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MULTI-DIMENSIONAL SPACE-TIME-FREQUENCY COMPONENT ANALYSISOF EVENT RELATED EEG DATA USING CLOSED-FORM PARAFAC
Martin Weis, Florian R¨ omer, Martin Haardt
Ilmenau University of TechnologyCommunications Research LaboratoryD-98684 Ilmenau, Germany
Dunja Jannek, Peter Husar
Ilmenau University of TechnologyBiosignal Processing GroupD-98684 Ilmenau, Germany
Abstract —
The ef
Þ
cient analysis of electroencephalographic(EEG) data is a long standing problem in neuroscience, whichhas regained new interest due to the possibilities of multidimen-sional signal processing. We analyze event related multi-channelEEG recordings on the basis of the time-varying spectrum foreach channel. It is a common approach to use wavelet trans-formations for the time-frequency analysis (TFA) of the data.To identify the signal components we decompose the data intotime-frequency-space atoms using Parallel Factor (PARAFAC)analysis. Inthispaper we showthat a TFAbased onthe Wigner-Ville distribution together with the recently developed closed-formPARAFACalgorithmenhancetheseparabilityof thesignalcomponents. This renders it an attractive approach for process-ing EEG data. Additionally, we introduce the new concept of component amplitudes, which resolve the scaling ambiguity inthe PARAFAC model and can be used to judge the relevance of the individual components.
Index Terms
—
Tensor, Multi-dimensional signal processing,PARAFAC, Event Related EEG, Wigner-Ville Distribution
1. INTRODUCTION
In this contribution we focus on analyzing measured electroen-cephalographic (EEG) data to
Þ
nd the components of speci
Þ
c ac-tivity on the scalp. This analysis can also be used to detect andlocalize epileptic seizure onset zones on the scalp as well as pro- jections of cognitive processing like speech or auditory handling.Unfortunately, different sources in the brain can produce the sameEEG pattern on the scalp, which renders them in general non-separable. Source localization algorithms, such as LORETA [11]or dipole
Þ
tting methods can resolve this ambiguity by imposingadditional assumptions. For further improvements of these methods, preprocessing in form of subspace decompositions, e.g., Princi- ple Component Analysis (PCA), Independent Component Analysis(ICA), Singular Value Decomposition (SVD), or beamforming al-gorithms [6] have been applied. However, these methods cannotexploit the multi-dimensional nature of the EEG data. Moreover,to obtain matrix decompositions like PCA or ICA, physically irrel-evant assumptions like orthogonality or independence have to beimposed. Therefore, tensor decompositions are a more promisingapproach to handle EEG signals. Especially the well known Par-allel Factor (PARAFAC) analysis is a powerful tool for analyzingEEG data, because it is essentially unique under mild conditions [1]without any arti
Þ
cial constraints, such as orthogonality. In the lastyears PARAFAC was applied to EEG signals, e.g., for estimatingsources of cognitive processing [9], for the analysis of event related potentials (ERP) [10], and for epileptic seizure localization [16].
In order to resolve the temporal evolution as well as the fre-quency content of the EEG recordings, a time-frequency analysis(TFA) is applied for each channel. Therefore, the data is analyzedover three dimensions, i.e., time, frequency, and space (channels).Different TFA algorithms have been studied for the analysis of EEGsignals [5]. The most common time-frequency decomposition is thecontinuous wavelet transformation (CWT) [14]. However, we haveshown in [7] that wavelet analysis may not provide adequate timeand frequency resolution for EEG data.In this contribution we use a TFA method based on the Wigner-Ville distribution. Thereby, we suppress the effect of cross terms byusing the reduced interference distribution [5]. This method showsa signi
Þ
cantly improved time-frequency resolution and thereforealso improves the PARAFAC analysis. We compare the resultsto the standard wavelet based techniques. For the computation of the PARAFAC decomposition the most common methods to dateare based on iterative alternating least squares (ALS) algorithms.However, these algorithms may require many iterations and arenot guaranteed to converge to the global minimum. The recentlydeveloped closed-form PARAFAC algorithm [12, 13] outperforms
the iterative approaches. Therefore, we use it to decompose thetime-frequency distributions into time-frequency-space atoms andto identify the different signal components of the EEG data.This paper is organized as follows: In Section 2 we clarify thenotation and de
Þ
ne the operators and symbols that are used. In Sec-tion 3 we discuss the signal processing steps to process EEG sig-nals. Then, Section 3.1 brie
ß
y presents the methods for the time-frequency analysis of the EEG data and Section 3.2 describes theclosed-form PARAFAC decomposition. Here we also show how toresolve the scaling ambiguities in the PARAFAC model. In Sec-tion 4 we present the results of the event related EEG analysis basedon measurements, before drawing the conclusions in Section 5.
