The empirical mode decomposition and the Hilbert spectrum for nonlinear

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黄老先生98年 提出的EMD全文,很实用The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis
Nonlinear and non-stationary time series analysis 905 the Fourier spectral analysis: the system must be linear; and the data must be strict ly periodic or stationary; otherwise, the resulting spectrum will make little physical sense The stationarity requireMent is Ilot particular to the Fourier spectral allalysis it is a general one for most of the available data analysis methods. Therefore, it behoves us to review the definitions of stationarity here. According to the traditional definition, a time series, X(t),is stationary in the wide sense, if, for all t, E(X(t)2)<∞, E(X(t)=m X(t1),X(t2))=C(X(t1+7),X(t2+7) in which E( is the expected value defined as the ensemble average of the quantity, and C( is the covariance function Stationarity in the wide sense is also known as weak stationarity, covariance stationarity or second-order stationarity(see, for example, Brockwell Davis 1991). A time series, X(t), is strictly stationary, if the joint distribution of [X(t1),X(t2),,X(t2)and[X(t1+7),X(t2+7),……,X(tn+7)(1.2) are the same for all ti and T. Thus, a strictly stationary process with finite second moments is also weakly stationary, but the inverse is not true. Both definitions are rigorous but idealized. Other less rigorous definitions for stationarity have also been used; for example, piecewise stationarity is for any random variable that is stationary within a limited time span, and asymptotically stationary is for any random variable that is stationary when T in equations(1.1)or(1. 2)approaches infinity. In practice, we can only have data for finite time spans; therefore, even to check these defini- tions. we have to make approximations. Few of the data sets, from either natural phenomena or artificial sources, can satisfy these definitions. It may be argued that the difficulty of invoking stationarity as well as ergodicity is not, on principle but phase plane; therefore, most of the cases facing us are transient in nature. This on practicality: we just cannot have enough data to cover all possible points in th the reality; we are forced to face it Other than stationarity, Fourier spectral analysis also requires linearity. Although many natural phenomena can be approximated by linear systems, they also have the tendency to be nonlinear whenever their variations become finite in anplitude Compounding these complications is the imperfection of our probes or numerical schemes; the interactions of the imperfect probes even with a perfect linear system can make the final data nonlinear. For the above reasons. the available data are usu- ally of finite duration. non-stationary and from systems that are frequently nonlinear either intrinsically or through interactions with the imperfect probes or numerica schemes. Under these conditions, Fourier spectral analysis is of limited use. For lack of alternatives, however, Fourier spectral analysis is still used to process such data The uncritical use of Fourier spectral analysis and the insouciant adoption of the stationary and linear assumptions may give misleading results; some of those are described as follows First, the Fourier spectrum defines uniform harmonic components globally; there fore. it needs many additional harmonic components to simulate non-stationary data that are non-uniform globally. as a result, it spreads the energy over a wide frequen- cy range. For example, using a delta function to represent a Hash of light will give Proc. I. Soc. Lond. A(1998) 906 N. E. Huang and others a phase-locked wide white Fourier spectrum. Here, many Fourier components are added to simulate the non-stationary nature of the data in the time domain, but their existence diverts energy to a much wider frequency domain. Constrained by the energy conservation, these spurious harmonics and the wide frequency spectrum cannot faithfully represent the true energy density in the frequency space More seri- ously, the Fourier representation also requires the existence of negative light intensity so that the components can cancel out one another to give the final delta function Thus, the Fourier components might make mathematical sense, but do not really make physical sense at all. Although no physical process can be represented exactly by a delta function, some data such as the near-field strong earthquake records are of extremely short durations lasting only a few seconds to tens of seconds at Imlost Such re almost approach a delta function, and they always give artificially wide trier spe Second, Fourier spectral analysis uses linear superposition of trigonometric func- tions: therefore it needs additiona l harmonic components to simulate the deformed wave-profiles. Such deformations, as will be shown later, are the direct consequence of nonlinear effects. Whenever the form of the data deviates from a pure sine or cosine function, the Fourier spectrum will contain harmonics. As explained above both non-stationarity and nonlinearity can induce spurious harmonic components that cause