406
Electroelwephalography and Clblical Neurophysiology, 1977, 42: 406--t 13
© Elsevier/North-Holland Scientific Publishers Ltd.
APPLICATIONS OF STATISTICAL EQUIVALENCE TO NEWBORN EEG RECORDINGS s. G]AQUINTO, F. MARCIANO, N. MONOD and G. NOLFE Laboratorio di Cibcrnctica del C.N.R., Arco Felice, Napoli (llaly), and Hdpital Porl Royal, Paris (France)
(Accepted for publication: July 5, 1976)
The newborn EEG differs noticeably from that recorded in the adult. In particular the basic activity is slower, the variations in time are more rapid, the stages of the waking--sleep cycle are rather unstable, the reaction to opening of the eyes is missing, an interhemispheric a s y m m e t r y is noticeable; certain features of the EEG are missing (e.g., spindles and K-complexes); finally the ratio of waking to sleep times is lower. According to Monod and Dreyfus-Brisac (1965) and Monod and Pajot (1965) most of the sleeping time of normal term newborns is composed of two sleep states: 1. Quiet sleep (QS), characterized by decreased motility, regular respiration, absence of rapid eye movements (REMs) and presence of tonic activity at the chin level. The electroencephalographic (EEG) pattern is either a "tracd a l t e r n a n t " (consisting of bursts of activity of 0.5--3 c/sec reaching 50--100 pV and appearing on a background of low voltage and faster activity of 4--8 c/sec) or a continuous high voltage slow activity. 2. A c t i v e sleep (AS) or REM sleep, characterized by increased motility, irregular respiretion, presence of REMs and absence of tonic activity at the chin level. The EEG pattern is either "activitd m o y e n n e " (consisting of continuous low voltage activity, below 50 pV with a d o min ant frequency of 5--6 c/sec) or mixed activity (high and low voltage). The remaining sleep may be classified as "intermediate sleep". The manual of standard terminology (Anders et al. 1971) accepts a
wider definition of QS and AS including for both states mixed activity and high voltage slow activity. The identification of sleep states, the study of their alternation and of their relative percentages, which are necessary in neuropaediatrics, are usually made by visual inspection, a qualitative method. The present article describes an a t t e m p t to introduce automatic discrimination of EEG records by means of spectral analysis, which is quantitative and reproducible. Methods of automatic analysis which have been employed, used mainly the autocorrelation function (Nolte et al. 1968; Prechtl 1968; Parmelee et al. 1969; Prechtl et al. 1969; Schulte and Bell 1973). Our procedure consists of a first order analysis followed by spectral analysis.
Methods E E G m e t h o d s . The research was carried out on 6 children born at term with an Apgar index of 10. EEGs were taken within the first 3 days of life after feeding and in controlled surroundings, by means of Ag--AgC1 cont act electrodes from right and left fronto-central (RFC, LFC) and right and left centro-occipital (RCO, LCO) derivations. The time constant was 0.3 sec, high frequencies were cut off at 70 c/sec. Records were made in parallel on paper and magnetic tape. In addition to the EEG, cardiac activity, respiration, eye movements and antigravition-
NEWBORN EEG RECORDINGS al tonic activity o f sub-mental muscles were r e c o r d e d on paper. Sleep states were classified o n the basis o f the clinical criteria described above. E E G patterns were c h o s e n b y visual analysis f r o m the p o r t i o n s o f the records with the m o s t h o m o geneous p a t t e r n s , e.g. " h i g h voltage s l o w " activity for QS and "activit~ moyenne" for AS. F o r the p u r p o s e o f the present research, a c o m p a r i s o n was t h e n m a d e b e t w e e n samples o f q u i e t sleep and o f REM phases. These samples, which are here o n w a r d s r e f e r r e d t o as " e p o c h s " , were a c c u r a t e l y selected and free f r o m artefacts. T h e analysis was carried o u t o n a 2116 HP c o m p u t e r with 16K m e m o r y , analogue to digital c o n v e r t e r , tape reader, t e l e t y p e , X--Y p l o t t e r , display and disk unit. Computing methods are described in the following paragraph. T h e analysis was p e r f o r m e d o n EEGs rec o r d e d f r o m f r o n t o - c e n t r a l and centro-occipital regions on b o t h right and left hemispheres. C o m p a r i s o n s were m a d e o f average spectra in the range 0.5--20 c/sec, this range was further divided in the f r e q u e n c y bands 0.5--4, 4--8, 8--12, 12--16, 1 6 - - 2 0 c/sec and comparisons were m a d e within each band.
