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Hilbert-Huang Transform and Its Applications.pdf
CONTENTS
PREFACE
CHAPTER 1 INTRODUCTION TO THE HILBERT–HUANG TRANSFORM AND ITS RELATED MATHEMATICAL PROBLEMS
1.1. Introduction
1.2. The Hilbert–Huang transform
1.2.1. The empirical mode decomposition method (the sifting process)
1.2.2. The Hilbert spectral analysis
1.3. Recent developments
1.3.1. Normalized Hilbert transform
1.3.2. Confidence limit
1.3.3. Statistical significance of IMFs
1.4. Mathematical problems related to the HHT
1.4.1. Adaptive data-analysis methodology
1.4.2. Nonlinear system identification
1.4.3. The prediction problem for nonstationary processes (the end effects of EMD)
1.4.4. Spline problems (the best spline implementation for HHT, convergence and 2-D)
1.4.5. The optimization problem (the best IMF selection and uniqueness mode mixing)
1.4.6. Approximation problems (the Hilbert transform and quadrature)
1.4.7. Miscellaneous statistical questions concerning HHT
1.5. Conclusion
References
CHAPTER 2 ENSEMBLE EMPIRICAL MODE DECOMPOSITION AND ITS MULTI-DIMENSIONAL EXTENSIONS
2.1. Introduction
2.2. The empirical mode decomposition
2.3. The ensemble empirical mode decomposition
2.4. The multi-dimensional ensemble empirical mode decomposition
2.5. Summary and discussions
Acknowledgments
References
CHAPTER 3 MULTIVARIATE EXTENSIONS OF EMPIRICAL MODE DECOMPOSITION
3.1. Introduction
3.2. Multivariate extensions of EMD
3.2.1. Complex extensions of EMD
3.2.1.1. Complex EMD (CEMD)
3.2.1.2. Rotation-invariant EMD
3.2.1.3. Bivariate EMD
3.2.1.4. Data-driven direction vectors in BEMD
3.2.2. Trivariate EMD
3.2.3. Multivariate EMD
3.3. Mode-alignment property of MEMD
3.4. Filter bank property of MEMD and noise-assisted MEMD
3.5. Applications
3.5.1. Speed estimation using Doppler radar data
3.5.2. Respiration study using NA-MEMD
3.5.3. Classification of motor imagery data
3.6. Discussion and conclusions
Referrence
CHAPTER 4 B-SPLINE BASED EMPIRICAL MODE DECOMPOSITION
4.1. Introduction
4.2. A B-spline algorithm for empirical mode decomposition
4.3. Some related mathematical results
4.4. Performance analysis of BS-EMD
4.5. Application examples
4.6. Conclusion and future research topics
Acknowledgements
References
CHAPTER 5 EMD EQUIVALENT FILTER BANKS, FROM INTERPRETATION TO APPLICATIONS
5.1. Introduction
5.2. A stochastic perspective in the frequency domain
5.2.1. Model and simulations
5.2.2. Equivalent transfer functions
5.3. A deterministic perspective in the time domain
5.3.1. Model and simulations
5.3.2. Equivalent impulse responses
5.4. Selected applications
5.4.1. EMD-based estimation of scaling exponents
5.4.2. EMD as a data-driven spectrum analyzer
5.4.3. Denoising and detrending with EMD
5.5. Concluding remarks
References
CHAPTER 6 HHT SIFTING AND FILTERING
6.1. Introduction
6.2. Objectives of HHT sifting
6.2.1. Restrictions on amplitude and phase functions
6.2.2. End-point analysis
6.3. Huang’s sifting algorithm
6.4. Incremental, real-time HHT sifting
6.4.1. Testing for iteration convergence
6.4.2. Time-warp analysis
6.4.3. Calculating warped filter characteristics
6.4.4. Separating amplitude and phase
6.5. Filtering in standard time
6.6. Case studies
6.6.1. Simple reference example
6.6.2. Amplitude modulated example
6.6.3. Frequency modulated example
6.6.4. Amplitude step example
6.6.5. Frequency shift example
6.7. Summary and conclusions
6.7.1. Summary of case study findings
6.7.2. Research directions
References
CHAPTER 7 STATISTICAL SIGNIFICANCE TEST OF INTRINSIC MODE FUNCTIONS
7.1. Introduction
7.2. Characteristics of Gaussian white noise in EMD
