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.