Hilbert-Huang Transform and Its Applications

The following sections are included:

• PREFACE

• CONTENTS

The Hilbert–Huang transform (HHT) is an empirically based data-analysis method. Its basis of expansion is adaptive, so that it can produce physically meaningful representations of data from nonlinear and non-stationary processes. The advantage of being adaptive has a price: the difficulty of laying a firm theoretical foundation. This chapter is an introduction to the basic method, which is followed by brief descriptions of the recent developments relating to the normalized Hilbert transform, a confidence limit for the Hilbert spectrum, and a statistical significance test for the intrinsic mode function (IMF). The mathematical problems associated with the HHT are then discussed. These problems include (i) the general method of adaptive data-analysis, (ii) the identification methods of nonlinear systems, (iii) the prediction problems in nonstationary processes, which is intimately related to the end effects in the empirical mode decomposition (EMD), (iv) the spline problems, which center on finding the best spline implementation for the HHT, the convergence of EMD, and two-dimensional EMD, (v) the optimization problem or the best IMF selection and the uniqueness of the EMD decomposition, (vi) the approximation problems involving the fidelity of the Hilbert transform and the true quadrature of the data, and (vii) a list of miscellaneous mathematical questions concerning the HHT.

This paper discusses some mathematical issues related to empirical mode decomposition (EMD). A B-spline EMD algorithm is introduced and developed for the convenience of mathematical studies. The numerical analysis using both simulated and practical signals and application examples from vibration analysis indicate that the B-spline algorithm has a comparable performance to that of the original EMD algorithm. It is also demonstrated that for white noise, the B-spline algorithm acts as a dyadic filter bank. Our mathematical results on EMD include Euler splines as intrinsic mode functions, the Hilbert transform of B-splines, and the necessary and sufficient conditions which ensure the validity of the Bedrosian identity of the Hilbert transform of product functions.

Huang's data-driven technique of empirical mode decomposition (EMD) is given a filter bank interpretation from two complementary perspectives. First, a stochastic approach operating in the frequency domain shows the spontaneous emergence of an equivalent dyadic filter bank structure when EMD is applied to the versatile class of fractional Gaussian noise processes. Second, a similar structure is observed when EMD is operated in the time domain on a deterministic pulse. A detailed statistical analysis of the observed behavior is carried out involving extensive numerical simulations that suggest a number of applications. New EMD-based approaches are used to estimate the scaling exponents in the case of self-similar processes, to perform a fully data-driven spectral analysis, and to denoise-detrend signals that contain noise.

Time-frequency analysis is the process of determining what frequencies are present in a signal, how strong they are, and how they change over time. Understanding how the frequencies in a signal change with time can explain much about the physical processes that generate or influence the signal. Better resolution of the details of frequency changes provides better insight into these underlying physical processes.

The Hilbert–Huang transform (HHT) offers higher frequency resolution and more accurate timing of transient and non-stationary signal events than conventional integral transform techniques. The HHT separates complex signals into simpler component signals, each of which has a single, well-defined, time-varying frequency. Real-time HHT algorithms enable this enhanced signal analysis capability to be used in process monitoring and control applications.

"Sifting" is the central signal separation process of the HHT algorithm. This chapter compares the component signal separations of Huang's sifting process with those produced by filtering techniques. Although intuition seems to suggest that filtering, with appropriate real-time adjustments to parameters, could be substituted for Huang's sifting process, our results did not support this suggestion. Five case studies present HHT and filtering results for stationary amplitude- and frequency-modulated signals, as well as signals with more dynamic transient behavior. These examples show that, in general, HHT sifting and filtering separate signal components quite differently. Our experiments with example signals led to the discovery of aliasing in the HHT sifting algorithm.

One of the preliminary tasks when analyzing a dataset is to determine whether it or its components contain useful information. The task is essentially a binary hypothesis testing problem in which a null hypothesis of pure noise is often pre-proposed. To test against the null hypothesis, the characteristics of noise need to be understood first, and often, these characteristics pertain to the analysis method used.

