iw ) s, ĝ (s) (w) = (1 e iw ) s. It can be easily shown that, for a compact K, almost surely and for 2 arbitrarily small ɛ > 0, sup jh t K S (H) j,[2 j t] (B H(t) + b 0 ) C2 j(h ɛ), where a random variable C depends on K, H, ɛ and Φ H. Vladas Pipiras University of North Carolina atadaptive Chapel Hill wavelet () decompositions of stochastic processes and some applications 8 / 31

9 Fractional Brownian motion 1.4 Convergence to fbm B H (t), H= Difference between consecutive approximations fbm log of difference t scale j Vladas Pipiras University of North Carolina atadaptive Chapel Hill wavelet () decompositions of stochastic processes and some applications 9 / 31

10 Fractional Brownian motion Interesting role of zero moments: Suppose that the orthogonal MRA has N zero moments, that is, û(w) = (1 + e iw ) N û 0,N (w) and v(w) = (1 e iw ) N v 0,N (w). Then, û (s) (w) = (1 + e iw ) N+s û 0,N (w) = f (N+s) (w)û 0,N (w) and similarly v (s) (w) = ĝ (s N) (w) v 0,N (w). When N is larger, this translates into a faster decay of fractional filters u (s) and v (s). Length of a truncated filter (s = 1.25) Filters Cutoff N = 1 N = 3 N = 10 u (s) v (s) Vladas Pipiras University of North Carolina atadaptive Chapel Hill wavelet () decompositions of stochastic processes and some applications 10 / 31

11 EXTENSION I: STATIONARY GAUSSIAN PROCESSES CONTINUOUS TIME Vladas Pipiras University of North Carolina atadaptive Chapel Hill wavelet () decompositions of stochastic processes and some applications 11 / 31

12 Stationary Gaussian processes: OU process Example: Ornstein-Uhlenbeck (OU) process: X (t) = t e λ(t u) db(u) = It is known (e.g. Zhang & Walter (1994)) that X (t) = k= a J,k θ J (t 2 J k) + where a J,k, d j,k are independent N(0, 1), and j=j k= e itx 1 λ + ix d B(x). d j,k Ψ j (t 2 j k), θ J (x) = (λ + ix) 1 2 J/2 φ(2 J x), Ψj (x) = (λ + ix) 1 2 j/2 ψ(2 j x). Vladas Pipiras University of North Carolina atadaptive Chapel Hill wavelet () decompositions of stochastic processes and some applications 12 / 31

13 Stationary Gaussian processes: OU process Adaptive wavelet decomposition (Didier & Pipiras (2008)): X (t) = X J,k Φ J (t 2 J k) + d j,k Ψ j (t 2 j k), k= j=j k= where X J (x) = 2 J (1 e (λ2 J +ix) ) 1 â J (x) is AR(1) time series, and The key here is that Φ J (x) = 2J/2 (1 e 2 J (λ+ix) ) φ(2 J x). λ + ix 2 J (1 e 2 J (λ+ix) ) λ + ix 1, as J. Vladas Pipiras University of North Carolina atadaptive Chapel Hill wavelet () decompositions of stochastic processes and some applications 13 / 31

14 Stationary Gaussian processes: OU process FWT-like algorithm (at reconstruction) X j+1 = u j 2 X j + v j 2 d j, where (with CMFs u, v) û j (x) = 1 2 (1 + +ix) e (λ2 (j+1) )û(x), v j (x) = 2 (j+1) (1 e (λ2 (j+1) +ix) ) 1 v(x). Because of zero moments, v j becomes essentially finite at small scales as j (if v is finite). It can be shown easily that sup t K 2 J/2 X J,[2 J t] X (t) C 2 Jγ a.s., where K is a compact interval and γ > 0 is a sample path smoothness parameter. Vladas Pipiras University of North Carolina atadaptive Chapel Hill wavelet () decompositions of stochastic processes and some applications 14 / 31

15 Stationary Gaussian processes: OU process Convergence to OU process 0.5 Difference between consecutive approximations 0 OU t log of difference scale j Vladas Pipiras University of North Carolina atadaptive Chapel Hill wavelet () decompositions of stochastic processes and some applications 15 / 31

16 Stationary Gaussian processes Adaptive wavelet decompositions (Didier & Pipiras (2008)): Stationary Gaussian processes: X (t) = R g(u)db(u) = R eitx ĝ(x)d B(x). where X (t) = Φ J (x) = k= X J,k Φ J (t 2 J k) + j=j k= d j,k Ψ j (t 2 j k), ĝ(x) ĝ J (2 J x) 2 J/2 φ(2 J x), ΨJ (x) = ĝ(x)2 J/2 ψ(2 J x) for a suitable sequence of discrete approximations g J = {g J,k } k Z (in particular, ĝ(x)/ĝ J (2 J x) 1 as J ). There are FWT-like algorithms, convergence of approximation coefficients, other examples (e.g. with general rational spectral densities). Vladas Pipiras University of North Carolina atadaptive Chapel Hill wavelet () decompositions of stochastic processes and some applications 16 / 31

