Derivative-Optimized Empirical Mode Decomposition (DEMD) for the Hilbert- Huang Transform

Transcription

1 Derivative-Optimized Empirical Mode Decomposition (DEMD) for the Hilbert- Huang Transform Peter C. Chu 1), Chenwu Fan 1), and Norden Huang 2) 1)Naval Postgraduate School Monterey, California, USA 2)National Central University Chungli, Taiwan

2 (1) General Description

3 The empirical mode decomposition and the Hilbert spectrum for nonlinear and non stationary time series analysis B y Norden E. Huang1, Zheng Shen2, Steven R. Long3, Manli C. Wu4, Hsing H. Shih5, Quanan Zheng6, Nai Chyuan Y en7, Chi Chao Tung8 and Henry H. Liu9 1 Laboratory for Hydrospheric Processes/Oceans and Ice Branch, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA 2 Department of Earth and Planetary Sciences, The John Hopkins University, Baltimore, MD 21218, USA 3 Laboratory for Hydrospheric Processes/Observational Science Branch, NASA Wallops Flight Facility, Wallops Island, VA 23337, USA 4 Laboratory for Atmospheres, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA 5 NOAA National Ocean Service, Silver Spring, MD 20910, USA 6 College of Marine Studies, University of Delaware, DE 19716, USA 7 Naval Research Laboratory, Washington, DC, , USA 8 Department of Civil Engineering, North Carolina State University, Raleigh, NC , USA 9 Naval Surface Warfare Center, Carderock Division, Bethesda, MD , USA Received 3 June 1996; accepted 4 November 1996 Google Scholar: 5746 Citations!!

4 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)

5 Empirical Mode Decomposition: Methodology : Test Data

6 Empirical Mode Decomposition: Methodology : data and m1

7 Empirical Mode Decomposition: Methodology : h1 & m2

8 Empirical Mode Decomposition: Methodology : h3 & m4

9 Empirical Mode Decomposition: Methodology : h4 & m5

10 Empirical Mode Decomposition Sifting : to get one IMF component x(t) m (t) c (t), 1 1 m (t) m (t) c (t), m (t) m (t) c (t). k 1 k k x(t) m k (t) c i (t). k i 1 c i (t) Intrinsic Mode Function (IMF) m k (t) Trend

11 The Stoppage Criteria The Cauchy type criterion: when SD is small than a pre-set value, where SD T t0 c (t) c (t) k 1 k T t0 c (t ) 2 k 1 2

12 Empirical Mode Decomposition: Methodology : IMFs EMD analysis on the speleothem δ 18 O time series from China s Furong Cave Chu et al Journal of Quaternary Science

13 Comparisons: Fourier, Hilbert & Wavelet

14 Hilbert Transform on Each IMF Hilbert-Huang Transform z () t c () t i cˆ (), t i 1 p p p 1 cp () s cˆ p () t CP ds, t s CP is the Cauchy principal value

15 Instantaneous Amplitude and Frequency z () t c () t i cˆ () t a ()exp[i t ()], t p p p p p () t d ()/ t dt p p Instantaneous Frequency a p (t) Instantaneous Amplitude

16 (2) EMD at Two End Points Unknown Local Maximum and Minimum at the Two End Points

17 Uncertain Upper/Lower Envelops at the Two End Points

18 Uncertain Upper/Lower Envelops at the Two End Points

19 Some Existing Methods (a) Frozen method Two end points are on the upper and lower envelops (b) Extension methods Linear extrapolation of local maximum (minimum) to the end points

20 Original Signal xt ( ) f( t) f( t), 4 i 0 i k i k 1 2 0( i) 0 i, f t At fk( ti) Ak sin( kti k), k 1, 2, 3, 4 k A k ω k 6π (3 Hz) 40π (20 Hz) 100π (50 Hz) 200π (100 Hz) φ k

21 Mode Mixing and Detrend Uncertainty

22 (3) Hermitian Polynomials for Upper (Lower) Envelop

23 Original Time Series: {x(t i ), i = 1, 2,, N} max max max 1 2 ( t, t,..., t ), K ( 1, 2,..., M ) e m local maxima (or minima) q m first derivative

24 Hermitian Polynomials for ( m m 2,3,..., M 1) p e e q q 3 1 m 2 m1 3 mm 4 m1 m, 1, ( t )/ m m m m m ( ) 13 2, ( ) 3 2, ( ) 2, ( )

