Abstract and Figures

This paper presents and discusses the challenge of mode mixing when using the Empirical Mode Decomposition (EMD) to identify intrinsic modes from EEG signals used for neural activity reconstruction. The standard version of the EMD poses some challenges when decomposing signals having intermittency and close spectral proximity in their bands. This is known as the Mode Mixing problem in EMD. Several approaches to solve the issue have been proposed in the literature, but no single technique seems to be universally effective in preserving independent modes after the EMD decomposition. This paper exposes the impact of mode mixing in the process of neural activity reconstruction and reports the results of a performance comparison between a well known strategy, the Ensemble EMD (EEMD), and a new strategy proposed by the authors for mitigating the mode mixing problem. The comparative evaluation shows a more accurate neural reconstruction when employing the strategy proposed by the authors, compared to the use of EEMD and its variants for neural activity reconstruction.
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The Mode Mixing Problem and its Influence in the
Neural Activity Reconstruction
Maximiliano Bueno-López, Eduardo Giraldo, Marta Molinas and Olav Bjarte Fosso
Abstract—This paper presents and discusses the challenge of
mode mixing when using the Empirical Mode Decomposition
(EMD) to identify intrinsic modes from EEG signals used
for neural activity reconstruction. The standard version of
the EMD poses some challenges when decomposing signals
having intermittency and close spectral proximity in their
bands. This is known as the Mode Mixing problem in EMD.
Several approaches to solve the issue have been proposed in
the literature, but no single technique seems to be universally
effective in preserving independent modes after the EMD
decomposition. This paper exposes the impact of mode mixing
in the process of neural activity reconstruction and reports the
results of a performance comparison between a well known
strategy, the Ensemble EMD (EEMD), and a new strategy
proposed by the authors for mitigating the mode mixing
problem. The comparative evaluation shows a more accurate
neural reconstruction when employing the strategy proposed
by the authors, compared to the use of EEMD and its variants
for neural activity reconstruction.
Index Terms—EEG signals, Empirical Mode Decomposition,
Mode Mixing.
I. INTRODUCTION
THE EEG is an indicator of neural activity and is
used to study cognitive processes, physiology, emotion
recognition and complex brain dynamics [1], [2]. Due to
the non-linear and non-stationary nature of EEG signals,
they are difficult to analyze in the time and frequency
domain. However, some important characteristics can be
extracted to assist early detection of different disorders by
using advanced signal analysis techniques [3], [4]–[6]. In
recent years, the Hilbert Huang Transform (HHT) has been
increasingly used in the analysis of such signals. However,
in some applications, the extraction of information has been
hampered by the mode mixing problem that appears in
the Empirical Mode Decomposition (EMD) when frequency
components are relatively close or exhibit intermittency.
A mode mixing problem appears when an Intrinsic Mode
Function (IMF) either consists of signals of widely disparate
Manuscript received July 31, 2018; revised January 21, 2019.
This work was carried out under the funding of the Departamento
Administrativo Nacional de Ciencia, Tecnología e Innovación (Colciencias).
Research project: 111077757982 "Sistema de identificación de fuentes
epileptogénicas basado en medidas de conectividad funcional usando
registros electroencefalográficos e imágenes de resonancia magnética en
pacientes con epilepsia refractaria: apoyo a la cirugía resectiva".
M. Bueno-López is with the Department of Electrical
Engineering, Universidad de La Salle, Bogotá, Colombia, e-mail:
maxbueno@unisalle.edu.co.
E. Giraldo is with the Department of Electrical Engineering, Universidad
Tecnológica de Pereira, Pereira, Colombia, e-mail: egiraldos@utp.edu.co.
M. Molinas is with the Department of Engineering Cybernetics,
Norwegian University of Science and Technology, Trondheim, Norway,
e-mail: marta.molinas@ntnu.no
O. B. Fosso is with the Department of Electric Power Engineering,
Norwegian University of Science and Technology, Trondheim, Norway,
e-mail: olav.fosso@ntnu.no
scales, or a signal of a similar scale resides in different IMF
components.