Statistical approach to EEG A sample f u n c t i o n x ( t ) o f a s t a t i o n a r y process (x(t)} can be sliced into adjacent segm e n t s x~(t) o f d u r a t i o n T, p r o v i d e d t h a t T is larger t h a n the largest signal p e r i o d p r e s e n t in x(t). In this case the Xiw(t) can also be considered sample f u n c t i o n s o f (x(t)} (Wang and U h l e n b e c k 1 9 5 4 ) . C o n s e q u e n t l y , all estimates .n . • J o f a statistical p a r a m e t e r 4 , o b t a i n e d f r o m a n y o f the x~(t), m u s t be statistically equivalent, which is to say t h a t t h e y d i f f e r one f r o m the o t h e r b y o n l y t h e n o r m a l i z e d standard e r r o r er. If the medical classification of sleep states c o r r e s p o n d s to the s e p a r a t i o n o f the E E G rec o r d e d f r o m a given region o f the c o r t e x into parts each c o r r e s p o n d i n g to a well d e f i n e d s t a t i o n a r y stochastic process, it is t h e n possi-
407 ble to c h a r a c t e r i z e t h o s e parts by means o f a particular p o w e r s p e c t r u m . C o n s e q u e n t l y , after calculation o f the spectra it is possible to verify if the spectra calculated f r o m r statistically i n d e p e n d e n t e p o c h s during a particular sleep state, and r e c o r d e d f r o m a particular area o f the c o r t e x , are statistically equivalent estimates o f the same theoretical s p e c t r u m characteristic of t h a t state and o f t h a t area. If the test gives positive results the following niWi(f) W(f)
-
i= 1
~ni i=1
can be t a k e n as the characteristic s p e c t r u m . In A the e x p r e s s i o n above Wi(f), (i = 1, 2, . .... r) are the calculated spectra and ni the degrees o f f r e e d o m associated with each s p e c t r u m .
Methods of analysis In o r d e r to investigate the applicability o f spectral analysis for the a u t o m a t i c classificat i o n o f the E E G , a m e t h o d o l o g i c a l p r o c e d u r e was f o l l o w e d consisting o f the tests described below. Analysed e p o c h s had a d u r a t i o n of T = 72 sec and were sampled at intervals At = 8 msec. Only 64 sec o u t o f 72 sec were actually used for reading data into the c o m p u t e r , the d i f f e r e n c e being s p e n t for transferring d a t a f r o m c o m p u t e r t o disk storage.
1. First order analysis This is a first a p p r o x i m a t i o n test o f h o m o geneity. T h e d a t a were classed as belonging to: (i) a single e p o c h ; (ii) d i f f e r e n t epochs, b u t in the same state o f sleep. In case (i) it was necessary to verify the h o m o g e n e i t y o f the data belonging to the same e p o c h . F o r this p u r p o s e each e p o c h was divided into r n o n - c o n t i g u o u s segments o f d u r a t i o n T~, with n u m b e r o f samples Nr. A run test was p e r f o r m e d on each o f the r seg-
~10~
,S. G I A Q U I N T O ET AI~.
m e n t s ( B e n d a t and Piersol 1971) in o r d e r to verify its s t a t i o n a r i t y . E a c h s e g m e n t was furt h e r divided into c equal b u t separate intervals of d u r a t i o n T¢, with n u m b e r of samples Nc. S e p a r a t i o n was required in o r d e r to satisfy the r e s t r i c t i o n necessary for assuring the i n d e p e n d e n c e o f the d a t a in each interval. In case (ii) it was necessary to verify the h o m o g e n e i t y of the d a t a b e l o n g i n g to different e p o c h s b u t in the s a m e state o f sleep. T h e a b o v e m e n t i o n e d r s e g m e n t s c o i n c i d e d in this case with r entire e p o c h s and each e p o c h was divided in c intervals. T h e r e was o b v i o u s l y no need for the run test on the r e p o c h s b e c a u s e their h o m o g e n e i t y , and t h e r e f o r e their stat i o n a r i t y , was already p r o v e d b y the previous test. S u b s e q u e n t l y , average and variance values, indicated r e s p e c t i v e l y b y xii and o~i were calculated for each interval j in each r e c o r d i(j = 1, 2 .... c), (i = 1, 2, ... r). Finally, by using one w a y variance analysis (Afifi and Azen 1972), it was established w h e t h e r average and variance values differed m o r e t h a n the a m o u n t t h a t c o u l d be explained b y the h y p o t h e s i s . T h e values of the param e t e r s used are s u m m a r i z e d in T a b l e I in b o t h cases (i) and (ii).
