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1
Philippe Naveau
Multi-Resolution Analysis An introduction to
wavelets
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Different perspectives on wavelets

• - Mathematics Coifman, Meyer, Daubechies,
• - Signal processing Mallat, Sweldens,
• - Statistics Dohono, Johnstone,
• - Physics trubulence Farge, Yano,

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Data Analysis
Two approaches with complex data sets

• Use a complex tool one time
• Use a simple tool many times

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Outline

• Introduction statistical issues examples
• (2) Lifting scheme A multi-resolution algorithm
• (3) Wavelets A little bit of theory
• (4) Applications
• - Solar irradiances
• - Stratospheric ozone
• (5) Extra topics

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Denoising
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Denoising
Abrupt jump -gt Lost of smoothness
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Denoising
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The problem
Our observations
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The problem
Our goal
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Polynomial fit of degree 3
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Polynomial fit of degree 10
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Polynomial fit of degree 20
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Polynomial fit of degree 50
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Sinusoidal fit linear fit
!!! Only perform on the first half of the data !!!
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Approach too specific
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Two periods polynomial fit of degree 10
!!! Perform on the whole data set !!!
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Wavelet fit (Daubechies)
How did you get this fit?
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Learned lessons from this example
(1) Even a simple but localized abrupt change can
strongly disturbed the global fit (2) Be careful
with automatic denoising procedures
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Outline

• Introduction statistical issues examples
• (2) Lifting scheme A multi-resolution algorithm
• (3) Wavelets A little bit of theory
• (4) Applications
• - Solar irradiances
• - Stratospheric ozone
• (5) Extra topics

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The Dividing Game
Good but too long 1024 points -gt 512 points
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The Dividing Game
Good but too long 1024 points -gt 512 points
time
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The Dividing Game
Good but too long 1024 points -gt 512 points
Dividing odds even numbers
time
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The Dividing Game
Good but too long 1024 points -gt 512 points
What is lost?
?
?
?
odd
odd
odd
time
even
even
even
even
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Prediction Error
Oddj,i
time
evenj,i
eveni,i1
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First scheme
Dividing the data set into 2 equal parts
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Repeating the scheme
An issue What is the last point from this
algorithm?
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Repeating the first trial scheme
We need to update the algorithm in order to
preserve the grande mean
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The Lifting Scheme (Sweldens)
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Lifting Scheme
Recall
UPADTING evenj1,i eveni,j oddj1,i/2
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Lifting Scheme
Recall
UPADTING evenj1,i eveni,j (oddj1,i
oddj1,i-1 )/4
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The Lifting Scheme
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The Lifting Scheme
Wavelets Coefficients
Wavelets coefficients
Wavelets coefficients
Scaling coefficients Averages Low pass
filter Wavelet coefficients Differences High
pass filter
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Lifting Scheme
wavelets
wavelets
wavelets
high frequency
lower frequency
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Simplest updating Haar wavelet
UPADTING evenj1,i eveni,j oddj1,i/2
Haar building block
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How to solve this problem?
Way 1 Change the predicting step by
increasing the length of support Way 2 Change
the updating step Way 3 Change both
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Discrete Wavelet Transform
How did you get this fit?
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The Lifting Scheme Decomposition
Original time series
Wavelet Coefficients level 1
Wavelet Coefficients level 2
Scaling Coefficients level 3
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An important remark
All wavelet coefficients are small but the
ones that contain some information
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Lifting Scheme Properties

• Localized in time (compact support)
• Easy to reconstruct (building block)
• Easy to implement (sequential 2)
• Very fast O(n)
• - Can preserve moments orthogonality
• Many different ways to choose
• the predicting and the updating steps

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Reconstruction
Wavelet coefficients (n/2)
Wavelet coefficients (n/4)
Wavelet coefficients (n/8)
Smooth coefficients (n/16)
Wavelet coefficients (n/16)
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Reconstruction
Original Signal (n data points)
Smooth coefficients (n/2)
Wavelet coefficients (n/2)
Smooth coefficients (n/4)
Wavelet coefficients (n/4)
Smooth coefficients (n/8)
Wavelet coefficients (n/8)
Smooth coefficients (n/16)
Wavelet coefficients (n/16)
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Consequences
If the Predict Update steps are linear
operators Then the lifting scheme can be
rewritten
Wavelet coefficients W ?Original data
with a matrix of size n ? n
Reconstruction Applying the inverse of W
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Lifting Scheme Questions

• How to choose the Predict
• Update steps?
• - Is the matrix W orthogonal?
• - How to perform a regression?