energy spreading. The consequence is the misleading energy frequency distribution for nonlinear and non-stationary data In this paper, we will present a new data analysis method based on the empirical mode decomposition(EMD) method, which will generate a collection of intrinsic mode functions(IMF). The decomposition is based on the direct extraction of the energy associated wit h various intrinsic time sca les, the most important parameters of the system. Expressed in IMFS, they have well-behaved Hilbert transforms, from which the instantaneous frequencies can be calculated. Thus, we can localize any event ol the tine as well as the frequency axis. The decomposition call also be viewed as an expansion of the data in terms of the IMFs. Then, these IMfs, based on and derived from the data, can serve as the basis of that expansion which can be linear or nonlinear as dictated by the data, and it is complete and almost orthogonal. Most important of a ll, it is adaptive. as will be shown later in more detail, locality and adaptivity are the necessary conditions for the basis for expanding nonlinear and non-stationary time series; orthogonality is not a necessary criterion for our basis lection for a nonlinear system. The principle of this basis construction is based on the physical time scales that characterize the oscillations of the phenomena. The ocal energy and the instantaneous frequency derived from the IMFs through the Hilbert transforM can give us a full energy-frequency-tine distribution of the data Such a representation is designated as the Hilbert spectrum; it would be ideal for nonlinear and non-stationary data analysis We have obtained good results and new insights by applying the combination of the emd and hilbert spectral analysis methods to various data: from the numerical results of the classical nonlinear equation systems to data representing natural phe- nomena. The classical nonlinear systems serve to illustrate the roles played by the nonlinear effects in the energy-frequency-tinle distribution. With the low degrees of freedom, they can train our eyes for more complicated cases. Some limitations of this method will also be discussed and the conclusions presented Before introducin the new method, we will first review the present available data analysis methods for non-stationary processes Proc. I. Soc. Lond. A(1998) Nonlinear and non-stationary time series analysis 907 2. Review of non-stationary data processing methods We will first give a brief survey of the methods available for processing non- stationary data. Since inost of the Inethods still depend on Fourier anlalysis, they are limited to linear systems only. Of the few methods available, the adoption of any method is almost strictly determined according to the special field in which the application is nade. The available Inethods are reviewed as follows (a) The spectrogram The spectrogram is the most basic method, which is nothing but a limited time window-width Fourier spectral analysis. By successively sliding the window along the time axis, one can get a time- frequency distribution. Since it relies on the tradition al Fourier spectral analysis, one has to assume the data to be piecewise stationary This assumption is not always justified in non-stationary data. Even if the data are piecewise stationary how can we guarantee that the window size adopted always coincides with the stationary time scales? What can we learn about the variations longer than the local stationary time scale? Will the collection of the locally station ary pieces cOnstitute soine longer period phenloinena? Furthermore, there are also practical difficulties in applying the method: in order to localize an event in time the window width must be narrow, but, on the other hand, the frequency resolu tioI requires longer tine series. These conflicting requirements render this nethod of limited usage. It is, however, extremely easy to implement with the fast Fourier transform; thus, it has attracted a wide following. Most applications of this method are for qualitative display of speech pattern analysis(see, for example, Oppenheim schafer 1989) (b) The wavelet analysis The wavelet approach is essentially an adjustable window Fourier spectral analysis with the following general definition W(a,bX,)=a-12/x( in which * is the basic wavelet function that satisfies certain very general conld tions. a is the dilation factor and b is the translation of the origin. Although time and frequency do not appear explicitly in the transformed result, the variable 1/a gives the frequency scale anld b, the temporal location of all event. An intuitive physica explanation of equation(2. 