MATH 3290 Mathematical Modeling Tutorial on the Empirical Mode Decomposition Method (EMD)

First, review of the procedure of EMD method The main idea of the EMD method is Sifting

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

Two Stoppage Criteria : SD Standard Deviation is small than a pre-set value, where

Stoppage Criteria It is critical that we use the correct stoppage criterion. Over shifting, we can prove that the envelopes defined has to be a straight line. If the data is not monotonically increasing or decreasing, the straight lines would be horizontal lines.

Empirical Mode Decomposition: Methodology : IMF c1

Definition of the Intrinsic Mode Function (IMF)

Empirical Mode Decomposition Sifting : to get all the IMF components

Empirical Mode Decomposition: data

Empirical Mode Decomposition: IMFs and residue

Definition of Instantaneous Frequency

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

Equivalence : The definition of frequency is equivalent to defining velocity as Velocity = Distance / Time

Instantaneous Frequency

Hilbert-Huang Transform (HHT vs. FFT) The combination of Hilbert Spectral Analysis and Empirical Mode Decomposition is designated as Hilbert-Huang Transform (HHT vs. FFT)

Comparison between FFT and HHT

The Idea behind EMD To be able to analyze data from the nonstationary and nonlinear processes and reveal their physical meaning, the method has to be Adaptive. Adaptive requires a posteriori (not a priori) basis. But the present established mathematical paradigm is based on a priori basis. Only a posteriori basis could fit the varieties of nonlinear and nonstationary data without resorting to the mathematically necessary (but physically nonsensical) harmonics.

The Idea behind EMD The method has to be local. Locality requires differential operation to define properties of a function. Take frequency, for example. The traditional established mathematical paradigm is based on Integral transform. But integral transform suffers the limitation of the uncertainty principle.

Global Temperature Anomaly Annual Data from 1856 to 2003

Global Temperature Anomaly 1856 to 2003

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

Data and Trend C6

Rate of Change Overall Trends : EMD and Linear

What This Means Instantaneous Frequency offers a total different view for nonlinear data: instantaneous frequency with no need for harmonics and unlimited by uncertainty. Adaptive basis is indispensable for nonstationary and nonlinear data analysis HHT establishes a new paradigm of data analysis

Comparisons Fourier Wavelet Hilbert Basis a priori Adaptive Frequency Integral transform: Global Integral transform: Regional Differentiation: Local Presentation Energy-frequency Energy-time-frequency Nonlinear no yes Non-stationary Uncertainty Harmonics

Conclusion Adaptive method is a scientifically meaningful way to analyze data. It is a way to find out the underlying physical processes; therefore, it is indispensable in scientific research. It is physical, direct, and simple.

History of EMD & 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, 460, 1597-1611. 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, 14,889-14,894. The correct adaptive trend determination method 2009: On Ensemble Empirical Mode Decomposition. Advances in Adaptive Data Analysis. (Advances in Adaptive data Analysis, 1, 1-41) 2009: On instantaneous Frequency. Advances in Adaptive Data Analysis (Advances in Adaptive Data Analysis. Advances in Adaptive data Analysis, 1, 177-229). 2009: Multi-Dimensional Ensemble Empirical Mode Decomposition. Advances in Adaptive Data Analysis (Advances in Adaptive Data Analysis. Advances in Adaptive data Analysis, 1, 339-372). 2010: The Time-Dependent Intrinsic Correlation based on the Empirical Mode Decomposition (Advances in Adaptive Data Analysis. Advances in Adaptive data Analysis, 2, 233-265). 2010: On Hilbert Spectral Analysis (to appear in AADA).

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)

Reference: Huang, M. L. Wu, S. R. Long, S. S. Shen, W. D. Qu, P. Gloersen, and K. L. Fan (1998)The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. Roy. Soc. Lond., 454A, 903-993. Flandrin, P., G. Rilling, and P. Gonçalves (2004) Empirical mode decomposition as a filter bank. IEEE Signal Proc Lett., 11, 112-114. Research Center for Adaptive Data Analysis, National Central University http://rcada.ncu.edu.tw/research1.htm