17 Stationary Gaussian processes Riesz basis, vaguelets: Let Ψ # j,k (x) = ĝ(x) ψ j,k (x), Ψj,k # (x) = ĝ(x) 1 ψ j,k (x) Ψ# j,k Ψ j,k = Ψ # j,k, Ψ j,k = Ψj,k # 2 Ψ j,k #. 2 When are these Riesz bases? Vaguelet bases? Sufficient conditions are provided in Didier & Pipiras (2012), as well as examples where the bases are not vaguelet. See also Benassi, Jaffard & Roux (1997). Vladas Pipiras University of North Carolina atadaptive Chapel Hill wavelet () decompositions of stochastic processes and some applications 17 / 31

18 EXTENSION I: STATIONARY SERIES DISCRETE TIME Vladas Pipiras University of North Carolina atadaptive Chapel Hill wavelet () decompositions of stochastic processes and some applications 18 / 31

19 Stationary series Adaptive wavelet decompositions (Didier & Pipiras (2010)): Let X 0 = a 0 ɛ 0 be a stationary time series with a 0 l 2 (Z) and a white noise ɛ 0. For j 1, define X j = 2 (U j d X j 1 ), ξ j = 2 (V j d X j 1 ), where ( Û j d (w) = âj (2w) ) â j 1 û(w), (w) ( j V d (w) = 1 ) â j 1 v(w) (w) for a collection of sequences a j l 2 (Z). Then, it can be shown that X j = a j ɛ j with a white noise ɛ j, and uncorrelated, white noise sequences ξ j, j 1. Vladas Pipiras University of North Carolina atadaptive Chapel Hill wavelet () decompositions of stochastic processes and some applications 19 / 31

20 Stationary series Example: FARIMA(0, s, 0) series: These are X = a ɛ with a white noise ɛ and â(w) = σ(1 e iw ) δ, where σ > 0 and δ ( 1/2, 1/2). Take so that â j (w) = â(w), j 1, â(2w) â(w) = (1 e i2w ) δ (1 e iw ) δ = (1 + e iw ) δ, 1 â(w) = (1 e iw ) δ. Again, the filters Û j d (w) = (1 + eiw ) δ û(w), V j d (w) = (1 eiw ) δ v(w) decay fast for larger number of zero moments. Vladas Pipiras University of North Carolina atadaptive Chapel Hill wavelet () decompositions of stochastic processes and some applications 20 / 31

21 Stationary series Parametric estimation: Consider a class of models with spectral densities f θ(w) = σ2 2π â θ(w) 2 =: σ 2 f θ (w) (a θ,0 = 1) indexed by unknown parameters θ = (σ 2, θ). Let θ 0 = (σ0 2, θ 0) be the true parameters of a time series to be estimated. It can be shown that 2π θ 0 = argmin θ 0 which suggests to estimate θ 0 as â θ0 (w) 2 dw = argmin â θ (w) 2 θ j=1 2 j E(ξ j,θ n ) 2, θ = argmin θ J j=1 2 j n j n j n=1 (ξ j,θ n ) 2. This can be thought as an approximate (Gaussian) ML estimator. Vladas Pipiras University of North Carolina atadaptive Chapel Hill wavelet () decompositions of stochastic processes and some applications 21 / 31

22 Stationary series Example: Estimation results for FARIMA(0, 0.4, 0) series with estimator bias given by BS 10 3, and standard deviation by SD Whittle AWD-based T BS SD N BS SD (256) (512) (1024) (2048) Vladas Pipiras University of North Carolina atadaptive Chapel Hill wavelet () decompositions of stochastic processes and some applications 22 / 31

23 EXTENSION II: ROSENBLATT PROCESS Vladas Pipiras University of North Carolina atadaptive Chapel Hill wavelet () decompositions of stochastic processes and some applications 23 / 31

24 Rosenblatt process Rosenblatt process: {Z H (t)} t R, H (3/4, 1): stationary increments, (1/2, 1) (2H 1) self-similar, non-gaussian with all its moments finite: R 2 { t 0 (s u 1 ) H (s u 2) H ds }db(u 1 )db(u 2 ). Non-CLT: If {X (H) k } k Z is a Gaussian FARIMA (0, H 1/2, 0) sequence, then [nt] 1 ( n 2H 1 (X (H) k ) 2 E(X (H) k ) 2) d ZH (t). k=1 Question: What about FARIMA sequence generated by wavelet method, that is, n = 2 j, X (H) k = X (H) j,k with X (H) j, = u (d) ( 2 X (H) j 1, ) + v (d) ( 2 ɛ j 1, ) and d = H 1/2? Is there almost sure convergence to Rosenblatt process as j? Vladas Pipiras University of North Carolina atadaptive Chapel Hill wavelet () decompositions of stochastic processes and some applications 24 / 31