2. Spectral analysis U l t i m a t e l y spectral analysis was applied to the p o r t i o n s of the E E G already selected b y visual o b s e r v a t i o n and a n a l y s e d by the a b o v e m e n t i o n e d test. T w o cases were c o n s i d e r e d , d e p e n d i n g on the d a t a chosen. (i) P o r t i o n s o f
the s a m e e p o c h . All selected e p o c h s were divided into t w o n o n - o v e r l a p p i n g r e c o r d s and f o r each r e c o r d the average p o w e r s p e c t r u m was calculated. (ii) D i f f e r e n t e p o c h s belonging to the same sleep state. The equivalence for the average spectra and t h e r e f o r e the second o r d e r h o m o g e n e i t y for each e p o c h were tested as described below.
Test of equivalence for spectral estimates. Spectral e s t i m a t e s were calculated by m e a n s of the fast F o u r i e r t r a n s f o r m ( F F T ) . E p o c h s x ( t ) o f d u r a t i o n T were divided in q s e g m e n t s of d u r a t i o n T'. Having chosen the c u t - o f f freq u e n c y fc a c c o r d i n g to the N y q u i s t criterion the s a m p l i n g interval is given b y A t = 1/2 f~,. T h e n u m b e r of s a m p l e s is given by N = T ' / A t . The spectral e s t i m a t e W(fk) is associated with a spectral r e s o l u t i o n of .Af = 2 f J N c/sec. An e s t i m a t e W(fk) of the entire e p o c h of d u r a t i o n T is o b t a i n e d by averaging the raw e s t i m a t e s Wi(fk) of the q s e g m e n t s of d u r a t i o n W': q
Wtfk) = 1 E Wi(fk} qi=l
with q = T / T ' . The spectral r e s o l u t i o n is still Af = 2 f J N and the degrees of f r e e d o m n are given b y n = 2q-= 2 T / T ' = 2 T / N • At = 2 . 2 f c . T / N = 2.Af.T T h e n o r m a l i z e d s t a n d a r d e r r o r w h i c h defines the r a n d o m p o r t i o n of the e s t i m a t i o n error is ( B e n d a t and Piersol 1971):
TABLE I Values of p a r a m e t e r s used in first order analysis. r
(i) (ii)
16 (x)
c
4 16
Tr
Tc
see
sec
4 64
0.8 4
Nr
Nc
512 8192
100 512
(x) Equal to the n u m b e r of available e p o c h s per subject.
The values chosen and t h o s e calculated for the p a r a m e t e r s i n t r o d u c e d a b o v e for the t w o cases to which spectral analysis was applied as m e n t i o n e d a b o v e ((a) w h e n each e p o c h was divided into t w o n o n - o v e r l a p p i n g parts; (b) w h e n an entire e p o c h was analysed) are summ a r i z e d in T a b l e If. Let ~/l(f) and W:(f) be t w o spectral estimates, o b t a i n e d either f r o m d i f f e r e n t p o r t i o n s
NEWBORN EEG RECORDINGS
409
TABLE II Values of parameters used in spectral analysis. T sec
T' sec
fc c/sec
At msec
N
Af c/sec
q
n
Cr
32 64
2 4
64 64
8 8
256 512
0.5 0.25
16 16
32 32
0.25 0.25
(a) (b)
o f the same e p o c h or f r o m d i f f e r e n t epochs. It is possible to test w h e t h e r t h e y are statistically e q u i v a l e n t estimates o f the same t h e o r e t ical s p e c t r u m characteristic o f the process f r o m w h i c h the E E G orginates. This can be c o n s i d e r e d a m o r e c o m p r e h e n s i v e test o f hom o g e n e i t y . F o r high values o f the degrees o f f r e e d o m n, and n2 (/> 30) relative to W,(f) and ~ : ( f ) the spectral estimates have an app r o x i m a t e l y n o r m a l sampling d i s t r i b u t i o n and the f o l l o w i n g a p p r o x i m a t i o n can be m a d e E[l~i(f)] = Wi(f ) V A R [ ~ / i ( f ) ] = 2 W~ (f)
i = 1, 2
11i-
Results
w h e r e W,(f) and ~/2(f) are estimates o f the same t h e o r e t i c a l s p e c t r u m . The r a n d o m variable ( B e n d a t and Piersol 1 9 7 1 ) : -I12
D =
E(
Nf ~
+ ~:
w h e r e fk = k • A f and Nt = (fmax--fmin)/Af, m u s t be n o r m a l l y d i s t r i b u t e d with average value zero and s t a n d a r d deviation equal to 1. Nf represents the n u m b e r o f f r e q u e n c y samples necessary in o r d e r t o cover the selected freq u e n c y range. A t a c h o s e n level o f significance ~ the h y p o t h e s i s W,(f) = W2(f) = W(f) is a c c e p t e d if --z,/2 ~< D ~< z -/2 and it is rejected f o r a value o f D outside this range. T h e values --zo h and z~h are read f r o m a statistical table o f areas u n d e r s t a n d a r d i z e d n o r m a l distribution. C h o o s i n g for o u r p u r p o s e a = 0.05, the region o f a c c e p t a n c e is - - 1 . 9 6 ~< D ~< 1.96.
