HILBERT-HUANG TRANSFORM ANALYSIS OF HYDROLOGICALAND ENVIRONMENTAL TIME SERIES

Water Science and Technology Library

VOLUME 60

Editor-in-Chief

V.P. Singh, Texas A&M University, College Station, U.S.A.

Editorial Advisory Board

M. Anderson, Bristol, U.K.L. Bengtsson, Lund, Sweden

J. F. Cruise, Huntsville, U.S.A.U. C. Kothyari, Roorkee, India

S. E. Serrano, Philadelphia, U.S.A.D. Stephenson, Johannesburg, South Africa

W. G. Strupczewski, Warsaw, Poland

The titles published in this series are listed at the end of this volume.

HILBERT-HUANG TRANSFORMANALYSIS OF

HYDROLOGICALAND ENVIRONMENTAL

TIME SERIES

by

A. RAMACHANDRA RAOSchool of Civil Engineering, Purdue University, West Lafayette, IN, U.S.A.

and

EN-CHING HSUSchool of Civil Engineering, Purdue University, West Lafayette, IN, U.S.A.

Library of Congress Control Number: 2007936514

ISBN 978-1-4020-6453-1 (HB)ISBN 978-1-4020-6454-8 (e-book)

Published by Springer,

P.O. Box 17, 3300 AA Dordrecht, The Netherlands.

www.springer.com

Cover Image: Time-frequency distribution of monthly streamflows in the Warta river (Fig 5.3.5 (b))

Printed on acid-free paper

All Rights Reserved© 2008 Springer Science+Business Media B.V.

No part of this work may be reproduced, stored in a retrieval system, or transmittedin any form or by any means, electronic, mechanical, photocopying, microfilming, recording

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and executed on a computer system, for exclusive use by the purchaser of the work.

DEDICATION

This book is respectfully dedicated tothe unique yogini of the twentieth century

Maatha Jayalakshmiand to her son the great Siddha Purusha

Sri Sri Sri Ganapathi Sachchidananda Swamijiof

Avadhootha Datta PeethamSri Ganapathi Sachchidananda Ashrama, Mysore 570 025, India

with namaskarams

CONTENTS

Preface xi

1. Introduction 1

2. Hilbert-Huang Transform (HHT) Spectral Analysis 52.1. Introduction 52.2. Conventional Spectral Analysis Methods 5

2.2.1. Fourier Transform Analysis 52.2.2. Multi-Taper Method (MTM) of Spectral Analysis 62.2.3. Spectrogram 7

2.3. Empirical Mode Decomposition 82.4. Hilbert-Huang Spectra 122.5. Relationship between HHT and Fourier Spectra 142.6. Volatility of Time Series 172.7. Degree of Stationarity of Time Series 192.8. Stationarity Tests 20

2.8.1. Modified Mann-Kendall Test 202.8.2. Trend Test of Segments Derived from IMFs 22

2.9. Concluding Comments 25

3. Hilbert-Huang Spectra of Simulated Data 273.1. Introduction 273.2. Synthetic Data Analysis 27

3.2.1. Introduction 273.2.2. Simple Harmonic Data 283.2.3. Decaying Signal 313.2.4. A Signal with Three Close Frequencies 323.2.5. Autoregressive Model 34

3.3. Simulation of Nonstationary Random Processes 383.3.1. Introduction 383.3.2. Simulation with Random Phases 383.3.3. Simulation with Random Phases and Amplitudes 443.3.4. Simulation by Wen-Yeh Method 57

vii

viii CONTENTS

3.4. Confidence Intervals for Marginal Hilbert Spectrum 763.5. Concluding Comments 81

4. Rainfall Data Analysis 834.1. Introduction and Data Used 83

4.1.1. U.S. Historical Climatology Network (U.S. HCN) 834.1.2. NCDC Average Divisional Rainfall Data 84

4.2. HCN Rainfall Data 864.2.1. Long-Term Oscillations 864.2.2. Time-Frequency Distribution 904.2.3. Frequency Domain Analysis 98

4.3. NCDC Rainfall Data 1034.3.1. Long-Term Oscillations 1034.3.2. Time-Frequency Distribution 1034.3.3. Frequency Domain Analysis 112

4.4. Concluding Comments 118

5. Streamflow Data Analysis 1215.1. Introduction and Data Used 121

5.1.1. USGS Streamflow Data from Indiana 1215.1.2. Streamflow Data from Warta, Godavari

and Krishna Rivers 1215.2. USGS Streamflow Data 125

5.2.1. Long-Term Oscillations 1255.2.2. Time-Frequency Distribution 1265.2.3. Comparison with MTM Spectra 130

5.3. Analysis of Warta, Godavari and Krishna River Flow Data 1355.3.1. Warta River Daily Streamflow Data 1355.3.2. Warta River Monthly Streamflow Data 1405.3.3. Godavari River Monthly Streamflow Data 1435.3.4. Krishna River Monthly Streamflow Data 144

