Norden E. Huang

The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis

A new method for analysing nonlinear and non-stationary data has been developed. The key part of the method is the ‘empirical mode decomposition’ method with which any complicated data set can be decomposed into a finite and often small number of ‘intrinsic mode functions’ that admit well-behaved Hilbert transforms. This decomposition method is adaptive, and, therefore, highly efficient. Since the decomposition is based on the local characteristic time scale of the data, it is applicable to nonlinear and non-stationary processes. With the Hilbert transform, the ‘instrinic mode functions’ yield instantaneous frequencies as functions of time that give sharp identifications of imbedded structures. The final presentation of the results is an energy-frequency-time distribution, designated as the Hilbert spectrum. In this method, the main conceptual innovations are the introduction of ‘intrinsic mode functions’ based on local properties of the signal, which make the instantaneous frequency meaningful; and the introduction of the instantaneous frequencies for complicated data sets, which eliminate the need for spurious harmonics to represent nonlinear and non-stationary signals. Examples from the numerical results of the classical nonlinear equation systems and data representing natural phenomena are given to demonstrate the power of this new method. Classical nonlinear system data are especially interesting, for they serve to illustrate the roles played by the nonlinear and non-stationary effects in the energy-frequency-time distribution.

ENSEMBLE EMPIRICAL MODE DECOMPOSITION: A NOISE-ASSISTED DATA ANALYSIS METHOD

A new Ensemble Empirical Mode Decomposition (EEMD) is presented. This new approach consists of sifting an ensemble of white noise-added signal (data) and treats the mean as the final true result. Finite, not infinitesimal, amplitude white noise is necessary to force the ensemble to exhaust all possible solutions in the sifting process, thus making the different scale signals to collate in the proper intrinsic mode functions (IMF) dictated by the dyadic filter banks. As EEMD is a time–space analysis method, the added white noise is averaged out with sufficient number of trials; the only persistent part that survives the averaging process is the component of the signal (original data), which is then treated as the true and more physical meaningful answer. The effect of the added white noise is to provide a uniform reference frame in the time–frequency space; therefore, the added noise collates the portion of the signal of comparable scale in one IMF. With this ensemble mean, one can separate scales naturall...

A study of the characteristics of white noise using the empirical mode decomposition method

Based on numerical experiments on white noise using the empirical mode decomposition (EMD) method, we find empirically that the EMD is effectively a dyadic filter, the intrinsic mode function (IMF) components are all normally distributed, and the Fourier spectra of the IMF components are all identical and cover the same area on a semi–logarithmic period scale. Expanding from these empirical findings, we further deduce that the product of the energy density of IMF and its corresponding averaged period is a constant, and that the energy–density function is chi–squared distributed. Furthermore, we derive the energy–density spread function of the IMF components. Through these results, we establish a method of assigning statistical significance of information content for IMF components from any noisy data. Southern Oscillation Index data are used to illustrate the methodology developed here.

A confidence limit for the empirical mode decomposition and Hilbert spectral analysis

The confidence limit is a standard measure of the accuracy of the result in any statistical analysis. Most of the confidence limits are derived as follows. The data are first divided into subsections and then, under the ergodic assumption, the temporal mean is substituted for the ensemble mean. Next, the confidence limit is defined as a range of standard deviations from this mean. However, such a confidence limit is valid only for linear and stationary processes. Furthermore, in order for the ergodic assumption to be valid, the subsections have to be statistically independent. For non‐stationary and nonlinear processes, such an analysis is no longer valid. The confidence limit of the method here termed EMD/HSA (for empirical mode decomposition/Hilbert spectral analysis) is introduced by using various adjustable stopping criteria in the sifting processes of the EMD step to generate a sample set of intrinsic mode functions (IMFs). The EMD technique acts as a pre‐processor for HSA on the original data, producing a set of components (IMFs) from the original data that equal the original data when added back together. Each IMF represents a scale in the data, from smallest to largest. The ensemble mean and standard deviation of the IMF sample sets obtained with different stopping criteria are calculated, and these form a simple random sample set. The confidence limit for EMD/HSA is then defined as a range of standard deviations from the ensemble mean. Without evoking the ergodic assumption, subdivision of the data stream into short sections is unnecessary; hence, the results and the confidence limit retain the full‐frequency resolution of the full dataset. This new confidence limit can be applied to the analysis of nonlinear and non‐stationary processes by these new techniques. Data from length‐of‐day measurements and a particularly violent recent earthquake are used to demonstrate how the confidence limit is obtained and applied. By providing a confidence limit for this new approach, a stable range of stopping criteria for the decomposition or sifting phase (EMD) has been established, making the results of the final processing with HSA, and the entire EMD/HSA method, more definitive.

On the trend, detrending, and variability of nonlinear and nonstationary time series

Determining trend and implementing detrending operations are important steps in data analysis. Yet there is no precise definition of “trend” nor any logical algorithm for extracting it. As a result, various ad hoc extrinsic methods have been used to determine trend and to facilitate a detrending operation. In this article, a simple and logical definition of trend is given for any nonlinear and nonstationary time series as an intrinsically determined monotonic function within a certain temporal span (most often that of the data span), or a function in which there can be at most one extremum within that temporal span. Being intrinsic, the method to derive the trend has to be adaptive. This definition of trend also presumes the existence of a natural time scale. All these requirements suggest the Empirical Mode Decomposition (EMD) method as the logical choice of algorithm for extracting various trends from a data set. Once the trend is determined, the corresponding detrending operation can be implemented. With this definition of trend, the variability of the data on various time scales also can be derived naturally. Climate data are used to illustrate the determination of the intrinsic trend and natural variability.