2. NOTATION
To facilitate the distinction between scalars, vectors, matrices, andhigher-order tensors, we use the following notation: scalars aredenoted by lower-case italic letters
(
a,b,...
)
, vectors by boldfacelower-case italic letters
(
a
,
b
,...
)
, matrices by boldface upper-case letters
(
A
,
B
,...
)
, and tensors are denoted as upper-case, boldface, calligraphic letters
(
A
,
B
,...
)
. This notation is consis-tently used for lower-order parts of a given structure. For example
A
∈
C
I
1
×
I
2
×···×
I
N
represents an
N
-dimensional tensor of size
I
n
along mode
n
. Its elements are referenced by
a
i
1
,i
2
,...,i
N
for
i
n
= 1
,
2
,...I
n
and
n
= 1
,
2
,...,N
. Furthermore, the
i
-th col-umn vector of a matrix
A
is denoted as
a
i
. For matrices we use thesuperscripts
T
,
H
,
−
1
,
+
for transposition, Hermitian transposition,matrix inverse, and Moore-Penrose pseudo-inverse, respectively.The Kronecker product and the Khatri-Rao product (column-wiseKronecker product) of two matrices
A
and
B
are expressed by
A
⊗
B
and
A
⋄
B
, respectively.The tensor operations we use are consistent with [8]. Thehigher-order norm of a tensor
A
, symbolized by
A
H
, is de-
349978-1-4244-2354-5/09/$25.00 ©2009 IEEE ICASSP 2009
Þ
ned as the square root of the sum of the squared magnitude of allelements in
A
. The
n
-mode vectors of a tensor
A
are obtained by varying the
n
-th index
i
n
of the tensor elements
a
i
1
,i
2
,...,i
N
within its range
(1
,
2
,...,I
n
)
while keeping all the other indices
Þ
xed. The matrix unfolding of the tensor
A
, denoted by
[
A
]
(
n
)
∈
C
I
n
×
I
1
·
...
·
I
n
−
1
·
I
n
+1
·
...
·
I
N
contains all the
n
-mode vectors of thetensor
A
. The
n
-mode product of a tensor
A
∈
C
I
1
×···×
I
N
and amatrix
U
∈
C
J
n
×
I
n
isdenoted as
(
A
×
n
U
)
∈
C
I
1
×···×
J
n
×···×
I
N
.It is obtained by multiplying all
n
-mode vectors of
A
from the lefthand side by the matrix
U
. The outer product of an
N
-dimensionaltensor
A
and a
K
-dimensional tensor
B
, denoted by
(
A
◦
B
)
,is a
(
N
+
K
)
-dimensional tensor whose elements are given by
(
A
◦
B
)
i
1
,...,i
N
,j
1
,...j
K
=
a
i
1
,...,i
N
·
b
j
1
,...j
K
. An
N
-dimensionaltensor
A
∈
C
I
1
×···×
I
N
is of rank one if and only if it can be writtenas the outer product between
N
non-zero vectors
c
(
n
)
∈
C
M
n
, suchthat
A
=
c
(1)
◦
...