The presence of mode mixing can hamper the physical
interpretation of the process which is intended to be
described by the individual IMFs [7], [8]. The mode mixing
problem has been studied in different applications, for
example in [9], where the authors discuss how the mode
mixing influences the EMD-based methods for hydrocarbon
detection. They use mode-mixing elimination methods,
specifically ensemble EMD (EEMD) and complete ensemble
EMD (CEEMD), as tools for identification of the peak
amplitude above average volume and the peak frequency
volume. In [10], a method was proposed based on the
morphological filter to remove the noise and the revised blind
source separation to deal with the mode mixing. The method
was tested with vibration signals from a mechanical system.
In [11], a sinusoidal-assisted EMD (SAEMD) for efficient
and effective HHT computation to solve mode-mixing
problems was proposed. The new tool was tested using the
Global Sea Surface Temperature (GSST) application from
1856 to 2003.
In the specific case of EEG signals, the EMD and
the Hilbert Huang Transform (HHT) have been used
to obtain a better signal representation and to detect
instantaneous frequencies [12], [13]. In [14], an approach
that combines EEMD and ICA for selection of artifactual
components and concentration of artifacts was presented. The
effectiveness of the proposed approach was examined using
semi-simulated data purposely contaminated with selected
artifacts. In [15], the authors quantified the interaction
between different electrodes using a nonlinear measure
known as synchronization likelihood (SL) which effectively
measures the synchronization between non-stationary signals
like EEG. The empirical mode decomposition (EMD)
is applied to decompose the EEG signal into intrinsic
oscillatory modes. In other cases, the EMD is used for
classification of electroencephalogram signals. The intrinsic
mode functions generated have been used as an input to
classifiers as least squares support vector machine (LS-SVM)
[16], [17]. Based on the above, it is arguable that the use
of EMD in applications involving EEG signals needs to be
combined with other techniques to handle the Mode Mixing
problem [13], [18]–[24].
The purpose of this paper is to contribute to a better
understanding of the challenges that EMD poses when
applied to EEG signals and to discuss solutions to the mode
mixing problem in this specific area. With this application,
the goal is to find an accurate brain reconstruction from
the EEG bands decomposition. This paper is organized as
follows: Section II gives an introduction to the essential
concepts of EMD. The mode mixing problem and some
established solutions are given in section III, and in section
IV we propose a new methodology to solve this problem.
Illustrative examples are shown in Section V, while the
discussion of the results is presented in Section VI. Finally,
conclusions are given in Section VII.
II. EMPIRICAL MO DE DECOMPOSITION
The Empirical Mode Decomposition has been proposed
as an adaptive time-frequency data analysis method [25].
The EMD does not require any restrictive assumption on the
underlying model (no basic function) of the process/system
under analysis and is able to handle both non-linear and
non-stationary signals. However, the algorithm has shown to
have some limitations in identifying closely spaced spectral
tones and components appearing intermittently in the signal.
The aim of the EMD method is to decompose the nonlinear
and nonstationary signal y(tk)into a sum of intrinsic mode
functions (IMFs) that satisfies two conditions [26]:
1) Symmetric upper/lower envelopes (zero mean).
2) The numbers of zero-crossing and extrema that are
either equal or differ by exactly one.
The EMD algorithm for the signal y(tk)can be
summarized as follows:
1) Identify all extrema (maxima and minima) in y(tk).
2) Interpolate between minima and maxima, generating
the envelopes el(tk)and em(tk).
3) Determine the local mean as m(t)=(el(tk)+em(tk))/2.
4) Obtain the residue r(tk)=y(tk) − m(tk)
5) Decide whether r(tk)is an IMF or not based on the
two basic conditions for IMFs mentioned above.
6) Repeat step 1 to 4 until r(tk)will be monotonic.
Empirical Mode Decomposition is applied over y(tk)to
obtain γi(tk)being ithe intrinsic mode function (IMF), and
y(tk)=
N
Õ
i=1
γi(tk)+r(tk)(1)
where Nis the number of IMFs and r(tk)a residual.
Recently, some optimization techniques have been proposed
to improve the performance of the EMD [27], [28].
Having obtained the intrinsic mode function components,
the Hilbert transform can be applied to each component and
the instantaneous frequency is found using equation (2).
fi(t),1
2π·dθi(t)
dt ,(2)
In (2), θi(t)is the instantaneous phase of each IMF calculated
from the associated analytical signal [29]. Finally, the
instantaneous frequency can be observed in the Hilbert
Spectrum.