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Outline

• Introduction statistical issues examples
• (2) Lifting scheme A multi-resolution algorithm
• (3) Wavelets A little bit of theory
• (4) Applications
• - Solar irradiances
• - Stratospheric ozone
• (5) Extra topics

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The Haar scaling function
?(x)1(0x 1)
1
0
1
?(x) ?(2x)?(2x-1)
Compact support!!
?(x) ?ck 21/2?(2x-k)
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The Haar mother wavelet
?(x)1(0x 1)
1
?(x) ?(2x)-?(2x-1)
0
1
Compact support!!
?(x) ?gk 21/2?(2x-k)
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N Vanishing moments
?xk?(x)dx 0 with k0,..,N-1
? (-1)n nk cn 0 with k0,..,N-1
Normalization
?(x) ?cn 21/2?(2x-n)
? cn 21/2
Orthogonality
??(x) ?(x) dx 0
? cn cn2k ?k with k0,..,N-1
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Example N2 (Daubechies D4)
c0 c1 c2 c3 21/2 c20 c21 c22 c23
1 -c1 2c2 -3c3 0 c0 c2 c1 c3 0
Solution
c0 (131/2)/(4?21/2) c1 (331/2)/(4?21/2)
c2 (1-31/2)/(4?21/2) c3 (3-31/2)/(4?21/2)
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Scaling functions
?(x) ?ck 21/2?(2x-k)
Lets find ?(x) in (0,3) and 0 otherwise
?(1) c-2?(4)c-1?(3)c0?(2)c1?(1)c2?(0)
?(1) c0?(2)c1?(1) ?(2) c2?(2)c3?(1)
We know ci, so we know ?(1) ?(2)
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Scaling functions via Mallats cascade algorithm
From ?(1), ?(2) and ? (x) ?ck 21/2?(2x-k) we
know any ?(1/2), ?(3/2), ?(5/2), etc because
?(k/2j) ?ck 21/2?(k(2j-1-1))
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Orthonormal basis
?j,k(x) 2j/2 ?(2jx-k)
Zooming from a building block
f(x) ?cj0,k ?j0,k(x) ? ? dj,k ?j,k(x)
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Summary

• - Compact supports provides localization
• Number of vanishing moments and
• the size of support link to
• the smoothness of the wavelet
• - A few coefficients are needed to
• compute wavelet coefficients (fast algorithm)
• - Our focus is on Discrete Wavelets
• - Lifting scheme can incorporate
• orthogonal wavelets

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Other wavelets
Continuous wavelets Symmlets (LA
Daubechies) Stationary wavelets
(Non-Decimated) Coiflets Biorthogonal
wavelets Wavelet packets Ridgelets Curvelets
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Fourier versus wavelets
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Denoising strategy
Data
Discrete Wavelet Transform (DWT)
Shrinkage
Inverse DWT
Denoised data
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Denoising framework
y(t) f(t) n(t) for t1,,n
with n(t) Gaussion I.I.D. noise and f in L2
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Denoising framework
Y X N
Multiplying by an orthogonal wavelet matrix W
W Y W X W N
D ? M
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Denoising framework
Mean Square Error MSE(f,f) Ef-f2/n
?f(t)-f(t)2/n with f an estimator of f
D ? M
MSE(f,f) MSE(d,d)
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An important remark
All wavelet coefficients are small but the
ones that contain some information
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Thresholding

• Keep the large wavelet coefficients and set the
  small ones equal to 0.

