Ensemble Empirical Mode Decomposition Zhaohua Wu Center for Ocean-Land-Atmosphere Studies And Norden E Huang National Central University

Jan. 10, 2008California Institute of Technology OUTLINE Non-stationarity Nonlinearity Temporal Locality Adaptativity Noise and Signal Time-frequency analysis –Fourier Transform –Windowed FT –Wavelets Hilbert-Huang Transform EEMD: Noise Assisted Data Analysis Applications

Jan. 10, 2008California Institute of Technology DATA ANALYSIS In nature, the later evolution can not change what have already happened in the past. In scientific research, the purpose of data analysis is to understand the data, i.e., the physical processes that are hidden in data Inferences –If we believe the results obtained from data analysis is physical, then, the analyses of data X t, for t=1,…N and X t,, for t=1,…M (M>N) should provide the same physical explanation for the physics behind data X t, for t=1,…N. –The First Principle of data analysis: The analysis should be temporally local and the analysis method should be based on temporally local properties of data. (Of course, the locality is not absolute, but relate to the timescales of phenomena)

Jan. 10, 2008California Institute of Technology LINEAR TREND

Jan. 10, 2008California Institute of Technology IMPLICATION

Jan. 10, 2008California Institute of Technology MERGE OF BLACK HOLES

Jan. 10, 2008California Institute of Technology TIME SERIES Chirp waves

Jan. 10, 2008California Institute of Technology FOURIER TRANSFORM Not related well to physical phenomena Lack of adaptation and locality Stationarity Chirp wave

Jan. 10, 2008California Institute of Technology TIME-FREQUENCY DOMAIN Chirp wave Advantages non-stationary, time-frequency Drawbacks window size discontinuity missing low frequencies Denis Gabor

Jan. 10, 2008California Institute of Technology WAVELETS Chirp Wave

Jan. 10, 2008California Institute of Technology WAVELET & SUB-SCALES (HARMONICS) Limited stretchiness and mobility of wavelets leading to sub- scales (sub-harmonics), and therefore, resulting in limited adaptation and locality

Jan. 10, 2008California Institute of Technology EXTREMA & ENVELOPES Norden E Huang Hilbert-Huang Transform (HHT) –Empirical Mode Decomposition –Hilbert (Instantaneous) Spectrum

Jan. 10, 2008California Institute of Technology EMPIRICAL MODE DECOMP. receiver signal source 1 signal source 2 Overall Signal

Jan. 10, 2008California Institute of Technology EMPIRICAL MODE DECOMP.

Jan. 10, 2008California Institute of Technology EMPIRICAL MODE DECOMPOSITION Sifting : to get all the IMF components

Jan. 10, 2008California Institute of Technology MIXING WAVES

Jan. 10, 2008California Institute of Technology SIFTING

Jan. 10, 2008California Institute of Technology SIFTING

Jan. 10, 2008California Institute of Technology SIFTING

Jan. 10, 2008California Institute of Technology SIFTING

Jan. 10, 2008California Institute of Technology SIFTING

Jan. 10, 2008California Institute of Technology SIFTING

Jan. 10, 2008California Institute of Technology SIFTING

Jan. 10, 2008California Institute of Technology SIFTING

Jan. 10, 2008California Institute of Technology SIFTING

Jan. 10, 2008California Institute of Technology SIFTING

Jan. 10, 2008California Institute of Technology SIFTING

Jan. 10, 2008California Institute of Technology DECOMPOSITION OF DIRAC DELTA DUNCTION EMD is, in this case, an adaptive wavelet.

Jan. 10, 2008California Institute of Technology DECOMPOSITION OF NOISE

Jan. 10, 2008California Institute of Technology A DYADIC FILTER EMD is a dyadic filter bank The spectra of IMFs (except the first one) collapse to a simple form in log(period) domain.

Jan. 10, 2008California Institute of Technology PDF OF IMF Each IMF of (all kind of) noise has a Gaussian distribution (inferred by the Central Limit Theorem)

Jan. 10, 2008California Institute of Technology EXAMPLE: SCALE MIXING

Jan. 10, 2008California Institute of Technology EXAMPLE

Jan. 10, 2008California Institute of Technology MATHEMATICAL UNIQUENESS The Mathematical Uniqueness (M-U) With all specifications in a decomposition method given, the results of the decomposition of a data set has one and only one set of components Does a decomposition method satisfy M-U? –All the methods currently available do, e.g., non-adaptive method such as Fourier Transform semi-adaptive method such as wavelet decomposition adaptive method such as EMD, principle component analysis (PCA)

Jan. 10, 2008California Institute of Technology PHYSICAL UNIQUENESS The Physical Uniqueness (P-U) the decompositions of a data set and of the same data set with added noise perturbation of small but not infinitesimal amplitude bear little quantitative and no qualitative change Does P-U Matter in Data Analysis? –Yes, since a data set from real world always contains random noise Does a method currently available satisfy P-U – non-adaptive method such as Fourier Transform does – semi-adaptive method such as wavelet decomposition does –adaptive method such as EMD, principle component analysis (PCA), often does not, thereby decomposition is not stable and hard to interpret

Jan. 10, 2008California Institute of Technology AGAIN, WHAT IS DATA ? Definition –A collection or representation of facts, concepts, or instructions in a manner suitable for communication, interpretation, analysis, or processing data = facts + distortion X(t) = S(t) + N(t) Observations –Observation I X 1 (t) = X(t) + N 1 (t) –Observation II X 2 (t) = X(t) + N 2 (t)

Jan. 10, 2008California Institute of Technology CONSIDERATIONS Two desirable qualities –The signals in data should not be affected by the observations –and more importantly, the signals being extracted by the analysis should remain the same if the analysis techniques are good enough Solution –adding noise to the targeted data during data analysis could be helpful — Noise-Assisted Data Analysis (NADA) ?!