III. MOD E MI XI NG P ROB LE M AN D IT S SO LU TI ON
Mode mixing, observed in the context of the EMD caused
by either intermittency of a signal component or by spectral
proximity, is a well recognized challenge of the method [7],
[9], [30]. In [8] and [31], the authors address the issue of
One or Two Frequencies?, and define a set of conditions that
must exist between the frequency components of a signal to
ensure that they can be recognized as independent modes
in the EMD decomposition. The mode mixing problem has
been analyzed in different areas. In [9], an application for
hydrocarbon detection is presented, while in [10], the authors
show an application in mechanical systems. In this paper, two
previously known methods and one new method proposed by
the authors in [23] are introduced to handle the mode mixing
in the detection of signal sources from different regions of
the brain. The signals are collected using EEG.
A. Masking Signal
The masking signal method reduces the problem of mode
mixing when frequency components are closely spaced (in
the same octave). The masking signal approach was proposed
in [30]. The basic idea is to add a new signal to the analyzed
signal. This signal will prevent lower frequency components
from being mixed together with the higher frequencies in
the same IMFs. Since the masking signal is known, it can
be removed from the IMF following the procedure indicated
below:
1) Construct a masking signal s(n), from the frequency
information of the original data, y(tk).
2) Perform EMD on y+(tk)=y(tk)+s(n)to obtain the
IMF z+(n). Similarly obtain z(n)from y(tk)=y(tk)−
s(n).
3) Define the IMF as z(n)=(z+(n)+z(n))/2.
The challenge with this method is the choice of the masking
signal s(n). How to select the frequency and amplitude of
this new signal? According to [30], an appropriate choice
would be to have each frequency within the signal separated
by at least a factor of 2, is s(n)=a0sin(2πfst). Although
some indications are given on how to choose a0and fs, this
process is usually empirical and the experience must guide
the selection of parameters for a particular problem.
In [23], a strategy to calculate a0and fswas proposed,
based on the relation between the frequencies and amplitudes
of each IMF. This new approach exploits the properties of
the boundary map shown in Figure 1, which was previously
developed in [8].
Fig. 1. Mode mixing boundary conditions map reproduced from [8] used
for defining parameters of the masking signal
B. Ensemble Empirical Mode Decomposition
Wu and Huang proposed a noise-assisted data analysis
(NADA), the Ensemble Empirical Mode Decomposition
(EEMD), which defines the true IMF components as the
mean of an ensemble of trials, each consisting of the signal
plus a white noise of finite amplitude [7]. The EEMD is
described as:
1) Add a white noise series to the data base y(tk).
2) Decompose the data with added white noise using
EMD to obtain the IMFs.
3) Repeat step 1 and step 2 again, but with different white
noise series each time.
4) Obtain the (ensemble) means of corresponding IMFs
of the decompositions as the final result.
The main effect of the decomposition using the EEMD
is that the added white noise series cancel each other
in the final mean of the corresponding IMFs. Modified
versions of the EEMD have been recently proposed. In [32],
the authors proposed one variation of EEMD, a Complete
Ensemble Empirical Mode Decomposition with Adaptive
Noise (CEEMDAN), and an improvement of CEEMDAN can
be found in [33].
IV. NEW MASKING SI GN AL STRATE GY
The new method applied is based on the combination
of the technique presented by Kaiser and the Boundary
Conditions Map presented by Flandrin [8]. The properties
of the map of boundary conditions between well separated
tones and mixed tones (Figure 1) guides the choice of the
masking signal’s frequency and amplitude. To be able to
extract a frequency by applying this principle, the ratio of
that frequency to the frequency of the masking signal, must
be located in the red area of the Boundary Map (mode mixing
area is higher than 0.67 and approaches 1.0), while the ratio
of the next frequency should be located in the blue area of
the map, where mode mixing does not occur. The amplitude
ratios need to be adopted from the map, with a ratio that will
ensure that the above conditions are preserved. It is therefore
necessary to operate with a frequency sufficiently close to the
first IMF mode and sufficiently distant from the next IMF
mode, to be successful. The map illustrates well how closely
spaced spectral tones attract each other in a mode mixing, the
colors representing the mode mixing degree; red for mode
mixing and blue for No-mode mixing. This same property
is exploited in this new masking method for constructing
effective masking signals to separate closely spaced spectral
tones.