◦
c
(
N
)
. The three-dimensional identity tensor
I
3
,d
is de
Þ
ned as
I
3
,d
=
d
n
=1
e
n,d
◦
e
n,d
◦
e
n,d
∈
R
d
×
d
×
d
,
(1)where
e
n,d
represents the
n
-th column of a
d
×
d
identity matrix(also termed the
n
-th pinning vector of size
d
).
3. SIGNAL PROCESSING STEPS
The processing of EEG data is a very challenging task due to thecomplexnatureof thesesignals, e.g., theyarenon-stationary andsuf-fer from very low signal to noise ratios. Moreover, they are affected by correlated noise with unknown distribution and artifacts srcinat-ing from eye blinks, eye movements, and muscle movements as wellas from diverse technical and biological distortions. Therefore, a
MeasuredEEG dataPrepro-cessingTime-frequencyanalysisPARAFACanalysis
Fig. 1
.
Signal processing steps for the identi
Þ
cation of signal componentsin event-related EEG data.
suitable preprocessing has to be applied in the form of
Þ
lters, ref-erence EEG channels, and averaging over numerous trials. Further-more, the EEG data has to be divided into smaller stationary timewindows. Afterwards, the time-frequency analysis is applied to eachchannel individually, in order to resolve the temporal evolution aswell as the frequency content of the EEG data. The components of the resulting three-way signal, which changes in frequency, space(channels), and time, are extracted via parallel factor (PARAFAC)analysis (see Figure 1).
3.1. Time-Frequency Analysis
There exist a large number of methods that can be applied to de-compose EEG signals into their time-frequency content [5]. An ap- proach that is very often used is the continuous wavelet transforma-tion (CWT). The continuous wavelet transform
C
(
a,τ
)
at scale
a
of a signal
x
(
t
)
is de
Þ
ned as
C
(
a,τ
) =
∞
−∞
x
(
t
)
ϕ
(
a,t,τ
) d
t ,
(2)where
ϕ
is the chosen wavelet. Common choices include the class of biorthogonal wavelets, Debauchy wavelets, and the Morlet wavelets[14]. The connection between the scale
a
and the frequency
f
isgiven by
f
≈
f
c
a
·
∆
t ,
(3)where
f
c
is the center frequency of the wavelet and
∆
t
is thesampling interval for
x
(
t
)
. The disadvantage of CWT-based time-frequency preprocessing is the limited resolution, especially in thelow-frequency region, which is very important in EEG signal analy-sis.A more powerful approach to time-frequency analysis is given by the family of Wigner-Ville distribution functions, based on theseminal work of Wigner [17] in 1932 and Ville [15] in 1948. The
distribution is based on the temporal correlation function (TCF)
q
x
(
t,τ
)
of the signal
x
(
t
)
which is de
Þ
ned as [5]
q
x
(
t,τ
) =
x
(
t
+
τ
2)
x
∗
(
t
−
τ
2)
.
(4)The Wigner-Villedistribution (WVD)
W
x
(
t,f
)
of
x
(
t
)
is de
Þ
ned asthe Fourier transform of the TCF with respect to the lag variable
τ W
x
(
t,f
) =
∞
−∞
q
x
(
t,τ
)
e
−
j
2
πfτ
d
τ .
(5)Therefore, the WVD is a quadratic, real-valued time-frequency dis-tribution (TFD). The ambiguity function
A
x
(
θ,τ
)
is symmetric in
τ
and is de
Þ
ned as the inverse Fourier transform of the TCF withrespect to the time
t
[5]
A
x
(
θ,τ
) =
∞
−∞
q
x
(
t,τ
)
e
j
2
πθt
d
t .