5.4. Concluding Comments 147

6. Temperature Data Analysis 1496.1. Introduction and Data Used 1496.2. European Long-Term Monthly Temperature Time Series 149

6.2.1. Original Data 1526.2.2. Linear-Trend Removed Data 1616.2.3. Annual-Cycle Removed Data 165

6.3. HCN and NCDC Monthly Temperature Time Series 1696.3.1. HCN Monthly Temperature Time Series 1696.3.2. NCDC Monthly Temperature Time Series 178

6.4. Concluding Comments 193

CONTENTS ix

7. Wind Data Analysis 1957.1. Introduction and Data Used 1957.2. Hourly Wind Speed Data 1957.3. Daily Average Wind Speed Data 2057.4. Daily Peak Wind Speed Data 2127.5. Concluding Comments 216

8. Lake Temperature Data Analysis 2198.1. Introduction and Data Used 2198.2. Lake Temperature Spatial Series Analysis 224

8.2.1. Spatial Series Analysis 2248.2.2. Time-Frequency Distribution 2258.2.3. Frequency Domain Analysis 229

9. Conclusions 235

References 239

Index 243

PREFACE

To accommodate the inherent non-linearity and non-stationarity of many naturaltime series, empirical mode decomposition (EMD) and Hilbert-Huang transform(HHT) provide an adaptive and efficient method. The HHT is based on thelocal characteristic time scale of the data. The HHT method provides not onlya precise definition in time-frequency representation than the other conventionalsignal processing methods, but also more physically meaningful interpretation ofthe underlying dynamic processes. The EMD also works as a filter to extract thevariability of signals with different scales and is applicable to non-linear and non-stationary processes. This promising algorithm has been applied in many fieldssince it was developed, but it has not been applied to hydrological and climatictime series.

The discussion in this book starts with several simulated data sets in order toinvestigate the capability of this method and to compare it to other conventionalfrequency-domain analysis methods that assume stationarity. Rainfall, streamflow,temperature, wind speed time series and lake temperature data are investigated inthis study. The aim of the work is to investigate periodicity, long term oscillationsand trends embedded in these data by using HHT. The analysis is performed in boththe time and frequency domains. The results from HHT are compared to those fromthe multi-taper method (MTM) which is based on Fourier Transform of the data.The results indicate that the HHT is clearly superior to MTM in delineating thestochastic structure of the data. Details about the data which cannot be investigatedby traditional methods are clearly seen with HHT. The nonstationarities of climaticand hydrologic data are also brought out. The HHT is seen to be an excellent toolto investigate the characteristics of environmental and hydrologic time series.

The details regarding the definition and application of Hilbert-Huang transform(Huang et al. (1998, 2005)) are discussed. It includes the sifting process usedfor empirical mode decomposition, Hilbert transform spectral analysis, some inthe time-frequency domain (degree of stationarity, volatility, and instantaneousenergy), and the trend tests (the modified Mann-Kendall test). Simulated data arefirst analyzed to investigate the performance of HHT analysis. Different types ofsynthetic data are discussed. One of the innovations based on HHT is the generationof nonstationary data. This aspect is of interest in time series analysis. Generationof data makes it possible to determine the confidence limits of the spectrum andfurthermore to identify the significant peaks in HHT spectra.

xi

xii PREFACE

The material herein puts much emphasis on the analysis of climatic, hydrologicaland environmental time series. Rainfall, temperature, streamflow and wind speeddata in the state of Indiana, U.S.A., are studied. Also, long historical temperaturerecords in Europe are investigated. The trends in European temperature data areclearly brought about by the results of EMD which compare well with the results ofparametric trend tests. The other issue which may be brought to readers’ attention isthat the HHT spectra are often characterized by power law equations. The detectionof periodicities, long-term oscillations, trends, nonstationarities embedded in thedata by using HHT technique is a promising approach in time series analysis.

We would like to thank Dr. Miki Hondzo of the St. Anthony Falls Laboratoryfor sending us the lake temperature data of Chapter 8. He also reviewed Chapter 8where the data acquisition is discussed. Dr. Tim Whalen of Purdue Universitycontributed the wind data discussed in Chapter 7. Dr. Whalen wrote a draft paperon Chapter 7 based on which Chapter 7 has been written.

Professor Rao would like to thank the numbers of his family, Mamatha Rao hiswife, Dr. Malini Rao Prasad his daughter, Dr. Sathya Prasad his son-in-law, KarthikA. Rao and Siddhartha S. Rao his sons and especially Shambhavi N. Prasad, hisdelightful grand daughter for their support. Dr. Hsu would like to give thanks toher family for their continuous love and support.