Assume a signal with the two frequencies f1and f2
(f1>f2), where the ratio between them will cause mode
mixing due to spectral proximity. A masking signal of
frequency fmlarger than f1will attract f1if the ratio f1/fm
falls into the attraction region of the map (red color). If the
ratio between f2/fmfalls in the region where there is no
attraction (bluecolor), adding a positive masking signal of
frequency fmwill separate the two signals f1and f2and the
first IMF will have a controlled mode mixing of the signals
f1and fm. To separate f1and fma negative masking signal
may be added, and by averaging the two first IMFs, the new
IMF will be a signal of frequency f1. However, depending on
how close the two frequencies f1and f2are, some amplitude
modulation may be observed between the signals f1and f2.
A way of identifying the frequencies involved in the original
signal, is to process a Fast Fourier Transform (FFT) of the
signal. In [23], a technique has been developed to identify the
involved instantaneous frequencies and amplitudes, to assist
in choosing the right masking signal. Assume a signal x
defined by:
x=Asin(2πf1t)+Bsin(2πf2t)(3)
After EMD is performed, these two signals will be
mixed into one IMF if they are sufficiently close in
frequency. A Hilbert-transform of the mode mixed IMF
(s=x+jy)followed by an amplitude and an instantaneous
frequency calculation, will provide the required information
for identifying the amplitudes and frequencies of the two
signals involved. The instantaneous frequency used here is
defined by:
f=1
2π
φ
t,(4)
where tan φ=(y/x)and φ=arctan(y/x).
Using this definition, the equations for the amplitudes and
the instantaneous frequencies can be derived. The amplitudes
of the Hilbert-transformed signal are given by:
K=qA2+B2+2AB cos(2π(f1f2)t)(5)
From this equation, the following expressions for the
extreme values can be derived:
Kmin =pA2+B22AB =(AB)(6)
Kmax =pA2+B2+2AB =(A+B)(7)
Similarly, the equation for the instantaneous frequencies
are given by:
f=A2f1+B2f2+AB(f1+f2)cos(2π(f1f2)t)
A2+B2+2AB cos(2π(f1f2)t)(8)
The expressions obtained for the extreme values are:
Fmin =Af
(A+B)+f2(9)
Fmax =Af
(AB)+f2(10)
where, Kmin is the Minimum value of the amplitude plot,
Kmax is the Maximum value of the amplitude plot, Fmin
is the Minimum value of the instantaneous frequency plot,
Fmax is the Maximum value of the instantaneous frequency
plot and fis the difference between the two frequencies
(f1f2).
It is also demonstrated that fis equal to the number
of peaks/second in the instantaneous frequency and the
amplitude plots. The frequencies f1and f2can now
be calculated and may be further validated with a FFT
calculation. In the case of synthetic signals, these calculations
are accurate and in principle the signal components
could have been obtained directly by applying the above
presented analytical technique. However, for real signals the
instantaneous amplitude and frequency functions are less
smooth, but still this procedure will reveal information about
the amplitudes and frequencies involved in the different
periods of a mode mixed signal. From this information,
an optimal masking signal based on the boundary map
can be constructed. The process of extracting the IMFs
is illustrated in the flowchart of Figure 2. An iterative
procedure is used where the IMFs are extracted one by
one by using the previously described procedure for optimal
design of a masking signal. After the first IMF is extracted
and accepted as an intrinsic mode, the residue (sn+1(t)) of
the signal is used in the next iteration. The acceptance of
the IMF is decided by observation of the obtained first IMF
instantaneous frequency profile and depends on the user
experience and his a-priory knowledge of the studied system.
The next iteration consists on performing a new EMD and
again observing the first IMF instantaneous frequency profile.
