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An Introduction to HHT: Instantaneous Frequency, Trend, Degree of Nonlinearity and Non-stationarity Norden E. Huang Research Center for Adaptive Data Analysis Center for Dynamical Biomarkers and Translational Medicine NCU, Zhongli, Taiwan, China Outline Rather than the implementation details, I will talk about the physics of the method. • What is frequency? • How to quantify the degree of nonlinearity? • How to define and determine trend? What is frequency? It seems to be trivial. But frequency is an important parameter for us to understand many physical phenomena. Definition of Frequency Given the period of a wave as T ; the frequency is defined as 1 T . Instantaneous Frequency Velocity distance ; mean velocity time Newton v Frequency dx dt 1 ; mean frequency period HHT defines the phase function d dt So that both v and can appear in differential equations. Other Definitions of Frequency : For any data from linear Processes 1. Fourier Analysis : T F ( ) x( t ) e j i jt dt . 0 2. Wavelet Analysis 3. Wigner Ville Analysis Definition of Power Spectral Density Since a signal with nonzero average power is not square integrable, the Fourier transforms do not exist in this case. Fortunately, the Wiener-Khinchin Theroem provides a simple alternative. The PSD is the Fourier transform of the auto-correlation function, R(τ), of the signal if the signal is treated as a widesense stationary random process: S( ) R( )e 2i d S( )d 2 ( t ) Fourier Spectrum Problem with Fourier Frequency • Limited to linear stationary cases: same spectrum for white noise and delta function. • Fourier is essentially a mean over the whole domain; therefore, information on temporal (or spatial) variations is all lost. • Phase information lost in Fourier Power spectrum: many surrogate signals having the same spectrum. Surrogate Signal: Non-uniqueness signal vs. Power Spectrum I. Hello The original data : Hello The surrogate data : Hello The Fourier Spectra : Hello The Importance of Phase To utilize the phase to define Instantaneous Frequency Prevailing Views on Instantaneous Frequency The term, Instantaneous Frequency, should be banished forever from the dictionary of the communication engineer. J. Shekel, 1953 The uncertainty principle makes the concept of an Instantaneous Frequency impossible. K. Gröchennig, 2001 The Idea and the need of Instantaneous Frequency According to the classic wave theory, the wave conservation law is based on a gradually changing φ(x,t) such that k , ; t k 0 . t Therefore, both wave number and frequency must have instantaneous values and differentiable. Hilbert Transform : Definition For any x( t ) Lp , y( t ) 1 x( ) d , t then, x( t )and y( t ) form the analytic pairs: z( t ) x( t ) i y( t ) a( t ) e i ( t ) , where a( t ) x 2 y 2 1 / 2 and ( t ) tan 1 y( t ) . x( t ) The Traditional View of the Hilbert Transform for Data Analysis Traditional View a la Hahn (1995) : Data LOD Traditional View a la Hahn (1995) : Hilbert Traditional Approach a la Hahn (1995) : Phase Angle Traditional Approach a la Hahn (1995) : Phase Angle Details Traditional Approach a la Hahn (1995) : Frequency The Real World Mathematics are well and good but nature keeps dragging us around by the nose. Albert Einstein Why the traditional approach does not work? Hilbert Transform a cos + b : Data Hilbert Transform a cos + b : Phase Diagram Hilbert Transform a cos + b : Phase Angle Details Hilbert Transform a cos + b : Frequency The Empirical Mode Decomposition Method and Hilbert Spectral Analysis Sifting (Other alternatives, e.g., Nonlinear Matching Pursuit) Empirical Mode Decomposition: Methodology : Test Data Empirical Mode Decomposition: Methodology : data and m1 Empirical Mode Decomposition: Methodology : data & h1 Empirical Mode Decomposition: Methodology : h1 & m2 Empirical Mode Decomposition: Methodology : h3 & m4 Empirical Mode Decomposition: Methodology : h4 & m5 Empirical Mode Decomposition Sifting : to get one IMF component x( t ) m 1 h1 , h1 m 2 h2 , ..... ..... hk 1 m k hk . hk c1 . The Stoppage Criteria The Cauchy type criterion: when SD is small than a preset value, where T SD h t 0 k 1 ( t ) hk ( t ) 2 T 2 h k 1 ( t ) t 0 Or, simply