904N.E.Huang and others
10.Discussion987
11.Conclusions991
References993 A new method for analysing has been devel-oped.The key part of the method
any complicated data t can be decompod into
of‘intrinsic mode functions’Hilbert trans-This decomposition method is adaptive,and,highly efficient.Since
applicable to nonlinear and non-stationary process.With the Hilbert transform,
Examples
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the classical nonlinear equation systems and data
are given to demonstrate the power new method.
章鱼哥图片data are especially interesting,for rve to illustrate the roles the
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nonlinear and non-stationary effects in the energy–frequency–time distribution.
Keywords:non-stationary time ries;nonlinear differential equations;
frequency–time spectrum;Hilbert spectral analysis;intrinsic time scale;
empirical mode decomposition
1.Introduction
nd by us;
data analysis rves two purpos:determine the parameters needed to construct the necessary model,and to confirm the model we constructed to reprent the phe-nomenon.Unfortunately,the data,whether from physical measurements or numerical modelling,most likely will have one or more of the following problems:(a)the total data span is too short;(b)the data are non-stationary;and(c)the data reprent nonlinear process.Although each of the above problems can be real by itlf,the first two are related,for a data ction shorter than the longest time scale of a sta-tionary process can appear to be non-stationary.Facing such data,we have limited options to u in the analysis.
Historically,Fourier spectral analysis has provided a general method for examin-
the data analysis has been applied to all kinds of data.Although the Fourier transform is valid under extremely general conditions(e,for example,Titchmarsh1948),there are some crucial restrictions of Proc.R.Soc.Lond.A(1998)
Nonlinear and non-stationary time ries analysis
905
the Fourier spectral analysis:the system must be linear;and the data must be strict-ly periodic or stationary;otherwi,the resulting spectrum will make little physical
< the Fourier spectral analysis methods.Therefore,behoves us review the definitions of stationarity here.According to the traditional definition,a time ries,X (t ),is stationary in the wide n,if,for all t ,E (|X (t )2|)<∞,E (X (t
))
=m,C (X (t 1),X (t 2))=C (X (
t 1+τ),X (t 2+τ))=C (t 1−t 2),
(1.1)
in which
E (·)is the expected value defined as the enmble average of the quantity,
and C (·)is the covariance function.Stationarity in the wide n is also known as weak stationarity,covariance stationarity or cond-order stationarity (e,for
example,Brockwell &Davis 1991).A time ries,X (t ),is strictly stationary,if the joint distribution of [X (t 1),X (t 2),...,X (t n )]and [X (t 1+τ),X (t 2+τ),...,X (t n +τ)](1.2)are the same for all t i and τ.Thus,a strictly stationary
process with finite cond moments is also
weakly stationary,but the inver is not true.Both definitions are
梦鬼rigorous but idealized.Other less rigorous definitions have also been
ud;for example,that is stationary within a limited time
span,
asymptotically stationary is for any random variable
is stationary when τin equations (1.1)or (1.2)approaches infinity.In practice,
we can only have data for finite time spans;the defini-
tions,we have
to make
approximations.Few of the data ts,from either natural phenomena or artificial sources,can satisfy the definitions.It may be argued that
the difficulty of invoking stationarity as well as ergodicity is not on principle
but on practicality:
we just cannot have enough data to cover all possible points in the
pha plane;therefore,most of the cas facing us are transient in nature.This is the reality;we are forced to face it.Fourier spectral analysis also requires linearity.can be approximated by linear systems,the tendency to
be nonlinear whenever their variations become finite Compounding the complications is the imperfection of or numerical schemes;the
interactions
of the imperfect probes even with a perfect linear system
can make the final data nonlinear.For the above the available data are ally of finite duration,non-stationary and from systems that are frequently nonlinear,either intrinsically
or through interactions with the imperfect probes or numerical schemes.Under the conditions,Fourier spectral analysis is of limited u.For lack of alternatives,
however,Fourier spectral analysis is still ud to process such data.The uncritical u of Fourier spectral analysis the insouciant adoption of the stationary and linear assumptions may give cy range.a delta function will give
Proc.R.Soc.Lond.A (1998)
906N.E.Huang and others
a pha-locked wide white Fourier spectrum.Here,
added to the data in the time domain,
Constrained by
the spurious harmonics the wide frequency spectrum cannot faithfully reprent the true energy density in the frequency space.More ri-ously,the Fourier reprentation also requires the existence of negative light intensity so that the components can cancel out one another to give thefinal delta function. Thus,the Fourier components might make mathematical n,but do not really make physical n at all.Although no physical process can be reprented exactly by a delta function,some data such as the near-field strong earthquake records are
Fourier spectra.