This profile will provide a hint about the presence of mode
mixing. Once the frequencies involved in the mode mixing
are identified, a new masking signal is designed according
to the procedure above described. After the masking method
is applied, the second IMF of this iterative process will be
obtained. Once this second IMF is accepted, a new residue of
the signal will be calculated by subtracting the two accepted
IMFs from the original signal. This new residue will then
be used in the next iteration. Iterations will be stopped once
mode mixing is not detected anymore. The last residue in
that case will contain the last IMF and/or the final residue
of the EMD process.
V. IL LU ST RATI VE EX AM PL ES
In order to evaluate the behavior of the methods presented
in section III and IV, three databases for EEG generation
are used. For the first and second databases the objective
is to be able to observe in each IMF, neural activities at
different frequencies. For the third database the objective is
to obtain in one IMF the relevant activity (alpha band). The
use of simulated databases for EEG source localization is a
common approach for evaluation of brain mapping methods
since the underlying source activity is known, and therefore,
having a benchmark it is possible to evaluate the quality of
the brain activity estimation.
The first simulated database (DB-1) contains neural
activity x(tk)by considering temporal localized sinusoidal
signals with two different frequencies (6Hz and 8Hz) by
using a Gaussian window, with sampling rate of 100Hz.
In this case, two sources randomly located into the brain
are selected, where the activity in each source is generated
according to the following expression:
xi(tk)=e1
2tkci
σ2
sin (2πfitk)(11)
being cithe center of the windowed signal in seconds,
and fithe frequency of the signal, with i=1,2. The ciis
selected in the following ranges ci:[0.5,3.5]seconds. The
second simulated database (DB-2) uses the same model
shown in (11) but with three different frequencies (4Hz,
8Hz and 10Hz). In DB-1 and DB-2, we have considered
information from 30 channels.
The third simulated database (DB-3) contains neural activity
generated by two sources randomly located into the brain
with activity x(tk)in the range of 8Hz to 13Hz, with
sampling rate of 100Hz. The time of the two distinguished
Original Signal

Mode
Mixing?
Empirical Mode
Decomposition
EMD
Applied Masking
signal 
The first IMF
has a good
decomposition?
    

=
YES
NO Final 
NO
YES
n: number of iterations
: original signal
Fig. 2. Flowchart of the iterative process for IMFs extraction in the presence
of Mode Mixing
sources in the vector x(tk)are modeled using bivariate
linear auto-regressive (AR) models with time-delayed
linear influences from one source to another. Sources are
bandpass-filtered in the alpha band using a causal third-order
Butterworth filter with zero phase delay. The generated
time series therefore represent alpha oscillations that are
either mutually statistically independent or characterized by
a clearly defined sender-receiver relationship. In addition,
500 mutually statistically independent brain noise time
series characterized by 1 / f-shaped (pink noise) power and
random phase spectra are generated, and placed randomly at
500 locations sampled from the entire cortical surface [34].
In DB-3, we have considered information from 108 channels.
For DB-1 and DB-2, the EEG signals are obtained as
a linear combination of the underlying neural activity as
follows:
y(tk)=M x(tk)+(tk)(12)
being y(tk) ∈ Rdthe EEG signal measured at each electrode
on the scalp, and x(tk) ∈ Rnthe neural activity or amplitude
of ncurrent dipoles (distributed sources inside the brain),
with tk=kh the time at sample kbeing k=1, . . . , Tthe
total number of samples, hthe sample time and MRd×n
the lead-field matrix that relates the neural activity with the
EEG. Specifically, each row of the lead-field matrix describes
the current flow for a given electrode through each dipole
position [35]. Here, the Gaussian additive noise is defined
by (). A Signal-to-Noise-Ratio (SNR) of 10dB is used for
both databases.
For the DB-1 and DB-2, the lead field matrix is described
in [36] based on a Boundary Element Method with a high
number of distributed sources n=20484 and d=30
electrodes over the scalp according to a 10-20 standard
BIOSEMI [37]. Positions of electrodes and sources are
shown in Fig. 3.
Fig. 3. Positions of electrodes and distributed sources for DB-1 and DB-2
Figure 4 shows the simulated DB-1 neural activity and
source locations for the two different frequencies (6Hz
and 8Hz) at two different positions into the brain and the
corresponding EEG, with f1=6Hz, f2=8Hz, c1=1s,
c2=2s and σ=0.12.