(6)Thus, the ambiguity function and the Wigner-Ville distribution arerelated by the two-dimensional Fourier transform. The main draw- back of the time-frequency analysis based on the TCF is that it pro-duces cross terms in
W
x
(
t,f
)
as well as in
A
x
(
θ,τ
)
. On the other hand, its advantage is that time and frequency resolution can be ad- justedseparately. In1966 Cohen introduced anoverall classof TFDs based on the WVD which allow the use of kernel functions for re-ducing cross terms [4]. This group of TFDs
P
x
(
t,f
)
is de
Þ
ned as
P
x
(
t,f
) =
∞
−∞∞
−∞
A
x
(
θ,τ
)Θ(
θ,τ
)
e
−
j
2
πθt
−
j
2
πτf
d
θ
d
τ ,
(7)where
Θ(
θ,τ
)
is the kernel function. A large number of TFDs have been proposed, each differing only in the choice of
Θ(
θ,τ
)
. Thesekernel functions can be used to suppress the effect of the cross termson the TFD. Choi and Williams [3] introduced the reduced interfer-ence distribution (RID), which is a TFD based on the exponentialkernel function
K
(
θ,τ
) =
e
−
θ
2
τ
2
σ
,
(8)where
σ >
0
is a scaling factor which in
ß
uences the cross termsuppression.
3.2. Three-Way PARAFAC Analysis
After the time-frequency analysis the EEG data is represented bytime-varying frequency distributions for every channel. This three-way data can be expressed in form of a tensor
X
∈
R
N
F
×
N
T
×
N
C
,
(9)where
N
F
and
N
T
are the number of samples in frequency and time,and
N
C
is the number channels, respectively. In order to separatethe signal components in this tensor, we use a multi-dimensionalextension of the singular value decomposition that is known as thePARAFAC decomposition [1]. Thereby, we decompose a tensor into
350
a minimal sum of rank one components. In the absence of noise, thePARAFAC model for the tensor (9) can be represented as
X
=
d
n
=1
a
n
◦
b
n
◦
c
n
,
(10)where the vectors
a
n
∈
R
N
F
,
b
n
∈
R
N
T
, and
c
n
∈
R
N
C
, repre-sent the frequency, time, and space (channel) signatures of the
n
-thPARAFAC component. Moreover,
d
represents the number of signalcomponents. In practice the PARAFAC model does not
Þ
t the dataexactly for a number of reasons:
•
The measured EEG data is affected by correlated noise withan unknown distribution. The signal to noise ratiois very low.
•
The signal components are not necessarily rank one. Thenumber of signal components is unknown.
•
The superposition of the components is not linear.Therefore, we require a robust algorithm for the computation of an approximate
Þ
t of the PARAFAC model to the data tensor
X
.Among the many existing PARAFAC methods we propose to use therecently developed closed-form PARAFAC algorithm [12, 13]. This
algorithm is based on the higher order singular value decomposition(HOSVD) of
X
which is de
Þ
ned as [8]
X
=
S
×
1
U
1
×
2
U
2
×
3
U
3
,
(11)where
S
∈
R
N
F
×
N
T
×
N
C
is the full core tensor of same size as
X
. The unitary matrices
U
1
∈
R
N
F
×
N
F
,
U
2
∈
R
N
T
×
N
T
and
U
3
∈
R
N
C
×
N
C
provide an orthonormal basis for the 1-, 2-, and 3-mode vector spaces of
X
, respectively. Thus, the HOSVDcan easily be obtained from the matrix singular value decomposition of the
n
-mode matrix unfoldings of
X
[8]. In the non-degenerate case (
d
≤
min
{
N
F
,N
T
,N
C
}
) the HOSVD of the tensor
X
can be truncatedto
X
=
S
[
d
]
×
1
U
[
d
]1
×
2
U
[
d
]2
×
3
U
[
d
]3
,
(12)where
S
[
d
]
∈
R
d
×
d
×
d
and where
U
[
d
]1
,
U
[
d
]2
and
U
[
d
]3
are of size
(
N
F
×
d
)
,
(
N
T
×
d
)
, and
(
N
C
×
d
)
, respectively. By de
Þ
ning theset of matrices
A
= [
a
1
,...
a
d
]
∈
R
N
F
×
d
,
B
= [
b
1
,...
b
d
]
∈
R
N
T
×
d
, and
C
= [
c
1
,...
c
d
]
∈
R
N
C
×
d
we can rewrite thePARAFAC model (10) in terms of the identity tensor
I
3
,d
X
=
I
3
,d
×
1
A
×
2
B
×
3
C
.
