using wavelet tools to estimate and assess trends in atmospheric data nrcse
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Using wavelet tools to estimate and assess trends
in atmospheric data
NRCSE
Wavelets
Fourier analysis uses big waves
Wavelets are small waves
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,...)
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?
€
A(λ,t) =1λ
x(u)dut−λ
2
t+λ2
∫
€
D(λ,t) = A(λ,t + λ2
) − A(λ,t − λ2
)
=1λ
x(u)du −t
t+λ∫
1λ
x(u)dut−λ
t∫
Haar wavelet
where
€
D(1,0) = 2 ψ(H) (u)x(u)du−∞
∞∫
€
ψ(H) (u) =
−12
, −1< u ≤ 0
12
, 0 < u ≤ 1
0, otherwise
⎧
⎨
⎪ ⎪
⎩
⎪ ⎪
Translation and scaling
€
ψ1,t(H) (u) = ψ(H) (u − t)
€
ψλ,t(H) (u) =
1λ
ψ(H) (u − t
λ)
Continuous Wavelet Transform
Haar CWT:
Same for other wavelets
where
€
) W (λ,t) = ψλ,t
(H) (u)x(u)du ∝ D(λ,t)−∞
∞∫
€
) W (λ,t) = ψλ,t (u)x(u)du
−∞
∞∫
€
ψλ,t (u) =1λ
ψu − t
λ
⎛ ⎝ ⎜
⎞ ⎠ ⎟
Basic facts
CWT is equivalent to x:
CWT decomposes energy:
€
x(t) =1
Cψ 0
∞∫ W(λ,u)ψλ,t (u)du
−∞
∞∫ ⎡
⎣ ⎢
⎤
⎦ ⎥dλ
λ2
€
x2 (t)dt =W2 (λ,t)
Cψλ2dtdλ
−∞
∞∫
0
∞∫
−∞
∞∫
energy
Discrete time
Observe samples from x(t): x0,x1,...,xN-1
Discrete wavelet transform (DWT) slices through CWT
λ restricted to dyadic scales j = 2j-1, j = 1,...,Jt 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.
€
Wj,t ∝) W (τ j,t)
€
rt = Wj,tj=1
J∑
€
st = xt −rt
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:
€
X 2 = xi2
i=0
N−1∑ = Wj
2
j=1
J∑ + S 2
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 tFor j=1,...,J calculate
t = 0,...,N 2-j-1€
Sj,t = glSj−1,2t+1−lmod(N2−j )l=0
L−1∑
€
Wj,t = hlSj−1,2t+1−lmod(N2−j )l=0
L−1∑
Daubachie’s LA(8)-wavelet
Oxygen isotope in coral cores at Malindi, Kenya
Cole et al. (Science, 2000): 194 yrs of monthly 18O-values in coral core.
Decreased oxygen corresponds to increased sea surface temperature
Decadal variability related to monsoon activity
Multiscale analysis of coral data
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 the process is stationaryd < 12
Nile river annual minima
d =0.40
Annual northern hemisphere
temperature anomalies
Coral data correlation
d =0.359 (CI [0.143,0.597])
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)
Nile river
1 yr
2 yr
4 yr
8 yr
≥16 yr
Wavelet variance
The wavelet coefficients pick up changes in the energy at different scales over timeThe variability of the coefficients at each scale is a variance decomposition (similar to Fourier analysis, although the frequency choices are different)The wavelet coefficients, even for a long-term memory process, behave (at each scale) like a sample from a mean zero white noise process (at least for large L)
Analysis of wavelet variance
In the Nile data there is a visual indication that the variability is changing after the first 100 observations at scales 1 and 2 years.
Let Xt be a time series with mean 0 and variance .To test
against
we use the statistics
H0 : σ12 =L σT
2
σ t2
HA : σ12 =L σk
2 ≠σk+12 =L σT
2
K t = Xi2
i=1
t
∑ Xj2
j=1
T
∑
Testing for changepoint of variability
Let and
D=max(D+,D-) measures the deviation of Kk from the 45° line expected if H0 is true.
Asymptotically, D converges to a Brownian bridge, which can be used to calculate critical values.
Alternatively Monte Carlo critical values, or simply Monte Carlo test.
D+ =max1≤k≤T−1
kT−K k
⎛⎝⎜
⎞⎠⎟
D− =max1≤k≤T−1 K k −k−1
T⎛⎝⎜
⎞⎠⎟
Nile river
Test result
Scale D MC 5% point
1 0.1559 0.1051
2 0.1754 0.1469
3 0.1000 0.2068
4 0.2313 0.2864
When did it change?
Why did it change?
Around 715 a nilometer was constructed and located at a mosque on Rawdah Island in the river. Replaced 850 after a huge flood.
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
Wavelet analysis of trend
where A is diagonal, picks out S and the boundary wavelet coefficients.
Write
where R=WTAW, so if X is Gaussian we have
X =W −1W=W TW
=W TAW+W T (I−A)W=T + Y
T =W TAW=W TAWX =RT +RY
T ~ N(RT,Rcov(X)RT )
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.
Tt
U =T−T
P(T −v ≤T ≤T + v) =1−2P(U > v)
Malindi trend
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
Flights
Measure temperature, pressure, humidity, air flow in East Pacific
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.
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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