We would like to thank a number of people for both direct and indirect supportduring the period that we worked on this book project. We would like to thankDr. V.P. Singh for his support. We thank the Publishing Editorial and Productionstaff at Springer Publishers (Dordrecht, The Netherlands) who helped to bringthis book project to a successful conclusion. Our special thanks to Petra D. vanSteenbergen (Publishing Editor).

A. Ramachandra Rao, Bangalore, India (April, 2007)En-Ching Hsu, West Lafayette, Indiana, USA (April 2007)

CHAPTER 1

INTRODUCTION

In earlier studies of climatic time series, long-term time series have been assumedto be either periodic or stationary to apply the time domain or frequency domainanalysis methods. Traditional frequency analysis techniques based on Fourier Trans-forms tend to spread the energy of the signal into several frequencies, whichsometimes leads to misinterpretations of the characteristics of the data. In particular,trends in data would seriously distort the low frequency characteristics of the data.

There are several methods used for analyzing the non-stationary processes, suchas spectrogram (short-time Fourier transform), Wagner-Ville distribution (Loutridis,2005), empirical orthogonal function (EOF) expansion for metrological and oceano-graphic data and wavelet transforms. The spectrogram or the fixed window Fourierspectral analysis is widely used for musical and speech signal analysis. To calculatea spectrogram, the Fourier transform is applied by sliding a window along the timeaxis and repeatedly calculating the Fourier transform to obtain a time-frequencydistribution. The disadvantage of this approach is that we have to ensure that thedata within the time window is stationary. Even if it is stationary, the spectrogrammethod has an additional problem of having a trade-off in time and frequencyresolution. The Wagner-Ville distribution is a quadratic-form time-frequency distri-bution with optimized resolution in both time and frequency domain, but it is notalways nonnegative. There are also miscellaneous other methods such as the leastsquared estimation of the trend (Brockwell and Davis, 1991), which have problemsand disadvantages.

Huang et al. (2001) and Flandrin (1999) point out that wavelets, though a goodtool to investigate features of data, is a poor method to analyze time-energy-frequency distributions. This lack of frequency resolution is also addressed in greatdetail in Huang et al. (1998). Loutridis (2005) points out that the time and frequencyresolution leads to compromises, as large scale wavelets are chosen for determininggeneral signal features and small scale wavelets for extracting the signal details.Consequently, time localization is poor for low frequency signals and frequencyresolution is poor for high frequency signals. Peng et al. (2005) demonstrate thatwavelet transforms may generate many small undesirable spikes over all frequencyscales and make the results confusing and difficult to interpret.

1

2 CHAPTER 1

None of these methods can simultaneously provide a good resolution in bothtime and frequency domain. Huang et al. (1998, 2003a) proposed a new techniqueto efficiently extract the information in both time and frequency domains directlyfrom the data. It is adaptive, efficient and without any prior assumptions. Thisscheme is called as Hilbert-Huang Transform (HHT), which is the combination ofempirical mode decomposition (EMD) and Hilbert spectral analysis (HSA). It offersa different approach to processing time-series data. The signal is decomposed intoseveral oscillation modes by extracting the characteristic scales embedded in thedata. Traditionally, filtering is carried out in frequency domain; however, frequencydomain filtering is difficult when the data are either nonstationary or nonlinear, orboth. Filtering can eliminate some of the harmonics. Empirical mode decompositioncan be treated as a time-frequency filtering method through the representation ofthe intrinsic mode function (IMF) components. Therefore the low-pass, high-passand band-pass filters can be designed from the IMF components.

This technique is widely applied in science, engineering and financial analysis.In mechanical system analysis, it has been used for gear fault detection (Loutridis,2004), fault diagnosis of roller bearings (Yu et al., 2005; Peng et al., 2005), andthe processing of rotor startup signals (Gai, 2006). In biomedical science andhealth monitoring, it is applied for analyzing neural data (Liang et al., 2005),indicial responses of pulmonary blood pressure to step change of oxygen tensionin the breathing gas (Huang et al., 1999a, 1999b), deriving the respiratory sinusarrhythmia from the heartbeat time series (Balocchi et al., 2004), and deriving mainrhythms of the human cardiovascular system from the heartbeat time series anddetecting their synchronization (Ponomarenko, 2005). Huang et al. (2003b) appliedthe empirical mode decomposition to financial market data analysis; they usedthe HHT algorithm to examine the changeability of the market, as a measure ofvolatility of the market. Montesinos et al. (2003) analyze the BWR neutron detectorsignals by using empirical mode decomposition and compare the result to thosebased on autoregressive models.