(13)Comparing the equations (12) and (13) indicates that there is a link between the PARAFAC model and the HOSVD. To exploit this con-nection we introduce the transformation matrices
T
1
∈
R
d
×
d
,
T
2
∈
R
d
×
d
, and
T
3
∈
R
d
×
d
such that
A
=
U
[
d
]1
·
T
1
,
B
=
U
[
d
]2
·
T
2
,
C
=
U
[
d
]3
·
T
3
.
(14)Inserting these equations into (13) and comparing it with (12) yields
S
[
d
]
×
1
T
−
11
×
2
T
−
12
×
3
T
−
13
=
I
3
,d
.
(15)Therefore, theclosed-form PARAFAC algorithmestimatesthe trans-formation matrices that diagonalize the truncated core tensor
S
[
d
]
to the identity tensor
I
3
,d
. In [12, 13] it is shown that this can
be accomplished very ef
Þ
ciently by means of joint matrix diago-nalizations, also in the degenerate case. The resulting closed-formalgorithm outperforms iterative approaches especially in critical sce-narios, since it does not require alternating least squares iterations.Moreover, it provides the opportunity to obtain a tradeoff betweenaccuracy and computational time.
Fig. 2
.
Time evolution of all 64 EEG channels. The data is averaged over 1600 trials of a 20 ms light
ß
ash to the right eye of a 23 years old healthywoman. We can see that the occipital channels show the response earlier than the frontal ones.
3.2.1. Scaling ambiguity in PARAFAC
The PARAFAC model (10) is unique under mild conditions up toa scaling ambiguity for the component vectors
a
n
,
b
n
, and
c
n
anda permutation of the components
a
n
◦
b
n
◦
c
n
. Due to the non-orthogonality of the PARAFAC decomposition, the higher order norms of the component tensors
a
n
◦
b
n
◦
c
n
do not add up to thehigher-order norm of
X
. Bro [1] suggested to judge the in
ß
uence of each of the components based on
X
−
a
n
◦
b
n
◦
c
n
H
. Becauseof the unknown dependency between the components, we suggestto
Þ
t all components jointly to the srcinal data tensor
X
in a leastsquares sense. Therefore, we normalize all component vectors tounit Frobenius norm, such that
a
n
=
a
n
a
n
F
,
b
n
=
b
n
b
n
F
,
c
n
=
c
n
c
n
F
∀
n
= 1
,...,d.
(16) Note that this normalization leads to
a
n
◦
b
n
◦
c
n
H
= 1
for all
n
= 1
,...,d
. Next we introduce the PARAFAC componentamplitudes
γ
n
for
n
= 1
,...,d
by
X
≈
d
n
=1
γ
n
·
a
n
◦
b
n
◦
c
n
.
(17)To determine all amplitudes jointly we rewrite this equation accord-ing to
vec(
X
) =
vec
a
1
◦
b
1
◦
c
1
,...,
vec
a
d
◦
b
d
◦
c
d
·
γ
=
C
⋄
B
⋄
A
γ
,
(18)where the matrices
A
= [
a
1
,...,
a
d
]
,
B
= [
b
1
,...,
b
d
]
, and
C
= [
c
1
,...,
c
d
]
contain the normalized component vectors. Thevector
γ
= [
γ
1
,...,γ
d
]
T
contains all component amplitudes. Theleast squares solution for the set of linear equations (18) is given by
γ
=
C
⋄
B
⋄
A
+
vec(
X
)
.
(19)In practical applications the PARAFAC model often does not exactly
Þ
t the data, and no apriori knowledge can be used to resolve thescaling and permutation ambiguity. In these cases we suggest to judge the in
ß
uence of the components based on the magnitudes of the component amplitudes
γ
n
. Please notice that for the real valuedcasethenormalized model (17) stillhas asign ambiguity, e.g., twoof the three component vectors can be multiplied by minus one withoutchanging the rank one component.