7.2.1. Numerical experiment
7.2.2. Mean periods of IMFs
7.2.3. The Fourier spectra of IMFs
7.2.4. Probability distributions of IMFs and their energy
7.3. Spread functions of mean energy density
7.4. Examples of a statistical significance test of noisy data
7.4.1. Testing of the IMFs of the NAOI
7.4.2. Testing of the IMFs of the SOI
7.4.3. Testing of the IMFs of the GASTA
7.4.4. A posteriori test
7.5. Summary and discussion
Acknowledgements
References
CHAPTER 8 THE TIME-DEPENDENT INTRINSIC CORRELATION
8.1. Introduction
8.2. Limitations of correlation coefficient analysis
8.3. TDIC based on EMD
8.3.1. TDIC computations
8.3.2. Interpretation of patterns in the TDIC plots
8.3.3. Time-dependent intrinsic cross-correlation
8.3.4. Time-dependent intrinsic auto-correlation
8.3.5. Alleviation to the limitations of correlation coefficient
8.4. Applications of TDIC for geophysical data
8.4.1. ENSO and IOD
8.4.2. Paleoclimate observations
8.5. Summary and conclusions
Acknowledgments
References
CHAPTER 9 THE APPLICATION OF HILBERT–HUANG TRANSFORMS TO METEOROLOGICAL DATASETS
9.1. Introduction
9.2. Procedure
9.3. Applications
9.3.1. Sea level heights
9.3.2. Solar radiation
9.3.3. Barographic observations
9.4. Conclusion
Acknowledgments
References
CHAPTER 10 EMPIRICAL MODE DECOMPOSITION AND CLIMATE VARIABILITY
10.1. Introduction
10.2. Data
10.3. Methodology
10.4. Statistical tests of confidence
10.5. Results and physical interpretations
10.5.1. Annual cycle
10.5.2. Quasi-Biennial Oscillation (QBO)
10.5.3. ENSO-like mode
10.5.4. Solar cycle signal in the stratosphere
10.5.5. Fifth mode
10.5.6. Trends
10.6. Conclusions
Acknowledgments
References
CHAPTER 11 EMD CORRECTION OF ORBITAL DRIFT ARTIFACTS IN SATELLITE DATA STREAM
11.1. Introduction
11.2. Processing of NDVI imagery
11.3. Empirical mode decomposition
11.4. Impact of orbital drift on NDVI and EMD-SZA filtering
11.5. Results and discussion
11.6. Extension to 8-km data
11.7. Integration of NOAA-16 data
11.8. Conclusions
References
CHAPTER 12 HHT ANALYSIS OF THE NONLINEAR AND NON-STATIONARY ANNUAL CYCLE OF DAILY SURFACE AIR TEMPERATURE DATA
12.1. Introduction
12.2. Analysis method and computational algorithms
12.3. Data
12.4. Time analysis
12.4.1. Examples of the TAC and the NAC
12.4.2. Temporal resolution of data
12.4.3. Robustness of the EMD method
12.4.3.1. EMD separation of a known signal in a synthetic dataset
12.4.3.2. Robustness with respect to data length
12.4.3.3. Robustness with respect to end conditions
12.5. Frequency analysis
12.5.1. Hilbert spectra of NAC
12.5.2. Variances of anomalies with respect to the NAC and TAC
12.5.3. Spectral power of the anomalies with respect to the NAC and TAC
12.6. Conclusions and discussion
Acknowledgements
References
CHAPTER 13 HILBERT SPECTRA OF NONLINEAR OCEAN WAVES
13.1. Introduction
13.2. The Hilbert–Huang spectral analysis
13.3. Spectrum of wind-generated waves
13.4. Statistical properties and group structure
13.5. Summary
Acknowledgements
References
CHAPTER 14 EMD AND INSTANTANEOUS PHASE DETECTION OF STRUCTURAL DAMAGE
14.1. Introduction to structural health monitoring
14.2. Instantaneous phase and EMD
14.2.1. Instantaneous phase
14.2.2. EMD and HHT
14.2.3. Extracting an instantaneous phase from measured data
14.3. Damage detection application
14.3.1. One-dimensional structures
14.3.2. Experimental validations
14.3.3. Instantaneous phase detection
14.4. Frame structure with multiple damage
14.4.1. Frame experiment
14.4.2. Detecting damage by using the HHT spectrum
14.4.3. Detecting damage by using instantaneous phase features
14.4.4. Auto-regressive modeling and prediction error