In testing structures, the HHT has been applied to detecting anomalies in beamsand plates (Quek et al., 2003), vibration signal analysis (Peng et al., 2004), time-frequency analysis of the free vibration response of a beam with a breathing crack(Douka and Hadjileontiadis, 2005), and investigating the dynamic response ofbridges to controlled pile damage (Zhang et al., 2005). Huang et al. (1998) concludethat HHT is a potential tool for cost-effective, efficient structural damage diagnosisprocedures and health-monitoring systems. In coastal engineering applications, A.D.Veltcheva (2002) discusses the wave and group transformation by the HHT. Hwanget al. (2002) compare the energy flux computation of shoaling waves by usingHilbert and wavelet spectral analysis techniques.

HHT has also been applied to analyze earthquake signals. Huang et al. (2001)apply the HHT spectral analysis to the earthquake data of 21 September 1999from Chi-Chi. Zhang et al. (2004) estimate the damping factor of non-linear soilsand their role in estimating seismic wave responses at soil sites from earthquakerecordings. Chen et al. (2004) tried to identify the natural frequencies and modal

INTRODUCTION 3

damping ratios of the Tsing Ma suspension bridge during Typhoon Victor usingthe HHT algorithm. In atmospheric and geophysical sciences, Pan et al. (2002) usethe intrinsic mode functions to interpret the scattermeter ocean surface wind vectorEOFs over the Northwestern Pacific. Gloersen and Huang (2003) compare interannualintrinsic modes in hemispheric sea ice cover and other geophysical parameters.

There are several extended studies and theoretical discussions of this method.Flandrin and Gonçalvès (2003, 2004) apply the empirical mode decomposition asan equivalent filter bank structure to analyze the fractional Gaussian noise andfurther rationalize the method as an alternative way to estimate the Hurst exponent.Yang et al. (2003) identified general linear structures with complex modes usingfree vibration response data polluted by noise.

As far as we can ascertain, results of analysis of climatic and hydrologic timeseries by using HHT have not been reported. The objective of the research discussedhere is to investigate climatic and hydrologic time series, such as those of rainfall,runoff, and temperature by using HHT analysis, and discuss the results. These timeseries may be nonstationary and nonlinear. The results obtained by conventionalspectral analysis cannot be well interpreted in such cases. The properties of thesedata are investigated in time, frequency and in time-frequency domains.

In this study, the rainfall (HCN and NCDC), streamflow (USGS data and threeother cases from Warta (Poland), Godavari (India) and Krishna (India) rivers),temperature (HCN, NCDC and long-term measurements in Europe), wind speed(in the state of Indiana), and lake temperature data (four stations in the state ofMinnesota) are analyzed. The results obtained by HHT analysis are compared tothose from Fourier and multi-taper methods. The trend and periodicity in the dataare studied by performing empirical mode decomposition to obtain the intrinsicmode functions. This procedure decomposes the data into several componentsrepresenting different frequencies which helps in the interpretation of the data moreefficiently and adaptively. The degree of stationarity is a statistic used to investigatethe variation in power spectral density in time. In addition, another measure, that ofvolatility, provides information of how the intrinsic mode functions are related tothe signal. When similar data are analyzed, common characteristics are of interestand are investigated.

The material herein is presented as follows. In Chapter 2, the details regarding thedefinition and application of Hilbert-Huang transform are discussed. It includes thesifting process used for empirical mode decomposition, Hilbert transform spectralanalysis, some statistics to evaluate the results in the time-frequency domain (degreeof stationarity, volatility, and instantaneous energy), and the trend tests (the modifiedMann-Kendall test) are discussed.

Simulated data are analyzed by using HHT in Chapter 3. The performance andresults of HHT analysis for different types of data are discussed first. One of theinnovations based on HHT is the synthetic generation of nonstationary data. Thisaspect is discussed in Chapter 3. These generated data are used to identify thesignificance of peaks in HHT spectra. They may also be used to generate syntheticdata commonly used in stochastic hydrology.

4 CHAPTER 1

The analysis and discussion of climatic and environmental data are discussed inChapters 4–8. Rainfall data from the state of Indiana are analyzed by using HHT andthe results are discussed in Chapter 4. Both the NCDC and HCN data are analyzed.The NCDC data, because it is averaged over a region, are more consistent thanthe HCN data and the results reflect this characteristic. The commonly occurringperiodicities are identified. Long term (greater than about 20 years) oscillations arenot present in these data.

Monthly streamflow data from Indiana are analyzed and reported in Chapter 5.These data show greater variability than the rainfall data, but the common periodsof oscillation correspond to those found in rainfall data. One other series, a verylong daily flow series from Warta River in Poland, has been analyzed and theresults discussed. The spectrum of this series is of considerable interest as this seriesis one of the longest streamflow sequences available. The Krishna and GodavariRiver data from India are analyzed. These data which give strong spurious peaksin spectra when Multi-taper and other methods are used do not give such peakswith HHT.

