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Towards computer-based analysis of clinical electroencephalograms Doyle, Daniel John 1974

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TOWARDS COMPUTER-BASED ANALYSIS OF CLINICAL ELECTROENCEPHALOGRAMS by D. JOHN DOYLE B.Sc. (Hons.)> St. Francis Xavier U n i v e r s i t y , 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of E l e c t r i c a l Engineering We accept t h i s t hesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1974 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y of B r i t i s h C olumbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date thbb&r 9,<??4 i ABSTRACT Two approaches to the automatic analysis of clinical electro-encephalograms (EEGs) are considered with a view towards classifying clin ical EEGs as normal or abnormal. The first approach examines the varia-bility of various EEG features in a population of astronaut candidates known to be free of neurological disorders by constructing histograms of these features; unclassified EEGs of subjects in the same age group are examined by comparison of their feature values to the histograms of this neurologically normal group. The second approach employs the techniques of automatic pattern recognition for classification of clinical EEGs. A set of 57 EEG records designated normal or abnormal by clinical electro-encephalographers are used to evaluate pattern recognition systems based on stepwise discriminant analysis. In particular, the efficacy of using various feature sets in such pattern recognition systems is evaluated in terms of estimated classification error probabilities (P e). The results of the study suggest a potential for the development of satisfactory automatic systems for the classification of clinical EEGs. i i TABLE OF CONTENTS j , Page ABSTRACT ; . . . j i 1 TABLE OF CONTENTS ... i i LIST OF ILLUSTRATIONS ... i i i LIST OF TABLES i v ACKNOWLEDGEMENT v 1. INTRODUCTION J; . 1 2. DATA ACQUISITION AND PREPROCESSING ' 4 2.1 Sources of EEG Data 4 2.2 Abnormalities Present i n the EEG Data 6 2.3 Instrumentation 8 2.4 Preprocessing 9 3. THE NORMATIVE MODEL APPROACH TO EEG ANALYSIS 11 3.1 Introduction 11 ^ *) Q n n r f n c r\f- " H o f - a . . . . . . ^ 2 3.3 Feature Selection 12 3.4 Results 13 3.5 Discussion 24 4. THE PATTERN RECOGNITION APPROACH TO EEG ANALYSIS 26 4.1 Introduction 26 4.2 Stepwise Discriminant Analysis 29 4.3 Feature S e l e c t i o n 29 4.4 Evaluation of Pattern Recognition Systems 34 4.5 Discussion 37 5. CONCLUSIONS ; 42 5.1 Conclusions 42 5.2 Suggestions f o r Further Work 42 APPENDIX A 45 APPENDIX B 46 APPENDIX C 50 APPENDIX D 53 APPENDIX E 61 REFERENCES 65 i i i LIST OF ILLUSTRATIONS Figure Page(s) 2.1 International 10-20 system of electrode placement . . . . 5 2.2 Examples of some EEG abnormalities . . . . . . . . . . . . 7 2.3 Data acquisition system . 10 3.1.. Histogram series for spectral features 14-19 3.2 Histogram series for coherence features 20-22 3.3 Relative delta band spectral power in occipital region for four EEG records having excessive occipital slow wave activity 23 4.1 Schematic diagram of pattern recognition system . . . . . 27 iv LIST OF TABLES Table Page I EEG channels used in this study . 4 II EEG channels used in the normative model . . . . . . . . 