WAVELET TRANSFORM. Convolution of time series x n’ with a scaled and translated version of a base...
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Transcript of WAVELET TRANSFORM. Convolution of time series x n’ with a scaled and translated version of a base...
WAVELET TRANSFORM
1
0
1
00
N
'n'n
N
'n'nn s
tn'n*xxsW
WAVELET TRANSFORM
Convolution of time series xn’ with a scaled and translated version of a base function: a wavelet 0 () – continuous function in time and frequency – “mother wavelet”Convolution needs to be effected N (# of points in time series) times for each scale s; n is a translational value
Much faster to do the calculation in Fourier space
Convolution theorem allows N convolutions to be done simultaneously with the Discrete Fourier Transform:
1
0
2'
1ˆN
n
Nknink ex
Nx
k is the frequency index
Convolution theorem: Fourier transform of convolution is the same as the pointwise product of Fourier transforms
Torrence and Compo (1998)
1
0
N
'n'nn s
tn'n*xsW
WAVELET TRANSFORM
1
0
2'
1ˆN
n
Nknink ex
Nx sˆ Fourier transform of
s
t
xW kInverse Fourier transform of is Wn (s)
1
0
N
k
tnikkn
kes*ˆxsW
2
22
2
Nk:
tN
k
Nk:
tN
k
k
With this relationship and a FFT routine, can calculate the continuous wavelet transform (for each s) at all n simultaneously
0
0
0
0
0
0
0
0
1
0
N
k
tnikkn
kes*ˆxsW
To ensure direct comparison from one s to the other, need to normalize wavelet function
kk sˆt
ssˆ
0
212
241 20 ee i
241 20 seH
11
2
2 mmm
i!m
!mi
sm
m
esH!mm
12
2
21
2
21
1
e
d
d
mm
mm 22
21
smm
es
m
i
12
0
'd'ˆ
i.e. each unscaled wavelet function has been normalized to have unit energy (daughter wavelets have same energy as mother)
1
0
2N
kk Nsˆ
and at each scale (N is total # of points):
Wavelet transform is weighted by amplitude of Fourier coefficients and not by
kx
1
0
N
k
tnikkn
kes*ˆxsW
Wavelet transform Wn(s) is complex because wavelet function is complex
Wn(s) has real and imaginary parts that give the amplitude and phase
and the wavelet power spectrum is |Wn(s)|2
for real the imaginary part is zero and there is no phase
22kn xNsW
for white noise
Nxk
22 22 sWnfor all n and s
Normalized wavelet power spectrum is 22 sWn
Seasonal SST averaged over Central Pacific
22 sWn
Power relative to white noise
0
0
0
0
0
0
0
0
Considerations for choice of wavelet function:
1) Orthogonal or non-orthogonal:Non-orthogonal (like those shown here) are useful for time series analysis. Orthogonal wavelets – Haar, Daubechies
2) Complex or real:Complex returns information on amplitude and phase; better adapted for oscillatory behavior. Real returns single component; isolates peaks
3) Width (e-folding time of 0):Narrow function -- good time resolutionBroad function – good frequency resolution
s2
s2
s2
2s
4) Shape:For time series with jumps or steps – use boxcar-like function (Haar)For smoothly varying time series – use a damped cosine (qualitatively similar results of wavelet power spectra).
Seasonal SST averaged over Central Pacific
22 sWn
22 sWn
Relationship between Wavelet Scale and Fourier period
Write scales as fractional powers of 2: J,,,j,ss jjj 1020
j
stNlogJ
02
smallest resolvable scale
largest scale
should be chosen so that the equivalent Fourier period is ~2 t
j ≤ 0.5 for Morlet wavelet; ≤ 1 for others
N = 506
t = 0.25 yrs0 = 2 t = 0.5 yr
j = 0.125 J = 56
57 scales ranging from 0.5 to 64 yr
Relationship between Wavelet Scale and Fourier period
0
0
0
0
0
0
0
0
Can be derived substituting a cosine wave of a known frequency into
1
0
N
k
tnikkn
kes*ˆxsW
and computing s at which Wn is maximum
200 2
4
s
6031 0 :s.
12
4
m
s
43961 m:s.
21
2
m
s
29743 m:s.
64652 m:s.
Seasonal SST averaged over Central Pacific
22 sWn
6031 0 :s.
29743 m:s.
How to determine the significance level?
s
2
Cone of Influence
Paul
Morlet & DOG
2s
Because the square of a normally distributed real variable is 2 distributed with 1 DOF
22
2 is xk
22
2 be should sWn
At each point of the wavelet power spectrum, there is a 2
2 distribution
For a real function (Mexican hat) there is a 1
2 distribution
Distribution for the local wavelet power spectrum:
22
2
2
1
P sW
k2n
Pk is the mean spectrum at Fourier frequency k, corresponding to wavelet scale s
SUMMARY OF WAVELET POWER SPECTRUM PROCEDURES
1) Find Fourier transform of time series (may need to pad it with zeros)
2) Choose wavelet function and a set of scales
3) For each scale, build the normalized wavelet function kk sˆt
ssˆ
0
212
4) Find wavelet transform at each scale
1
0
N
k
tnikkn
kes*ˆxsW
5) Determine cone of influence and Fourier wavelength at each scale
6) Contour plot wavelet power spectrum
7) Compute and draw 95% significance level contour
Seasonal SST averaged over Central Pacific
22 sWn
Power relative to white noise