Quantification of Nonlinearity and Nonstationarity

Norden Huang
National Central University

After fifteen years of research in time series analysis, it has dawned on me finally that quantifications of nonlinearity and nonstationarity depend critically on a proper definition of frequency. Ever since Fourier Transform is widely applied in data analysis, practitioners tend to think all possible signals in terms of a sum of independent simple harmonic waves. As a result, data are invariably transformed into frequency domain and the notion of frequency has become an indispensible quantity to specify the characteristics of complicate data. Although the Fourier analysis has been established rigorously in mathematics, the question begs answer is this: are the frequency so defined physically meaningful? Unfortunately, the answer to this question has to be negative, for Fourier analysis is based on stationary and linear assumptions: the results from Fourier analysis are always independent of temporal or spatial variations and sometimes also violate causality. Ameaningful specification of frequency for nonlinear and nonstationary processes has to reflect the property of locality, i.e., frequency has to have 'instantaneous' values. This is impossible for Fourier analysis or any analysis relies on integral transform, which would result in the limitation imposed by the uncertainty principle from the integration operation in the transform. Indeed, in the theoretical study of wave motion, the most important governing equations are certain conservation laws. The simplest and the most fundamental one is the kinematic conservation: In general, for any wave motion, there must be a smooth phase function, ?, so that we can define wave number, k, and frequency as k = , and = - xt??????? . (1) Therefore, by cross differentiation, we have k + = 0 tx.????? (2) No wave motion can violate it. For it to hold, the necessary condition is that both the wave number and the frequency have to have instantaneous and local values and 2 also be differentiable. The constant wave number and frequency defined through Fourier analysis certainly satisfy the kinematic conservation, but that would be a trivial condition. The only frequency that could satisfy the requirement for the kinematic conservation would be the true instantaneous frequency (Huang et al., 1998, 2009, 2012). Only with instantaneous frequency can we use Time-frequency analysis to describe the richness of variation in frequency of the nonlinear and nonstationary data, where the intra-wave frequency modulation is the rule rather than the exception. Obviously, the instantaneous frequency depends on the existence of a phase function. In order to find the proper phase function, we have to give up the comfort of a priori basis (such as Fourier or wavelet analyses) and develop a totally new adaptive data analysis approach, with which the time series can be decomposed adaptively and based on a posteriori basis. The crucial idea is to have a sum of 'mono-components' to represent the original data. Only through mono-component function would one obtain the unique phase function from which the instantaneous frequency could be defined. Recent research reveals that the instantaneous frequency values contain rich information of the underlying properties of the processes that produce the final data. Once the frequency is properly extracted from the data, quantification of degree of nonlinearity and nonstationarity could all be achieved easily. Only precisely defined instantaneous frequency can describe the intra-wave frequency modulation in nonlinear oscillatory phenomena. Let us take the simplest nonlinear system given by: () ,2n122n2dxxxf(t)couldbewrittenasdtdxxxf(t)dtaeae+++=++= (3) in which a and e are constants. For such nonlinear oscillatory system, the motion is equivalent to a spring with variable spring constant. In fact, the term in the parenthesis is equivalent to the frequency squared. The frequency of the system is ever changing even within a single swing of the pendulum, which is defined by Huang et al. (1998, 2012) as intra-wave frequency modulation. For example, when a = 1 and n = 1, Equation (3) becomes a quadratic nonlinear oscillator. This wave would have up-down asymmetry with the sharp crests but rounded troughs wave form are the consequence of odd value of n and positive e. The intra-wave frequency modulation pattern would be once each wave, thus the nonlinearity order to be 3 quadratic. On the other hand, when a = 1 and n = 2, Equation (3) becomes a cubic nonlinear oscillator. This wave would have up-down symmetry with the sharp crests and troughs wave form as the consequence of even value of n and positive e. The intra-wave frequency modulation pattern would be twice each wave, thus the nonlinearity order to be cubic. With Fourier analysis, we would be forced to resort to harmonics, a mathematical artifact. Let us consider a nonlinear dispersive wave train, such as deepwater surface waves. We need harmonics to fit the nonlinearly deformed wave forms. To maintain the integrity of the wave form, the harmonics would cease to be dispersive: No matter how high the order of the harmonics is, the phase velocity of that component will have to commensurate with whatever the fundamental frequency it is associated with. This would force the wave with a given frequency, but for being harmonics of different fundaments, to propagate at different phase velocity. Thus, none of the harmonics can be physical waves, but are mathematical artifacts rather than true physical wave components. Therefore, harmonics should not be meaningful physical quantities. Only intra-wave frequency modulation to describe the frequency change within one cycle could faithful represent the crucial physics for deep understanding. A rigorous and logic definition of frequency will be given, and definitions of degree of nonlinearity and nonstationary will be introduced. Examples of various nonstationary and nonlinear process would be used to illustrate the new view of data that would conform with physical perception rather than abstract mathematical views. References: Huang, N. E., Shen, Z., Long, S. R., Wu, M. C., Shih, S. H., Zheng, Q., Tung, C. C. and Liu, H. H. 1998: The empirical mode decomposition method and the Hilbert spectrum for non-stationary time series analysis, Proc. Roy. Soc. London, A454, 903-995. Huang, N. E., Wu, Z., Long, S. R., Arnold, K. C., Blank, K., Liu, T. W. 2009: On instantaneous frequency, Adv. Adap. Data Analy. 1,. 177-229. Huang, N. E., Z. Wu, M. T. Lo, X. Y. Chen and C. K. Peng: 2012: Method For Quantifying And Modeling Degree Of Nonlinearity, Combined Nonlinearity, and Nonstationarity (On the Degree of Nonlinearity) US Patent Pending 13/241,565. Wu, Z., N. E. Huang, S. R. Long, and C.-K. Peng, 2007: On the trend, detrending, and variability of nonlinear and nonstationary time series. Proc. Natl. Acad. Sci., 104, 14,889-14,894.

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