Nf
W~
k=l
W2(fk)
F o r case (i), defined in the previous paragraphs, the results are s u m m a r i z e d in Table III, w h i c h shows the n u m b e r o f times t h a t each o f the a b o v e m e n t i o n e d tests gave positive results. The tests were run in the following s e q u e n c e : first o r d e r analysis and spectral equivalence.
TABLE III Percentage number of cases relative to all subjects which passed the two tests. Derivations
RFC RCO LFC LCO
First order analysis
Spectral equivalence
QS
AS
QS
AS
89 95 79 79
93 79 80 70
74 (84) 79 (95) 71 (79) 64 (79)
86 (93) 64 (79) 80 60 (70)
410
S. (;IAQUINTO ET AL.
TABLE IV Average (over all subjects) percentage number of cases oJ' spectral equivalence in the same regions but in difl~erent epochs of the same sleep state Frequency band (c/sec)
0.5--20 0.5-
Electroelwephalography and Clblical Neurophysiology, 1977, 42: 406--t 13
© Elsevier/North-Holland Scientific Publishers Ltd.
APPLICATIONS OF STATISTICAL EQUIVALENCE TO NEWBORN EEG RECORDINGS s. G]AQUINTO, F. MARCIANO, N. MONOD and G. NOLFE Laboratorio di Cibcrnctica del C.N.R., Arco Felice, Napoli (llaly), and Hdpital Porl Royal, Paris (France)
(Accepted for publication: July 5, 1976)
The newborn EEG differs noticeably from that recorded in the adult. In particular the basic activity is slower, the variations in time are more rapid, the stages of the waking--sleep cycle are rather unstable, the reaction to opening of the eyes is missing, an interhemispheric a s y m m e t r y is noticeable; certain features of the EEG are missing (e.g., spindles and K-complexes); finally the ratio of waking to sleep times is lower. According to Monod and Dreyfus-Brisac (1965) and Monod and Pajot (1965) most of the sleeping time of normal term newborns is composed of two sleep states: 1. Quiet sleep (QS), characterized by decreased motility, regular respiration, absence of rapid eye movements (REMs) and presence of tonic activity at the chin level. The electroencephalographic (EEG) pattern is either a "tracd a l t e r n a n t " (consisting of bursts of activity of 0.5--3 c/sec reaching 50--100 pV and appearing on a background of low voltage and faster activity of 4--8 c/sec) or a continuous high voltage slow activity. 2. A c t i v e sleep (AS) or REM sleep, characterized by increased motility, irregular respiretion, presence of REMs and absence of tonic activity at the chin level. The EEG pattern is either "activitd m o y e n n e " (consisting of continuous low voltage activity, below 50 pV with a d o min ant frequency of 5--6 c/sec) or mixed activity (high and low voltage). The remaining sleep may be classified as "intermediate sleep". The manual of standard terminology (Anders et al. 1971) accepts a
wider definition of QS and AS including for both states mixed activity and high voltage slow activity. The identification of sleep states, the study of their alternation and of their relative percentages, which are necessary in neuropaediatrics, are usually made by visual inspection, a qualitative method. The present article describes an a t t e m p t to introduce automatic discrimination of EEG records by means of spectral analysis, which is quantitative and reproducible. Methods of automatic analysis which have been employed, used mainly the autocorrelation function (Nolte et al. 1968; Prechtl 1968; Parmelee et al. 1969; Prechtl et al. 1969; Schulte and Bell 1973). Our procedure consists of a first order analysis followed by spectral analysis.
