16-08-2012, 05:10 PM
ON INTRINSIC MODE FUNCTION
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Introduction
Intrinsic Mode Function (IMF) was introduced by Huang et al. [1998] as the result
of the Empirical Mode Decomposition (EMD). It is a necessary intermediate step
toward computing instantaneous frequency through the Hilbert Transform or any
other methods [Huang et al. 2009]. Therefore, it is a key part of the Hilbert Spectral
Analysis in the NASA designated Hilbert-Huang Transform (HHT). Since its
introduction, HHT has found a wide range of applications: voice [Khaldi et al.
(2010)], image [Nunes and Del´echelle (2009); Linderhed (2009); Chen et al. (2008);
Sinclair and Pegram (2005) and Wu et al. (2009)], medicine [Chen (2009); Nunes
et al. (2005); Liang et al. (2005)], climate or atmosphere [Ruzmaikin and Feynman
(2009); Huang et al. (2009);Molla et al. (2007); Iyengar and Kanth (2006),Wu et al.
(2008), Qian et al. (2009)], gravitational wave [Camp et al. (2009)], ocean [Huang
and Wu (2008); Huang et al. (1999)], and geography [Wilson et al. (2007); Battista
et al. (2007); Zhang et al. (2003)]. Many of the applications are actually based
on IMFs. As a result, in the subsequent development of HHT, most of the efforts
had been concentrated in the improvements of EMD, such as the intermittence test
[Huang et al., 1999; 2003], Ensemble EMD [EEMD, Wu and Huang, 2009], Complementary
EEMD [CEEMD, Yeh et al. (2010)]. The EMD method has recently also
been extended to multi-dimensional data by Wu et al. [2009] and others [Shi et al.
(2009), Fauchereau et al. (2008), Liu et al. (2007)].
With these advances, the implementation of EMD seemed to have satisfied the
requirements of most practical applications. Left untreated, however, is the rigorous
mathematic foundation, which is absolutely necessary for making the empirical
approach standing on a more solid foundation. Unfortunately, the progress is
painfully slow. Among the urgently needed definitive work are on the definition
of IMF and the stoppage criterion for EMD. Indeed, the recent study by Wu and
Huang [2010] has shown empirically that the sifting process used in EMD could
be equivalent to a bank of dyadic filter only when the number of sifting iterations
is fixed to ten. The ratio of mean frequency between neighboring components
would decrease below 2 as the numbers of iterations increase. In the limit when
EMD is carried toward the limit with infinite many iteration of sifting, the ratio
for mean frequency between neighboring components would approach unit. Then,
EMD would produce IMFs of constant amplitude. Even though the limiting case
actually conforms to the definition of IMFs better, such results would produce frequency
modulated (FM) functions for IMFs, which might even approach Fourier
expansion and lost all their intrinsic physical significance. Based on this empirical
study, they suggested a fixed number of sifting for EMD implementation.
On Intrinsic Mode Function 279
The observations by Wu and Huang [2010] suggest that there might be a conflict
between the definition of IMF and the method proposed to implement it. To
clarify these potential conflicts and to support the empirical approach of Wu and
Huang [2010], we have embarked on the present study. In this paper, we will first
revisit the EMD procedure, as originally introduced by [Huang et al. (1998)], and
the definition of IMF. Then, a theorem would be proven to highlight the limitation
and the consequence of the spline approach. By this theorem, we hope we
could arouse the interest of theoretically oriented investigation toward the rigorous
foundation of HHT. Finally, we will discuss the meaning of IMF in light of this
theorem.
Review of EMD
EMD as originally proposed is implemented through a sifting process that is summarized
as follows:
(1) For any given data, x(t), we identify all the local extrema.
(2) Separately connect all the maxima and minima with natural cubic spline lines
to form the upper, u(t), and lower, l(t), envelopes.
(3) Find the mean of the envelopes as m(t) = [u(t) + l(t)]/2.
(4) Take the difference between the data and the mean as the proto-IMF, h(t) =
x(t) − m(t).
(5) Check the proto-IMF against the definition of IMF and the stoppage criterion
to determine if it is an IMF.
(6) If the proto-IMF does not satisfy the definition, repeat step 1 to 5 on h(t) as
many time as needed till it satisfies the definition.
(7) If the proto-IMF does satisfy the definition, assign the proto-IMF as an IMF
component, c(t).
(8) Repeat the operation step 1 to 7 on the residue, r(t) = x(t)−c(t), as the data.
(9) The operation ends when the residue contains no more than one extremum.
The S-number criterion
This form was proposed by Huang et al. [2003], which is related to another aspect
of the definition of IMF. To implement this definition, one needs to count the
number of extrema and zero-crossings. The S-number is defined as the number of
consecutive sifting iterations in which the number of zero-crossings and extrema
stay the same and are equal or differ by one.
The fixed sifting time criterion
In separate studies, Flandrin [2004], and Wu and Huang [2004] established that
EMD is in fact a bank of dyadic filters. In a more recent study through systematic
empirical trials by Wu and Huang [2010], they reached the conclusion that the
dyadic property is valid only if one kept the iterations of sifting process around
10 times. The dyadic property would break down if the iteration number is too
high or too low. The asymptotic state of infinite number of iterations in sifting
would produce a result of frequency modulate (FM) waves with constant amplitude,
almost approaching the result of the Fourier decomposition.
As we can see, none of the stoppage criteria is totally satisfactory, but all of
them would serve the purpose if applied judiciously. The problem is that there
is no rigorous mathematic standard for us to make decision. The determination
of a stopping criterion to produce physically meaning IMFs is still a challenging
objective to be reached in the implementation of EMD. At the present time, all the
criteria will have to be implemented with some degree of fussiness and to satisfy
the definition of IMF only approximately. Indeed, the fussiness in the stoppage
criteria arises from a conflict between IMF definition and the presently used EMD
implementation algorithm. This would be the subject of the next section.
Discussion and Conclusion
Now, we have two results, one empirical, in the form of Wu and Huang [2010], and
one theoretical, in the form of the present theorem, to highlight the conflict of EMD
and it eventual goal: to produce physical meaningful IMFs. Both the empirical and
the theoretical results point to the asymptotic state of infinite number of sifting
iterations. The empirical result further established the processes how the final IMFs
of constant amplitude are reached. These two results together help us to understand
an oddity of the sifting operation: the ability of sifting operation to split a perfect
IMF further. By understand the processes, we could gain further understanding of
the meaning of IMFs and the operation in obtaining them.
The interesting oddity of the sifting operation is its ability of splitting a function
that seems to be a perfect IMF into more separate components. Consider the
following trigonometric identity: