Transcript Slide 1

# 669 M-AM
Parallel Factor Analysis as an exploratory tool for wavelet transformed event related EEG
Morten Mørup1, Lars Kai Hansen1, Sidse M. Arnfred2
1) Informatics and Mathematical Modeling, Technical University of Denmark
e-mail: [email protected]
2) Department of Psychiatry, Hvidovre Hospital, University Hospital of Copenhagen, Denmark.
Intelligent Signal Processing
Abstract
In the decomposition of multi-channel EEG signals principal component analysis (PCA) and independent component analysis (ICA) have widely been used. However, as both methods
are based on handling two-way data, i.e. two-dimensional matrices - multi-way methods might improve the interpretation of frequency transformed multi-channel EEG of channel x
frequency x time data. On this poster the multi-way decomposition method Parallel Factor (PARAFAC) also named Canonical Decomposition (CANDECOMP) is used for the first time
to decompose wavelet transformed event related EEG. The PARAFAC decomposition is able to extract the expected features of a previously reported ERP paradigm: namely a
quantitative difference of coherent occipital gamma activity. Furthermore, a scheme on how to do data exploration using PARAFAC is given.
PARAFAC
Wavelet Analysis
The use of PARAFAC in the analysis of EEG and ERP is not new. In his original paper on PARAFAC Harshman in
1970 suggested PARAFAC to be used to decompose the EEG (Harshman, 1970). But it was not before 1988, when
Möcks reinvented the model, naming it topographic component analysis, that PARAFAC was used to analyze the
ERP of channel x time x subject (Möcks, 1988) an idea that was further pursued by Field (Field and Graupe, 1991).
In 2004 Miwakeichi suggested the use of PARAFAC on the wavelet transformed ongoing EEG of channel x
frequency x time. It was here shown how PARAFAC was capable of successfully identifying the theta and alpha
atoms of a cognitive task and that the decomposition method could identify eye blinks (Miwakeichi et al., 2004). In
this poster it is however to our knowledge the first time PARAFAC is used to analyze the wavelet transformed data
of the event related EEG.
Consider the complex Morlet wavelet:
2

t
~
 t  
exp i 2 c t  exp  
 b
 b
1



- Real part
- Imaginary part
The Parallel Factor (PARAFAC) model was independently proposed by Harshman (Harshman, 1970) and by Carrol
and Chang (Carrol and Chang, 1970), the latter naming it Canonical Decomposition (CANDECOMP). The model is
a parsimonious extension of the factor analysis to higher orders
The Complex Morlet wavelet
The continuous wavelet transform of a signal x(t) by the wavelet  is given by:
1 ~ t  p  
 a, p, t  


a  a 

C ( scale, shift) 
 xt  scale, shift, t dt
*
where

Shifting a wavelet by p means delaying or hastening its onset. The scale, a, is related to the frequency, , of a signal by:
c

a
Wavelet transform
3-way array of channel x frequency x time
Multi channel EEG of channel x time
Like the factor analysis, PARAFAC decomposes the data into factor effects pertaining to each modality. F
denotes the number of factors. For higher orders than 3 the PARAFAC model can be stated as:
channel
channel
F
frequency
xi1i2iN   ai1 ai2  aiN   ei1i2iN
time
time
 1
Consider the 3-way array XI x J x K, and let XI x j x K be the matrix corresponding to the jth slice of X. In matrix the
factor analysis and PARAFAC can be expressed as:
X
I J
A
I F
S
J F T
E
I J
X
I  jK
A
I F
F F
Dj
S
K F T
E
The continuous wavelet transform converts an EEG-data matrix of channel x time into a 3-way array of
channel x frequency x time here shown as a 16x4 array where each array corresponds to the time-frequency plot of
one of the EEG channels shown to the left.
I  jK
Where Dj is a diagonal matrix corresponding to the jth row of D. Rotating the factor analysis solution by P and
the PARAFAC solution by the matrices P and Q gives:

X  AS  AP P S
T
1 T

~ ~T
 AS
X
I  j K
 AD j S  X
T
I  j K


 AP  P D j Q Q S
1
1 T

Let Xe(c,f,t) be the coefficient of the wavelet transform at channel c at frequency f and time t for epoch e, and let there be a total of n
~ ~ ~T
 AD j S
ITPC(c, f , t ) 
~
However, since D j has to be a diagonal matrix for all j, P-1DjQ has to be diagonal which in practise restricts P and
Q to only be each a scaling and permutation matrix. As a result, the PARAFAC model is in general unique apart
from scaling and permutation indeterminacies. Kruskal gave in 1977 a very mild uniqueness condition. The result
makes use of the k-rank, kB, which is given by the smallest subset of columns of the matrix B that are linearly
independent (Kruskal, 1977):
k A  k D  k S  2F  2
The most common way of estimating the PARAFAC model is by alternating least squares. In this approach a cost
function (normally the squared error) is minimized in order to explain most of the variation in the data. This is done
by alternating between re-estimating each parameter given the estimation of the other parameters. The algorithm can
be initialized in several ways i.e. by randomly defining all parameters and stopped when all parameters have
converged. For a description of this simple but popular algorithm confer (Bro, 1998).
n
1
n

e1
X e c, f , t 
X e c, f , t 
The inter trial phase coherence is a measure of evoked activity. An ITPC value of one indicates perfect phase coherence in all epochs
1
n
An ANOVA will be used to investigate the difference between the evoked activities given by the ITPC of two
conditions. Let ITPC(c,f,t,k,s) be the ITPC at channel c at frequency f at time t for subject s in condition k. Let there
be a total of K conditions and S subjects. The ANOVA F-test value Z(c,f,t) of difference between the two conditions
at channel c at frequency f at time t is then given by:
K
Z ( c, f , t ) 
2