Methods E E G m e t h o d s . The research was carried out on 6 children born at term with an Apgar index of 10. EEGs were taken within the first 3 days of life after feeding and in controlled surroundings, by means of Ag--AgC1 cont act electrodes from right and left fronto-central (RFC, LFC) and right and left centro-occipital (RCO, LCO) derivations. The time constant was 0.3 sec, high frequencies were cut off at 70 c/sec. Records were made in parallel on paper and magnetic tape. In addition to the EEG, cardiac activity, respiration, eye movements and antigravition-
NEWBORN EEG RECORDINGS al tonic activity o f sub-mental muscles were r e c o r d e d on paper. Sleep states were classified o n the basis o f the clinical criteria described above. E E G patterns were c h o s e n b y visual analysis f r o m the p o r t i o n s o f the records with the m o s t h o m o geneous p a t t e r n s , e.g. " h i g h voltage s l o w " activity for QS and "activit~ moyenne" for AS. F o r the p u r p o s e o f the present research, a c o m p a r i s o n was t h e n m a d e b e t w e e n samples o f q u i e t sleep and o f REM phases. These samples, which are here o n w a r d s r e f e r r e d t o as " e p o c h s " , were a c c u r a t e l y selected and free f r o m artefacts. T h e analysis was carried o u t o n a 2116 HP c o m p u t e r with 16K m e m o r y , analogue to digital c o n v e r t e r , tape reader, t e l e t y p e , X--Y p l o t t e r , display and disk unit. Computing methods are described in the following paragraph. T h e analysis was p e r f o r m e d o n EEGs rec o r d e d f r o m f r o n t o - c e n t r a l and centro-occipital regions on b o t h right and left hemispheres. C o m p a r i s o n s were m a d e o f average spectra in the range 0.5--20 c/sec, this range was further divided in the f r e q u e n c y bands 0.5--4, 4--8, 8--12, 12--16, 1 6 - - 2 0 c/sec and comparisons were m a d e within each band.
Statistical approach to EEG A sample f u n c t i o n x ( t ) o f a s t a t i o n a r y process (x(t)} can be sliced into adjacent segm e n t s x~(t) o f d u r a t i o n T, p r o v i d e d t h a t T is larger t h a n the largest signal p e r i o d p r e s e n t in x(t). In this case the Xiw(t) can also be considered sample f u n c t i o n s o f (x(t)} (Wang and U h l e n b e c k 1 9 5 4 ) . C o n s e q u e n t l y , all estimates .n . • J o f a statistical p a r a m e t e r 4 , o b t a i n e d f r o m a n y o f the x~(t), m u s t be statistically equivalent, which is to say t h a t t h e y d i f f e r one f r o m the o t h e r b y o n l y t h e n o r m a l i z e d standard e r r o r er. If the medical classification of sleep states c o r r e s p o n d s to the s e p a r a t i o n o f the E E G rec o r d e d f r o m a given region o f the c o r t e x into parts each c o r r e s p o n d i n g to a well d e f i n e d s t a t i o n a r y stochastic process, it is t h e n possi-
407 ble to c h a r a c t e r i z e t h o s e parts by means o f a particular p o w e r s p e c t r u m . C o n s e q u e n t l y , after calculation o f the spectra it is possible to verify if the spectra calculated f r o m r statistically i n d e p e n d e n t e p o c h s during a particular sleep state, and r e c o r d e d f r o m a particular area o f the c o r t e x , are statistically equivalent estimates o f the same theoretical s p e c t r u m characteristic of t h a t state and o f t h a t area. If the test gives positive results the following niWi(f) W(f)
-
i= 1
~ni i=1
can be t a k e n as the characteristic s p e c t r u m . In A the e x p r e s s i o n above Wi(f), (i = 1, 2, . .... r) are the calculated spectra and ni the degrees o f f r e e d o m associated with each s p e c t r u m .