The monthly temperature data from Indiana and some long historical data fromEurope are analyzed and discussed in Chapter 6. The monthly data from Indianaindicate variability corresponding to 1, 2, 4 and 11 years. The trends in Europeantemperature data are clearly brought about by the results of EMD which comparewell with the results of parametric trend tests.

Daily wind speed data from four stations in Indiana are analyzed in Chapter 7.The HHT spectra of these data exhibit considerable similarity, thereby indicatingthe potential of HHT spectra to characterize similar regions of wind velocity. Thespectra of wind speed data are also characterized by power law equations. Thereare some significant periodicities in wind speed data also.

The lake temperature and PAR data are obviously nonstationary. Previousattempts to analyze these data used the technique of segmenting the data. Thesesegments were approximately stationary. The Fourier spectra of these segmentswere computed. Naturally there are considerable variations in these spectra and inthe results based on HHT. These nonstationary data are analyzed by the HHT andthe results are discussed.

CHAPTER 2

HILBERT-HUANG TRANSFORM (HHT)SPECTRAL ANALYSIS

2.1. INTRODUCTION

Huang et al. (1998) introduced a general signal-analysis technique, calledHilbert-Huang Transform (HHT). It is a two-step algorithm, combining empiricalmode decomposition (EMD) and Hilbert spectral analysis, to accommodate the non-linear and non-stationary processes. This method is not based on a priori selectionof kernel functions, but instead it decomposes the signal into intrinsic oscillationmodes derived from the succession of extrema.

Before discussing the Hilbert-Huang Transform algorithm, traditional spectralanalysis methods which are used for comparison are reviewed. They are the Fouriertransform and Multi-Taper methods. In addition, statistical measures to investigatethe time series and spectral properties are discussed. These include the time-frequency representation known as spectrogram, degree of stationarity, volatility,and trend tests.

2.2. CONVENTIONAL SPECTRAL ANALYSIS METHODS

2.2.1 Fourier Transform Analysis

The common definition for Fourier transform of a continuous-time signal x�t� isgiven in Eq. (2.2.1).

X��� =∫ �

−�x�t�e−j�tdt�� ∈ �−���� (2.2.1)

Almost all data analysis is carried out not with functions in continuous time butwith discrete-time data. Hence discrete-time Fourier transform (DFT) is used indata analysis. The DFT replaces the infinite integral in Eq. (2.2.1) with a finitesummation representation,

X��k� �N−1∑n=0

x�tn�e−j�ktn � k = 0� 1� 2� � � � � N −1 (2.2.2)

5

6 CHAPTER 2

where N is the number of time samples, and �k is the kth frequency. Thisformula has finite summation limits. DFT is implemented by using the Fast FourierTransform (FFT) algorithm when possible. The FFT yields an efficient way tocalculate the DFT.

Fourier amplitude spectrum defines harmonic components globally and thusyields average characteristics over the entire duration of the data. In order toinvestigate nonstationary data and investigate the time-frequency characteristics,the Fourier transform can be utilized with segments of data to produce the so-calledspectrogram. In using this technique, each segment should be stationary so thatwe can minimize the non-stationarity in the signal caused by different types ofpropagating waves. However, the frequency resolution is reduced when the length ofthe window is shortened. A trade-off situation is faced in this approach: the shorterthe window, the better the temporal localization of Fourier amplitude spectrum, butthe poorer the resolution in frequency.

2.2.2 Multi-Taper Method (MTM) of Spectral Analysis

The MTM method (Thomson, 1982) makes use of an extended version of conven-tional spectral representation. The process x�t� may include a number of periodiccomponents in addition to an underlying stationary process,

x�t� =∑j

cj cos�2�fjt +�j�+ t =∑j

jei2�fj t +∗

j e−i2�fj t + t (2.2.3)

where t is a zero mean stationary process with spectral density S�f�, cj and fj

are the amplitude and frequency of periodic or line components j, j = �cj/2�ei�j

is the complex amplitude corresponding to the real amplitude cj . These types ofprocesses are known as centered stationary or conditional stationary processes andoften have mixed spectra.

The basic idea of MTM spectral analysis is using multiple data windows knownas “discrete prolate sheroidal sequences” or “Slepian sequences”, which are definedas the solution of the symmetric Toeplitz matrix eigenvalue problem in Eq. (2.2.4),

�kv�k�n �N�W� =

N−1∑m=0

sin�2�W�n−m�

��n−m�v�k�

m �N�W� (2.2.4)

where N is the number of data points, W is the spectral band width and �k are theeigenvalues associated with the Slepian sequences v�k�

n �N�W�. The values of Slepiansequences can be calculated numerically by using methods given by Percival andWalden (1993) and Thomson (1982). The Fourier transform of these sequences aregiven in Eq. (2.2.5),