25 Hermitian Polynomials for ( m m 2,3,..., M 1) p 0 e, p (1) e, dp 0 / dt q, dp 1 / dt q 3 m 3 m1 3 m 3 m1 de/ dt de/ dt ( 0) ( 0) m m em 1 e m mqm 12m1mqm m 1qm1 3 m m 1, m 1 m e e e m m1 m

26 Hermitian Polynomials near the Left End Point (Initial Time Instance t 1 ) p ( ) ( ) e ( ) q ( ) q, L L 3 1 L ( tt )/, t 1 L L ( ) 1, ( ) 12 /2, ( ) 1 / p (1) e, dp (0) / dt q, dp (1) / dt q L 2 1 e2 e1 1qL 4L 1q1 2Lq2 6 L, 1

27 Hermitian Polynomials near the Right End Point (Last Time Instance t N ) em em 1 2RqM 14R M 1qM M 1qR 6 R, M 1 t R N M

28 Tri-diagonal Linear Equations for (q 1, q 2,, q M ) em 1 e m mqm 12m1mqm m 1qm1 3 m m 1, m 1 m m = 2, 3,, M-1 e2 e1 1qL 4L 1q1 2Lq2 6 L, 1 em em 1 2RqM 14R M 1qM M 1qR 6 R, M 1

29 Assumption Upper and lower envelops have the same first derivatives at the two end points

30 Solutions of the Tri-diagonal Linear Equations q a b q c q u u u u k k k L k R q a b q c q l l l l j j j L j R First derivatives at the local maximum (minimum) points of the upper (lower) envelop depend on the first derivatives at the two end points (q L, q R ).

31 (4) Optimization (q L, q R )

32 At each step, the EMD is to decompose the signal into high and low frequency components with the average of upper and lower envelops, i.e., m(t), as the low frequency component and the anomaly from the low frequency component as the high frequency component. Thus, the low frequency component, m(t), should have minimum temporal variability. Usually, small absolute values of derivatives mean small temporal variation. Since the first and second derivatives are already used in obtaining the upper and lower envelops, minimization of integrated squared values of the third derivatives,

33 t tn 3 N du dl S dt dt min 3 3 dt dt t t du uk uk 1 1 u u u u u u ak ak 1bk bk 1qL ck ck 1qR Fk( ql, qr) dt u u k k k 3 dl lj l j1 1 l l l l l l aj aj 1bj bj 1qL cj cj 1qR Gj( ql, qr) dt l l j j j K 2 S F ( q, q ) G ( q, q ) k L R k j L R j k1 j1 J 2

34 Optimization to determine (q L, q R ) t tn 3 N du dl S dt dt min 3 3 dt dt t t 1 1 S q L S 0, 0 q R A11 A12 ql B1. A A q B R 2

35 2 J 1 1 u u l l k u k1 l j j1 j k j K1 1 2 A b b b b k1 1 K1 J1 1 u u u u 1 l l l l k k 1 k k 1 j j 1 j u l j1 k1 j1 k j 1 2 J 1 u u l l k u k1 l j j1 j k j A A c c b b c c b b K1 1 2 A c c c c k1 1 1 B b b K 1 u u k1 k k 2 u k1 u k 1 k k 1 3 u u l l u u J 1 1 u u u u l l j1 j l l a a b b 2 a a k k1 l 3 j j1 l j j1 j1 j j l l K1 1 u u B2 c c a a c c a a 3 3 k1 1 u u 1 u u J 1 u u k 1 j 1 j k u u l l l l 2 2 k k 1 u k u k1 l j j1 l j j1 k j k j j

36 Original Signal xt ( ) f( t) f( t), 4 i 0 i k i k 1 2 0( i) 0 i, f t At fk( ti) Ak sin( kti k), k 1, 2, 3, 4 k A k ω k 6π (3 Hz) 40π (20 Hz) 100π (50 Hz) 200π (100 Hz) φ k

37

38 Correlation Coefficient (CC) and Relative Root-Mean Square Error (RRMSE) between the original and OEMD Components CC 1 = 0.979, CC 2 = 0.983, CC 3 = 0.998, CC 4 = 1.000, CC T = RRMSE 1 = , RRMSE 2 = , RRMSE 3 = , RRMSE 4 = , RRMSE T =

39 Conclusions DEMD eliminates mode mixing, end point effect, and detrend uncertainty.