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Wavelet Estimation of a Local Wavelet Estimation of a Local Long Memory ParameterLong Memory Parameter
B. WhitcherEURANDOM, The Netherlands
M. J. JensenUniversity of Missouri - Columbia
March 15, 2000ASEG 2000, Perth, Western Australia
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EURANDOM B. Whitcher 2
OutlineOutline
• Motivation
• Locally Stationary Time Series
• Discrete Wavelet Transforms
• Local Wavelet Variance
• Vertical Ocean Shear Measurements
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EURANDOM B. Whitcher 3
MotivationMotivation
• Long-range dependence is everywhere.• Want to generalise current time series models
– fractional ARIMA
• Popular method of estimation is ordinary least-squares (OLS).
• Propose local version of the OLS estimator based on wavelet coefficients.
• Compare it to an adapted global estimator.
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EURANDOM B. Whitcher 4
Locally Stationary Long-Memory Locally Stationary Long-Memory ModelModel
• Define to be a stochastic process given by
– Time-varying generalisation of Box & Jenkins model.
– Long-memory parameter:
• Spectrum for has the property
– Log-linear relation between spectrum and frequency.
.1 )(, t
tdTt BBXB
TtX ,
tdtS 2,
TtX ,
.0as
.2121 td
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EURANDOM B. Whitcher 5
Discrete Wavelet TransformDiscrete Wavelet Transform
• Project observations onto wavelet functions.– Common wavelets are the Haar and Daubechies.
• Decompose process on a scale-by-scale basis.– Multiresolution analysis.
– Appealing for the physical sciences.
• Also captures features locally in time.– Allows us to estimate time-varying structure.
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EURANDOM B. Whitcher 6
Wavelet Basis FunctionsWavelet Basis Functions
Haar D(4) D(8) LA(8)
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EURANDOM B. Whitcher 7
Comparison of TransformsComparison of Transforms
• DWT
– Orthonormal transform
– Filter and downsample
– Decorrelates LMPs
– Poor time resolution
– Inferior statistical properties
– Not used here
• Maximal Overlap DWT
– NOT orthogonal
– Filter, no downsample
– Correlated coefficients
– Better time resolution
– Better statistical properties
– Used to construct local wavelet variance
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EURANDOM B. Whitcher 8
Local Wavelet VarianceLocal Wavelet Variance
• Intuitive definition of the wavelet variance
• Local wavelet variance is estimated by
– is the width of the “central portion”.
– is the offset of the “central portion”.
j
j
K
sstj
jjX w
Kt
1
2,
2 ~1,~
jK
j
2,,
2 Var tjtjjX wEw
jX t ,2
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EURANDOM B. Whitcher 9
Vertical Ocean ShearVertical Ocean Shear
-0.75
-0.55
-0.35
-0.15
0.05
0.25
0.45
0.65
0.85
350 450 550 650 750 850 950
depth (metres)
-6.5
-4.5
-2.5
-0.5
1.5
3.5
5.5
350 450 550 650 750 850 950depth (metres)
1/s
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EURANDOM B. Whitcher 10
Parameter EstimationParameter Estimation
-0.4
-0.2
0
0.2
0.4
0.6
0.8
350 450 550 650 750 850 950
depth (metres)
d(t)
-0.4
-0.2
0
0.2
0.4
0.6
0.8
350 450 550 650 750 850 950
depth (metres)
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EURANDOM B. Whitcher 11
ConclusionsConclusions
• Methodology– Introduced new time series model.
– Developed wavelet-based estimation procedure.
• Results– Quantified time-varying persistence in vertical ocean
shear measurements.
– Outperformed global estimator on partitioned data
• Future Research– Quantify variability of estimator.
– Weighted least squares or Maximum Likelihood.