In this paper, the characteristics of Gaussian white noise are studied by using the empirical mode decomposition (EMD) method. Statistical testing methods for Gaussian white noise for the intrinsic mode functions (IMFs) are designed based on the characteristics of Gaussian white noise by using EMD. These methods are applied to well-studied geophysical datasets to demonstrate the method's validity and effectiveness.

Recently a new spectral technique has been developed for the analysis of aperiodic signals from nonlinear systems – the Hilbert–Huang transform. It is shown how this transform can be used to discover synoptic and climatic features: For sea level data, the transforms capture the oceanic tides as well as variations in precipitation patterns. In the case of solar radiation, variations in the diurnal and seasonal cycles are observed. Finally, from barographic data, the Hilbert–Huang transform reveals the passage of extratropical cyclones, fronts, and troughs. Thus, this technique can detect signals on synoptic to interannual time scales.

Empirical mode decomposition (EMD) is used to extract inter-annual atmospheric signals. Our climate contains components that are often distinguished from one another on temporal, rather than spatial, scales, and EMD is found to be a useful tool in extracting physically meaningful information from the data. The National Centers for Environmental Prediction has provided reanalyzed data, which are used here to demonstrate that dynamical variables in the zonally averaged troposphere and lower stratosphere contain only five oscillating modes and a trend. These oscillations contain interesting amplitude and frequency modulations which are cumbersome and difficult to interpret when viewed in the traditional Fourier frequency domain. EMD analysis provides a unique, relatively unbiased and useful way to decompose the time series. Statistical tests are developed to determine the confidence in our analysis. Physical interpretations of the significant modes are also discussed with corresponding justification. By combining the information found for the time series of each horizontal strip into a spatial matrix, valuable spatial information about these physical processes have been obtained.

Orbital drift results in late equatorial crossing times for the afternoon National Oceanic and Atmospheric Administration (NOAA) polar satellites and in changes of illumination that affect measurements made by the advanced very high resolution radiometer (AVHRR). Processing and correcting for calibration variation(s) and atmospheric effects have improved, but one of the standard AVHRR products, the normalized difference vegetation index (NDVI), may still contain variations due to orbital drift or changes in sun-target-sensor geometry. In this study, the solar zenith angle (SZA) trends associated with orbital drift are identified and analyzed with respect to their effects on the NDVI. The adaptive empirical mode decomposition (EMD) method is used to identify and remove the induced artifacts from the NDVI time series. The EMD is based on the local characteristic time scale of the data and is used to identify embedded nonlinear and nonstationary variation. Trend artifacts associated with drift were uncoupled from the surface signal, and their contributions were quantified at all latitudes. The approach was tested on 1 degree and 8-km NDVI global datasets, and showed that it is very suitable for addressing the long-standing issues of orbital shifting or the inconsistencies of the AVHRR data among sensors. The results showed that the interference of satellite drift artifacts with the surface signal was (i) large in a tropical forest, (ii) moderate in the tropics for less-densely vegetated areas, and (iii) lowest at higher northern and lower southern latitudes.

The empirical mode decomposition (EMD) method is used to analyze the nonlinear and non-stationary annual cycle (NAC) in climate data. The NAC is defined as an intrinsic mode function generated in the EMD process and has a mean period equal to a year. Both Hilbert and Fourier spectra of the NAC are examined to validate the power density at the frequency of one cycle per year. The NAC differs from the strictly periodic thirty-year-mean annual cycle (TAC), which is now commonly used in climate anomaly analysis. It is shown that the NAC has a stronger signal of the annual cycle than that of the TAC. Thus, the anomalies derived from the NAC have less spectral power at the frequency of one cycle per year than those from the TAC. The marginal Hilbert spectra of the maximum daily surface air temperature of ten North America stations demonstrate the expected characteristics of an annual cycle: the NAC's strength is proportional to the latitude of the station location and inversely proportional to the heat capacity of the Earth's surface materials around a station. The NAC of the Niño3.4 sea surface temperature analysis indicates that El Niño events correspond to a weaker annual cycle, as suggested by wavelet analysis.

