Wavelet Estimation of a Local Wavelet Estimation of a Local Long Memory ParameterLong Memory Parameter

B. WhitcherEURANDOM, The Netherlands

whitcher@eurandom.tue.nl

M. J. JensenUniversity of Missouri - Columbia

March 15, 2000ASEG 2000, Perth, Western Australia

EURANDOM B. Whitcher 2

OutlineOutline

• Motivation

• Locally Stationary Time Series

• Discrete Wavelet Transforms

• Local Wavelet Variance

• Vertical Ocean Shear Measurements

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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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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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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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Wavelet Basis FunctionsWavelet Basis Functions

Haar D(4) D(8) LA(8)

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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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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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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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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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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.