14.4.5. Chaotic-attractor-based prediction error
14.5. Summary and conclusions
References
CHAPTER 15 HHT-BASED BRIDGE STRUCTURAL HEALTH-MONITORING METHOD
15.1. Introduction
15.2. A review of the present state-of-the-art methods
15.2.1. Data-processing methods
15.2.2. Loading conditions
15.2.3. The transient load
15.3. The Hilbert Huang transform
15.4. Damage-detection criteria
15.5. Case study of damage detection
15.6. Conclusions
Acknowledgements
References
CHAPTER 16 APPLICATIONS OF HHT IN IMAGE ANALYSIS
16.1. Introduction
16.2. Overview
16.3. The analysis of digital slope images
16.3.1. The NASA laboratory
16.3.2. The digital camera and set-up
16.3.3. Acquiring experimental images
16.3.4. Using EMD/HHT analysis on images
16.3.5. The digital camera and set-up
16.3.5.1. Volume computations and isosurface visualization
16.3.5.2. Use of EMD/HHT in image decomposition
16.4. Summary
Acknowledgments
References
INDEX
Hilbert–Huang Transform
and Its Applications
2nd Edition
8804hc_9789814508230_tp.indd 1
27/3/14 2:25 pm
INTERDISCIPLINARY MATHEMATICAL SCIENCES*
Series Editor: Jinqiao Duan (Illinois Institute of Technology, Chicago, USA)
Editorial Board: Ludwig Arnold, Roberto Camassa, Peter Constantin,
Charles Doering, Paul Fischer, Andrei V. Fursikov, Xiaofan Li,
Sergey V. Lototsky, Fred R. McMorris, Daniel Schertzer,
Bjorn Schmalfuss, Yuefei Wang, Xiangdong Ye, and Jerzy Zabczyk
Published
Yanheng Ding
eds. Jinqiao Duan, Shunlong Luo & Caishi Wang
eds. Yisong Yang, Xinchu Fu & Jinqiao Duan
Melvin F. Janowitz
eds. Hemanshu Kaul & Henry Martyn Mulder
A Volume in Honour of Professor K D Elworthy
eds. Huaizhong Zhao & Aubrey Truman
Vol. 7: Variational Methods for Strongly Indefinite Problems
Vol. 8: Recent Development in Stochastic Dynamics and Stochastic Analysis
Vol. 9: Perspectives in Mathematical Sciences
Vol. 10: Ordinal and Relational Clustering (with CD-ROM)
Vol. 11: Advances in Interdisciplinary Applied Discrete Mathematics
Vol. 12: New Trends in Stochastic Analysis and Related Topics:
Vol. 13: Stochastic Analysis and Applications to Finance:
Vol. 14 Recent Developments in Computational Finance:
Vol. 15 Recent Advances in Applied Nonlinear Dynamics with Numerical
Vol. 16 Hilbert–Huang Transform and Its Applications (2nd Edition)
Analysis: Fractional Dynamics, Network Dynamics, Classical
Dynamics and Fractal Dynamics with Their Numerical Simulations
eds. Changpin Li, Yujiang Wu & Ruisong Ye
Foundations, Algorithms and Applications
eds. Thomas Gerstner & Peter Kloeden
Essays in Honour of Jia-an Yan
eds. Tusheng Zhang & Xunyu Zhou
eds. Norden E Huang & Samuel S P Shen
*For the complete list of titles in this series, please go to
http://www.worldscientific.com/series/ims
Interdisciplinary Mathematical Sciences – Vol. 16
Hilbert–Huang Transform
and Its Applications
2nd Edition
Editors
Norden E Huang
National Central University, Taiwan
Samuel S P Shen
San Diego State University, USA
N E W J E R S E Y • L O N D O N • S I N G A P O R E • B E I J I N G • S H A N G H A I • H O N G K O N G • TA I P E I • C H E N N A I
World Scientific
8804hc_9789814508230_tp.indd 2
27/3/14 2:25 pm
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Interdisciplinary Mathematical Sciences — Vol. 16
HILBERT–HUANG TRANSFORM AND ITS APPLICATIONS
Second Edition
Copyright © 2014 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or
mechanical, including photocopying, recording or any information storage and retrieval system now known or to
be invented, without written permission from the publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center,
Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from
the publisher.