Fig. 4. Neural Activity of simulated DB-1 with their corresponding source
locations and simulated EEG.
Figure 5 shows the Fourier spectrum of one channel for
DB-1 and DB-2. In this figure it is possible to see that
different frequencies appear but they are not appropriate to
determine at which instant of time they occur.
Fig. 5. Fourier spectrum for one channel using DB-1 and DB-2.
For the DB-3, the lead-field matrix is obtained from the
so-called New York Head model as used in [34], which
combines a highly detailed magnetic resonance (MR) image
of an average adult human head with state-of-the-art finite
element electrical modeling. In particular, the New York
Head model holds n=2004 sources and d=108 electrodes.
Figure 6 shows the simulated DB-3 neural activity and
source locations for the two different sources and their
locations into the brain with their corresponding EEG.
Figure 7 shows the Fourier spectrum of one channel of
DB-3.
The brain activity estimation is performed by using a
dynamic inverse problem considering that only the EEG y(tk)
and the lead-field matrix Mare known. The dynamic inverse
problem of brain activity estimation bx(tk)can be formulated,
according to [36], as:
bx(tk)=arg min
x(tk)ky(tk) − M x(tk)k2
2+λkkx(tk) − bx(tk1)k 2
2
+αkkx(tk)k1
(13)
where λkand αkare the regularization parameters computed
by generalized cross validation. It can be noticed that as
Fig. 6. Neural Activity of simulated DB-3 with their corresponding source
locations and simulated EEG.
Fig. 7. Fourier spectrum for one channel using DB-3.
a result of the EMD decomposition, the brain activity
estimation of x(tk)can be obtained for each one of the
resulting IMFs. That means that according to (1), the
dynamic inverse problem of (13) is solved for γi(tk)instead
of y(tk), as described in [38], as follows
bχi(tk)=arg min
χ(tk)kγi(tk) − M χi(tk)k2
2
+λkkχi(tk) − bχi(tk1)k2
2
+αkkχi(tk)k1
(14)
where χi(tk)is the neural activity estimation for the
corresponding IMF γi(tK). In addition, as shown in (12),
the forward problem of EEG generation is a linear problem,
then it can be seen that x(tk)can be rewritten as a linear
combination of the estimated neural activity for each IMF
x(tk)=ÍN
i=1χi(tk). Therefore, equation (14) is a sub-band
brain mapping based on an EMD decomposition. Finally,
Fig. 8 shows the methodology used, which starts with the
decomposition of the EEG using three different methods to
finally do the mapping of the neural activity.
Multi Channel
EMD
EEMD
Masking Signal
IMF1
2
B i M ppi g
IMFs Selection
Fig. 8. Schematic representation of the methodology with its different
stages
A. Analysis for DB-1
The IMFs obtained for 1 of the 30 channels with the EMD
proposed in [25] and the EEMD proposed in [7] are shown in
Fig. 9. The decomposition obtained for one channel with the
novel method proposed in this paper is shown in Figure 10.
The noise standard deviation of EEMD is set to 0.1 in this
work. According to the experiments carried out, the noise
standard deviation is the most relevant parameter in this
algorithm. The number of iterations had been set to 1000.
This process is repeated for each channel and in this way,
the IMFs are obtained for the entire database.
In this case, the results of EMD shows clear mode mixing
in all the IMFs. In the first and second IMF, it is possible
to observe the two frequency components (6Hz in t=1s and
8Hz in t=2s). In the first IMF, the frequency component of
6Hz appears more clearly. Normally, we would expect to
find the highest frequency component,in this case 8Hz, in
the first IMF. For the EEMD, it is possible to observe the
frequency component for 6Hz and 8Hz in the first and second
IMF respectively. The instantaneous frequency shows similar
behavior with the EMD and the EEMD. In the instantaneous
frequency corresponding to the first IMF of the EMD, it
is possible to distinguish one frequency component around
t=1s and other frequency component around t=2s. In the
EEMD, it is possible to distinguish one frequency component
around t=2s. However, although it is possible to identify the
frequencies of our interest, some other elements are observed,
especially in the IMF1; these elements correspond to noise
components. The results obtained with the new masking
signal method, allow us to observe a better decomposition
of the signal. In Figure 10, it is possible to see how the
signal of one-channel EEG is decomposed correctly in 3
IMFs. At the top of the figure, the original signal appears.