Methods of analysis In o r d e r to investigate the applicability o f spectral analysis for the a u t o m a t i c classificat i o n o f the E E G , a m e t h o d o l o g i c a l p r o c e d u r e was f o l l o w e d consisting o f the tests described below. Analysed e p o c h s had a d u r a t i o n of T = 72 sec and were sampled at intervals At = 8 msec. Only 64 sec o u t o f 72 sec were actually used for reading data into the c o m p u t e r , the d i f f e r e n c e being s p e n t for transferring d a t a f r o m c o m p u t e r t o disk storage.
1. First order analysis This is a first a p p r o x i m a t i o n test o f h o m o geneity. T h e d a t a were classed as belonging to: (i) a single e p o c h ; (ii) d i f f e r e n t epochs, b u t in the same state o f sleep. In case (i) it was necessary to verify the h o m o g e n e i t y o f the data belonging to the same e p o c h . F o r this p u r p o s e each e p o c h was divided into r n o n - c o n t i g u o u s segments o f d u r a t i o n T~, with n u m b e r o f samples Nr. A run test was p e r f o r m e d on each o f the r seg-
~10~
,S. G I A Q U I N T O ET AI~.
m e n t s ( B e n d a t and Piersol 1971) in o r d e r to verify its s t a t i o n a r i t y . E a c h s e g m e n t was furt h e r divided into c equal b u t separate intervals of d u r a t i o n T¢, with n u m b e r of samples Nc. S e p a r a t i o n was required in o r d e r to satisfy the r e s t r i c t i o n necessary for assuring the i n d e p e n d e n c e o f the d a t a in each interval. In case (ii) it was necessary to verify the h o m o g e n e i t y of the d a t a b e l o n g i n g to different e p o c h s b u t in the s a m e state o f sleep. T h e a b o v e m e n t i o n e d r s e g m e n t s c o i n c i d e d in this case with r entire e p o c h s and each e p o c h was divided in c intervals. T h e r e was o b v i o u s l y no need for the run test on the r e p o c h s b e c a u s e their h o m o g e n e i t y , and t h e r e f o r e their stat i o n a r i t y , was already p r o v e d b y the previous test. S u b s e q u e n t l y , average and variance values, indicated r e s p e c t i v e l y b y xii and o~i were calculated for each interval j in each r e c o r d i(j = 1, 2 .... c), (i = 1, 2, ... r). Finally, by using one w a y variance analysis (Afifi and Azen 1972), it was established w h e t h e r average and variance values differed m o r e t h a n the a m o u n t t h a t c o u l d be explained b y the h y p o t h e s i s . T h e values of the param e t e r s used are s u m m a r i z e d in T a b l e I in b o t h cases (i) and (ii).
2. Spectral analysis U l t i m a t e l y spectral analysis was applied to the p o r t i o n s of the E E G already selected b y visual o b s e r v a t i o n and a n a l y s e d by the a b o v e m e n t i o n e d test. T w o cases were c o n s i d e r e d , d e p e n d i n g on the d a t a chosen. (i) P o r t i o n s o f
the s a m e e p o c h . All selected e p o c h s were divided into t w o n o n - o v e r l a p p i n g r e c o r d s and f o r each r e c o r d the average p o w e r s p e c t r u m was calculated. (ii) D i f f e r e n t e p o c h s belonging to the same sleep state. The equivalence for the average spectra and t h e r e f o r e the second o r d e r h o m o g e n e i t y for each e p o c h were tested as described below.
Test of equivalence for spectral estimates. Spectral e s t i m a t e s were calculated by m e a n s of the fast F o u r i e r t r a n s f o r m ( F F T ) . E p o c h s x ( t ) o f d u r a t i o n T were divided in q s e g m e n t s of d u r a t i o n T'. Having chosen the c u t - o f f freq u e n c y fc a c c o r d i n g to the N y q u i s t criterion the s a m p l i n g interval is given b y A t = 1/2 f~,. T h e n u m b e r of s a m p l e s is given by N = T ' / A t . The spectral e s t i m a t e W(fk) is associated with a spectral r e s o l u t i o n of .Af = 2 f J N c/sec. An e s t i m a t e W(fk) of the entire e p o c h of d u r a t i o n T is o b t a i n e d by averaging the raw e s t i m a t e s Wi(fk) of the q s e g m e n t s of d u r a t i o n W': q
Wtfk) = 1 E Wi(fk} qi=l
with q = T / T ' . The spectral r e s o l u t i o n is still Af = 2 f J N and the degrees of f r e e d o m n are given b y n = 2q-= 2 T / T ' = 2 T / N • At = 2 . 2 f c . T / N = 2.Af.T T h e n o r m a l i z e d s t a n d a r d e r r o r w h i c h defines the r a n d o m p o r t i o n of the e s t i m a t i o n error is ( B e n d a t and Piersol 1971):
TABLE I Values of p a r a m e t e r s used in first order analysis. r
(i) (ii)
16 (x)
c
4 16
Tr
Tc
see
sec
4 64
0.8 4
Nr
Nc
512 8192
100 512
(x) Equal to the n u m b e r of available e p o c h s per subject.
