Multi-variate EMD
Extension of time series analysis to
multi-variable sets
Multivariate
• Multivariate is not multi-dimension.
• Multivariate variable could be a collection of
totally different signals; it could be related
ones too: for example, the three components
of a velocity vector.
• It is useful for image fusion as the three colors
of a color picture.
History
• Tanaka and Mandic first initiated complex, and
Rilling et al bivariate EMD.
– Tanaka, T. & Mandic, D. P. 2006 Complex empirical mode decomposition.
IEEE Signal Process. Lett. 14, 101–104.
– Rilling, G., P. Flandrin, P. Gonçalves, and J. M. Lilly, 2007: Bivariate
Empirical Mode Decomposition, IEEE SIGNAL PROCESSING LETTERS, VOL.
14, 936-939.
• Mandic extended it to Tri- and Multi-variate.
– N. Rehman, D. P. Mandic, 2009: Empirical Mode Decomposition for
Trivariate Signals. IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58,
1059-1068.
– N. Rehman, D. P. Mandic, 2009: Multivariate empirical mode
decomposition. Proc. Roy. Soc. Lond., doi:10.1098/rspa.2009.0502 N.
– Rehman, D. P. Mandic, 2011: Filter Bank Property of Multivariate
Empirical Mode Decomposition. IEEE TRANSACTIONS ON SIGNAL
PROCESSING, VOL. 59, 2421-2426.
Initial Example by Rilling et al.
Free Matlab/C codes are available at
http://perso.ens-lyon.fr/patrick.flandrin.
Space Curve
Centroid of Space Curve
Example : 64 Direction 10 iterations
Example : Results
Extension from Bivariate to
Tri- and Multivariate is non-trivial
We need a new mathematical tool:
Quaternion and beyond
Technical prerequisites
• Defining the local extrema for multivariate signals is
not straightforward; for instance, the complex and
quaternion fields are not ordered : using multiple realvalued projections of the signal; the extrema of such
projected signals are then interpolated componentwise
to yield the multidimensional envelopes of the signal.
• The computation of the local mean for multivariate
signals: to use real-valued projections along multiple
directions on hyperspheres (n-spheres) in order to
calculate the envelopes and the local mean of
multivariate signals.
• To generate a suitable set of direction vectors, unit
hyperspheres are sampled based on both uniform
angular sampling methods and quasi-Monte Carlobased low-discrepancy sequences.
3D Data
• For simple 3D data plus time, we need the
quaternion with algebra well developed.
• The quaternions are a number system that
extends the complex numbers to higher
dimensions. They were first described by Irish
mathematician Sir William Rowan Hamilton in
1843 and applied to mechanics in 3D space. To
extend it to multi dimension is very complicated.
Quaternion
Mathematics: Any number of the form
q = a + bi + cj + dk
where a, b, c, and d are real numbers,
i2 = j2 =k2 =-1, and ij =k= -ji, jk=i and ki=j;
Under addition and multiplication, quaternions
have all the properties of a field, except
multiplication is not commutative.
The norm of q is defined as :
q  a2  b2  c 2  d 2
Unit vector : e u  cos   u sin 
Quaternion Memorial
• Broom Bridge, also known as Brougham Bridge,
is a bridge along Broombridge Road which
crosses the Royal Canal in Cabra, Dublin, Ireland.
• Here as he walked by on the 16th of October
1843Sir William Rowan Hamilton in a flash of
genius discovered the fundamental formula for
quaternion multiplication i² = j² = k² = ijk = −1 &
cut it on a stone of this bridge.
Quaternion Memorial
Quaternion Memorial
Rotation
Generalized to n-Dimension
• For multivariate signals, the local maxima and
minima may not be defined directly.
• Moreover, the notion of ‘oscillatory modes’
defining an IMF is rather confusing for
multivariate signals.
• To deal with these problems, Rehman and
Mandic (2009) proposed to generate multiple ndimensional envelopes by taking signal
projections along different directions in ndimensional spaces; these are then averaged to
obtain the local mean.
Projection of trivariate signal
(a) The direction vector OA in 3D space, can also be represented by a point
on the surface of a unit sphere.
(b) Multiple direction vectors represented by points on a longitude line.
Multiple direction vectors in a 3D space
(a) For projections along longitudinal lines on a sphere, multiple axes represented by a set
of vectors are chosen in the xy plane, with angle taken with respect to x-axis.
(b) projections of the input signal can be taken by rotating the input signal along rotation
axes represented by a set of unit quaternions .
n-Dimensional Case
• A simple and practically convenient choice for a
set of direction vectors is to employ uniform
angular sampling of a unit sphere in an ndimensional hyperspherical coordinate system.
The resulting set of direction vectors covers the
whole (n − 1) sphere.
• A coordinate system in an n-dimensional
Euclidean space can then be defined to serve as a
pointset (and the corresponding set of direction
vectors) on an (n − 1) sphere.
Generalization to n-dimension case
Let {θ1, θ2, . . . , θ(n−1)} be the (n − 1) angular coordinates,
then an n-dimensional coordinate system having  x n as the
i i1
n coordinates on a unit (n − 1) sphere is given by
Generalization to n-dimension case
• The pointset corresponding to the n-dimensional
coordinate system is now very convenient to
generate; however, for n >1, it does not provide a
uniform sampling distribution having a higher
density of the points when approaching the poles.
