Using wavelet tools to estimate and assess trends in atmospheric data Peter Guttorp University of Washington [e-mail]

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Using wavelet tools to estimate and assess trends
in atmospheric dataPeter GuttorpUniversity of
Washingtonpeter_at_stat.washington.edu
NRCSE
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Wavelets

• Fourier analysis uses big waves
• Wavelets are small waves

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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 (m2, continents,...)

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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?

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Haar wavelet

• where

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Translation and scaling
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Continuous Wavelet Transform

• Haar CWT
• Same for other wavelets
• where

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Basic facts

• CWT is equivalent to x
• CWT decomposes energy

energy
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Discrete time

• Observe samples from x(t) x0,x1,...,xN-1Discrete
  wavelet transform (DWT) slices through CWT
• restricted to dyadic scales tj 2j-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.

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Properties

• Let Wj (Wj,0,...,Wj,N-1) S (s0,...,sN-1).
• Then W (W1,...,WJ,S ) is the DWT of X
  (x0,...,xN-1).
• (1) We can recover X perfectly from its DWT W, X
  W-1W.
• (2) The energy in X is preserved in its DWT

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The pyramid scheme

• Recursive calculation of wavelet coefficients
  hl wavelet filter of even length L gl
  (-1)lhL-1-l scaling filter
• Let S0,t xt for each t
• For j1,...,J calculate
• t 0,...,N 2-j-1

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Daubachies LA(8)-wavelet
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Oxygen isotope in coral cores at Malindi, Kenya

• Cole et al. (Science, 2000) 194 yrs of monthly
  d18O-values in coral core.
• Decreased oxygen corresponds to increased sea
  surface temperature
• Decadal variability related to monsoon activity

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Multiscale analysis of coral data
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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 lt 1/2 the process is stationary

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Nile river annual minima
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Annual northern hemisphere temperature anomalies
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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)

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Coral data correlation
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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
• Xt Tt Yt

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Wavelet analysis of trend

• where A is diagonal, picks out S and the boundary
  wavelet coefficients.
• Write
• where RWTAW, so if X is Gaussian we have

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Confidence band calculation

• Let v be the vector of sds 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.

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Malindi trend
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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

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Flights

• Measure temperature, pressure, humidity, air flow
  in East Pacific

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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.

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Wavelet variability

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

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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
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W-transform
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Karhunen-Loeve expansion

• and
• where Ai are iid N(0,1)
• Idea use wavelet basis instead of
  eigenfunctions, allow for dependent Ai

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Covariance expansion

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

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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.

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ROM covariance