12 III Features used in the normative mode 12 IV Different feature sets used in pattern recognition systems 33 V Performance of cl a s s i f i e r using NASA data base 35 VI Peformance of cl a s s i f i e r using NASA data plus hospital data 36 ACKNOWLEDGEMENT I would like to thank my supervisor, Dr. G.B. Anderson, for his continuous support and guidance throughout the course of this research. I would like to express my sincere gratitude to Dr. M.D. Low of the Department of Electroencephalography, Vancouver General Hospital for providing the necessary c l i n i c a l f a c i l i t i e s and making many valuable sug-gestions. I would also like to thank my friend Mr. M.E. Koombes for pro-viding the software and hardware for the A/D conversion of the analog EEG data. I am indebted to Dr. J.D. Frost, Jr. of the LBJ Space Center, Houston, Texas for providing 49 of the 57 sets of EEG records used in this study, to Gord McConnell for rerecording the data provided by Dr. Frost, to Jim McEwen for invaluable criticisms and comments and to Ms. Shelagh Lund for her efficient and patient typing of the manuscript. Finally, I would like to thank a l l my colleagues at the E.E. Department, particularly Ms. Carol Lane and Mr. Doug Wasson for invaluable assistance, Ms. Margret Perry of the EEG Department at Vancouver General Hospital for her help in recording the hospital EEG data, and f i n a l l y , a l l the patients who, unbeknownst to them, provided the data for this study. 1 CHAPTER 1 INTRODUCTION The electroencephalogram (EEG) of man has been used c l i n i c a l l y as an indicator of possible cerebral pathologies since about World War II. Fundamental to this application, however, i s a criterion by which a given EEG can be labeled normal or abnormal. The criterion most frequently used to date in such studies has been to visually compare the EEGs ob-tained from a subject with a suspected cerebral disorder with those ob-tained from normal subjects. In so far as this process i s usually time consuming and requires the s k i l l s of a professional with many years of specialized training, the payoffs involved i n schemes for automatic or semi-automatic analysis of the EEG are manifest. For example, computer • .. _ .1 n n n . . . _ • ! , * . . « . . -t 1 i ......1 • • . . % f • „ . . I - _ ' . U - «-•»_.. „ 1_ CiO JL LCU 1_IJ_I O CXLl 0--L- y DXO L.UUJ.U U C- ci S^ysZ C l-CU L.O O -1. glix ±. J. l^ CUt I— J~J O O VJ . - I - • L . I I J . ^ ugli ' put in c l i n i c a l EEG laboratories or to make feasible such programs as mass EEG screening for applicants for private p i l o t licences. To date few systems have been proposed whereby a given EEG record can be tested and labeled as normal or abnormal with a specified confidence of correct decision, although many quantitative methods of analysis have been proposed [1-6]. This thesis considers two approaches to this problem. The f i r s t approach examines the v a r i a b i l i t y of various EEG features in a population of subjects supposedly free of neurological disorders by constructing histograms of these features. This allows for a quantitative assessment of the var i a b i l i t y of EEG features over a neurologically normal population. In order to examine a particu-lar EEG i t s feature values are compared with.the histograms of the neuro-logically normal group. This allows us to determine whether a particular 2 feature of an EEG i s w i t h i n the range of normal v a r i a t i o n . This technique does not produce an absolute c a t e g o r i z a t i o n of an EEG record, but rather examines the extent a given EEG record conforms to features of a normal population. In so f a r as c l i n i c a l EEG analysis i s done l a r g e l y on t h i s b a s i s , that i s , by examining an EEG record with reference to the l i m i t s of normal v a r i a b i l i t y , t h i s technique may be regarded as a semi-quanti-t a t i v e method of EEG assessment which mimics the method of the c l i n i c a l electroencephalographer. The second approach i s to employ techniques of automatic pat-tern recognition for c l a s s i f i c a t i o n of c l i n i c a l EEGs. One such a v a i l a b l e technique uses a number of EEG records known to belong to s p e c i f i c c l a s -ses (e.g. known to be c l i n i c a l l y normal or abnormal) to construct a d i s -criminant function which permits c l a s s i f i c a t i o n of u n c l a s s i f i e d EEGs. The c l a s s i f i c a t i o n procedure i s , of course, absolute,. and i s i n contrast to the above approach. The c r i t e r i o n u sually adopted for assessing the o v e r a l l performance of such a scheme i s i t s p r o b a b i l i t y of m i s c l a s s i f i -c a t i o n e r r o r . The features chosen to characterize the EEG i n t h i s study are. r e s t r i c t e d to those known to be both c l i n i c a l l y meaningful and computa-t i o n a l l y t r a c t a b l e . C l e a r l y expressions such as a "spikey p a t t e r n " or "pattern of immature character", which are common to EEG parlance, are too Imprecise to be amenable to computer a n a l y s i s , so the motivation to use c l i n i c a l l y proven features which are also computationally straightforward i s c l e a r . In p a r t i c u l a r , the use of power s p e c t r a l information i n the analysis of the EEG even predates the c l i n i c a l use of EEGs [7], so spec-t r a l features were obvious candidates for EEG c h a r a c t e r i z a t i o n . As w e l l , 3 -* - ; ffr » • • . . . " , . coherence features are useful i n d i s c r i m i n a t i n g dyslexic abnormalities [8] and thus are included i n t h i s study. F i n a l l y , the computational s i m p l i c i t y of features based on zero-crossings of the EEG and success with features of t h i s kind i n various contexts [2,3,9] motivated the i n c l u s i o n of t h i s t h i r d class of EEG features f o r EEG c h a r a c t e r i z a t i o n . The work of this study may be considered to be composed of two parts. The f i r s t part consists of the a c q u i s i t i o n and preprocessing of the EEG data and i s the subject of chapter 2. The remainder of the thesis concerns i t s e l f with feature s e l e c t i o n and the use of these fea-tures i n the two schemes of EEG analysis described. Chapter 3 discusses the use of a normative.model employing feature histograms from a neuro-l o g i c a l l y normal group while chapter 4 considers the use of automatic further work are given i n chapter 5. A number of appendices are included f o r reference purposes. Appendix A contains a glossary of neurophysical terms, appendix B o u t l i n e s the neurophysiology of EEG generation, appendix C describes the funda-mentals of v i s u a l analysis of c l i n i c a l EEGs, appendix D discusses power s p e c t r a l analysis of EEGs, and.appendix E covers, the mathematical formu-l a t i o n of. the stepwise discriminant analysis algorithm. 4 CHAPTER 2 Data A c q u i s i t i o n and Preprocessing 2.1 Sources of EEG Data The EEG records used i n th i s study were drawn from two sources. EEG records of 49 astronaut candidates from the National Aeronautics and Space Administration's Normative Electroencephalographic Data Reference L i b r a r y [10, 11] were used as a source of data corresponding to a sup-posedly n e u r o l o g i c a l l y normal group of i n d i v i d u a l s . The remaining records were recorded at the Department of Electroencephalography at Vancouver General H o s p i t a l . From these h o s p i t a l records a set of eight records was chosen which had been determined to be epileptogenic by a c l i n i c a l electroencephalographer. Both the NASA and c l i n i c a l data were recorded from subjects r e s t i n g with t h e i r eyes closed. The electrode placement scheme used i n recording both data sets conformed to the International "10-20 System" [12]. Although 18 channels of EEG were recorded for the NASA study, only 8 of these 18 channels were used i n this study. The channels used for the NASA and h o s p i t a l data sets are given i n Table I. The top o l o g i c a l o r i e n t a t i o n of electrode placement positions i n the 10-20 System i s shown i n F i g . 