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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

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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? How to quantify the degree of nonlinearity? How to define and determine trend?

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

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

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Instantaneous Frequency

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Other Definitions of Frequency : For any data from linear Processes

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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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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.

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Surrogate Signal: Non-uniqueness signal vs. Power Spectrum 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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The Importance of Phase

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To utilize the phase to define 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.

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Hilbert Transform : Definition

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The Traditional View of the Hilbert Transform for Data Analysis

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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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The Real World Mathematics are well and good but nature keeps dragging us around by the nose. Albert Einstein

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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 ( Other alternatives, e.g., Nonlinear Matching Pursuit)

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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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Definition of Instantaneous Frequency

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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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Mean Hilbert Spectrum : All CEs

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