ISBN 978-981-4508-23-0
Printed in Singapore
March 11, 2014 13:30
The Hilbert-Huang (8804) - 9.75 x 6.5
book page v
PREFACE
Eight years have elapsed since the first edition was published in 2005. During this pe-
riod, the HHT applications have experienced an explosive growth. Hundreds of new
papers have been published each year, covering a wide spectrum of fields ranging
from cosmological waves to biomedical diagnosis. Three international HHT con-
ferences have been convened: the first at National Central University, Chung-Li,
Taiwan, in 2006; the second at Sun Yat-Sen University, Guangzhou, China, in 2008;
and the third at the First Institute of Oceanography, Qingdao, China, in 2011.
More than 100 research scientists and engineers from all over the world attended
each conference.
This second edition updates some HHT methodological developments since 2005
and includes three additional chapters on ensemble empirical mode decomposition
(EEMD), multivariate EMD (MEMD), and time-dependent intrinsic correlation
(TDIC). These new procedures introduced to HHT methodology have effectively
expanded the HHT applicability and user groups. The new edition also includes
more index items to facilitate users quickly finding appropriate places in the book
for specific theories and procedures.
In addition to the methodological progress documented in these three new chap-
ters, significant progress has also been made in EMD mathematics. Mathematicians
have an increasing interest in developing new, efficient algorithms to represent mul-
tiscale signals and images by exploiting the sparsity of these signals or images. It
turns out that EMD can serve as a nonlinear version of the sparse time-frequency
representation of data. This EMD data representation has led to the most excit-
ing mathematical development since the introduction of EMD. Professor Thomas Y.
Hou and his group at the California Institute of Technology and Professor Zuoqiang
Shi of Tsinghua University, Beijing, demonstrated that an arbitrary function could
be decomposed into a sum of finite number of Intrinsic Mode Functions through
a nonlinear optimization process based on sparsity. Their results are great break-
throughs in the long evasive theoretical foundation, since their work justifies that
EMD is a systematic generalization of the Fourier expansion with defined and adap-
tive computational procedures.
These advances motivated two special international meetings organized by the
applied mathematics community. The first was the 2011 Hot Topic Conference spon-
sored by the Institute of Mathematics and its Applications (IMA) of the University
of Minnesota, which focused on “Instantaneous Frequency and Trend for Nonlinear
and Nonstationary Data.” The second was a weeklong workshop on “Adaptive Data
v
March 11, 2014 13:30
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book page vi
vi
Preface
Analysis and Sparsity” sponsored by the Institute of Pure and Applied Mathemat-
ics, UCLA, in 2013. The goal of these workshops was to explore the potential of
adaptive data analysis methodologies. A topic was to determine trends and instan-
taneous frequency in nonlinear and nonstationary data, which is a challenging, yet
critical, problem in climate and financial studies. The meetings discussed recent ex-
citing progress in adaptive data analysis and other new mathematical theories and
effective computational algorithms to define trends and instantaneous frequency.