2.1. NASA data Hospital Data P3/01 P4/02 P3/01 P4/02 C3/T3 C4/T4 C3/T3 C4/P4 F7/T3 F8/T4 F7/T3 F8/T4 C3/P3 C4/P4 Table I: EEG Channels Used i n this Study Homologous channels for both data sets are l i s t e d on the same l i n e . 5 Fig. 2.1 International 10-20 System of Electrode Placement 6 2.2 Abnormalities Present i n the EEG Data Of the 49 records selected from the NASA data l i b r a r y , " which j included records from 200 astronaut candidates i n t o t a l , 27 had been determined by a panel of electroencephalographers [11] to contain no ab-normalities. The remaining 22 records were s p e c i f i c a l l y selected be-cause they contained c l i n i c a l l y recognizable EEG abnormalities. In par-t i c u l a r , the following abnormalities were observed: (i ) Much rhythmic theta a c t i v i t y , c h i e f l y i n temporal and anterior leads (7 records) ( i i ) Bursts of high voltage alpha-like a c t i v i t y i n cen t r a l and temporal leads (1 record) ( i i i ) Excessive o c c i p i t a l slow wave a c t i v i t y (4 records) (iv) Psychomotor variant discharges (4 records) (v) Focal abnormalities (3 records) (vi) Paroxysmal slow dysrhythmia (2 records) Some examples of some^of these abnormalities are shown i n F i g . 2.2. In ad--e d i t i o n , the eight h o s p i t a l records a l l had some abnormalities o f an epi l e p -togenic nature as determined by a c l i n i c a l electroencephalqgrapher [13]. It was f e l t that the NASA EEG records which had been judged to be abnormal"were "generally representative" of abnormal c l i n i c a l input of abnormal EEG records with the exceptions that the occurence of psycho-motor variant discharges was more frequent than usually encountered c l i n -i c a l l y and that no epileptogenic a c t i v i t y was found [13]. I t i s par-t i c u l a r l y d i f f i c u l t to e s t a b l i s h baselines concerning the frequency of occurence of the various abnormal EEG waveforms because t h e i r d i s t r i b u -tions vary from c l i n i c to c l i n i c and because l i t t l e normative data has i sec - Fig 2.2a Excessive occipital slow wave activity I t t C F, - F 7 — ^M/W^ ~ ~ ~ — 4 — v — — . . JU. F^ -T j —*-—-VAA/W*"— • — • — - v — - — - • — — — T3-T5 v^yv^ A^r^ "^ Tj -0| —-MA^V\/W—-— 0 2 - T 6 — '~ -— '— T 6 " T 4 F 8 - F 2 ~ — f Fig 2.2b Paroxysmal frontal dominant slow dysrhythmia I S E C F r - T 3 -T3-T5 _ ^ A J S ^ - A * T5-O, *Aw*~v~~ 0 2 - T 6 ^ T 6 - T « T 4 - F 8 F 8 - F 2 Fig 2.2 c Focal slow activity in l e f t temporal region F, - F 7 F 7 - T 3 T 3 - T 5 T 5 - O 1 0 2 - T 6 T 6 - T 4 T 4 - F 8 Fig 2.2 d Psychomotor variant discharges' 8 been accumulated other than the work of Gibbs and Gibbs {14, 15]. Never-theless, the incidence of psychomotor variant discharges was much higher (4/49 NASA records) than one would expect i n e i t h e r n e u r o l o g i c a l l y nor-mal or abnormal subjects. Gibbs and Gibbs set the incidence of psycho-motor variant discharges at under 0.5% [15]. The lack of epileptogenic records was remedied by adding the h o s p i t a l records to the data base. That the NASA data, would contain no epileptogenic abnormalities should be expected i n so f a r as the NASA subjects were astronaut candidates and would have been screened f o r any possible e p i l e p t i f o r m EEG a c t i v i t y p r i o r to being admitted to candidacy. One f i n a l comment should be made. The usual''incidence of nor-mal EEG records i n an EEG laboratory i s about 30-40% of the patients referred f or EEG analysis [14] while the incidence of normal records i n the data base used i n th i s study on the other hand was 55% without i n -cluding the h o s p i t a l data and was 47.4% when the h o s p i t a l data was i n -cluded. 