1) is very simple: W(a, b X, y) is the" energy'of X of scale a at t=b Because of this basic form of at+b involved in the transformation. it is also known as affine wavelet analysis. For specific applications, the basic wavelet function, v*() can be modified according to special needs, but the form has to be given before the analysis. In most common applications, however, the morlet wavelet is defined as Gaussian enveloped sine and cosine wave groups with 5.5 waves(see, for example, han 1995). Generally, l/ *( is not orthogonal for different a for continuous wavelets Alt hough one can make the wavelet, orthogonal by selecting a discrete set, of a, this discrete wavelet analysis will miss physical signals having scale different from the selected discrete set of a. Continuous or discrete, the wavelet analysis is basically linear analysis. A very appealing feature of the wavelet analysis is that it provides Proc. I. Soc. Lond. A(1998) 908 N. E. Huang and others uniform resolution for all the scales. Limited by the size of the basic wavelet function the downside of the uniform resolution is uniformly poor resolution Although wavelet analysis has been available only in the last ten years or so, it has become extremely popular. Indeed, it is very useful in analysing data with gradual frequency changes. Since it has an analytic form for the result, it has attracted extensive attention of the applied mathematicians. Most of its applications have been in edge detection and image colnpressiOIl. Linited applications have also been made to the time-frequency distribution in time series(see, for example, Farge 1992 Long et al. 1993) and two-dimensional images(Spedding et al. 1993 Versatile as the wavelet analysis is, the problem with the most commonly used Morlet wavelet is its leakage generated by the limited length of the basic wavelet iunction, which makes the quantitative definition of the energy-frequency-time dis- tribution difficult. Sometimes, the interpretation of the wavelet can also be counter intuitive. For example, to define a change occurring locally, one must look for the result in the high-frequency range, for the higher the frequency the more localized the basic wavelet will be. If a local event occurs only in the low-frequency range, one will still be forced to look for its effects in the high-frequency range. Such interpretation will be difficult if it is possible at all(see, for example, Huang et al. 1996).Another difficulty of the wavelet analysis is its non-adaptive nature. Once the basic wavelet is selected, one will have to use it to analyse all the data. Since the most commonl used morlet wavelet is Fourier based, it also suffers the many shortcomings of Fouri- er spectral analysis: it can only give a physically meaningful interpretation to linear phenomena; it can resolve the interwave frequency modulation provided the frequen cy variation is gradual, but it cannot resolve the intrawave frequency modulation because the basic wavelet has a length of 5.5 waves. In spite of all these problems wavelet analysis is still the best available non-stationary data analysis method so far therefore, we will use it in this paper as a reference to establish the validity and the calibration of the Hilbert spectrum (c) The Wigner-Ville di stribution The Wigner-Ville distribution is sometimes also referred to as the Heisenberg wavelet. by definition. it is the fourier transform of the central covariance function For any time series, X(t), we can define the central variance as C(T,t)=X(t-7)X“(t+是T) 2.2 Then the Wigner-Ville distribution is Cc(T, te-d This transform has been treated extensively by Claasen Mecklenbrauker(1980a b, c) and by Cohen(1995 ). It has been extremely popular with the electrical engi- neering community The difficulty with this method is the severe cross terms as indicated by the exis- tence of negative power for some frequency ranges. Although this shortcoming can be elimina ted by using the Kernel method( see, for example, Cohen 1995). the result is, then, basically that of a windowed Fourier analvsis; therefore it suffers all the lim- itations of the Fourier analysis. An extension of this method has been made by ye (1994), who used the Wigner-Ville distribution to define wave packets that reduce Proc. I. Soc. Lond. A(1998) Nonlinear and non-stationary time series analysis 909 a complicated data set to a finite number of simple components. This extension is very powerful and can be applied to a variety of problems. The applications to complicated data, however, require a great amount of judgement (d) Evolutionary spectrum The evolutionary spectrum was first proposed by Priestley(1965). The basic idea is to extend the classic Fourier spectral analysis to a. more genera lized basis: from sine or cosine to a family of orthogonal functions p(w, t) indexed by time, t, and defined for all real w, the frequency. Then, any real random variable, X(t), can be expressed as X(t) P(w, t)dA(w, t) in which dA(w, t), the Stieltjes function for the amplitude, is related to the spectrum E(dA(w, t )2)=du(w, t)=S(w, t) (25) where u(w, t) is the spectrum, and S(w, t)is the spectral density at a specific time t, also designated as the evolutionary spectrum. If for each fixed w, p(w, t)has a Fourier transform 2(u)t (26) then the function of aw, t)is the envelope of u, t), and (w) is the frequency. If further, we can treat (w) as a single valued function of w, then o(wt)=a(w, t)e iwt (27) Thus, the original data can be expanded in a family of amplitude modulated trigono metric functions The evolutionary spectral analysis is very popular in the earthquake communi- ty(see, for example, Liu 1970, 1971, 1973: Lin