pre-determine the number of iterations. Empirical Mode Decomposition: Methodology : IMF c1 Definition of the Intrinsic Mode Function (IMF): a necessary condition only! Any function having the same numbers of zero cros sin gs and extrema ,and also having symmetric envelopes defined by local max ima and min ima respectively is defined as an Intrinsic Mode Function ( IMF ). All IMF enjoys good Hilbert Transform : c( t ) a( t )e i ( t ) Empirical Mode Decomposition: Methodology : data, r1 and m1 Empirical Mode Decomposition Sifting : to get all the IMF components x( t ) c1 r1 , r1 c2 r2 , . . . rn 1 cn rn . x( t ) n c j1 j rn . Definition of Instantaneous Frequency The Fourier Transform of the Instrinsic Mode Funnction, c( t ), gives W ( ) i ( t ) a( t ) e dt t By Stationary phase approximation we have d ( t ) , dt This is defined as the Ins tan tan eous Frequency . An Example of Sifting & Time-Frequency Analysis Length Of Day Data LOD : IMF Orthogonality Check • Pair-wise % • Overall % • • • • • • • • • • • 0.0003 0.0001 0.0215 0.0117 0.0022 0.0031 0.0026 0.0083 0.0042 0.0369 0.0400 • 0.0452 LOD : Data & c12 LOD : Data & Sum c11-12 LOD : Data & sum c10-12 LOD : Data & c9 - 12 LOD : Data & c8 - 12 LOD : Detailed Data and Sum c8-c12 LOD : Data & c7 - 12 LOD : Detail Data and Sum IMF c7-c12 LOD : Difference Data – sum all IMFs Traditional View a la Hahn (1995) : Hilbert Mean Annual Cycle & Envelope: 9 CEI Cases Mean Hilbert Spectrum : All CEs Properties of EMD Basis The Adaptive Basis based on and derived from the data by the empirical method satisfy nearly all the traditional requirements for basis empirically and a posteriori: Complete Convergent Orthogonal Unique The combination of Hilbert Spectral Analysis and Empirical Mode Decomposition has been designated by NASA as HHT (HHT vs. FFT) Comparison between FFT and HHT 1. FFT : x( t ) aj e i jt . j 2. HHT : x( t ) a j ( t ) e j i j ( t )d . Comparisons: Fourier, Hilbert & Wavelet Surrogate Signal: Non-uniqueness signal vs. Spectrum I. Hello The original data : Hello The IMF of original data : Hello The surrogate data : Hello The IMF of Surrogate data : Hello The Hilbert spectrum of original data : Hello The Hilbert spectrum of Surrogate data : Hello Additionally To quantify nonlinearity we also need instantaneous frequency. How to define Nonlinearity? How to quantify it through data alone (model independent)? The term, ‘Nonlinearity,’ has been loosely used, most of the time, simply as a fig leaf to cover our ignorance. Can we be more precise? How is nonlinearity defined? Based on Linear Algebra: nonlinearity is defined based on input vs. output. But in reality, such an approach is not practical: natural system are not clearly defined; inputs and out puts are hard to ascertain and quantify. Furthermore, without the governing equations, the order of nonlinearity is not known. In the autonomous systems the results could depend on initial conditions rather than the magnitude of the ‘inputs.’ The small parameter criteria could be misleading: sometimes, the smaller the parameter, the more nonlinear. Linear Systems Linear systems satisfy the properties of superposition and scaling. Given two valid inputs to a system H, x1 ( t ) and x2 ( t ) as well as their respective outputs y1 ( t ) H{ x1 ( t )} and y2 (t) = H{ x2 ( t )} then a linear system, H, must satisfy y1 ( t ) y1 ( t ) H{ x1 ( t ) x2 ( t )} for any scalar values α and β. How is nonlinearity defined? Based on Linear Algebra: nonlinearity is defined based on input vs. output. But in reality, such an approach is not practical: natural system are not clearly defined; inputs and out puts are hard to ascertain and quantify. Furthermore, without the governing equations, the order of nonlinearity is not known. In the autonomous systems the results could depend on initial conditions rather than the magnitude of the ‘inputs.’ The small parameter criteria could be misleading: sometimes, the smaller the parameter, the more nonlinear. How should nonlinearity be defined? The alternative is to define nonlinearity based on data characteristics: Intra-wave frequency modulation. Intra-wave frequency modulation is known as the harmonic distortion of the wave forms. But it could be better measured through the deviation of the instantaneous frequency from the mean frequency (based on the zero crossing period). Characteristics of Data from Nonlinear Processes d2 x 3 x x cos t 2 dt d2 x x 2 dt 1x 2 cos t Spring with position dependent cons tan t , int ra wave frequency mod ulation; therefore , we need ins tan tan eous frequency. Duffing Pendulum x d2x 2 x ( 1 x ) cos t . 2 dt Duffing Equation : Data Hilbert’s View on Nonlinear Data Intra-wave Frequency Modulation A simple mathematical model x( t ) cos t sin2t Duffing Type Wave Data: x = cos(wt+0.3 sin2wt) Duffing Type Wave Perturbation Expansion For 1 , we can have x( t ) cos t sin 2 t cos t cos sin 2 t sin t sin sin 2 t cos t sin t sin 2 t .... 1 cos t cos 3 t .... 2 2 This is very similar to the solution of Duffing equation . Duffing Type Wave Wavelet Spectrum Duffing Type Wave Hilbert Spectrum Degree of nonlinearity Let us consider a generalized intra-wave frequency modulation model as: d x( t ) cos( t sin t ) IF= 1 cos t dt IF IFz DN (Degree of Nolinearity ) should be I F z 2 1/ 2 . 2 Depending on the value of , we can have either a up-down symmetric or a asymmetric wave form. Degree of Nonlinearity • DN is determined by the combination of δη precisely with Hilbert Spectral Analysis. Either of them equals zero means linearity. • We can determine δ and η separately: – η can be determined from the instantaneous frequency modulations relative to the mean frequency. – δ can be determined from DN with known η. NB: from any IMF, the value of δη cannot be greater than 1. • The combination of δ and η gives us not only the Degree of Nonlinearity, but also some indications of the basic properties of the controlling Differential Equation, the Order of Nonlinearity. Stokes Models d2x 2 2 x x cos t with ; =0.1. 2 dt 25 Stokes I: is positive ranging from 0.1 to 0.375; beyond 0.375, there is no solution. Stokes II: is negative ranging from 0.1 to 0.391; beyond 0.391, there is no solution. Data and IFs : C1 Stokes Model c1: e=0.375; DN=0.2959 2 1.5 1 Frequency : Hz 0.5 0 -0.5 -1 -1.5 IFq IFz Data IFq-IFz -2 -2.5 -3 0 20 40 60 T ime: Second 80 100 Summary Stokes I Summary Stokes I 0.35 C1 C2 CDN Combined Degree of Stationarity 0.3 0.25 0.2 0.15 0.1 0.05 0 0.1 0.15 0.2 0.25 0.3 Epsilon 0.35 0.4 0.45 Lorenz Model • Lorenz is highly nonlinear; it is the model equation that initiated chaotic studies. • Again it has three parameters. We decided to fix two and varying only one. • There is no small perturbation parameter. • We will present the results for ρ=28, the classic chaotic case. Phase Diagram for ro=28 Lorenz Phase : ro=28, sig=10, b=8/3 30 20 z 10 0 -10 -20 -30 20 10 50 40 0 30 20 -10 y 10 -20 0 x X-Component DN1=0.5147 CDN=0.5027 Data and IF Lorenz X : ro=28, sig=10, b=8/3 50 IFq IFz Data IFq-IFz Frequency : Hz 40 30 20 10 0 -10 0 5 10 15 T ime : second 20 25 30 Comparisons Fourier Wavelet Hilbert Basis a priori a priori Adaptive Frequency Integral transform: Global Integral transform: Regional Differentiation: Local Presentation Energy-frequency Energy-timefrequency Energy-timefrequency Nonlinear no no yes, quantifying Non-stationary no yes Yes, quantifying Uncertainty yes yes no Harmonics yes yes no How to define and determine Trend ? Parametric or Non-parametric? Intrinsic vs. extrinsic approach? The State-of-the arts: Trend “One economist’s trend is another economist’s cycle” Watson : Engle, R. F. and Granger, C. W. J. 1991 Long-run Economic Relationships. Cambridge University Press. Philosophical Problem Anticipated 名不正則言不順 言不順則事不成 ——孔夫子 Definition of the Trend Proceeding Royal Society of London, 1998 Proceedings of National Academy of Science, 2007 Within the given data span, the trend is an intrinsically fitted monotonic function, or a function in which there can be at most one extremum. The trend should be an intrinsic and local property of the data; it is determined by the same mechanisms that generate the data. Being local, it has to associate with a local length scale, and be valid only within that length span, and be part of a full wave length. The method determining the trend should be intrinsic. Being intrinsic, the method for defining the trend has to be adaptive. All traditional trend