Vk�f� =N−1∑n=0

v�k�n �N�W�e−i2�fn (2.2.5)

HILBERT-HUANG TRANSFORM (HHT) SPECTRAL ANALYSIS 7

The Slepian functions have the maximum energy concentration within the interval(f − W , f + W ). In this method, the bias from all frequencies remote from thefrequency range of interest decreases exponentially as a function of NW; thus thismethod very effectively eliminates window leakage. The first step in MTM spectralestimation is the expansion of the time series x�t� as Eq. (2.2.6),

yk�f� =N−1∑t=0

x�t�v�k�t �N�W�e−i2�ft (2.2.6)

where k = 0� 1� � � � �K−1 and K is usually taken as 2NW–1. The band width W isusually chosen between 2/N and 20/N , with 4/N a good initial choice. If W is toosmall, the resulting spectral estimate is unstable, but if W is too large, it results inpoor resolution. The spectrum can be estimated by Eq. (2.2.7).

S�f� = 1K

K−1∑k=0

�yk�f��2 (2.2.7)

Priestley (1965) suggests a method for calculating the evolutionary spectra S(t,f)of nonstationary time series based on a double windowing technique, which canreduce the variance of the estimate of the evolutionary spectrum. This technique issimilar to that applied in the multitaper method for spectral analysis. The differenceis that in MTM, the variance is reduced by averaging the spectra from the samedata segment using multiple data tapers. MTM is used as an alternative to thedouble window technique to evaluate the evolutionary spectra. To apply multi-tapermethod to study the time-frequency spectra, the signal is divided into a number ofsegments (possibly overlapping as a sliding window) each of length T and MTMspectra are calculated for each segment to obtain S(t,f).

2.2.3 Spectrogram

Spectrograms are usually created in one of two ways; either with a series of bandpassfilters or they are calculated from time signals by using the short-time Fouriertransform (STFT). Piece-wise stationarity is assumed and sliding a window acrossthe time series and performing Fourier analysis to construct the spectrograms. STFTis simply described in a continuous case. A window function, which is nonzerofor a short period of time, is convolved with the function to be transformed andFourier transformed. The resulting signal is taken as the window sliding along thetime axis and written as

STFT�t��� =�∫

−�x���w�t − ��e−j��d� (2.2.8)

where w�t� is the window function and x�t� is the signal to be transformed.STFT(t��) is then a complex function representing the phase and magnitude of

8 CHAPTER 2

the signal over time and frequency. The spectrogram, SP�t���, is given by themagnitude of the STFT function:

SP�t��� = �STFT�t����2 (2.2.9)

To calculate the spectrogram, the digital sampled data in time domain is broken upinto several segments, which usually overlap, and Fourier transformed to calculatethe magnitude of the power spectrum of each segment. Each segment then corre-sponds to a vertical line in the image of time-frequency representation- a represen-tation of magnitude versus frequency at a specific moment in time.

2.3. EMPIRICAL MODE DECOMPOSITION

In the traditional Fourier analysis, the frequency is defined by using the sine andcosine functions spanning the entire length of data. Such a definition would notmake sense for non-stationary data in which changes occur with time. This difficultyis overcome by the introduction of the approaches based on the Hilbert transform.For an arbitrary time series, x�t�, its Hilbert transform, y�t�, is obtained by

y�t� = 1�

P

�∫

−�

x�t′�t − t′ dt′ (2.3.1)

where P indicates the Cauchy principal value. It is the convolution of x�t� with 1/t;hence, the transform emphasizes the local properties of x�t�. x�t� and y�t� form thecomplex conjugate pair by definition, so we can have an analytical signal, z�t� asshown in Eq. (2.3.2),

z�t� = x�t�+ iy�t� = a�t�ei��t� (2.3.2)

in which

a�t� = �x2�t�+y2�t��1/2� ��t� = arctan(

y�t�

x�t�

)(2.3.3)

The polar coordinate expression is the local fit of an amplitude and phase varyingtrigonometric function to x�t�. Based on Hilbert transform, the instantaneousfrequency is defined as

��t� = d��t�

dt(2.3.4)

In practice, at any time, it is quite possible that the signal may involve morethan one oscillation mode, and consequently the signal has more than one localinstantaneous frequency at a time. There is still considerable controversy in definingthe instantaneous frequency with Hilbert transform. A detailed discussion is found

HILBERT-HUANG TRANSFORM (HHT) SPECTRAL ANALYSIS 9

in Huang et al. (1998). Restrictive conditions have to be imposed on the data in orderto obtain meaningful instantaneous frequency. For this purpose, Huang et al. (1998)suggest modifying the restrictive condition from a global to a local one so thatwe can translate the requirement into physically implementable steps. Furthermore,this local restriction also suggests a method to decompose the data into componentsfor which the instantaneous frequency can be defined. Hence, EMD is needed.Otherwise negative amplitudes may appear in Hilbert transform. Intrinsic modefunction (IMF) is thus designated as a class of functions so that the instantaneousfrequency can be defined everywhere based on the local properties. As a result, thelimitation of interest here is not on the existence of the Hilbert transform which isgeneral and global, but on the existence of a meaningful instantaneous frequencywhich is restrictive and local.

