Hilbert-Huang transform

Hilbert-Huang transform

The Hilbert-Huang Transform (HHT) is a way to decompose a signal into so-called intrinsic mode functions (IMF), and obtain instantaneous frequency data. It is designed to work well for data that are nonstationary and nonlinear. In contrast to other common transforms like the Fourier Transform, the HHT is more like an algorithm (an empirical approach) that can be applied to a data set, rather than a theoretical tool.Almost all the case studies reveal that the HHT gives results much sharper than any of the traditional analysis methods in time-frequency-energy representation. Additionally, it reveals true physical meanings in many of the data examined.

Introduction

The Hilbert-Huang transform (HHT), a NASA’s designated name, is proposed by Huang et al.(1996, 1998, 1999, 2003).It is the result of the empirical mode decomposition (EMD) and the Hilbert spectral analysis (HSA). The HHT uses the EMD method to decompose a signal into so-called intrinsic mode function, and uses the HSA method to obtain instantaneous frequency data. The HHT provides a new method of analyzing nonstationary and nonlinear time series data.

Introduction to EMD and IMF

The fundamental part of the HHT is the empirical mode decomposition (EMD) method. Using the EMD method, any complicated data set can be decomposed into a finite and often small number of components, which is a collection of intrinsic mode functions (IMF). An IMF represents a generally simple oscillatory mode as a counterpart to the simple harmonic function. By definition, an IMF is any function with the same number of extrema and zero crossings, with its envelopes being symmetric with respect to zero. The definition of an IMF guarantees a well-behaved Hilbert transform of the IMF. This decomposition method operating in the time domain is adaptive and highly efficient. Since the decomposition is based on the local characteristic time scale of the data, it can be applied to nonlinear and nonstationary processes.

Introduction to HSA

The Hilbert spectral analysis (HSA) provides a method for examining the IMF's instantaneous frequency data 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.

Techniques

The empirical mode decomposition (EMD)

The EMD method is a necessary step to reduce any given data into a collection of intrinsic mode functions (IMF) to which the Hilbert spectral analysis can be applied. An IMF is defined as a function that satisfies the following requirements:
*1. In the whole data set, the number of extrema and the number of zero-crossings must either be equal or differ at most by one.

*2. At any point, the mean value of the envelope defined by the local maxima and the envelope defined by the local minima is zero.

Therefore, an IMF represents a simple oscillatory mode as a counterpart to the simple harmonic function, but it is much more general: instead of constant amplitude and frequency in a simple harmonic component, an IMF can have variable amplitude and frequency along the time axis.
The procedure of extracting an IMF is called sifting. The sifting process is as follows:

*1. Identify all the local extrema in the test data.

*2. Connect all the local maxima by a cubic spline line as the upper envelope.

*3. Repeat the procedure for the local minima to produce the lower envelope.
The upper and lower envelopes should cover all the data between them. Their mean is m1. The difference between the data and m1 is the first component h1:
:$X\left(t\right)-m_1=h_1.,$

Ideally, h1 should satisfy the definition of an IMF, for the construction of h1 described above should have made it symmetric and having all maxima positive and all minima negative. After the first round of sifting, the crest may become a local maximum. New extrema generated in this way actually reveal the proper modes lost in the initial examination. In the subsequent sifting process, h1 can only be treated as a proto-IMF. In the next step, it is treated as the data, then
:$h_\left\{1\right\}-m_\left\{11\right\}=h_\left\{11\right\}.,$

After repeated sifting up to k times, h1 becomes an IMF, that is
:$h_\left\{1\left(k-1\right)\right\}-m_\left\{1k\right\}=h_\left\{1k\right\}.,$Then, it is designated as the first IMF component from the data: :$c_1=h_\left\{1k\right\}.,$

The stoppage criteria of the sifting process

The stoppage criterion determines the number of sifting steps to produce an IMF. Two different stoppage criteria have been used traditionally:
*1. The first criterion is proposed by Huang et al. (1998). It similar to the Cauchy convergence test, and we define a sum of the difference, SD, as:$SD_k=frac\left\{sum_\left\{t=0\right\}^\left\{T\right\}|h_\left\{k-1\right\}\left(t\right)-h_k\left(t\right)|^2\right\}\left\{sum_\left\{t=0\right\}^\left\{T\right\} h_\left\{k-1\right\}^2 \left(t\right)\right\}.,$:Then the sifting process is stop when SD is smaller than a pre-given value.
*2. A second criterion is based on the number called the S-number, which is defined as the number of consecutive siftings when the numbers of zero-crossings and extrema are equal or at most differing by one. Specifically, an S-number is pre-selected. The sifting process will stop only if for S consecutive times the numbers of zero-crossings and extrema stay the same, and are equal or at most differ by one.

Once a stoppage criterion is selected, the first IMF, c1, can be obtained. Overall, c1 should contain the finest scale or the shortest period component of the signal. We can, then, separate c1 from the rest of the data by $X\left(t\right)-c_1=r_1.,$ Since the residue, r1, still contains longer period variations in the data, it is treated as the new data and subjected to the same sifting process as described above.

This procedure can be repeated to all the subsequent rj's, and the result is :$r_\left\{n-1\right\}-c_n=r_n.,$The sifting process stops finally when the residue, rn, becomes a monotonic function from which no more IMF can be extracted. From the above equations, we can induce that:$X\left(t\right)=sum_\left\{j=1\right\}^n c_j+r_n.,$

Thus, a decomposition of the data into n-empirical modes is achieved. The components of the EMD are usually physically meaningful, for the characteristic scales are defined by the physical data. Flandrin et al. (2003) and Wu and Huang (2004) have shown that the EMD is equivalent to a dyadic filter bank.

