Poster Session II
ARMA ORDER SELECTION FOR EEG - AN EMPIRICAL COMPARISON
O F THREE ORDER SELECTION ALGORITHMS
Prasad Gannabathula
I.S.N.
Murthy
Department of E l e c t r i c a l Engineering
I n d i a n I n s t i t u t e of S c i e n c e
B a n g a l o r e 560 0 1 2 , I N D I A
ABSTRACT
The p e r f o r m a n c e o f t h r e e ARMA o r d e r e s t i mation algorithms, i)Canonical Correlation
a n a l y s i s b)S-array and c)Franke algorithm
on s i m u l a t e d a n d r e a l EEG s i g n a l s i s p r e sented.
I t i s shown t h a t t h e S - a r r a y a l ways c o r r e c t l y i n d i c a t e s t h e AR o r d e r b u t
makes i n c o r r e c t e s t i m a t e s o f t h e MA o r d e r .
The c a n o n i c a l c o r r e l a t i o n method i d e n t i f i e s t h e model o r d e r c o r r e c t l y f o r simul a t e d d a t a , atid o v e r e s t i m a t e s t h e AR o r d e r
on r e a l d a t a . The F r a n k e a l g o r i t h m i s
shown t o p e r f o r m p o o r l y i n c o m p a r i s o n t o
t h e o t h e r two a l g o r i t h m s .
INTRODUCTION
Modelling
t h e EEG s i g n a l s u s i n g ARMA
models r e q u i r e s a p r i o r i knowledge o f t h e
o r d e r s o f t h e polynomials and have t o be
A simple
e s t i m a t e d from t h e d a t a i t s e l f .
method would b e t o e x a m i n e t h e s p e c t r u m o f
A n o r m a l EEG h a s a t most f o u r
the signal.
component w a v e s ,
the delta, alpha,
beta
a n d t h e t a waves w i t h f r e q u e n c y r a n g e s of
0-lOHz,
5-14Hz, 10-28Hz a n d 2-7Hz r e s p e c tively.
F o r a n EEG s i g n a l w i t h a p e a k a t
t h e o r i g i n a n d two p e a k s b e t w e e n 1 t o 30
Hz i n t h e s p e c t r u m ,
Zetterberg suggested
t h e u s e of a n ARMA (5,4) m o d e l .
However,
i n p r a c t i c e t h e s p e c t r u m o f r e a l EEG may
n o t show
a l l t h e component w a v e s .
This
may h a p p e n when a component wave h a s low
e n e r g y and wide bandwidth, o v e r l a p p i n g
w i t h t h e o t h e r w a v e s f o r is e n t i r e l y a b s ent.
I n such c a s e s t h e o r d e r s of t h e AR
a n d MA p o l y n o m i a l s a r e n o t i m m e d i a t e l y
obvious or a p p a r e n t .
I n t h i s paper w e c o n s i d e r t h i s problem
of
ARMA model i d e n t i f i c a t i o n when a p p l i e d t o
s i m u l a t e d a n d r e a l EEG s i g n a l s a n d s t u d y
the
performance
of
three
general
a l g o r i t h m s t o estimate t h e o r d e r .
The
a l g o r i t h m s allow t h e s i g n a l t o be weakly
s t a t i o n a r y ( i . e ) allow t h e system function
t o h a v e p o l e s on t h e u n i t c i r c l e . W e a r e
concerned only with order estimation,
the
a c t u a l e s t i m a t i o n of t h e c o e f f i c i e n t s is
not d e a l t with.
ARMA IDENTIFICATION
A l l the algorithms that identify the
ARMA
model u s e two d i f f e r e n t a p p r o a c h e s . I n t h e
f i r s t , t h e o r d e r is e s t i m a t e d by examining
some f u n c t i o n o f t h e r e s i d u a l v a r i a n c e o f
t h e s i g n a l , e s t i m a t e d by maximizing t h e
l i k e l i h o o d f u n c t i o n o f a p t h o r d e r AR a n d
q t h o r d e r MA m o d e l . T h e s e a l g o r i t h m s a r e
c o m p u t a t i o n a l l y e x p e n s i v e a s a number o f
o p t i m a l ARMA m o d e l s o f d i f f e r e n t o r d e r s
have t o b e examined. I n s p i t e o f t h i s ,
t h e r e is no a s s u r a n c e t h a t t h e correct
model w i l l b e i d e n t i f i e d . T h e s e a l g o r i t h m s
were n o t u s e d i n t h e s t u d y .
I n t h e second class of algorithms,
two
t y p e s o f a l g o r i t h m s , which d o n o t r e q u i r e
o p t i m a l estimates o f t h e r e s i d u a l v a r i a n c e
or t h e c o e f f i c i e n t s are used. I n t h e
Franke a l g o r i t h m [ 2 ] t h e r e s i d u a l v a r i a n c e
is
r e c u r s i v e l y e s t i m a t e d by
Levinson
recursion.
The s u b o p t i m a l e s t i m a t e s t h u s
o b t a i n e d a r e used i n t h e A I C or BIC.
The
o t h e r two a l g o r i t h m s u s e d , e x p l o i t t h e
c o r r e l a t i o n p r o p e r t i e s of t h e d a t a t o
estimate t h e o r d e r . I n a l l t h e s e a l g o r i t is assumed t h a t t h e s i g n a l is
ithms,
weakly s t a t i o n a r y ( i . e . ) h a s a l l i t s p o l e s
w i t h i n t h e u n i t c i r c l e . T h i s is t o t a k e
i n t o account the f i n i t e precision a r i t h metic u s e d t o c o m p u t e t h e c o r r e l a t i o n
c o e f f i c i e n t s and t h e i r f u n c t i o n s f o r s i g n a l s w i t h p o l e s which a r e a r b i t r a r i l y
close to but still inside the u n i t circle.