Second,
tions;
破折号的作用是什么wave-profiles.Such deformations,later,are the direct conquence 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 cau energy spreading.The conquence is the misleading energy–frequency distribution for
In this paper,
mode
mode functions The decomposition is bad on the direct extraction of the
event on the time the frequency The decomposition be viewed as an expansion of the data in terms of the IMFs.Then,bad on and derived from the data,can rve as the basis of that expansion linear or nonlinear as dictated by the data,Most important of all,it is adaptive.As will locality and adaptivity are the necessary conditions for the basis for expanding nonlinear and non-stationary time orthogonality is not a necessary criterion
lection for a nonlinear
on the physical time scales
local energy and the instantaneous frequency
Hilbert transform can give us a full energy–frequency–time distribution of the data. Such a reprentation 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 reprenting natural phe-nomena.The classical nonlinear systems rve to illustrate the roles played by the nonlinear effects in the energy–frequency–time distribution.With the low degrees of freedom,they can train our eyes for more complicated cas.Some limitations of this method will also be discusd and the conclusions prented.Before introducing the new method,we willfirst review the prent available data analysis methods for non-stationary process.
Proc.R.Soc.Lond.A(1998)
Nonlinear and non-stationary time ries analysis
907
2.Review of non-stationary data processing methods
We will
first
give
a brief survey of the
methods
stationary data.are limited to linear systems any method is almost strictly determined according to the special field in which the application is made.The available methods are reviewed as follows.(a )The spectrogram
nothing but a limited time window-width Fourier spectral analysis.the a distribution.Since it relies on the tradition-
al Fourier spectral analysis,one has to assume the data to be piecewi stationary.
This assumption is not always justified in non-stationary data.Even if the data are piecewi 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 some longer period phenomena?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-tion requires longer time ries.The conflicting requirements render this method 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 (e,for example,Oppenheim &Schafer 1989).(b )The wavelet analysis
The wavelet approach is esntially an adjustable window Fourier spectral analysis
with the following general definition:W (a,b ;X,ψ)=|a |−1/2
∞−∞X (t )ψ∗ t −b a
d t,(2.1)
in which
ψ∗
(·)is the basic wavelet function that satisfies certain very general condi-tions,a is the dilation factor and b is the translation
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origin.
Although time and
frequency do not appear explicitly in the transformed result,the variable 1/a gives
苏泊尔电磁炉e0the frequency scale and b ,the temporal location of an event.An intuitive physical explanation of equation (2.1)is very simple:W (a,b ;X,ψ)is the ‘energy’of X of
scale a at t =b .Becau of this basic form of at +如何设置屏保
b involved
in the
transformation,
it is also known
as affine
wavelet analysis.For specific applications,the basic wavelet function,ψ∗(·),
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 5.5waves (e,for example,Chan 1995).Generally,ψ∗(·)
is not orthogonal
for
different a for continuous wavelets.
Although one can make the wavelet orthogonal by lecting a discrete t of a ,this
discrete wavelet analysis will miss physical signals having scale different from the
lected discrete t of a .Continuous or discrete,the wavelet analysis is basically a linear analysis.A very appealing feature of the wavelet analysis is that it provides a
Proc.R.Soc.Lond.A (1998)