25 Rosenblatt process Convergence to frm Z H (t), H= Difference between consecutive approximations frm log of difference t t Vladas Pipiras University of North Carolina atadaptive Chapel Hill wavelet () decompositions of stochastic processes and some applications 25 / 31

26 Rosenblatt process Wavelet-based decomposition (Pipiras (2004), Abry & Pipiras (2006)): The Rosenblatt process Z H admits a wavelet-based decomposition with a nonstandard approximation term at scale 2 j : 2 j(2h 1) n= k= Φ H,n (2 j t k)s (H,n) j,k, where Φ H,n are suitable deterministic functions, and = ( S (H,n) j,k 0<i k X (H) j,i X (H) j,i+n ) (H) EX j,i X (H) j,i+n. As j increases, 2 j(2h 1) S (H,0) j,[2 j t] Z H(t) a.s. pointwise (but conjectured and observed uniformly on compact intervals) and exponentially fast. Vladas Pipiras University of North Carolina atadaptive Chapel Hill wavelet () decompositions of stochastic processes and some applications 26 / 31

27 SOME RELATED WORK Vladas Pipiras University of North Carolina atadaptive Chapel Hill wavelet () decompositions of stochastic processes and some applications 27 / 31

28 Some related work Unser, M. and Blu, T. (2007) Self-Similarity: Part I Splines and Operators. IEEE Transactions on Signal Processing 55, pp Blu, T. and Unser, M. (2007) Self-Similarity: Part II Optimal Estimation of Fractal Processes. IEEE Transactions on Signal Processing 55, pp Unser, M. and Blu, T. (2005) Cardinal Exponential Splines: Part I Theory and Filtering Algorithms. IEEE Transactions on Signal Processing 53, pp Unser, M. (2005) Cardinal Exponential Splines: Part II Think Analog, Act Digital. IEEE Transactions on Signal Processing 53, pp Vladas Pipiras University of North Carolina atadaptive Chapel Hill wavelet () decompositions of stochastic processes and some applications 28 / 31

29 THANK YOU! Vladas Pipiras University of North Carolina atadaptive Chapel Hill wavelet () decompositions of stochastic processes and some applications 29 / 31

30 References Abry, P. and Sellan, F. (1996) The wavelet-based synthesis for the fractional Brownian motion proposed by F. Sellan and Y. Meyer: Remarks and fast implementation. Applied and Computational Harmonic Analysis 3, pp Benassi, A., Jaffard, S. and Roux, D. (1997) Elliptic Gaussian Random Processes. Revista Matematica Iberoamericana 13, pp Jaffard, S., Lashermes, B. and Abry, P. (2006) Wavelet leaders in multifractal analysis. In Wavelet Analysis and Applications, T. Qian et al. (eds.), Springer, pp Meyer, Y., Sellan, F. and Taqqu, M. S. (1999) Wavelets, generalized white noise and fractional integration: the synthesis of fractional Brownian motion. The Journal of Fourier Analysis and Applications 5, pp Zhang, J. and Walter, G. (1994) A wavelet-based KL-like expansion for wide-sense stationary random processes. IEEE Transactions on Signal Processing 42, pp Vladas Pipiras University of North Carolina atadaptive Chapel Hill wavelet () decompositions of stochastic processes and some applications 30 / 31

31 References Abry, P. and Pipiras, V. (2006) Wavelet-based synthesis of the Rosenblatt process. Signal Processing 86, pp Didier, G. and Pipiras, V. (2008) Gaussian stationary processes: adaptive wavelet decompositions, discrete approximations and their convergence. Journal of Fourier Analysis and Applications 14, pp Didier, G. and Pipiras, V. (2010) Adaptive wavelet decompositions of stationary time series. Journal of Time Series Analysis 31, pp Didier, G. and Pipiras, V. (2012) On the Riesz property of L 2 -unbounded transformations of orthogonal wavelet bases, Preprint. Pipiras, V. (2004) Wavelet-type expansion of the Rosenblatt process. The Journal of Fourier Analysis and Applications 10, pp Pipiras, V. (2005) Wavelet-based simulation of fractional Brownian motion revisited. Applied and Computational Harmonic Analysis 19, pp Vladas Pipiras University of North Carolina atadaptive Chapel Hill wavelet () decompositions of stochastic processes and some applications 31 / 31