Hilbert spectral analysis

Having obtained the intrinsic mode function components, the instantaneous frequency can be computed using the Hilbert Transform. After performing the Hilbert transform on each IMF component, the original data can be expressed as the real part, Real, in the following form::$X\left(t\right)= ext\left\{Real\right\}\left\{sum_\left\{j=1\right\}^n a_j\left(t\right)e^\left\{iintomega_j\left(t\right)\right\}dt\right\}.,$

Current applications

*Biomedical applications: Huang et al. [1999b] analyzed the pulmonary arterial pressure on conscious and unrestrained rats.

*Chemistry and chemical engineering: Phillips et al. [2003] investigated a conformational change in Brownian dynamics(BD) and molecular dynamics(MD) simulations using a comparative analysis of HHT and wavelet methods. Wiley et al. [2004] used HHT to investigate the effect of reversible digitally filtered molecular dynamics(RDFMD) which can enhance or suppress specific frequencies of motion. Montesinos et al. [2002] applied HHT to signals obtained from BWR neuron stability.

*Financial applications: Huang et al. [2003b] applied HHT to nonstationary financial time series and used a weekly mortgage rate data.

*Meteorological and atmospheric applications: Salisbury and Wimbush [2002] , using Southern Oscillation Index(SOI) data, applied the HHT technique to determine whether the SOI data are sufficiently noise free that useful predictions can be made and whether future El Nino southern oscillation(ENSO) events can be predicted from SOI data. Pan et al. [2002] used HHT to analyze satellite scatterometer wind data over the northwestern Pacific and compared the results to vector empirical orthogonal function(VEOF) results.

*Ocean engineering:Schlurmann [2002] introduced the application of HHT to characterize nonlinear water waves from two different perspectives, using laboratory experiments. Veltcheva [2002] applied HHT to wave data from nearshore sea. Larsen et al. [2004] used HHT to characterize the underwater electromagnetic environment and identify transient manmade electromagnetic disturbances.

*Seismic studies: Huang et al. [2001] used HHT to develop a spectral representation of earthquake data. Chen et al. [2002a] used HHT to determined the dispersion curves of seismic surface waves and compared their results to Fourier-based time-frequency analysis. Shen et al. [2003] applied HHT to ground motion and compared the HHT result with the Fourier spectrum.

*Structural applications: Quek et al. [2003] illustrate the feasibility of the HHT as a signal processing tool for locating an anomaly in the form of a crack, delamination, or stiffness loss in beams and plates based on physically acquired propagating wave signals. Using HHT, Li et al. [2003] analyzed the results of a pseudodynamic test of two rectangular reinforced concrete bridge columns.

*Health monitoring: Pines and Salvino [2002] applied HHT in structural health monitoring. Yang et al. [2004] used HHT for damage detection, applying EMD to extract damage spikes due to sudden changes in structural stiffness. Yu et al. [2003] used HHT for fault diagnosis of roller bearings.

*System identification: Chen and Xu [2002] explored the possibility of using HHT to identify the modal damping ratios of a structure with closely spaced modal frequencies and compared their results to FFT. Xu et al. [2003] compared the modal frequencies and damping ratios in various time increments and different winds for one of the tallest composite buildings in the world.

Limitations

Chen and Feng [undated] proposed a technique to improve the HHT procedure. The authors noted that the EMD is limited in distinguishing different components in narrow-band signals. The narrow band may contain either (a) components that have adjacent frequencies or (b) components that are not adjacent in frequency but for which one of the components has a much higher energy intensity than the other components. The improved technique is based on beating-phenomenon waves.

Datig and Schlurmann [2004] did the most comprehensive studies on the performance and limitations of HHT with particular applications to irregular waves. The authors did extensive investigation into the spline interpolation. The authors discussed using additional points, both forward and backward, to determine better envelopes. They also performed a parametric study on the proposed improvement and showed significant improvement in the overall EMD computations. The authors noted that HHT is capable of differentiating between time-variant components from any given data. Their study also showed that HHT was able to distinguish between riding and carrier waves.

*Hilbert transform
*Hilbert spectral analysis
*Hilbert spectrum
*Instantaneous frequency
*Nonlinear
*Wavelet transform
*Fourier transform

References

*Huang, et al. "The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis." "Proc. R. Soc. Lond. A" (1998) 454, 903–995 ( [http://keck.ucsf.edu/~schenk/Huang_etal98.pdf Link] )

*Norden Huang, Nii O. Attoh-Okine, "The Hilbert-Huang transform in engineering"," Taylor & Francis, 2005.

*Norden E. Huang, Samuel S.P. Shen, "Hilbert-Huang transform and its applications"," London : World Scientific, c2005.

*Flandrin, P., Rilling, G. and Gonçalves, P., 2003: Empirical mode decomposition as a filterbank. IEEE Signal Proc Lett. 11 : 112-114.

*Huang, N. E., Long, S. R.and Shen, Z. 1996: The mechanism for frequency downshift in nonlinear wave eolution. Adv. Appl. Mech., 32, 59-111.

*Huang, et al. 1998: The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. Roy. Soc. Lond., 454, 903-993.

*Huang, N. E., Z. Shen, R. S. Long, 1999: A New View of Nonlinear Water Waves -- The Hilbert Spectrum, Ann. Rev. Fluid Mech. 31, 417-457.

*Huang, N. E., M. L. Wu, S. R. Long, S. S. Shen, W. D. Qu, p. Gloersen, and K. L. Fan, 2003: A confidence limit for the Empirical Mode Decomposition and Hilbert Spectral Analysis. Proceedings Royal Society of London, A459, 2,317-2,345.

*Wu, Z. and N. E. Huang, 2004: A study of the characteristics of white noise using the empirical mode decomposition method, Proceedings Royal Society of London, A,460,1597-1611.

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