Ensemble Empirical Mode Decomposition

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

OUTLINE • Time-frequency analysis • Fourier Transform • Windowed FT • Wavelets • Hilbert-Huang Transform • EEMD: Noise Assisted Data Analysis • Applications • Non-stationarity • Nonlinearity • Temporal Locality • Adaptativity • Noise and Signal California 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 Xt, for t=1,…N and Xt,, for t=1,…M (M>N) should provide the same physical explanation for the physics behind data Xt, 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) California Institute of Technology

LINEAR TREND California Institute of Technology

IMPLICATION California Institute of Technology

MERGE OF BLACK HOLES California Institute of Technology

Chirp waves TIME SERIES California Institute of Technology

FOURIER TRANSFORM Chirp wave • Not related well to physical phenomena • Lack of adaptationand locality • Stationarity California Institute of Technology

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

Chirp Wave WAVELETS California Institute of Technology

WAVELET & SUB-SCALES (HARMONICS) Limitedstretchiness and mobility of wavelets leading to sub-scales (sub-harmonics), and therefore, resulting in limited adaptation and locality California Institute of Technology

Norden E Huang EXTREMA & ENVELOPES • Hilbert-Huang Transform (HHT) • Empirical Mode Decomposition • Hilbert (Instantaneous) Spectrum California Institute of Technology

signal source 1 signal source 2 receiver Overall Signal EMPIRICAL MODE DECOMP. California Institute of Technology

EMPIRICAL MODE DECOMP. California Institute of Technology

EMPIRICAL MODE DECOMPOSITIONSifting : to get all the IMF components California Institute of Technology

MIXING WAVES California Institute of Technology

SIFTING California Institute of Technology

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

DECOMPOSITION OF NOISE California 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. California Institute of Technology

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

EXAMPLE: SCALE MIXING California Institute of Technology

EXAMPLE California 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) California 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 California 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 X1(t) = X(t) + N1(t) • Observation II X2(t) = X(t) + N2(t) California 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) ?! California 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) California Institute of Technology

NADA— PRELIMINARY TEST (I) California Institute of Technology

NADA— PRELIMINARY TEST (II) California Institute of Technology

EEMD — NADA (I) California Institute of Technology

EEMD — NADA (II) California Institute of Technology

EXAMPLE California 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) California Institute of Technology

VOICE: WPD California Institute of Technology

VOICE: EMD California Institute of Technology

VOICE: EEMD California Institute of Technology

VOICES California Institute of Technology

MERGE OF BLACK HOLES California Institute of Technology

Ensemble Empirical Mode Decomposition