S
I
(
c
,
f
,
t
,
k
)

I
(
c
,
f
,
t
)
/ K  1

k 1
K
S
where
 ITPC(c, f , t , s, k )  I c, f , t , k  / S  K 
k 1 s 1
2
1 S
I c, f , t , k    ITPCc, f , t , s, k 
S s 1
1
I c, f , t  
KS
K
S
 ITPCc, f , t , s, k 
k 1 s 1
Ob
Flow chart of the analysis. The ERP was wavelet transformed using a complex Morlet wavelet with center
frequency c=1 and bandwidth parameter b=2. The measure of interest (MOI) i.e. the ITPC was calculated for
each subject under each condition and a 3-way array of the ANOVA F-test value calculated. The F-test array was
analyzed using PARAFAC and the region of interest (ROI, here in the time-frequency domain) of most difference
between the two conditions identified. The ITPC in the ROI was then analyzed using PARAFAC on the 5-way array
given by channel x frequency x time x subject x condition of ITPC values. Finally, the ITPC of the ROI of each
subject under each condition given by the 3-way array of channel x frequency x time was also analyzed by
PARAFAC and the individual peaks of the ITPC identified for each subject under each condition.
Nob
The grand average of the topographies corresponding to the individual peak time and frequency for each condition of each subject is
Discussion
• From the PARAFAC analysis of the ANOVA F-test values it is evident that the difference between the Ob and
Nob condition is in the gamma band of the occipital region around 100 ms as expected by the paradigm. It seems
Experimental details
as ifdrawings:
the explorative
application
of PARAFAC
the ANOVAevery-day
F-test values
give a solid
if and
whereorina pipe, and 2. Non-ob
The stimulus paradigm has been described in detail previously (Herrmann et al., 2004). Briefly it consists of two types of black and white
1. Objects
(Ob), which
are easilyon
recognizable
typecan
of objects
like aidea
chair,
a number
the three-dimensional array a region of difference between conditions are present.
•The PARAFAC decomposition of the 5-way array of ITPC values reveals both quantitative and qualitative effects.
Furthermore, the main effect is as expected the quantitative difference between Ob and Nob. Consequently, the 5way PARAFAC endorse novel and easy applicable comprehensive view of data, which has not previously been
seen.
Results
•The PARAFAC decomposition plot of the ITPC of each individual subject in each condition enables easy read off;
a procedure which here enables the construction of a cross modality grand average topography again substantiating
the previously reported condition difference.
•In this poster evoked activity was of interest, consequently the measure of interest (MOI) was the ITPC. Yet the
PARAFAC is expected to perform just as successful on other MOI. As a result, the presented flow chart is
considered applicable to a wide range of analyses of the wavelet transformed event related EEG.
•PARAFAC might become an important tool in the analysis of a wide range of brain data. The scheme developed
here is obviously applicable also to MEG data, and we think it is worthwhile to mention that PARAFAC has
previously
been
applied
to fMRI
et al., 2005).
Consequently,
lie ahead
of thethat the main differe
The 3-way array analysis of the ANOVA test values Z(c,f,t) of difference between the ITPC of the Ob and Nob conditions taken over the
11 subjects
yielded
a good
fit in(Beckmann
a one component
PARAFAC
model. Asgreat
seenpotentials
on the figure
above,initterms
is evident
analysis of brain data using PARAFAC.
Hz
Following this indication of region of interest (ROI) subsequent analyses were performed in a narrower frequency and
latency window (Frequency 30-80 Hz; Time 0-200ms); the number of channels were not restricted.
Hz
ms
Subject
Hz
ms
Condition
ms
Presently, we work on applying PARAFAC to other paradigms. The figure above shows the result of a PARAFAC
analysis of the ITPC of a subject that during even conditions had his right hand pulled and during odd the left
The ITPC 5-way array ITPC(c,f,t,s,k) of channel x frequency x time x subject x condition, shown to the left of the above figure, could only entail a two component PARAFAC model. As seen to the right, the first component encompassed occipital activity at app. 30 Hz
hand. The analysis was here performed to yield time-frequency plots. Consequently, the analyzed multi-way had
the form channel x time-frequency x condition. From the PARAFAC analysis clear evoked gamma activity around
60 ms and 40 Hz is seen in the parietal right region during stimulation of left hand (odd conditions) whereas the
parietal left region was stimulated primarily during even conditions where right hand was pulled.
Hz
ms
Subject
Condition
Conclusion
PARAFAC decomposition is a promising data exploratory tool in the analysis of the wavelets transformed event related EEG. The m
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When the ITPC was decomposed in each condition of every subject (a 3-way array of channel x frequency x time), a one component PARAFAC model yielded a good fit. This resulted in 22 decomposition plots of a subjects ITPC given by channel x frequency x time.