2 .3 Ins t r umen t at i on In recording the h o s p i t a l data set, standard disc electrodes applied to the scalp were connected to a Beckman 16-channel electroen-cephalograph with a time constant of 0.3 sec and lowpass f i l t e r s e t t i n g of 50 Hz. A Hewlett-Packard 3960 4-channel FM instrumentation tape re-corder was used to store the analog EEG data on magnetic tape for trans-fer to the University of B r i t i s h Columbia (UBC). The NASA data was not p r e f i l t e r e d at the time of recording and was i n i t i a l l y recorded on two P r e c i s i o n Instrument Model 414 24 channel tape transport units [11]. This data was l a t e r replayed onto 9 the Hewlett-Packard 3960 instrumentation tape recorder at the LBJ Space Center i n Houston, Texas and subsequently transfered to UBC. 2.4 Preprocessing As indicated i n F i g . 2.3, a l l analog EEG records were sub-sequently reproduced and bandpass f i l t e r e d from 0.5 to 45.0 Hz. The lowpass f i l t e r i n g was done to prevent any a l i a s i n g errors during d i g i -t i z a t i o n while the highpass f i l t e r i n g eliminated any possible D.C. o f f s e t which might have existed on the reproduced EEG records. A Data General Nova 840 computer system was used to d i g i t i z e the f i l t e r e d EEGs and to store the d i g i t i z e d EEG records on 9-track IBM compatible computer tapes. The d i g i t i z e r consisted of a multiplexer which sampled each EEG channel at 100 samples/sec and a 12 b i t a n a l o g - t o - d i g i t a l converter which converted each sample value to binary form. The computer tapes were then used as input to a computer program on the Data General Nova 840 which sorted out the data on the basis of an automatic coding system employed during the EEG recording [11]. V i s u a l screening of the NASA data during d i g i t i z a t i o n indicated this data to be e s s e n t i a l l y free of a r t i f a c t s but this was not t r u e f f o r the h o s p i t a l data. Hospital records containing EEGs with obvious a r t i f a c t were eliminated. EEG DATA (ON MAG TAPE) s Fig. 2.3 Data Aquisition System CHAPTER 3 The Normative Model Approach to EEG Analysis 3.1 Introduction An important aspect of the v i s u a l analysis of c l i n i c a l EEGs i s whether or not a given EEG record i s within the "range of normal v a r i a -t i o n " i n view of the patient's age and state of consciousness. Conse-quently an assessment of an EEG can only be made when the EEGer has a knowledge, a l b e i t only q u a l i t a t i v e , of how much v a r i a t i o n i n EEG features one can expect from a n e u r o l o g i c a l l y normal population. Of obvious advan-r tage to v i s u a l EEG analysis would be quantitative descriptions of the v a r i a b i l i t y of various EEG features obtained by constructing feature h i s t -ograms from a population of subjects free of EEG abnormalities. With this data the vagaries of a subjective assessment of whether or not an EEG record i s "within normal l i m i t s " could be replaced by a measure of the p r o b a b i l i t y that a given feature value i n an EEG record arose from a normal population. This idea i s not new. The concept of a normative model appears i n many f i e l d s of medicine and i n p a r t i c u l a r has been exploited by Bennett et. a l . [16, 17] who developed a normative model for the v i s u a l evoked p o t e n t i a l . However, a p p l i c a t i o n of a s t a t i s t i c a l model to the character-i z a t i o n of the normal EEG has not been reported p r i o r to this- study, a l -though work i n t h i s d i r e c t i o n has been done [18-23]. There are several classes of EEG features which could be used to produce a normative EEG model but the most important class of computation-a l l y t r a c t a b l e features from a c l i n i c a l viewpoint are those derived from the power spectrum of the EEG [13]. Consequently the normative model 12 established i n this study was for the power spectrum of the EEG and the re l a t e d quantity, the coherence.spectrum between homologous EEG channels. 