ON INSTANTANEOUS FREQUENCY

Instantaneous frequency (IF) is necessary for understanding the detailed mechanisms for nonlinear and nonstationary processes. Historically, IF was computed from analytic signal (AS) through the Hilbert transform. This paper offers an overview of the difficulties involved in using AS, and two new methods to overcome the difficulties for computing IF. The first approach is to compute the quadrature (defined here as a simple 90° shift of phase angle) directly. The second approach is designated as the normalized Hilbert transform (NHT), which consists of applying the Hilbert transform to the empirically determined FM signals. Additionally, we have also introduced alternative methods to compute local frequency, the generalized zero-crossing (GZC), and the teager energy operator (TEO) methods. Through careful comparisons, we found that the NHT and direct quadrature gave the best overall performance. While the TEO method is the most localized, it is limited to data from linear processes, the GZC method is the m...

Travelling waves in the occurrence of dengue haemorrhagic fever in Thailand

Dengue fever is a mosquito-borne virus that infects 50–100 million people each year. Of these infections, 200,000–500,000 occur as the severe, life-threatening form of the disease, dengue haemorrhagic fever (DHF). Large, unanticipated epidemics of DHF often overwhelm health systems. An understanding of the spatial–temporal pattern of DHF incidence would aid the allocation of resources to combat these epidemics. Here we examine the spatial–temporal dynamics of DHF incidence in a data set describing 850,000 infections occurring in 72 provinces of Thailand during the period 1983 to 1997. We use the method of empirical mode decomposition to show the existence of a spatial–temporal travelling wave in the incidence of DHF. We observe this wave in a three-year periodic component of variance, which is thought to reflect host–pathogen population dynamics. The wave emanates from Bangkok, the largest city in Thailand, moving radially at a speed of 148 km per month. This finding provides an important starting point for detecting and characterizing the key processes that contribute to the spatial–temporal dynamics of DHF in Thailand.

THE MULTI-DIMENSIONAL ENSEMBLE EMPIRICAL MODE DECOMPOSITION METHOD

A multi-dimensional ensemble empirical mode decomposition (MEEMD) for multi-dimensional data (such as images or solid with variable density) is proposed here. The decomposition is based on the applications of ensemble empirical mode decomposition (EEMD) to slices of data in each and every dimension involved. The final reconstruction of the corresponding intrinsic mode function (IMF) is based on a comparable minimal scale combination principle. For two-dimensional spatial data or images, f(x,y), we consider the data (or image) as a collection of one-dimensional series in both x-direction and y-direction. Each of the one-dimensional slices is decomposed through EEMD with the slice of the similar scale reconstructed in resulting two-dimensional pseudo-IMF-like components. This new two-dimensional data is further decomposed, but the data is considered as a collection of one-dimensional series in y-direction along locations in x-direction. In this way, we obtain a collection of two-dimensional components. Thes...

The Hilbert-Huang Transform in Engineering

Introduction to Hilbert-Huang Transform and Some Recent Developments Norden E. Huang Carrier and Riding Wave Structure of Rogue Waves Torsten Schlurmann and Marcus Datig Applications of Hilbert-Huang Transform to Ocean-Atmosphere Remote Sensing Research Xiao-Hai Yan, Young-Heon Jo, Brian Dzwonkowski, and Lide Jiang A Comparison of the Energy Flux Computation of Shoaling Waves Using Hilbert and Wavelet Spectral Analysis Techniques Paul A. Hwang, David W. Wang, and James M. Kaihatu An Application of HHT Method to Nearshore Sea Waves Albena Dimitrova Veltcheva Transient Signal Detection Using the Empirical Mode Decomposition Michael L. Larsen, Jeffrey Ridgway, Cye H. Waldman, Michael Gabbay, Rodney R. Buntzen, and Brad Battista Coherent Structures Analysis in Turbulent Open Channel Flow Using Hilbert-Huang and Wavelets Transforms Athanasios Zeris and Panayotis Prinos An HHT-Based Approach to Quantify Nonlinear Soil Amplification and Damping Ray Ruichong Zhang Simulation of Nonstationary Random Processes Using Instantaneous Frequency and Amplitude from Hilbert-Huang Transform Ping Gu and Y. Kwei Wen Comparison of Hilbert-Huang, Wavelet, and Fourier Transforms for Selected Applications Ser-Tong Quek, Puat-Siong Tua, and Quan Wang The Analysis of Molecular Dynamics Simulations by the Hilbert Huang Transform Adrian P. Wiley, Robert J. Gledhill, Stephen C. Phillips, Martin T. Swain, Colin M. Edge, and Jonathan W. Essex Decomposition of Wave Groups with EMD Method Wei Wang Perspectives on the Theory and Practices of the Hilbert-Huang Transform Nii O. Attoh-Okine Index

A B-spline approach for empirical mode decompositions

We propose an alternative B-spline approach for empirical mode decompositions for nonlinear and nonstationary signals. Motivated by this new approach, we derive recursive formulas of the Hilbert transform of B-splines and discuss Euler splines as spline intrinsic mode functions in the decomposition. We also develop the Bedrosian identity for signals having vanishing moments. We present numerical implementations of the B-spline algorithm for an earthquake signal and compare the numerical performance of this approach with that given by the standard empirical mode decomposition. Finally, we discuss several open mathematical problems related to the empirical mode decomposition.