In the first IMF, the noise component is well isolated and in
the remaining IMFs, the expected two frequency components
are clearly identified. Figure 11 shows a comparison among
the proposed methods by using the neural activity mapping
for the first two IMFs for DB-1. It can be seen that in the
Fig. 9. IMFs and Instantaneous Frequency with EMD and EEMD using DB-1
Fig. 10. IMFs with EMD+Mask using DB-1
first IMF, the EMD with a Mask has less spurious activity
than the standard EMD and EEMD in comparison with the
Ground truth.
B. Analysis for DB-2
The IMFs obtained for 1 of the 30 channels with the EMD
proposed in [25] and the EEMD proposed in [7] are shown
in Fig. 12. Finally the decomposition obtained with the novel
method proposed in this paper is shown in Figure 13.
In this case, the results of EMD shows clear mode mixing
in all the IMFs. In the second and third IMF, it is possible to
observe the three frequency components (4Hz in t=1s, 8Hz
in t=2s and 10Hz in t=3s). The first IMF corresponds to the
noise of the signal. For the EEMD is possible to observe
all the information of interest in the second and third IMFs,
where the second IMF is one clear case of mode mixing.
The instantaneous frequency has a similar behavior with
the EMD and the EEMD. In the instantaneous frequency
corresponding to the second IMF of the EMD, it is possible
to distinguish three frequency components around t=1s, t=2s
and t=3s. In the EEMD, it is possible to distinguish two
frequency components around t=2s and t=3s in the second
IMF. The results obtained with the new masking signal
method, allow us to observe a better decomposition of the
signal. In Figure 13, it is possible to see how the signal of
one-channel EEG is decomposed correctly in 3 IMFs. At the
top of the figure, the original signal appears. In the first IMF,
the noise component is well isolated and in the remaining
IMFs the expected three frequency components are clearly
identified.
Figure 14 shows a comparison among the proposed
methods by using the neural activity mapping for the first
two IMFs for DB-2. It can be seen that in the first IMF the
EMD with a Mask has less spurious activity than the standard
EMD and EEMD in comparison with the Ground truth.
C. Analysis for DB-3
The IMFs obtained for 1 of the 108 channels with the
EMD proposed in [25] and the EEMD proposed in [7] are
shown in Fig. 15. For this case, the decomposition obtained
with the new masking signal is shown in Fig. 16.
The EMD and the EEMD have clear mode mixing in
all the IMFs. However, the frequency components of our
interest, 8 Hz and 13 Hz, appear in 2 of the 7 IMFs. The
IMF1 in both cases corresponds to noise. In IF2 and IF3,
it is possible to see two red lines that corresponds to the
frequencies of 8 Hz and 13 Hz.
Fig. 17 shows a comparison among the proposed methods
by using the neural activity mapping for the IMF2 and IMF3
for DB-3 (according to the IMFs instantaneous frequencies
the IMF2 and IMF3 hold the relevant information). The
reconstruction of neural activity in this case was very close
to the ground truth.
VI. DISCUSSION
The mode mixing problem has been studied in different
fields and in most cases the suggested solution has been to
address the problem by using EEMD or to a lesser extent
a masking signal (the idea is to separate two components
whose frequencies are close). Noteworthy is the fact that
while EEMD has been conceived to solve the mode mixing
problem caused by intermittency, it has been generally used
to solve all types of mode mixing problems regardless
of their origin. The new masking method presented in
this paper is specifically designed to reduce the mode
mixing problem caused by the presence of modes with
close spectral proximity. The main challenge when using
the masking signal is the selection of its amplitude and
Fig. 11. Brain mapping comparison of the proposed methods for each IMFs for DB-1.
Fig. 12. IMFs and Instantaneous Frequency with EMD and EEMD using DB-2
Fig. 13. IMFs with EMD+Mask using DB-2
frequency, but after selecting adequate values, the method
shows to work very well. Another alternative that has been
used to reduce mode mixing was to combine EMD or
EEMD with some statistical method that allows to establish
a criterion for selection of the IMFs according to the
application. For example, in [39], the authors decompose
EEG signal segments using EEMD, and then they extract
statistical moment based features from the resulting IMFs.