3.2 Sources of Data The EEG data used to e s t a b l i s h the normative model described i n t h i s chapter consisted of EEG records from 27 astronaut candidates whose EEGs had been determined by a panel of electroencephalographers to be free of EEG abnormalities [11]. The EEG channels employed are given i n Table I I . P4/02 P3/01 C4/T4 C3/T3 C4/P4 C3/P3 F8/T4 F7/T3 Table II EEG channels used i n the normative model (Channels l i s t e d on same l i n e form homologous pairs) .3..3.^Feature Selection Table III l i s t s the features used to e s t a b l i s h the normative model for EEG power spectra and coherence. The various frequency bands (6 ,0 ,a, 6^ are defined i n section 4.3 where the computational aspects of the s p e c t r a l estimation procedure are also discussed. (i) Relative s p e c t r a l energy i n 6 band "\ ( i i ) Relative s p e c t r a l energy i n 0 band la-Q channel 1 ( i i i ) Relative .spectral energy i n a band t (iv) Relative s p e c t r a l energy i n 3^ band J (v) Average coherence i n 6 band (vi) Average coherence i n 0 band / a l l homologous ( v i i ) Average coherence i n a band f channel pairs ( v i i i ) Average coherence i n bandj Table III Features used i n the normative model 13 3.4 Results Histograms for the spectral features i n Table I I I are given i n Fig. 3.1. Each histogram has 'an associated "PROBLEM CODE" i n the upper left-hand corner which i d e n t i f i e s i t . The f i r s t four characters of the problem code give the EEG channel used and the l a s t two give the spectral band involved. For example, the problem code P402B1 specifies channel P402 and spectral band BI. The band i d e n t i f i c a t i o n i s as follows: BI = 6 band B2 = 6 band B3 = a band B4 = $ 1 band Fig. 3.2 i s composed of 16 histograms of the coherence features i n Table I I I . The f i r s t two characters of the problem code for these histograms specify the channel pairs used to calculate the coherence estimates. The i d e n t i f i c a t i o n i s as follows: PO = P402/P301 CT = C4T4/C3T3 CP = C4P4/C3T3 FT = F8T4/F7T3 The l a s t four characters give the coherence band involved, where C0H1 = 6 band C0H2 = 9 band C0H3 = a band C0H4 = &± band As an example, the problem code P0C0H1 specifies the <5 band average co-herence for the channel pair P402/P301. PROBLEM C0DE-P4O2B1 CDS' cn i— U J t J ce UJo PROBLEM C0DE-P402B3 IJJ,-: cn t— UJ CJ U J . 0.0 40.0 RELATIVE ao.o POWER ~T 120.0 PROBLEM C0DE=P301B1 Si-cc c_> U J o Cu, i i 1 1 1 0.0 50.0 100.0 R E L A T I V E POWER —\—— tso.o AS. IV.!. 9 4 . . 1 1 3 . 6 6 . If!. a o . 3 8 . B O . 1 3 0 . 6 4 . £ 3 6 . 4 1 8 . S 0 3 6 3 1 107 533 , 153 110 £ 2 1 9 5 151 499 6 7voo 5')M 9 89 8 OH S I 5 8 2 2 88 3 6 798 6 7 1 40 048 5 539 0 569 0' . 4S8 5 . 9 9 6 7 , 5 0 4 8 . 8 5 9 8 ' . 538 1 ' . 59 29 ./16 56 . 709 5 . 2 8 6 0 .89 52 1 1 1 1 0.0 BO.O 163.0 240.3 320.0 RELATIVE POWER 403.0 91. 42 7 . 8 6 6. 2 6 3 . 5 5 . 1 0 1 . 1 49 . ; i oH . 318 . 105 . 3 5 7 -I 70 105/1. BB 2 3 4 309 65/1 1010 2 3 5 3 9 4 3 551 1 1 3 4 0 2 7 3 O 50 3 5 2 2 9 9 7 51! 6 5 5 3 9 8 9 9 *>302 !! 1 58 o:;r.i 29 1 1 3 6 28 61 40 2 2 3 6 69 6 1 7(5 4 2 7 2 0 0 9 0 1 6 9 3 5 8 4 6 4 8 . 7 4 0 2 160.3 , ( X l O 1 1 200.0 } 6 8 . 2 2 . 9 2 . 4 2 2 . 41 . 4 7 9 8 42 1 18 BO 79 109 3 3 4 122 114 160 73 t>6 62 2 1 7 59 9 3 . 0 1 3 1 . 2 3 70 . 7 9 1 9 . 3 0 9 1 . 6 5 0 1 .41(06 . 4 4 4 5 . 3 51! 1 . 0 6 7 0 . 3 3 6 5 . 4 9 4 4 . 4 52 7 . 0 7 5 9 . 28 58 . 3 1 43 . 2 7 7 7 . 4 1 9 1 5 4 1 3 . 4 4 1 9 7 0 7 4 . 0 6 7 7 4 8 2 2 "2OO.O 250.0 PROBLEM C0DE-P4CI2B2 PROBLEM C0DE=P402B4 (X 0.0 40.0 83.3 R E L A T I V E POWER 1 120.0 PROBLEM C0DE=P301B2 CDS" CX O Cf. UJo 2 6 . 0 8 6 4 I H . 48 56 I! 4(1. 5311 7 3 1 9 . 1 3 2 8 1 8 . ::o 1 6 1 6 . 2 9 9 0 3 3 . 8 0 6 6 29 . 39 3 5. 9 2 . 5 0 4 4 8 0 . ( 8 30 1 0 5 . 1 3 8 7 8 4 . 5 9 2 3 2 1 0 . 