We h a v e u s e d two a l g o r i t h m s , o n e S - a r r a y
a s proposed by Gray e t . a 1 . [ 3 ]
and t h e
other using canonical correlat,ion analysis
[ 4 ] . The S - a r r a y o f G r a y a l s o i n d i c a t e s
t h e number o f p o l e s w h i c h a r e on t h e u n i t
circle.
D e t a i l s o f t h e s e two a l g o r i t h m s
are given i n [ 5 ] . I n t h e next s e c t i o n t h e
p e r f o r m a n c e o f t h e s e t h r e e a l g o r i t h m s when
a p p l i e d t o s i m u l a t e d a n d r e a l EEG d a t a i s
presented. For simulated d a t a , t h e e f f e c t
of d a t a s i z e , t h e p o s i t i o n of p o l e s and
z e r o s is a l s o i n v e s t i g a t e d .
APPLICATION TO SIMULATED EEG
The EEG s i g n a l s were g e n e r a t e d a ) b y p a s s ing Gaussian white noise through a s i n g l e
f i f t h o r d ? r A R a n d f o u r t h o r d e r MA f i l t e r
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and b)as the sum of the outputs of three
independent filters each driven by an
independent white noise. A sampling rate
of 100 was used. The EEG signals were
simulated with various centre frequencies
and bandwidths and with varying energy
contents in the component waves.
The S array identifies the AR order correctly even when the data size is as small
as 2.5 seconds and the poles are close to
the unit circle. The MA order is however,
not always correctly identified. In particular, when the EEG signal is filtered to
remove the effect of a pole(s) on or very
close to the unit circle, the identified
MA order is far too low. Further, when the
pole has large bandwidth and low energy
the AR order is incorrectly identified.
When the bandwidths of alpha and beta wave
are between 0.65 and 1.275 and atleast 12%
energy in each wave, the S array gives a
correct estimate of the AR order but under
estimates the MA order by one. When band
widths of both alpha and beta waves are
large (>.2Hz), and the energy in either
wave is less than half the energy in the
other, the two waves are not separately
identifiable and the S array underestimates the AR order.
The canonical correlation analysis correctly identifies the AR portion of the
model for data of 5 seconds duration or
more. The estimated AR coefficients were
consistent and stable. For the MA part the
order was correctly estimated for data of
more than 5 seconds duration. For records
of 5 second duration, the MA order was
underestimated by one. The MA order is
also incorrectly identified when the component waves are very close to each other,
or if they have wide bandwidths, and if
one wave masks the presence of other.
Otherwise the algorithm is insensitive to
the percentage energy under the curve and
the location of the poles.
An interesting feature of this algorithm
is that it is insensitive to the method of
simulation of EEG when the half power
points of the component waves are small(
(0.7Hz
for the alpha and beta waves and
(1.25Hz for the delta wave).
Franke's algorithm overestimated the AR
order and underestimated the MA order. For
all data lengths of 125, 250 and 500
samples, the estimated AR order is either
7 or 8 while the MA order varied from 0 to
3, and is uneffected by the pole location
or the energy under the component wave.
4 REAL DATA
The algorithms were applied on a number of
normal EEG signals. The maximum length of
data which can be considered stationary
was determined by using the run test[61.
This length was found to be 11 seconds.
The three algorithms were run on records
of 11,10,7.5, and 5 seconds duration.
The S array works well for records a s
short a s 2.5 seconds. The AR orders were
either 3 or 5. An order of 5 implies the
presence of a beta wave, which was not
visible in the spectra of the signals. The
MA orders are highly variable and range
from 1 to 4. In most cases the S array
identified a pole very close to the unit
circle.
The canonical correlation overestimates
tha AR order. The MA order as 4. In most
cases the AR order identified is 6 . This
is higher than the order identified by the
S array. Further, the minimum date size
needed to get an estimate of the order is
7.5 seconds, compared to the 2.5 seconds
required by the S array. Assuming the
presence of a beta component, that could
not be identified by the S array, an extra
pole has been identified. In all cases the
estimated
AR coefficients were found to
be consistent. Filtering the data to remove the effect of frequencies beyond 25Hz
identifies models that are in close agreement with the models identified by the S
array. The AR coefficients, thus estimated
are however inconsistent.
The Levinson-Durbin recursion in all cases
performed poorly. A 7th or 8th order AR
model was identified. The MA portion, if
identified,had an order of one or two.
CONCLUSIONS
Three algorithms, Franke's algorithm for
ARMA models, the S array and the canonical
correlation analysis were used for identification of the ARMA model for the EEG
signal. The algorithms were tested on a
large number of simulated and real EEG
data.
REFERENCES
1. L .H. Zetterberg ,
"Estimation of parameters for linear difference
equation
with
application
to
EEG",
Math.
Biosici, 5, pp227-275,1969
2. J.Franke, "A Levinson-Durbin recursion
for
Autoregressive-Moving
Average
process"
Biometrical
72,
3,pp573581,1985
3. A.Gray et.al,"A new approach to ARMA
modelling", Comm. of Stast. ,B,7,ppl77 ,1978
4. R.S.Tsay and G.C.Tiao, "Use of Canonical analysis in time
series model
identification", BiometricaI72,2,pp299315,1985.
5. G.S.S.Durga
Prasad and 1.S.N.Murthy:
Technical report, Indian Institute of
Science. 1988
6. J.B.Bendot and A.G.Pearso1, Random data
analysis, Wiley, New York, 1974.
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