These efforts involve a number of mathematical tools, including nonlinear varia-
tional methods, optimization, sparse representation of data, compressed sensing,
total variation denoising methods, multiscale analysis, randomized algorithms, and
statistical methods. The workshops brought together area experts to exchange ideas,
identify new research opportunities, and develop emerging directions in HHT re-
search. New opportunities may include exploring intrinsic underlying processes to
build predictive models and extending the trend study in regression, the latter of
which is of great interest to other research communities, such as econometrics and
finance.
To summarize the ever-increasing research results, a new journal entitled Ad-
vances in Adaptive Data Analysis (AADA) was inaugurated in 2009 under joint
editors-in-chief Thomas Y. Hou (Caltech) and Norden E. Huang (National Central
University). AADA is intended to be an interdisciplinary journal dedicated to re-
porting original research results on data analysis methods and their applications,
with a special emphasis on adaptive approaches. The mission of the journal is to
elevate data analysis above the routine data processing level and make it a tool
for scientific explorations and engineering applications. During the last five years,
AADA has published more than 200 technical papers, which have substantially con-
tributed to popularizing current HHT research frontiers. For example, the highly
cited and widely used EEMD method appeared in AADA’s inaugural issue.
The HHT field has become much richer since 2005 and it is still experiencing
vigorous growth. No book summarizing many salient HHT results has yet appeared.
In a sense, since 2005, this present volume has served as an introduction to the HHT
method, along with the original 1998 paper by Huang et al. (1998): The empirical
mode decomposition method and the Hilbert spectrum for non-stationary time se-
ries analysis, Proc. Roy. Soc. London, A454, 903–995. With the three additional
chapters, we hope that this new edition will continue to serve as an introduction
until books that are more comprehensive appear.
Norden E. Huang and Samuel S. P. Shen
Chungli, Taiwan, and San Diego, USA
September 2013
March 11, 2014 13:30
The Hilbert-Huang (8804) - 9.75 x 6.5
book page vii
CONTENTS
PREFACE
CHAPTER 1 INTRODUCTION TO THE
HILBERT–HUANG TRANSFORM AND ITS
RELATED MATHEMATICAL PROBLEMS
1.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 The Hilbert–Huang transform . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 The empirical mode decomposition method (the sifting process)
1.2.2 The Hilbert spectral analysis . . . . . . . . . . . . . . . . . . . .
1.3 Recent developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Normalized Hilbert transform . . . . . . . . . . . . . . . . . . . .
1.3.2 Confidence limit
. . . . . . . . . . . . . . . . . . . . . . . . . . .
Statistical significance of IMFs . . . . . . . . . . . . . . . . . . .
1.3.3
1.4 Mathematical problems related to the HHT . . . . . . . . . . . . . . . .
1.4.1 Adaptive data-analysis methodology . . . . . . . . . . . . . . . .
1.4.2 Nonlinear system identification . . . . . . . . . . . . . . . . . . .
1.4.3 The prediction problem for nonstationary processes (the end
1.4.4
effects of EMD) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Spline problems (the best spline implementation for HHT,
convergence and 2-D)
. . . . . . . . . . . . . . . . . . . . . . . .
1.4.5 The optimization problem (the best IMF selection and
uniqueness mode mixing) . . . . . . . . . . . . . . . . . . . . . .
1.4.6 Approximation problems (the Hilbert transform and quadrature)
1.4.7 Miscellaneous statistical questions concerning HHT . . . . . . . .
1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CHAPTER 2 ENSEMBLE EMPIRICAL MODE DECOMPOSI-
TION AND ITS MULTI-DIMENSIONAL EXTENSIONS
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 The empirical mode decomposition . . . . . . . . . . . . . . . . . . . . .
2.3 The ensemble empirical mode decomposition . . . . . . . . . . . . . . .
2.4 The multi-dimensional ensemble empirical mode decomposition . . . . .
2.5 Summary and discussions . . . . . . . . . . . . . . . . . . . . . . . . . .
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