Physically, the required conditions to define a meaningful instantaneousfrequency are that the functions are symmetric with respect to the local zero meanand have the same number of zero crossings and extrema. An intrinsic modefunction (IMF) is defined as a function that satisfies two conditions: (1) the numberof extrema and the number of zero crossings must either equal or differ at most byone in the whole data set, and (2) the mean value of the envelope defined by thelocal maxima and the envelope defined by the local minima is zero everywhere. Anillustration of local mean, and envelopes of local maxima and minima are shownin Figure 2.3.1.

Figure 2.3.1. Definition of sifting

10 CHAPTER 2

Knowing the well-behaved Hilbert transform of the IMF components is only astarting point. In most cases, we have to decompose the data into several IMFssince most time series involve more than one oscillatory mode. A systematic wayto extract the IMFs, designated as a sifting process, is based on the followingassumptions: (1) the signal has at least two extrema, and (2) the characteristictime scale is defined by the time lapse between the extrema. The sifting process isdescribed as follows.(1) Identify all extrema (maxima and minima) of the signal x�t�.(2) Connect these maxima with the cubic spline lines to construct an upper

envelope, emax�t�; use the same procedure for minima and to construct a lowerenvelope, emin�t�.

(3) Compute the mean of the upper and lower envelope: m�t� = �emin�t� +emax�t��/ 2.

(4) Calculate d�t� = x�t�−m�t�.(5) Let d�t� be the new signal x�t�. Follow the previous procedure again until d�t�

becomes a zero-mean process according to a stopping criterion. These iterationsare shown in Figure 2.3.2.

(6) Once we have the zero-mean d�t�, it is designated as the first intrinsic modefunction(IMF 1), c1.

(7) The IMF 1 is subtracted from the original signal and the residual is used as anew signal x�t�. The sifting process is repeated to get IMF 2.

(8) Continuing like this, we obtain c3, c4, and so on. This process is stoppedwhen the residual is a monotonic function having only one minimum or onemaximum.

In practice, after a certain number of iterations, the resulting signal does notcarry significant physical information. The sifting process is stopped by limitingthe standard deviation, which is computed from the two consecutive sifting results.The threshold is usually set as 0.2 and 0.3. Also, the number of extrema decreaseswhile moving to the higher order IMF, and this guarantees that the sifting processends with a finite number of intrinsic mode functions. Basically, the sifting processeliminates the riding waves and makes the IMF profiles mode symmetrical in orderto obtain meaningful results for instantaneous frequency.

IMF components represent simple oscillatory modes embedded in the signal andis much more general compared to the simple harmonic functions. As a check ofthe completeness of using Eq. (2.3.5), we can reconstruct the data by adding all theIMF components and the residual trend. Assume that we have n IMF components(c1� c2� � � � � cn) and one residual (rn), which follow the order from the shortest tothe longest period. Hence it implies that they range from the highest frequency tothe lowest frequency. The characteristic scale is physical which helps us to examinethe physical meaning of each IMF component.

x�t� =n∑

j=1

cj + rn (2.3.5)

HILBERT-HUANG TRANSFORM (HHT) SPECTRAL ANALYSIS 11

Figure 2.3.2. A demonstration of the iteration process to obtain a zero-mean process, i.e. an intrinsicmode function. The thin-solid line is the time series before sifting, the dot-dashed lines are the upperand lower envelopes from the local maxima and minima, and the thick solid line is the local mean valueof the envelopes

A new identified use of the IMF component is filtering. For example, a low passfiltered result of a signal having n IMFs can be expressed by

xLP�t� =n∑

j=k

cj + rn (2.3.6)

and high pass filtered results can be expressed as

xHP�t� =k∑

j=1

cj (2.3.7)

12 CHAPTER 2

Further, a band pass filtered result can be expressed as

xBP�t� =k∑

j=b

cj (2.3.8)

In order words, we can add the long period components to get a lowpass filterresult, or we can add all components with selected omissions to get the band-passresult.