In [40], they proposed one technique to separate the sources
from single-channel EEG signals by combining EEMD and
Independent component analysis (ICA). For the case study
in this paper, the objective is to show that it is possible to
achieve an accurate mapping of neuronal activity using the
IMFs obtained directly from the EMD with masking.
In Fig. 12, it is evident that the IMFs from the EMD
have mode mixing. This is detected by observing different
oscillations in the same IMF. As shown in Figures 11 and 14,
it can be seen that the sub-band reconstruction effectively
splits the brain activity into frequency bands. In fact, for
signals with low SNR, it is clear that the noise is adaptively
filtered in a specific IMF. However, a drawback of the
method can be observed in the reconstructions obtained at
low frequencies, where the activity is identifiable in several
IMFs. On the other hand, the Instantaneous Frequency (IF)
shown in Fig. 9, 12 and 15 is useful to detect instantaneous
variations of the frequency. For EEG signals resulting from
Fig. 14. Brain mapping comparison of the proposed methods for each IMFs for DB-2.
Fig. 15. IMFs and Instantaneous Frequency with EMD and EEMD using DB-3
Fig. 16. IMFs with EMD+Mask using DB-3
Evoked-Related-Potentials or Epilepsy, the IF can be used
to automatically detect the neural response to an external
stimulus, or the beginning of an epileptic seizure. To improve
the results with the new masking signal proposed, it is
possible to use some statistical methods and combine the
results of both strategies.
VII. CONCLUSIONS
In this paper, we have proposed a method for decomposing
an EEG signal into a set of intrinsic mode functions (IMF)
for mapping and reconstruction of the neuronal activity
of the brain. The use of EMD is motivated by the fact
that EEG signals are non-stationary and EMD is a data
dependent method exhibiting a better adaptability towards the
non-stationary nature of the EEG signals. The decomposition
Fig. 17. Brain mapping comparison of the proposed methods for the IMF-4 for DB-3.
allows to identify behaviors in each of the frequency
bands. Although the reconstruction has some undesirable
components, the results show clearly the two sources where
the activity was added in DB-1 and DB-2. For DB-3, it
was possible to identify the frequency range of interest. Due
to the mode mixing problem of EMD, it is possible that
one frequency component appears in different bands, which
is one of the drawbacks when using EMD. However, the
results obtained show a significantly improved reconstruction
of neural activity compared to conventional methods used for
the same purpose. Future research efforts will be dedicated
to the development of tools for mitigating the problem
of mode mixing in EEG signals to obtain an even better
sub-bands separation. The proposed methodology shows a
clear potential to separate EEG signals in sub-bands that
will facilitate the analysis of brain activity reconstruction,
by adaptively separating noise from signals. In addition,
based on the obtained results by applying a sub-band neural
reconstruction, it can be seen that the reconstructed neural
activity can be used for functional connectivity analysis by
sub-bands, which implies a great potential of the proposed
methodology as a tool for assisted diagnosis in brain related
disorders. The case studies shown in this paper have allowed
us to demonstrate that it is not always necessary to combine
EMD or EEMD with some statistical method (energy or
entropy function) to obtain a good reconstruction of the
neuronal activity.
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Noise-assisted multivariate empirical mode decomposition (NA-MEMD) is suitable to analyze multichannel electroencephalography (EEG) signals of non-stationarity and non-linearity natures due to the fact that it can provide a highly localized time-frequency representation. For a finite set of multivariate intrinsic mode functions (IMFs) decomposed by NA-MEMD, it still raises the question on how to identify IMFs that contain the information of inertest in an efficient way, and conventional approaches address it by use of prior knowledge. In this work, a novel identification method of relevant IMFs without prior information was proposed based on NA-MEMD and Jensen-Shannon distance (JSD) measure. A criterion of effective factor based on JSD was applied to select significant IMF scales. At each decomposition scale, three kinds of JSDs associated with the effective factor were evaluated: between IMF components from data and themselves, between IMF components from noise and themselves, and between IMF components from data and noise. The efficacy of the proposed method has been demonstrated by both computer simulations and motor imagery EEG data from BCI competition IV datasets.
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