0 0 4 7 1 0 0 . 4 0 2 9 2 1 0 . 6 7 1 7 . 9 9 . 9 69 1 3 5 . S 9 7 7 59 . 649 7 1 4 3 . 4 75 5 1 1 9 . 4 9 7 3 5 5 . 2 0 0 3 6 6 . 1 2 0 3 _ r , 1 , . 0.0 50.0 100.0 150.0 200.0 253.D RELATIVE POWER 1 4 . 0 6 2 4 I 5 . 4.1 10 1:!. 6 I 9 7 7 5 . 1 1 5 1 1 2 . 9 9 0 6 9 . 8 1:!! 3 0 . 11 S3 6 . 9 0 2 4 1 3 - C O 16 S . 0 8 2:! 2 7 . 3 3 4 4 1 7 . 79 24 1 7 . 2 3 7 0 7 . 3 179 18 . 9 69 3 6 . 2 6 0 6 6 . 3 1 C 2 1 5 . 0 S 4 7 3 6 . 9 74-1 4 3 . 2 2 6 9 4 5 . 6 4 8 8 "l60.0 200.0 ~l l " 0.0 40.0 80.0 R E L A T I V E POWER 3 1 . 0 3 3 9 1 4 . 5 7 7 1 2 8 8 . 7 8 64 3 2 8 . 6 1 5 1 9 . 8 559 1 2 . 7 9 58 2 2 . 1 6 5 5 3 7 . 2 3 0 1 7 0 . 8 4 0 4 7 3 . 5 4 4 4 11 I . 3 0 0 1 69 . 7 9 1 ! 1 7 0 . 7 7 4 3 8 8 . 6 7 6 1 6 9 . 9 9 4 3 7 8 . 8 3 7 7 5 0 . 0 5 9 8 6 6 . 4 9 64 1 3 .3 .45C 7 1 1 3 . 1 1 6 7 3 4 . 9 9 6 8 £ 5 . 2 6 6 2 120.0 "I 160.0 -200.0 Figure 3.1 PROBLEM C0DE-P301B3 u_) -cr. i— UJ 1 1 0.0 20.3 40.0 RELATIVE POWER 60.0 ISO. 16. nsA. 6 6 3 . 1.10. 61 . S3 . 153. 0/1. 126. 236. 1 63 . /<ou. 9 5 . 03 6. 1 16. 1 40. 311. 1049. 761 • 163. 150. 67 5) Viii!/I 1 6 S3 4H60 /Hi 33 HIS 73 037 1 0431 7/I S 4 9PR1 r>7 4:>_ 9 42 r AO 5 3 6565 6 / 1 6 6 S731 3 7 55 013? 1 51 6 808 s) 0 508 83.0 ' .X10 ] 100.0 PROBLEM CnDE=P301B4 UJ„-cr i — UJ' (_) 0.0 20.0 RELRTIVE 40.0 POWER 60.0 1 7 9 . 10. n'" 4. ! 10. I ,8* I 7. 1 I 7-! 17-; io. i's! r 16. 9 : 6 06 ; 46 31 < I B 13 II300 0661 319: : 51! 61 H60'.1. 1 77(1 78 33 0756 . 39 0 7 ,on «9 .5317 . 4003 . « i : : o . 6499 .2122 . 1 470 . 7 8 7 9 . 7 1 4 9 2 78 4 628 6 83 66 2518 RO.O 103.0 PS-PROBLEM CDDE=C4T4B1 cr. CJ cn U J C 3 0.0 50.0 100.0 R E L A T I V E POWER ~i 150.0 PROBLEM C0DE=C4T4B3 cn o 9 0 . 7 7 2 8 9/1.4447 115.2003 553.9524 120.2730 129.1579 210.9081 77 .1019 123.3311 9 5 . 75 64 58 .0882 8 0 . 3 6 6 7 219.1471 73 - 109 7 4 3 9 . 9 543 8 4 . 6 5 7 4 104.9454 145-3731 48.2038 119.559 6 2 1 2 . 2 6 8 2 -T 200.0 250.0 80 55. 133. 124 16. 58. 116. 63. 75. ' 42. 161 . 97. 151 . 45. 51 1 . 39. 60. 2S5. 141. . 1 3 7 . 81 . 6080 9476 0631 0227 9 440 4451 0451 622 7 6827 0357 5769 6988 0653 9 623 612S 8641 9473 0528 209 3 2175 7304 3i. tuo 1 1 1 — 1 1 : — 1 0.0 83.0 IGO.O 240.0 320.0 403-0 R E L A T I V E POWER , PROBLEM C0DE=C4T4B2 UJ,-; CD-5-cc 1— 2: UJ CJ cr. U J C 3 0.0 T 20.0 R E L A T I V E 1 40.3 POWER GO. 3-1 a C D S " UJ O_o. PROBLEM C0DE=C4T4B4 0.0 a.o R E L A T I V E 16.0 POWER 24.0 t 31. ! 147. j 375. i 342. 19. •• 54. ' 48. 37. 52. 33. 105. 27 . 100. 73. 2 0 6 . , 3 0 . ' 46. ' 59 . ' 106. • 55-2 2 6 . . 154C 8271 1936 6730 5306 0577 2066 2751 99 59 8330 2855 6369 0670 8 459 4646 7054 0668 09 52 7760 91R5 08 55 BO.3 103.0 13 22 14 3 1 . 5. 1 5 . 2 6 . 5 . 11-9 . 17-10. 23. 4. 34. 13 23 16 19 25 12 1 541 6513 2842 6827 6060 .8219 , 1009 . 69 53 .4639 .429 6 .0224 .6572 .7200 .489 7 . 9 8 0 7 .089 6 .7742 .308 7 . 3 1 6 6 3741 399 7 - r 32.0 40.0 Figure 3.1 (cont'd) cr. h--pr U J (_> dr. UJCT , a. -A 89.4341 69.9713 1 8 1 . £ 9 6 7 7 0 . 79 38 1 42.41 1 1 2 0 8 . 0 6 2 6 6 0 . 4 645 142.3109 123.4955 58.9100 69 .9Ut>t> 125.0443 .09 62 1 1 7. 0.0 5 0 . 0 1 0 0 . 0 RELATIVE POWER ~ i iso.o PROBLEM C0DE--C3T3B3 cc UJ CJ or UJo PROBLEM C3DE=C4P4B1 cn C J cc UJo 0.0 40.0 80.3 RELATIVE POWER ~ l 120.0 367.8987 108.5491 485.6999 121.3 602 65.8881 141.0898 i 1 7.4351 139.8943 " I 203.0 250.0 69 17 71 92 10 64 26 1 14 80 E7 136 162 124 78 404 90 106 "i 1 r-—•—i— 0.0 50.0 103.0 150.0 200.0 RELATIVE POWER 176 47 103 SS 37 • 268 7 2018 • 4836 • 2821 .451 5 . 8 4 6 3 • 2793 . 8 558 • 9070 0701 9269 9448 0052 54 72 . 7832 9437 49 7 6 9869 9043 2304 • 5814 5503 250.0 46, 89. ' 165, 41, 53, 83. 60. • 75. 142. 50. 39. 141. 262. 9 1. 887. 84. 56. 48. 85. 2S3. 35. 131. 1307 2793 4085 6210 6807 4608 558 7 9090 6307 6269 0565 1831 448 5 1908 698 0 39 73 7160 048 3 9563 2209 4540 5564 . 163.0 I — 200.0 cc. I— •2-. UJ C J CK UJa 1 T 3 . 3 2 0 . 3 4 0 . 