•
• Proposed a quasi-Monte Carlo methods to
generate a ‘low discrepancy pointset’, providing a
more uniform distribution on the n sphere.
Low Discrepancy Sampling
• By definition, the discrepancy of a sequence is
low if the number of points in the sequence
falling into an arbitrary set B and it is close to
uniform distribution.
• A low-discrepancy sequence is a sequence with
the property that for all values of N, its
subsequence x1, ..., xN has a low discrepancy.
• Low-discrepancy sequences are also called quasirandom or sub-random sequences, due to their
common use as a replacement of uniformly
distributed random numbers.
Low Discrepancy Sampling
• The "quasi" modifier is used to denote more clearly
that the values of a low-discrepancy sequence are
neither random nor pseudorandom, but such
sequences share some properties of random variables.
• There are many ways to generate such sequence. In
certain applications such as the quasi-Monte Carlo
method their lower discrepancy is an important
advantage.
• Rehman and Mandic used Hammersley-Halton
sequence (involved primary numbers etc) to generate
the quasi-Monte Carlo sampling, which is almot
uniformly distributed.
Multiple direction vectors in a n-D space
Direction vectors for taking projections of trivariate signals on a two-sphere
generated by using (a) spherical coordinate system and (b) a low-discrepancy
Hammersley sequence.
Empirical Mode Decomposition,
Multivariate EMD, Matlab and data
See below for our recent
contributions in this field.
You can also download EEG Data used in some of the simulations in
the work below. The EEG data contain EOG and EMG artifacts
coming from eye blinks, eyebrow raising and eye rolling, and are
used in our BSE, BSS, and EMD algorithms. The recordings also
contain a 50Hz power line noise. Enjoy!
Matlab code:
www.commsp.ee.ic.ac.uk/mandic
Some examples
3-Dimensional
Wind vector
Wind vector
Multi-variate
6-Dimensional
6-variates
As a Filtering Bank
Rehman, N and D. P. Mandic, 2011: Filter Bank
Property of Multivariate Empirical Mode Decomposition.
IEEE Transactions on Signal Processing, vol. 59, 24212426
MVEMD as Filtering Bank
MVEMD as Filtering Bank : 500 Average
MVEMD as Filtering Bank
Observations
• MVEMD indeed is a super-filter bank.
• The filter is much sharper than EMD as shown.
• Unfortunately, the sharpness of the filter bank
has a fatal flaw: it could strip the harmonics
off a nonlinear signal. See example.
It is possible to implement
Noise Assisted MVEMD
Adding several noise signals as additional variates to
the real data to form a multivariate data set.
N. B. the data never mixed with the noises.
Noise Assisted MVEMD
Bi-variate (left) VS Noise Assisted MVEMD (right)
Effects of noise level and
sharpness of the filter
Duffing Type : t=1:1000
duff_s= cos(2*pi*4*t/2000+0.3*sin(2*pi*8*t/2000));
EMD
EEMD
1
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MEMD
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EMD
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EEMD
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MEMD
0.1
NA-MVEMD applied on a Duffy signal
• MVEMD parameters used:
n = 3 (2 extra noise channels)
dir = 32; stopping_criteria=‘stop’ ; threshold=[0.1 1 0.1]
• Power of Duffy signal was set to 1, and the power (variance) of
two noise channels was varied to get a range of SNR = 10log (P
duffy/P noise) values.
• NA-MVEMD has been shown to perform well for a range of SNR
values but for lower SNR (higher noise power), the obtained
decomposition for this input signal is not very accurate.
• In NA-MVEMD, increase in the noise power tends to enforce filter
bank structure on the input signal. In such cases NA-MVEMD will
decompose a nonlinear signal with its harmonics across multiple
IMFs.
NA-MEMD with SNR = inf (both noise channels set to zero)
• NA-MEMD acts similarly to standard EMD as expected
NA-MEMD with SNR = 40db
• NA-MEMD extracts the desired signal in IMF 3
NA-MEMD with SNR = 17db
• NA-MEMD extracts the desired signal in IMF 6
NA-MEMD with SNR = 14db
• NA-MEMD extracts the desired signal in IMF 6
NA-MEMD with SNR = 12db
• NA-MEMD extracts the desired signal in IMF 6
NA-MEMD with SNR = 10db
• NA-MEMD extracts the desired signal in IMF 7 but some
distortion can be seen in IMF 6 in this case.
NA-MEMD with SNR = 7db
• NA-MEMD extracts the desired signal in IMF 7 but some
distortion can be seen in IMF 6 in this case.
NA-MEMD with SNR = 1db
• Distortion is found to increase by increasing the noise
level as seen in IMF 6 and IMF 8 in this case
Application to EEG
2
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2000
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MEMD/tw o noise channels
EEMD
EMD
-7
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EMD
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EEMD
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MEMD
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Summary
• MVEMD is a very useful and intriguing new
development in EMD methodology.
• It offers an uniformity (same scale and same number)
to the IMF from the multivariate data set so that the
results will be easier to interpret.
• The NA-MVEMD is a beautiful idea.
• The fatal flaw of MVEMD and NA-MVEMD is that the
filter is too sharp, which leads to separation of
harmonics from the fundamental and destroy the
nonlinearity.