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What is Frequency? Norden E. Huang Research Center for Adaptive Data Analysis Center for Dynamical Biomarkers and translational Medicine National Central University Zhongli, Taiwan, ROC The 15 th Wu Chien Shiung Science Camp, 2012

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Definition of Frequency Given the period of a wave as T ; the frequency is defined as

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T This definition is easy for regular sine wave, but not very practical for complicated oscillations.

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Impossible for complicated waves Need Decomposition

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Jean-Baptiste-Joseph Fourier 1807 “On the Propagation of Heat in Solid Bodies” 1812 Grand Prize of Paris Institute 1817 Elected to Académie des Sciences 1822 Appointed as Secretary of Math Section paper published Fourier’s work is a great mathematical poem. Lord Kelvin Ever since Fourier’s ground breaking work, people always think of any changes in terms of waves. Fourier’s work is a great mathematical poem. Lord Kelvin

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Definitions of Fourier Type Frequency : For any data from linear Processes

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Fourier Analysis Fourier proved that the Fourier expansion in terms of trigonometric function is complete, convergent, orthogonal and unique. Therefore, every signal could be think as combination of sinusoidal waves each with constant amplitude and frequency. Are those frequency meaningful?

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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 wide- sense stationary random process:

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Fourier Spectrum

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Surrogate Signal I. Hello

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The original data : Hello

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The surrogate data : Hello

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The Fourier Spectra : Hello

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Surrogate Signal II. delta function and white noise Non-causality: Event involves both past and future

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Random and Delta Functions

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Fourier Components : Random Function

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Fourier Components : Delta Function

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The Importance of Phase

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Can we just use phase to explore physical processes? Yes, to some extent.

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Then: In search of frequency, I found the methods to quantify nonlinearity and determine the trend. After more than 15 years of searching, I found the key to nonlinear and nonstationary data analysis is the proper definition for frequency.

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Instantaneous Frequencies and Trends for Nonstationary Nonlinear Data Hot Topic Conference University of Minnesota, Institute for Mathematics and Its Applications, 2011

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How to define frequency? It seems to be trivial. But frequency is an important parameter for us to understand many physical phenomena.

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Traditional Definition of Frequency frequency = 1/period. Definition too crude Only work for simple sinusoidal waves Does not apply to nonstationary processes Does not work for nonlinear processes Does not satisfy the need for wave equations

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Instantaneous Frequency

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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

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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 Therefore, both wave number and frequency must have instantaneous values and differentiable. This is the true definition of frequency.

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Hilbert Transform : Definition

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The Traditional use of the Hilbert Transform for Data Analysis failed miserably and gave IF a bad break.

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Traditional View a la Hahn (1995) : Data LOD

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Traditional View a la Hahn (1995) : Hilbert

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Traditional Approach a la Hahn (1995) : Phase Angle

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Traditional Approach a la Hahn (1995) : Phase Angle Details

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Traditional Approach a la Hahn (1995) : Frequency

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Why the traditional approach does not work?

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Hilbert Transform a cos + b : Data

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Hilbert Transform a cos + b : Phase Diagram

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Hilbert Transform a cos + b : Phase Angle Details

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Hilbert Transform a cos + b : Frequency

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The Empirical Mode Decomposition Method and Hilbert Spectral Analysis Sifting

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Empirical Mode Decomposition: Methodology : Test Data

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Empirical Mode Decomposition: Methodology : data and m1

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Empirical Mode Decomposition: Methodology : data & h1

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Empirical Mode Decomposition: Methodology : h1 & m2

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Empirical Mode Decomposition: Methodology : h3 & m4

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Empirical Mode Decomposition: Methodology : h4 & m5

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Empirical Mode Decomposition Sifting : to get one IMF component

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The Stoppage Criteria The Cauchy type criterion: when SD is small than a pre- set value, where Or, simply pre-determine the number of iterations.

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Empirical Mode Decomposition: Methodology : IMF c1

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Definition of the Intrinsic Mode Function (IMF): a necessary condition only!

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Empirical Mode Decomposition: Methodology : data, r1 and m1

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Empirical Mode Decomposition Sifting : to get all the IMF components

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An Example of Sifting & Time-Frequency Analysis

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Length Of Day Data

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LOD : IMF

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Orthogonality Check Pair-wise % 0.0003 0.0001 0.0215 0.0117 0.0022 0.0031 0.0026 0.0083 0.0042 0.0369 0.0400 Overall % 0.0452

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LOD : Data & c12

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LOD : Data & Sum c11-12

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LOD : Data & sum c10-12

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LOD : Data & c9 - 12

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LOD : Data & c8 - 12

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LOD : Detailed Data and Sum c8-c12

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LOD : Data & c7 - 12

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LOD : Detail Data and Sum IMF c7-c12

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LOD : Difference Data – sum all IMFs

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Traditional View a la Hahn (1995) : Hilbert

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Mean Annual Cycle & Envelope: 9 CEI Cases

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Uniqueness It should be pointed out that in general is not unique. But the EMD determines the envelope based on the data to make the expansion unique. Phase Function computation no longer based on Hilbert Transform, but through quadrature to account for the full nonlinear wave form distortion and other complications.

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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

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The combination of Hilbert Spectral Analysis and Empirical Mode Decomposition has been designated by NASA as HHT (HHT vs. FFT)

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Comparison between FFT and HHT

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Comparisons: Fourier, Hilbert & Wavelet

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Speech Analysis Hello : Data

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Four comparsions D

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The original data : Hello

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To quantify nonlinearity we also need instantaneous frequency. Additionally

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How to define Nonlinearity? How to quantify it through data alone?

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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?

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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.

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Linear Systems Linear systems satisfy the properties of superposition and scaling. Given two valid inputs to a system H, as well as their respective outputs then a linear system, H, must satisfy for any scalar values α and β.

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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.

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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).

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Characteristics of Data from Nonlinear Processes

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Duffing Pendulum x

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Duffing Equation : Data

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Hilbert ’ s View on Nonlinear Data Intra-wave Frequency Modulation

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A simple mathematical model

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Duffing Type Wave Data: x = cos(wt+0.3 sin2wt)

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Duffing Type Wave Perturbation Expansion

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Duffing Type Wave Wavelet Spectrum

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Duffing Type Wave Hilbert Spectrum

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Duffing Type Wave Marginal Spectra

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Quantify nonlinearity

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Degree of nonlinearity

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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.

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Stokes Models

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Data and IFs : C1

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Duffing Models

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Data and Ifs Details

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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.

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Phase Diagram for ro=28

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X-Component DN1=0.5147 CDN=0.5027

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Data and IF

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Lorenz Model

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Comparisons FourierWaveletHilbert Basisa priori Adaptive FrequencyIntegral transform: Global Integral transform: Regional Differentiation: Local PresentationEnergy-frequencyEnergy-time- frequency Nonlinearno yes, quantifying Non-stationarynoyesYes, quantifying Uncertaintyyes no Harmonicsyes no

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How to define Trend ? Parametric or Non-parametric? Intrinsic vs. extrinsic approach?

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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.

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Philosophical Problem Anticipated 名不正則言不順 言不順則事不成 —— 孔夫子

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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.

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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.

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Global Temperature Anomaly Annual Data from 1856 to 2003

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GSTA

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IMF Mean of 10 Sifts : CC(1000, I)

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Statistical Significance Test

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IPCC Global Mean Temperature Trend

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Comparison between non-linear rate with multi-rate of IPCC 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.

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GSTA Summary

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Extend to N-dimension Wave number in spatial data

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The numerically simulated vertical velocity at 500 hPa level of Hurricane Rita of 2005. The warm color represents the upward motion and the cold color downward motion. The unit is m/s. I. Hurricane

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The EEMD components of the vertical velocity. In each panel, the color scales are different, the blue lines corresponding to zeros.

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The final components of hurricane Andrew.

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Comparisons FourierWaveletHilbert Basisa priori Adaptive FrequencyIntegral-Transform: Global Integral-Transform Regional Differentiation: Local PresentationEnergy-frequencyEnergy-time- frequency Nonlinearno Yes: quantify Non-stationarynoyesYes: quantify Uncertaintyyes no Harmonicsyes no

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Scientific Activities Collecting, analyzing, synthesizing, and theorizing are the core of scientific activities. Theory without data to prove is just hypothesis. Therefore, data analysis is a key link in this continuous loop.

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Data Processing and Data Analysis Processing [proces < L. Processus < pp of Procedere = Proceed: pro- forward + cedere, to go] : A particular method of doing something. –Data Processing –Data Processing >>>> Mathematically meaningful parameters Analysis [Gr. ana, up, throughout + lysis, a loosing] : A separating of any whole into its parts, especially with an examination of the parts to find out their nature, proportion, function, interrelationship etc. –Data Analysis –Data Analysis >>>> Physical understandings

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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 methods and let the data sing, and not to force the data to fit preconceived modes. The Job of a Scientist

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Thanks

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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: dynamical biomarkers –(Harvard, Johns Hopkins, NTU, VHT, …) Climate changes –(NASA Goddard, NOAA, CCSP) Cosmological Gravity Wave –(NASA Goddard) Financial market data analysis –(NCU, AS) Theoretical foundations –(Princeton University and Caltech)

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Conclusions Yes, we can still think of any change in terms of waves, but no the simple harmonic kind. With EMD, we can define the true frequency, quantify nonlinearity and extract trend from any data. Among other applications, the degree of nonlinearity could be used to set an objective criterion in health monitoring and in natural phenomena; the trend could be used in financial as well as natural sciences. These are all possible because of adaptive data analysis method.

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Data

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Fourier Spectra : Stations #3,4,5,& 6

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Phase Angle

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Phase Angle Reference to Station # 1

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Wave Fusion Station #5

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The most precise way is utilizing the phase to define Instantaneous Frequency

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Hilbert Spectrum : Station #1

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