The Hilbert–Huang transform (HHT) analysis interprets wave nonlinearity in terms of frequency modulation instead of harmonic generation. The resulting spectrum contains much higher spectral energy at low frequency and sharper drop off at high frequency in comparison with the spectra derived from Fourier-based analysis methods. The high energy level in the low-frequency components of the Hilbert spectrum seems to be consistent with the rich group structure apparent in typical ocean-wave records. For wind-generated waves, the spectral level of the Fourier spectrum is about two orders of magnitude smaller than that of the Hilbert spectrum at the first subharmonic of the peak frequency. The mean frequency of the Fourier spectrm is 20% higher than that of the Hilbert spectrum. Furthermore, the frequency of the wave groups is also more likely to be properly identified in the Hilbert spectrum than in the Fourier spectrum. The implications for ocean engineering and air-sea interaction are discussed.

In this chapter, a new structural health-monitoring and damage-detection method is presented. A general time-frequency data analysis technique (empirical mode decomposition and the Hilbert–Huang spectrum) in conjunction with a wave-mechanics-based concept is developed to provide a diagnostic tool for detecting and interpreting adverse changes in a structure. Sets of simple basis function components, known as intrinsic mode functions (IMF), are extracted adaptively from the measured structural response time series data. These IMFs are amplitude- and phase-modulated signals and are used to define the instantaneous phases of structural waves. The state of a structure is evaluated, and damage is identified based on these instantaneous phase features. Furthermore, fundamental relationships are developed connecting the instantaneous phases to a local physics-based structural representation in order to infer damage in terms of physical parameters, such as structural mass, stiffness, and damping. Damage-detection applications are investigated by using numerical simulations and a variety of laboratory experiments with simple structures. Several different types of excitation mechanisms are used for dynamic input to the structures. The time series output of the structural response is then analyzed by using the new method. The instantaneous phase relationships are extracted and examined for changes which may have occurred due to damage. These results are compared to those from other newly developed detection methods, such as an algorithm based on the geometric properties of a chaotic attractor. The studies presented here show that our method, without linear-system or stationary-process assumptions, can identify and locate structural damage and permit the further development of a reliable real-time structural health-monitoring and damage-detection system.

A new Hilbert–Huang transform (HHT)-based method for nondestructive instrument structure health monitoring is developed. When applied to bridges, this new method depends on a transient test load and simple data collection. The essense of the method is the newly developed HHT for nonstationary and nonlinear time series analysis, which consists of the empirical mode decomposition and Hilbert spectral analysis. The final decision on the health of the bridge structure is based on two criteria. The first criterion detects the nonlinear characteristic of the intra-wave frequency modulations of the bridge response, which usually appears during comparisons of light with heavy loads. The second criterion detects the frequency downshift as an indication of structural yield. This new method enjoys many advantages: no a priori data required, simple data collection, minimum traffic disruption, and precise and nuance quantitative answers. The result of a case study is presented, which establishes the feasibility of this new approach for structural health monitoring.

The recently developed empirical mode decomposition/Hilbert–Huang transform (EMD/HHT) for the analysis of nonlinear and non-stationary data has been extended to include the analysis of image data. Because image data can be expressed in terms of an array of rows and columns, this robust concept is applied to these arrays row by row. Each slice of the data image, either row or column-wise, represents local variations of the image being analyzed. Just as much of the data from natural phenomena are either nonlinear or non-stationary, or both, so are the data that form images of natural processes. Thus, the EMD/HHT approach (or HHT in abbreviated form) is especially well-suited for image data, giving frequencies, inverse distances or wavenumbers as a function of time or distance, along with the amplitudes or energy values associated with these, as well as a sharp identification of embedded structures. The varied products of this new analysis approach are joint and marginal distributions to be viewed as isosurfaces, contour plots, and surfaces that contain information on frequency, inverse wavelength, amplitude, energy and location in time, space, or both. Additionally, component images representing the intrinsic scales and structures embedded in the data will be described, as well as a technique for obtaining frequency variations.

The following sections are included:

• INDEX