Poster Session II
ARMA ORDER SELECTION FOR EEG
-
AN EMPIRICAL COMPARISONO F THREE ORDER SELECTION ALGORITHMS
Prasad Gannabathula I.S.N. Murthy
D e p a r t m e n t o f E l e c t r i c a l E n g i n e e r i n g I n d i a n I n s t i t u t e o f 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 - m a t i o n a l g o r i t h m s , i ) C a n o n i c a l C o r r e l a t i o n a n a l y s i s b ) S - a r r a y a n d c ) F r a n k e a l g o r i t h m o n 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 - s e n t e d . I t i s shown t h a t t h e S - a r r a y a l - w a y s 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 m a k e s 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 m e t h o d i d e n t i - f i e s t h e m o d e l o r d e r c o r r e c t l y f o r s i m u - l 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 o n r e a l d a t a . T h e 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
M o d e l l i n g t h e EEG s i g n a l s u s i n g ARMA m o d e l s r e q u i r e s a p r i o r i k n o w l e d g e o f t h e o r d e r s o f t h e p o l y n o m i a l s a n d h a v e t o b e e s t i m a t e d f r o m t h e d a t a i t s e l f . A s i m p l e m e t h o d w o u l d b e t o e x a m i n e t h e s p e c t r u m o f t h e s i g n a l . A n o r m a l EEG h a s a t most f o u r c o m p o n e n t w a v e s , t h e d e l t a , a l p h a , b e t a a n d t h e t a w a v e s 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 - t i v e l y . 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 , Z e t t e r b e r g s u g g e s t e d t h e u s e o f 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 c o m p o n e n t w a v e s . T h i s may h a p p e n when a c o m p o n e n t wave h a s low e n e r g y a n d w i d e b a n d w i d t h , 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 i s e n t i r e l y a b s - e n t . I n s u c h c a s e s t h e o r d e r s o f t h e A R 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 o b v i o u s o r a p p a r e n t .
I n t h i s p a p e r w e c o n s i d e r t h i s p r o b l e m o f ARMA m o d e l 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 t h e p e r f o r m a n c e o f t h r e e g e n e r a l a l g o r i t h m s t o e s t i m a t e t h e o r d e r . The a l g o r i t h m s a l l o w t h e s i g n a l t o b e w e a k l y s t a t i o n a r y ( i . e ) a l l o w t h e s y s t e m f u n c t i o n t o h a v e p o l e s o n t h e u n i t c i r c l e . W e a r e c o n c e r n e d o n l y w i t h o r d e r e s t i m a t i o n , t h e a c t u a l e s t i m a t i o n o f t h e c o e f f i c i e n t s i s n o t d e a l t w i t h .
ARMA IDENTIFICATION
A l l t h e a l g o r i t h m s t h a t i d e n t i f y t h e 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 i s e s t i m a t e d b y e x a m i n i n g 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 b y m a x i m i z i n g 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 h a v e t o b e e x a m i n e d . I n s p i t e o f t h i s , t h e r e i s n o a s s u r a n c e t h a t t h e c o r r e c t 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 s e c o n d c l a s s o f a l g o r i t h m s , two t y p e s o f a l g o r i t h m s , w h i c h d o n o t r e q u i r e o p t i m a l e s t i m a t e s o f t h e r e s i d u a l v a r i a n c e o r t h e c o e f f i c i e n t s a r e u s e d . I n t h e F r a n k e 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 i s r e c u r s i v e l y e s t i m a t e d b y L e v i n s o n r e c u r s i o n . T h e 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 u s e d i n t h e A I C o r B I C . T h e 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 e s t i m a t e 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 h m s , i t i s a s s u m e d t h a t t h e s i g n a l i s w e a k l y 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 i s t o t a k e i n t o a c c o u n t t h e f i n i t e p r e c i s i o n a r i t h - m e t i c 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 a n d 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 w h i c h a r e a r b i t r a r i l y c l o s e t o b u t s t i l l i n s i d e t h e u n i t c i r c l e . 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 p r o p o s e d b y G r a y e t . a 1 . [ 3 ] a n d t h e o t h e r u s i n g c a n o n i c a l c o r r e l a t , i o n a n a l y s i s [ 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 c i r c l e . 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 a r e g i v e n i n [ 5 ] . I n t h e n e x t 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 p r e s e n t e d . F o r s i m u l a t e d d a t a , t h e e f f e c t o f d a t a s i z e , t h e p o s i t i o n o f p o l e s a n d z e r o s i s 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 - i n g G a u s s i a n w h i t e n o i s e t h r o u g h 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
1686--1EEE ElOaINBEPIMG I11 XEDICINB & BIOLOGY SOCIETY llTH ANNUAL IHTERNATIOHAL ~ " F E ~ C B CH2770-6/89/0000-1686 $01.00 C 1989 IEEE
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 cor- rectly even when the data size is a s small as 2.5 seconds and the poles are close to the unit circle. The MA order is however, not always correctly identified. In parti- cular, 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 underesti- mates the AR order.
The canonical correlation analysis cor- rectly 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 com- ponent waves are very close to each other, or if they have wide bandwidths, and i f 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 a s 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 re- move the effect of frequencies beyond 25Hz identifies models that are in close agree- ment 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 identi- fication of the ARMA model for the EEG signal. The algorithms were tested on a large number of simulated and real EEG data.
1.
2.
3.
4.
5.
6.
REFERENCES
L .H. Zetterberg
,
"Estimation of para- meters for linear difference equation with application to EEG", Math.Biosici, 5, pp227-275,1969
J.Franke, "A Levinson-Durbin recursion for Autoregressive-Moving Average process" Biometrical 72, 3,pp573- 581,1985
A.Gray et.al,"A new approach to ARMA modelling", Comm. of Stast. ,B,7,ppl- 77
,
1978R.S.Tsay and G.C.Tiao, "Use of Canoni- cal analysis in time series model identification", BiometricaI72,2,pp299- 315,1985.
G.S.S.Durga Prasad and 1.S.N.Murthy:
Technical report, Indian Institute of Science. 1988
J.B.Bendot and A.G.Pearso1, Random data analysis, Wiley, New York, 1974.
IEEE EBIGIISEEPI#Q I# MBDICIISE L BIOLOGY BOCIETY llTB ANNUAL I#TERNATIO#AL COB!FERENCE--l687
