1. Using wavelet tools to estimate and assess trends in atmospheric data Peter Guttorp University of Washington peter@stat.washington.edu NRCSE

2. Wavelets Fourier analysis uses big waves Wavelets are small waves

3. Requirements for wavelets Integrate to zero Square integrate to one Measure variation in local averages Describe how time series evolve in time for different scales (hour, year,...) or how images change from one place to the next on different scales (m 2, continents,...)

4. Continuous wavelets Consider a time series x(t). For a scale l and time t, look at the average How much do averages change over time?

5. Haar wavelet where

6. Translation and scaling

7. Continuous Wavelet Transform Haar CWT: Same for other wavelets where

8. Basic facts CWT is equivalent to x: CWT decomposes energy: energy

9. Discrete time Observe samples from x(t): x 0,x 1,...,x N-1 Discrete wavelet transform (DWT) slices through CWT restricted to dyadic scales  j = 2 j-1, j = 1,...,J t restricted to integers 0,1,...,N-1 Yields wavelet coefficients Think of as the rough of the series, so is the smooth (also called the scaling filter). A multiscale analysis shows the wavelet coefficients for each scale, and the smooth.

10. Properties Let W j = (W j,0,...,W j,N-1 ); S = (s 0,...,s N-1 ). Then W = (W 1,...,W J,S ) is the DWT of X = (x 0,...,x N-1 ). (1) We can recover X perfectly from its DWT W, X = W -1 W. (2) The energy in X is preserved in its DWT:

11. The pyramid scheme Recursive calculation of wavelet coefficients: {h l } wavelet filter of even length L; {g l = (-1) l h L-1-l } scaling filter Let S 0,t = x t for each t For j=1,...,J calculate t = 0,...,N 2 -j -1

12. Daubachie’s LA(8)-wavelet

13. Oxygen isotope in coral cores at Malindi, Kenya Cole et al. (Science, 2000): 194 yrs of monthly  18 O-values in coral core. Decreased oxygen corresponds to increased sea surface temperature Decadal variability related to monsoon activity

14. Multiscale analysis of coral data

15. Long term memory A process has long term memory if the autocorrelation decays very slowly with lag May still look stationary Example: Fractionally differenced Gaussian process, has parameter d related to spectral decay If |d| < 1/2 the process is stationary

16. Nile river annual minima

17. Annual northern hemisphere temperature anomalies

18. Decorrelation properties of wavelet transform Periodogram values are approximately uncorrelated at Fourier frequencies for stationary processes (but not for long memory processes) Wavelet coefficient at different scales are also approximately uncorrelated, even for long memory processes (approximation better for larger L)

19. Coral data correlation

20. What is a trend? “The essential idea of trend is that it shall be smooth” (Kendall,1973) Trend is due to non-stochastic mechanisms, typically modeled independently of the stochastic portion of the series: X t = T t + Y t

21. Wavelet analysis of trend where A is diagonal, picks out S and the boundary wavelet coefficients. Write where R= W T A W, so if X is Gaussian we have

22. Confidence band calculation Let v be the vector of sd’s of and. Then which we can make 1-  by choosing d by Monte Carlo (simulating the distribution of U). Note that this confidence band will be simultaneous, not pointwise.

23. Malindi trend

24. Air turbulence EPIC: East Pacific Investigation of Climate Processes in the Coupled Ocean-Atmosphere System Objectives: (1) To observe and understand the ocean-atmosphere processes involved in large-scale atmospheric heating gradients (2) To observe and understand the dynamical, radiative, and microphysical properties of the extensive boundary layer cloud decks

25. Flights Measure temperature, pressure, humidity, air flow in East Pacific

26. Flight pattern The airplane flies some high legs (1500 m) and some low legs (30 m). The transition between these (somewhat stationary) legs is of main interest in studying boundary layer turbulence.

28. Wavelet variability The variability at each scale constitutes an analysis of variance. One can clearly distinguish turbulent and non-turbulent regions.

29. Estimating nonstationary covariance using wavelets 2-dimensional wavelet basis obtained from two functions  and  : First generation scaled translates of all four; subsequent generations scaled translates of the detail functions. Subsequent generations on finer grids. detail functions

30. W-transform

31. Karhunen-Loeve expansion and where A i are iid N(0,1) Idea: use wavelet basis instead of eigenfunctions, allow for dependent A i

32. Covariance expansion For covariance matrix  write Useful if D close to diagonal. Enforce by thresholding off-diagonal elements (set all zero on finest scales)

33. Surface ozone model ROM, daily average ozone 48 x 48 grid of 26 km x 26 km centered on Illinois and Ohio. 79 days summer 1987. 3x3 coarsest level (correlation length is about 300 km) Decimate leading 12 x 12 block of D by 90%, retain only diagonal elements for remaining levels.

34. ROM covariance