Hard thresholding hard(d,?)d 1(d
gt?) with 1(A) equal to 1 if A true and 0
otherwise Soft thresholding soft(d,?)sgn(d)
(d-?)
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Soft thresholding
Wavelet decomposition
Thresholding
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Our example
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How to choose the threshold?
Hard thresholding hard(d,?)d 1(d
gt?) Soft thresholding soft(d,?)sgn(d)
(d-?)
Universal threshold (Donoho Johnstone) ? ? (2
log n)1/2
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How to choose the threshold?
- Universal threshold (Donoho Johnstone) ? ?
(2 log n)1/2

• Minimax estimation
• Risk inffsupf MSE(f,f)
• Steins Unbiased Risk estimate (Sure)
• Hypothesis testing
• Bayesian methodology
• Cross-validation

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Outline

• Introduction statistical issues examples
• (2) Lifting scheme A multi-resolution algorithm
• (3) Wavelets A little bit of theory
• (4) Applications
• - Solar irradiances
• - Stratospheric ozone
• (5) Extra topics

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Scientific Questions
? Did the sun leave a fingerprint in climate of
the recent past? ? Is it possible to extract
the solar forcing from proxies coupled model
outputs?
A statistical question an old problem
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John A. Eddy
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Possible link between the Sun the climate?
Late Maunder Minimum Very cold winters,
particularly in Europe
Aert van der Neer
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Why do we want to extract the solar forcing?

• Successful climate change studies depend on
  knowing the climate sensitivity to anthropogenic
  and natural forcings (e.g. sun, volcanoes).
• (B) Still lack solid knowledge of how exactly
  solar variations influence the earth and how the
  atmosphere translates it into a climate forcing
• (C) Although solar irradiance changes are global,
  the impacts should differ spatially and over
  time!!

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Solar irradiances
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Temperature reconstructions An example
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Climate Reconstructions Past Millennium

• centennial and multi-decadal variability 3 time
  periods (MWP-LIA-Present)

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Wavelet decomposition for 10-Be
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Multi-Resolution Analysis
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Multi-Resolution Analysis
Two main components of Solar irradiance
Gleissberg cycle 85 years
Schwabe cycle 11 years
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Multi-Resolution Analysis Comparison
Just a descriptive tool!!
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Correlations (level D2)
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850 AD - presentT31x3 model with Solar
Forcing (various magnitudes)Volcanic Forcing
(ice core based history)GHG (ice core based
history)Fixed Ozone and nat. Sulfate Climatology
(12 months)After 1870 either natural only
(GHG, sulfate fixed at 1870)ramped after
observations
CSM 1.4 outputs over the last Millennium
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Multi-resolution analysis with the NWT Annual
global average temperature from a CSM run
Years
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Comparing extraction between model proxies for
D3 200 years
D3 from original irradiance (Bard et al 2000)
Irradiance Years Surface temp
anomaly
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Lessons learnt from this solar example

• - Multi-resolution analysis can be used as a
  descriptive tool to visualize localized changes
  in time at a given frequency band
• - Cautiousness is needed when comparing time
  series (wavelet cross spectrum)
• - Understanding the impact of solar forcing on
  climate remains a challenge

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Stratospheric Arctic Ozone
Open PDF FILE!! ozone.pdf
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Outline

• Introduction statistical issues examples
• (2) Lifting scheme A multi-resolution algorithm
• (3) Wavelets A little bit of theory
• (4) Applications
• - Solar irradiances
• - Stratospheric ozone
• (5) Other topics

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Statistical Issues
Example III
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Outline

• Geostatistics
• (2) Extreme value theory the univariate case
• 2-i Lichenometry
• 2-ii Ice core volcanic signals
• (3) Extreme value theory the bivariate case
• (4) Extreme value theory the multivariate case
• (5) Conclusions

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Conclusions

• EVT There exist mathematically sound tools to
  deal with extremes and exceedances
• Lichenometry Taking advantages of the data type
  improves the error analysis
• Volcanic signals in cores Heavy tail
  distributions can be used to characterize
  volcanic signatures
• Spatial extremes Madogram captures some
  dependence in max-stable fields

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The main message about wavelets
Like real estate, there are 3 important things
with wavelets location, location, location
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Work in progress and future research

• Lichenometry Extension of the methods to many
  glaciers look at the LIA in South America
• Volcanic signals in cores Compare the intensity
  of the volcanic signal between forcing climate
  response
• Spatial extremes Developing spatial
  interpolation schemes apply to downscaling of
  extremes

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Statistical Issues
Example III Change-point detection
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Lifting Scheme Wavelets
Matrix
Smooth coefficients
Smooth coefficients
wavelets
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Comparing the original and extracted solar forcing
Comparison for D3 200 years
Irradiance Years Surface temp
anomaly