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
x( t )  m 1  h1 ,
h1  m 2  h2 ,
.....
.....
hk  1  m k  hk .

hk  c1 .
Two Stoppage Criteria : SD
Standard Deviation is small than a pre-set value,
where
T
SD 
h
t 0
k 1
T
( t )  hk ( t )
2
h
 k 1 ( t )
t 0
2
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)
Any function having the same numbers of
zero  cros sin gs and extrema ,and also having
symmetric envelopes defined by local max ima
and min ima respectively is defined as an
Intrinsic Mode Function ( IMF ).
All IMF enjoys good Hilbert Transform :
  c( t )  a( t )e i  ( t )
Empirical Mode Decomposition
Sifting : to get all the IMF components
x( t )  c1  r1 ,
r1  c2  r2 ,
. . .
rn  1  cn  rn .
 x( t ) 
n
c
j 1
j
 rn .
Empirical Mode Decomposition:
data
Empirical Mode Decomposition:
IMFs and residue
Definition of Instantaneous Frequency
The Fourier Transform of the Instrinsic Mode
Funnction, c( t ), gives
W ( ) 
i (   t )
a(
t
)
e
dt

t
By Stationary phase approximation we have
d ( t )
 ,
dt
This is defined as the Ins tan tan eous Frequency .
Definition of Frequency
Given the period of a wave as T ; the frequency is
defined as
1

.
T
Equivalence :
The definition of frequency is equivalent to
defining velocity as
Velocity = Distance / Time
Instantaneous Frequency
Velocity 
distance
; mean velocity
time
Newton  v 
Frequency 
dx
dt
1
; mean frequency
period
HHT defines the phase function   
d
dt
So that both v and  can appear in differential equations.
The combination of Hilbert Spectral Analysis and
Empirical Mode Decomposition is designated as
Hilbert-Huang Transform
(HHT vs. FFT)
Comparison between FFT and HHT
1. FFT :
x( t )  
aj e
i jt
.
j
2. HHT :
x( t )    a j ( t ) e
j
i
  j (
t
)d
.
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
a priori
Adaptive
Frequency
Integral transform:
Global
Integral transform:
Regional
Differentiation:
Local
Presentation
Energy-frequency
Energy-timefrequency
Energy-timefrequency
Nonlinear
no
no
yes
Non-stationary
no
yes
yes
Uncertainty
yes
yes
no
Harmonics
yes
yes
no
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