determination methods are extrinsic. Algorithm for Trend • Trend should be defined neither parametrically nor non-parametrically. • It should be the residual obtained by removing cycles of all time scales from the data intrinsically. • Through EMD. Global Temperature Anomaly Annual Data from 1856 to 2003 GSTA IMF Mean of 10 Sifts : CC(1000, I) Statistical Significance Test IPCC Global Mean Temperature Trend Comparison between non-linear rate with multi-rate of IPCC Various Rates of Global Warming 0.025 0.02 Warming Rate: oC/Yr 0.015 0.01 0.005 0 -0.005 -0.01 -0.015 -0.02 1840 1860 1880 1900 1920 1940 Time : Year 1960 1980 2000 2020 Blue shadow and blue line are the warming rate of non-linear trend. Magenta shadow and magenta line are the rate of combination of non-linear trend and AMO-like components. Dashed lines are IPCC rates. Conclusion • With HHT, we can define the true instantaneous frequency and extract trend from any data. • We can quantify the degree of nonlinearity. Among the applications, the degree of nonlinearity could be used to set an objective criterion in biomedical and structural health monitoring, and to quantify the degree of nonlinearity in natural phenomena. • We can also determine the trend, which could be used in financial as well as natural sciences. • These are all possible because of adaptive data analysis method. History of HHT 1998: The Empirical Mode Decomposition Method and the Hilbert Spectrum for Non-stationary Time Series Analysis, Proc. Roy. Soc. London, A454, 903-995. The invention of the basic method of EMD, and Hilbert transform for determining the Instantaneous Frequency and energy. 1999: A New View of Nonlinear Water Waves – The Hilbert Spectrum, Ann. Rev. Fluid Mech. 31, 417-457. Introduction of the intermittence in decomposition. 2003: A confidence Limit for the Empirical mode decomposition and the Hilbert spectral analysis, Proc. of Roy. Soc. London, A459, 2317-2345. Establishment of a confidence limit without the ergodic assumption. 2004: A Study of the Characteristics of White Noise Using the Empirical Mode Decomposition Method, Proc. Roy. Soc. London, A460, 1179-1611. Defined statistical significance and predictability. Recent Developments in HHT 2007: On the trend, detrending, and variability of nonlinear and nonstationary time series. Proc. Natl. Acad. Sci., 104, 14,88914,894. 2009: On Ensemble Empirical Mode Decomposition. Adv. Adaptive data Anal., 1, 1-41 2009: On instantaneous Frequency. Adv. Adaptive Data Anal., 2, 177229. 2010: The Time-Dependent Intrinsic Correlation Based on the Empirical Mode Decomposition. Adv. Adaptive Data Anal., 2. 233-266. 2010: Multi-Dimensional Ensemble Empirical Mode Decomposition. Adv. Adaptive Data Anal., 3, 339-372. 2011: Degree of Nonlinearity. (Patent and Paper) 2012: Data-Driven Time-Frequency Analysis (Applied and Computational Harmonic Analysis, T. Hou and Z. Shi) Current Efforts and Applications • Non-destructive Evaluation for Structural Health Monitoring – (DOT, NSWC, DFRC/NASA, KSC/NASA Shuttle, THSR) • Vibration, speech, and acoustic signal analyses – (FBI, and DARPA) • Earthquake Engineering – (DOT) • Bio-medical applications – (Harvard, Johns Hopkins, UCSD, NIH, NTU, VHT, AS) • Climate changes – (NASA Goddard, NOAA, CCSP) • Cosmological Gravity Wave – (NASA Goddard) • Financial market data analysis – (NCU) • Theoretical foundations – (Princeton University and Caltech) Outstanding Mathematical Problems 1. Mathematic rigor on everything we do. (tighten the definitions of IMF,….) 2. Adaptive data analysis (no a priori basis methodology in general) 3. Prediction problem for nonstationary processes (end effects) 4. Optimization problem (the best stoppage criterion and the unique decomposition ….) 5. Convergence problem (finite step of sifting, best spline implement, …) Do mathematical results make physical sense? Do mathematical results make physical sense? Good math, yes; Bad math ,no. The Job of a Scientist The job of a scientist is to listen carefully to nature, not to tell nature how to behave. Richard Feynman To listen is to use adaptive method and let the data sing, and not to force the data to fit preconceived modes. All these results depends on adaptive approach. Thanks