The values chosen and t h o s e calculated for the p a r a m e t e r s i n t r o d u c e d a b o v e for the t w o cases to which spectral analysis was applied as m e n t i o n e d a b o v e ((a) w h e n each e p o c h was divided into t w o n o n - o v e r l a p p i n g parts; (b) w h e n an entire e p o c h was analysed) are summ a r i z e d in T a b l e If. Let ~/l(f) and W:(f) be t w o spectral estimates, o b t a i n e d either f r o m d i f f e r e n t p o r t i o n s
NEWBORN EEG RECORDINGS
409
TABLE II Values of parameters used in spectral analysis. T sec
T' sec
fc c/sec
At msec
N
Af c/sec
q
n
Cr
32 64
2 4
64 64
8 8
256 512
0.5 0.25
16 16
32 32
0.25 0.25
(a) (b)
o f the same e p o c h or f r o m d i f f e r e n t epochs. It is possible to test w h e t h e r t h e y are statistically e q u i v a l e n t estimates o f the same t h e o r e t ical s p e c t r u m characteristic o f the process f r o m w h i c h the E E G orginates. This can be c o n s i d e r e d a m o r e c o m p r e h e n s i v e test o f hom o g e n e i t y . F o r high values o f the degrees o f f r e e d o m n, and n2 (/> 30) relative to W,(f) and ~ : ( f ) the spectral estimates have an app r o x i m a t e l y n o r m a l sampling d i s t r i b u t i o n and the f o l l o w i n g a p p r o x i m a t i o n can be m a d e E[l~i(f)] = Wi(f ) V A R [ ~ / i ( f ) ] = 2 W~ (f)
i = 1, 2
11i-
Results
w h e r e W,(f) and ~/2(f) are estimates o f the same t h e o r e t i c a l s p e c t r u m . The r a n d o m variable ( B e n d a t and Piersol 1 9 7 1 ) : -I12
D =
E(
Nf ~
+ ~:
w h e r e fk = k • A f and Nt = (fmax--fmin)/Af, m u s t be n o r m a l l y d i s t r i b u t e d with average value zero and s t a n d a r d deviation equal to 1. Nf represents the n u m b e r o f f r e q u e n c y samples necessary in o r d e r t o cover the selected freq u e n c y range. A t a c h o s e n level o f significance ~ the h y p o t h e s i s W,(f) = W2(f) = W(f) is a c c e p t e d if --z,/2 ~< D ~< z -/2 and it is rejected f o r a value o f D outside this range. T h e values --zo h and z~h are read f r o m a statistical table o f areas u n d e r s t a n d a r d i z e d n o r m a l distribution. C h o o s i n g for o u r p u r p o s e a = 0.05, the region o f a c c e p t a n c e is - - 1 . 9 6 ~< D ~< 1.96.
Nf
W~
k=l
W2(fk)
F o r case (i), defined in the previous paragraphs, the results are s u m m a r i z e d in Table III, w h i c h shows the n u m b e r o f times t h a t each o f the a b o v e m e n t i o n e d tests gave positive results. The tests were run in the following s e q u e n c e : first o r d e r analysis and spectral equivalence.
TABLE III Percentage number of cases relative to all subjects which passed the two tests. Derivations
RFC RCO LFC LCO
First order analysis
Spectral equivalence
QS
AS
QS
AS
89 95 79 79
93 79 80 70
74 (84) 79 (95) 71 (79) 64 (79)
86 (93) 64 (79) 80 60 (70)
410
S. (;IAQUINTO ET AL.
TABLE IV Average (over all subjects) percentage number of cases oJ' spectral equivalence in the same regions but in difl~erent epochs of the same sleep state Frequency band (c/sec)
0.5--20 0.5-