Jan. 10, 2008California Institute of Technology SOLUTION Ensemble EMD –STEP 1: add a noise series to the targeted data –STEP 2: decompose the data with added noise into IMFs –STEP 3: repeat STEP 1 and STEP 2 again and again, but with different noise series each time –STEP 4: obtain the (ensemble) means of corresponding IMFs of the decompositions as the final result Effects –In the mean IMFs, the added noise canceled with each other –The mean IMFs stays within the natural filter period windows (significantly reducing the chance of scale mixing and preserving dyadic property)

Jan. 10, 2008California Institute of Technology NADA — PRELIMINARY TEST (I)

Jan. 10, 2008California Institute of Technology NADA — PRELIMINARY TEST (II)

Jan. 10, 2008California Institute of Technology EEMD — NADA (I)

Jan. 10, 2008California Institute of Technology EEMD — NADA (II)

Jan. 10, 2008California Institute of Technology EXAMPLE

Jan. 10, 2008California Institute of Technology SOME APPLICATIONS Climate Sciences (Many Institutions) Cosmology (NASA) Voice and Image Analysis (FBI, NCU) Medical Sciences (Harvard, Oxford, NCU) Financial Data (NCU) Engineering (CARS)

Jan. 10, 2008California Institute of Technology VOICE: WPD

Jan. 10, 2008California Institute of Technology VOICE: EMD

Jan. 10, 2008California Institute of Technology VOICE: EEMD

Jan. 10, 2008California Institute of Technology VOICES

Jan. 10, 2008California Institute of Technology MERGE OF BLACK HOLES

Jan. 10, 2008California Institute of Technology MERGE OF BLACK HOLES

Jan. 10, 2008California Institute of Technology MERGE OF BLACK HOLES

Jan. 10, 2008California Institute of Technology SOI and CTI

Jan. 10, 2008California Institute of Technology DECOMPOSITIONS

Jan. 10, 2008California Institute of Technology CORRELATIONS

Jan. 10, 2008California Institute of Technology A BETTER DECOMPOSITION ?

Jan. 10, 2008California Institute of Technology A BETTER DECOMPOSITION ?

Jan. 10, 2008California Institute of Technology MAUNA LOA CO2

Jan. 10, 2008California Institute of Technology CO2 COMPONENTS

Jan. 10, 2008California Institute of Technology WAVELET NANAC

Jan. 10, 2008California Institute of Technology EEMD DECOMP.

Jan. 10, 2008California Institute of Technology STATISTICS

Jan. 10, 2008California Institute of Technology GROWING SEASON

Jan. 10, 2008California Institute of Technology SUMMARY Noise is a KEY to unlock the signals in data

Jan. 10, 2008California Institute of Technology METHODOLOGICAL DEVELOPMENT Huang et al., 1998: The Empirical Mode Decomposition Method and the Hilbert Spectrum for Non-stationary Time Series Analysis, Proc. Roy. Soc. London, A454, The invention of the basic method of EMD, and Hilbert transform for determining the Instantaneous Frequency and energy. Huang et al., 1999: A New View of Nonlinear Water Waves – The Hilbert Spectrum, Ann. Rev. Fluid Mech. 31, Introduction of the intermittence in EMD decomposition. Huang et al., 2003: A confidence Limit for the Empirical mode decomposition and the Hilbert spectral analysis, Proc. of Roy. Soc. London, A459, Establishment of a confidence limit without the ergodic assumption. Wu and Huang, 2004: A Study of the Characteristics of White Noise Using the Empirical Mode Decomposition Method, Proc. Roy. Soc. London, A460 Found the fundamental properties of the EMD and established statistical significance test method for IMF. Wu and Huang, 2007: On the Trend, Detrending and the Variability of Nonlinear and Non-stationary Time Series. Proc. Natl. Aca. USA., 104, Developed adaptive method for extracting nonlinear trend Huang and Wu, 2007: A Review on Hilbert-Huang Transform: the Method and Its Applications to Geophysical Studies, Reviews of Geophysics, (Accepted) A review of HHT covering the most recent developments of the method and its novel applications. Huang and Wu, 2007: On the Instantaneous Frequency, Advance Adaptive Data Analysis, (Accepted) Removal of the limitations posted by Bedrosian and Nuttall theorems for Instantaneous Frequency computations. Wu and Huang, 2008: Ensemble Empirical Mode Decomposition: A Noise-Assisted Data Analysis Method. Advance Adaptive Data Analysis. (Accepted) Solved the decomposition instability problem and scale (frequency) mixing problem Wu et al., 2008: Temporal Axial Dilation for Chirp Signals. Advance Adaptive Data Analysis. (Accepted) Introduced time dilation method for extracting chirp signals