The orthogonality of the IMF components should be checked a posteriori inorder to investigate the goodness of the decomposition process. Let the residue r�t�be the last IMF, i.e., cn +1 = r�t�. Then Eq. (2.3.9) can be rewritten as

x�t� =n+1∑j=1

cj�t� (2.3.9)

Then taking square of this signal x�t� we have

x2�t� =n+1∑j=1

c2j �t�+2

n+1∑j=1

n+1∑k=1

cj�t�ck�t� (2.3.10)

If the decomposition is orthogonal, the cross terms given in the second part ofthe right-hand side should be zero when they are integrated along time. Therefore,an overall index of the orthogonality, IO, is defined as

IO =T∑

t=0

(n+1∑j=1

n+1∑k=1

cj�t�ck�t�/x2�t�

)(2.3.11)

T is the time interval under consideration. The index IO should be very small inorder to have a good decomposition of IMF. Typically, values between 0.01 and0.001 are acceptable.

2.4. HILBERT-HUANG SPECTRA

Once we have these intrinsic mode function components, Hilbert transform can beapplied to each component to get the amplitudes, and meanwhile the instantaneousfrequency is calculated using Eq. (2.3.4). Therefore, Eq. (2.3.5) is rewritten in thefollowing expression,

x�t� = �n∑

j=1

aj�t� exp(i∫

�j���d�)

(2.4.1)

where � is the real part of the complex number.The time-frequency distribution of the amplitude is designated as the Hilbert

amplitude spectrum, H��� t�, or simply the Hilbert spectrum. At a given time t, the

HILBERT-HUANG TRANSFORM (HHT) SPECTRAL ANALYSIS 13

instantaneous frequency � and the amplitude are calculated simultaneously so thatthese values are assigned to Hilbert spectrum, H��� t�. aj�t� is a time-dependentexpansion coefficient similar to the constant in the Fourier expansion and �j isthe instantaneous frequency at a time � which differs from the constant frequency�j in Fourier transform Eq. (2.4.2). It represents a generalized form of Fourierexpansion.

x�t� = �n∑

j=1

aj�t� exp�i�jt� (2.4.2)

With the Hilbert spectrum defined, the marginal Hilbert spectrum, h���, isdefined in Eq. (2.4.3). It is a measure of total energy contribution from eachfrequency over the entire data span in a probabilistic sense. It provides a quantitativeway to describe the time-frequency-energy representation by integrating the Hilbertspectrum over the entire time span,

h��� =T∫

0

H��� t�dt (2.4.3)

where T is the total data length.Another integration over the frequency span is the instantaneous energy IE(t),

which is defined as Eq. (2.4.4). It provides information about the time variation ofthe energy.

IE�t� =∫

�

H��� t�d� (2.4.4)

The raw Hilbert spectrum presentation gives desirable and quantitative results.But, the higher resolution representation and small scattered points in time-frequency-energy plot are not easy to interpret in raw Hilbert spectrum. Hence, aGaussian weighted Laplacian filter is applied to the Hilbert spectrum. The schematicof this filter is shown in Figure 2.4.1. A “fuzzy” or “smoothed” view thus canbe derived from the original presentation by using two-dimensional filtering. Theproperties of four spectral analysis methods based on the capability of handling thenonlinear and nonstationary time series are listed in Table 2.4.1.

A flowchart in Figure 2.4.2 summarizes the calculation procedure of this two-stepHHT algorithm. The left-hand side of Figure 2.4.2 is basically the procedure forusing sifting process to define the intrinsic mode functions or the empirical modedecomposition while the right-hand side is the procedure to construct the Hilbertspectrum.

14 CHAPTER 2

Figure 2.4.1. Schematic of a 2D Gaussian weighted Laplacian filter

Table 2.4.1. Comparison of different spectral analysis methods (Huang et al., 1998)

Fourier MTM Wavelet HHT

Basic A priori A priori A priori Adaptive

Frequency Convolution:Global

Convolution:Global

Convolution:Regional

Differentiation:Local

Presentation Energy-frequency

Energy-frequency

Energy-time-frequency

Energy-time-frequency

Nonlinear No No No Yes

Nonstationary No Yes Yes Yes

Feature extraction No No No (discrete) Yes(Continuous)

Yes

2.5. RELATIONSHIP BETWEEN HHT AND FOURIER SPECTRA

As mentioned previously, the representations of a signal by using Fourier seriesor Hilbert transform are given in Eq. (2.5.1). It is clear that Hilbert transform is amore general representation than Fourier transform.

x�t� = � n∑j=1

aj�t� exp�i�jt� (Fourier transform)

= � n∑j=1

aj�t� exp(i∫

�j���d�)

(Hilbert transform)(2.5.1)

A way to investigate the relationship between HHT and Fourier transform is bycalculating the energy of HHT and Fourier spectra. Fourier transform of a time

HILBERT-HUANG TRANSFORM (HHT) SPECTRAL ANALYSIS 15

Figure 2.4.2. Flowchart of the empirical mode decomposition and Hilbert spectrum analysis

series x�t� is:

X��� =∫ �

−�x�t�e−i�tdt (2.5.2)

Its complex conjugate is

X∗��� =∫ �

−�x�t�ei�tdt (2.5.3)