351
4. EXPERIMENTAL RESULTS
The EEG signal is recorded from a 23 year old woman, healthy andright-handed. The position of the 64 EEG electrodes is based onthe international 10-10-system [2] with earlobe references
[(
A
1 +
A
2)
/
2]
. The sampling frequency is chosen to 1000 Hz. For the pre- processing of the raw signal, several
Þ
lters are applied: a 7 Hz high- pass, a135 Hzlow-pass and a band-stop
Þ
lterbetween 45 and 55 Hz.For the investigation of effects in the
Þ
eld of event related potentials,we record EEG data triggered as a function of a visual stimulus. Thesubject sits in front of a hemispherical perimeter. The stimulus is a20 ms central light
ß
ash from a white LED to the right eye. The trig-gered EEG responses to this stimulus are averaged over 1600 trialsfor all channels (see Figure 2). For the signal component analysis
time [ms]
f r e q u e n c y [ H z ]
time [ms]
f r e q u e n c y [ H z ]
time [ms]
f r e q u e n c y [ H z ]
008000800080
808080
Fig.3
.
Signal components for the TFA based on CWT with Morlet wavelets.The components are represented as topographic plots of the space signatures(top), together with the time-frequency signatures (bottom). The bars left toeach component represent the relative maginitude of the PARAFAC ampli-tudes. The analysis window reaches from 101 to 180 ms.
time [ms]
f r e q u e n c y [ H z ]
time [ms]
f r e q u e n c y [ H z ]
time [ms]
f r e q u e n c y [ H z ]
000000808080500500500
Fig. 4
.
Signal components for the TFA based on the reduced interferencedistribution. The analysis window reaches from 101 to 180 ms. Due to theincreased time-frequency resolution the desired components are clearly re-vealed.
we divide the recorded EEG data into windows of length 80 ms toassure stationarity. In each window we apply two different methodsfor the time-frequency analysis: the CWT based on Morlet waveletsand the reduced interference distribution (RID, see Section 3.1). Af-terwards, we use the closed-form PARAFAC algorithm (Section 3.2)to identify the signal components. The PARAFAC model is nor-malized according to equation (17). The number of components isset to three, and they are ordered according to the magnitude of thePARAFAC amplitudes de
Þ
ned in Section 3.2.1. Figure 3 shows theresults for the time window between 100 ms and 180 ms based onthe CWT with Morlet wavelets. The components are represented asa topographic plot of the space signatures, together with the associ-atedtime-frequency signature. Thebar ontheleftof eachcomponentrepresents the relative magnitude of its PARAFAC amplitude. From previous studies based on potential mapping, a strong component inthe lower right hemisphere (visual cortex) is expected. Because of the insuf
Þ
cient time resolution for low frequencies and the small fre-quency resolution for high frequencies, the desired component can-not be identi
Þ
ed with the CWT. However, the desired component isclearly represented in the results based on the RID time-frequencyanalysis (Figure 4). The comparison of the time-frequency signa-tures of both results clearly reveals the improved time and frequencyresolution of the RID. This leads to an increased spatial resolutionof the signal components.
5. CONCLUSIONS
In this contribution we have shown that an appropriate time-frequency analysis (TFA) scheme is an important factor for theidenti
Þ
cation of signal components in EEG data. We have shownthat Wigner-Ville distribution based TFA methods provide an in-creased time-frequency resolution, which leads to an increasedspatial resolution of the signal components. The effect of crossterms can be suppressed by using the reduced interference distribu-tion (RID). This technique provides particularly instructive results incombination with the closed-form PARAFAC algorithm to identifythe signal components in measured event related EEG data. In order to judge the in
ß
uence of the different components we have intro-duced the novel component amplitudes, which resolve the scalingambiguity in the PARAFAC model.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the partial support of the inter-nal excellence initiative at Ilmenau University of Technology.
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