Cai 1995). The difficulty of its application is to find a method to define the basis, i(u, t). In principle, for this method to work, the basis has to be defined a posteriori. So far, no systematic way has been offered; therefore, constructing an evolutionary spectrum from the given data is impossible. As a result, in the earthquake community, the applications of this method have changed the problem from data analysis to data simulation: an evo- lutionary spectrum will be assumed, then the signal will be reconstituted based on the assumed spectrum. Although there is some general resemblance to the simulated earthquake signal with the real data, it is not the data that generated the spectrum Consequently, evolutionary spectrum analysis has never been very useful. As will be shiowll, the EMD call replace the evolutionary spectrum with a truly adaptive representation for the non-stationary processes (e The empirical orthogonal function expansion(EOF The empirical orthogonal function expansion(EOF) is also known as the principal component analysis, or singular value decomposition method. The essence of EOF is briefly summarized as follows: for any real z(a, t). the eof will reduce it to 1)=∑a(t)(a), Proc. I. Soc. Lond. A(1998) 910 N. E. Huang and others in which f;·k=6k 29) The orthOnlOrlllal basis, ik), is the collection of the eMpirical eigenfunctions defined (210 where C is the sum of the inner products of the variable EOF represents a radical departure from all the above methods, for the expansion basis is derived from the data; therefore, it is a posteriori, and highly efficient. The critical flaw of EOF is that it only gives a distribution of the variance in the modes defined by fk, but this distribution by itself does not suggest scales or frequency content of the signal. Although it is tempting to interpret each Inode as inldepeI- dent variations, this interpretation should be viewed with great care, for the eOF decomposition is not unique. A single component out of a non-unique decomposition even if the basis is orthogonal, does not usually contain physical meaning. Recently, Vautard &e ghil (1989) proposed the singular spectral analysis method, which is the Fourier transform of the eof. here again, we have to be sure that each eof com ponent is stationary, otherwise the Fourier spectral analysis will Inlake little sellse on the EOF components. Unfortunately, there is no guarantee that EOF compo nents from a nonlinear and non-stationary data set will all be linear and stationary Consequently, singular spectral analysis is not a real improvement. Because of it adaptive nature, however, the EOF method has been very popular, especially in the oceanography and meteorology communities(see, for example, Simpson 1991) U Other miscellaneous methods Other than the above methods. there are also some miscellaneous methods such as least square estimation of the trend, smoothing by moving averaging, and differencing to generate stationary data. Methods like these, though useful, are too specialized to be of general use. They will not be discussed any further here. Additional details can be found in many standard data processing books(see, for example, Brockwel Davis 1991) All the above methods are designed to modify the global representation of the Fourier analysis, but they all failed in one way or the other. Having reviewed the methods, we can summarize the necessary conditions for the basis to represent a nonlinear and non-stationary time series: (a)complete; (b)orthogonal; (c) local; and (d) adaptive The first condition guarantees the degree of precision of the expansion; the second condition guarantees positivity of energy and avoids leakage. They are the standard requirements for all the linear expansion methods. For nonlinear expansions, the orthogonality condition needs to be modified. The details will be discussed later. But even these basic conditions are not satisfied by some of the above mentioned meth- ds. The additional conditions are particular to the nonlinear and non-stationary data. The requirement for locality is the most crucial for non-stationarity, for in such data there is no time scale; therefore, all events have to be identified by the time of their occurences. Consequently, we require both the amplitude(or energy) and the frequency to be functions of time. The requirement for adaptivity is also crucial for both nonlinear and non-stationary data, for only by adapting to the local variations of the data can the decomposition fully account for the underlying physics Proc. I. Soc. Lond. A(1998) Nonlinear and non-stationary time series analysis 911 of the processes and not just to fulfil the mathematical requirements for fitting the data. This is especially important for the nonlinear phenomena, for a manifestation of nonlinearity is the 'harmonic distortion'in the Fourier analysis. The degree of distortion depends on the severity of nonlinearity; therefore, One cannot expect a predetermined basis to fit all the phenomena. An easy way to generate the necessary daptive basis is to derive the basis from the data In this paper, we will introduce a general method which requires two steps in analysing the dat a as follows. The first step is to preprocess the data by the empirica. mode decomposition method, with which the data are decomposed into a number of intrinsic mode function components. Thus, we will expand the data in a basis derived froin the data. The second step is to apply the hilbert transform to the decomposed IMFs and construct the energy-frequency-time distribution, designated as the hilbert spectrum, from which the time localities of events will be preserved. In other words, we need the instantaneous frequency and energy rather than the global frequency and energy defined by the Fourier spectral analysis. Therefore, before going any further, we have to clarify the definition of the instantaneous frequency 3. Instantaneous frequency The notion of the instantaneous energy or the instantaneous envelope of the signal is well accepted; the notion of the instantaneous frequency, on the other hand, has been highly controversial. Existing opinions range from editing it out of existence Shekel 1953) to accepting it but only for special "monocomponent signals(Boashash 1992; Cohen1995) There are two basic difficulties with accepting the idea of an instantaneous fre quency as follows. The first one arises from the deeply entrenched influence of the Fourier spectral analysis. In the traditional Fourier analysis, the frequency is defined for the sine or cosine function spanning the whole data length with constant ampli tude. As an extension of this definition, the instantaneous frequencies also have to relate to either a sine or a cosine function. Thus. we need at least one full oscillation of a sine or a cosine wave to define the local frequency value. according to this logic nothing shorter than a full wave will do. Such a definition would not make sense for non-stationary data for which the frequency has to change values from time to time The second difficulty arises froIn the IloIl-unlique way in defiling the instantaneous frequency. Nevertheless, this difficulty is no longer serious since the introduction of the means to make the data analytical through the Hilbert transform. Difficultic however, still exist as paradoxes discussed by Cohen(1995 ). For an arbitrary time series, X (t), we can always have its Hilbert Transform, Y(t),as Y(t) 3. here P indicates the Cauchy principal value. This transform exists for all functions of class LP(see, for example, Titchmarsh 1948 ). With this definition, X(t) and Y(t) form the complex conjugate pair, so we can have an analytic signal. Z(t),as (t)=X(t)+i}(t)=a(t) in which (t)=X2()+Y2( 0() (3 Proc. I. Soc. Lond. A(1998) 912 N. E. Huang and others Theoretically, there are infinitely many ways of defining the imaginary part, but the Hilbert transform provides a unique way of defining the imaginary part so that the result is an analytic function. a brief tutorial on the hilbert transform with the emphasis on its physical interpretation can be found in Bendat Piersol(1986 Essentially equation(3. 1) defines the Hilbert transform as the convolution of X(t with 1/t; therefore. it emphasizes the local properties of X(t). In equation(3.2),the polar coordinate expression further clarifies the local nlature of this representatio: it is the best local fit of an amplitude and phase varying trigonometric function to X(t) Even with the Hilbert transform, there is still considerable controversy in defining the instantaneous frequency as (34) This leads Cohen(1995) to introduce the term, "monocomponent function. In prin- ciple, some limitations on the data are necessary, for the instantaneous frequency given in equation (3. 4) is a single value function of time. At any given time, there is only one frequency value: therefore, it can only represent one component, hence monocomponent,. Unfortunately, no clear definition of the 'monocomponent'signaI was given to judge whether a function is or is not 'monocomponent. For lack of a precise definition, narrow band' was adopted as a limitation on the data for the instantaneous frequency to make sense( Schwartz et al. 1966) There are two definitions for bandwidth. The first one is used in the study of the probability properties of the signals and waves, where the processes are assumed to be stationary and gaussian. Then, the bandwidth can be defined in terms of spectra. moments as follows. The expected number of zero crossings per unit time is given by m (35 while the expected number of extrema per unit time is given by 1/2 丌(m2 in which mi is the ith moment of the spectrum. Therefore, the parameter, v, defined m4m10-m12 2 rEno ffers a standard bandwidth measure(see, for example, Rice 1944a, b, 1945a, b Lonlguet-Higgins 1957). For a marrow band signal v =0, the expected Iluinber of extrema and zero crossings have to equal The second definition is a more general one; it is again based on the moments of thle spectrun, but in a different way. Let us take a complex valued function in polar coordinates as (t)=a(t)e (38 with both a(t) and A(t) being functions of time. If this function has a spectrum S(w), then the mean frequency is given by u)=/s(o)2d, (39) Proc. I. Soc. Lond. A(1998)

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