3 RELATIVE POWER 60.0 E-n a - l U-i*-C cn U J C J U J C . . PROBLEM C0DE=C3T3B4 1=L 3.0 r 40.0 RELATIVE 80.0 POWER 120.0 PROBLEM C0DE=C4P4B2 CDf' d UJ CJ ce_ 34. 303. 198. 12. 83. 23. 28. 51 46 107 41 S9 78 123 48 60 57 36 103 i oi: 102 . 6032 .3259 .009 6 . 7350 . 7285 .4260 .9243 .9432 .0682 . 1 535 .8661 .9031 .089 5 . 9 4 4 6 . 531 1 . 0 9 3 5 . 5898 . 3 4 4 6 • 28 60 • 6fc2D .9168 16 n — BO.O —1 133 0 10.9589 15-3268 9 . 2 0 0 5 2 6 . 5 8 2 9 3 . 0 3 9 2 9 . 7 4 3 7 1 4 . 0 6 6 2 7.2209 16.669 3 6.2574 1 1 . 0 9 7 6 17.7539 9 . 1 6 5 7 5.9798 2 9 . 9 4 2 7 2 4 . 4 5 2 5 36.1051 12.548 6 10.18 79 2 3 . 2 5 0 2 15-5747 9 . b 6 3 £ ~1 160.0 -1 200.0 15. 206. 96. 10. 8. 26. 40. 66. i 148. I 79, !' I 5 , 51 i 179 60 121 64 78 38 92 142 23 26 6740 58 56 8897 8891 7343 .2608 .8 389 .2413 . 7430 .7100 .9416 .7472 .1212 .0991 .3504 .7191 .3709 . 5761 .3739 .2609 . 6822 . 4 1 2 6 T"1 1 1 1 ~ L n 0.0 40.0 83.3 120.0 160.0 200-0 RELATIVE POWER Figure 3.1 (cont'd) &2-(X H-Lu' C J CK U J o 190 re 1 61 £1 89 SOS 216 818 230 185. 316 747 128 ' 819 45 22S 100 352 1249 88 117, .7741 • 2955 • 0172 .8643 . 3 56 2 • 2362 .8793 .6721 .6840 • 8127 • 9 788 • 5371 .7146 .318 1 .3933 • 5156 . 1 1 67 .9734 .279 1 .9422 .2219 0.0 10.0 20.3 R E L A T I V E POWER 30.0 •40.0 '.XlO1 so.o PI-P R O B L E M C 0 D E = C 3 P 3 B 1 •&2-cr. CJ cc UJo 0.0 40.3 03.0 R E L A T I V E P O W E R T 120.3 36.9094 66-9545 136.2141 61-0802 • 64.4931 57.9360 87 .9 1 58 82.4099 ! 200.5988 49.5522 67.0330 . 104.9472 146.8601 142.4006 • 314.2134 ' 74.9 752 ! 62 .3 679 73.8327 8 9 . S 8 I 4 188.49 30 40.5471 129.508 4 -r 1G3.0 _1  200.0 SI- P R O B L E M C 0 D E = C 3 P 3 B 3 cn UJ CJ cc U J o „ 0 - « H I 0.0 1 20.3 R E L A T I V E 14. 120. 233. 62. 36. 73. 301. 117. 62. 247. 174. 396. 506. 125. 601 . 34. 160. 144. 365. 949 , 116. 254, 1 127 1 626 9925 728 1 67H9 401 1 369 1 4174 5031 61 65 6916 1614 4961 9038 0210 0007 9 338 8262 1 528 3604 9 653 0O3S 40.0 P O W E R 60.3 BO.O ( X l O 1 —I 100.0 ) cn l— UJ <_j Cf. UJo PROBLEM C0DE -MF -MB4 o.o fl.O 16.0 R E L A T I V E P O W E R - r — 24.0 5' P R O B L E M C 0 D E = C 3 P 3 B 2 S i U J o" cn C J U J C Q-E n L n o.o —• i 40.0 R E L A T I V E — i — 80.0 P O W E R —l 120.0 P R O B L E M C 0 D E = C 3 P 3 B 4 L<-|_; co°-cc UJ CJ £ ° Q.O. (M T— • — — r 1 — 0.0 20.0 40.3 63.0 R E L A T I V E P O W E R 1 3.V 108 11.66 19 26.1460 6.8840 5.7133 14.3594 6 .7923 1fi.6564 20.9247 16.1578 12.3783 1 4 .2074 16.9820 5.6457 39 . 49 31 5.4587 9.3278 1 1 -1025 18.5372 38.4728 14.4925 13.7747 1 7 32.0 40.3 13 196 1 1 7 14 17 25. 40. 38. 57. 84 . 21 . 55. 105. 67. 127. 47. 50. 46 . 114. I Oi l . 21 . /if.-228 6 5806 • 2842 . 1982 • 2432 .62 74 . 69 1 7 . 7000 1 319 0019 1 749 S8 3 5 0*7 61 7 739 4331 3689 1214 199 1 599 3 89 1 6 'V38 M O O T ~T 160.0 -1 200.0 BO.O -1 100.0 Figure 3 . 1 (coht•! d) g_ PROBLEM CODE=FOT<1Bl-UJ CJ o.o 20.0 40.0 RELATIVE POWER co.o PROBLEM C0DE=F8T4B3 fe'-cc CJ UJo 0.0 50.0 100.0 RELATIVE POWER 150.0 PROBLEM C0DF.=F7T3B1 t a ? -cn i— U J r~ I 0.3 50.0 RELATIVE 130.0 POWER r 150.0 102 118 128 199 i 479 169 £63 77 460 95 65 426 1 1 1 298 404 217 285. 125. 132. 423. I 66, 359. 1 610 . 3 539 • 509 5 . 3 9 8 6 . 1902 • 1 441 . .0251 . 8 3 2 7 .5439 .3569 . 7899 . 7236 .631 7 • 2C9 3 • 69 58 .8528 . 1 650 • 1 544 • 2714 .89 60 19 3 5 7 5557 ~I 80.0 (X102 —-1 133.0 ) 59 18 63 98 20. 20. 17. 42. 44. 37. 91 . 9 3 . 221 . 9 2 . 367. 2 2 . 6 6 . 4 6 . 177. 182. SI . 3 4 . • 7340 • 4137 • 3420 • 4265 • 19 19 • 8544 • 0 3 8 0 • 2 533 • 5807 .9 603 .2051 • B376 .598 1 • 1830 , 3 669 ,0841 , 1 69 9 0 6 1 1 12 78 6 236 1 1 79 7547 —I 200.0 250.0 98 71 : 124 ! £ 8 8 j 4 2 6 i 1 8 7 : 3 8 1 8 6 , 6 6 4 • 7 5 1 4 8 . 1 8 8 , 2 0 2 . 1 3 3 . 367. 1 7 2 . 3 9 2 . 1 6 7 . 2 7 6 . £ 2 3 . £ 1 1 3 7 1 7764 3877 0370 3923 • 0330 • 3426 • 4314 1096. • 9795 • 3580 • 7272 • 1801 . 8 2 0 6 •9653 • 9124 •9062 •2903 ,4044 • 9238 4641 8668 3591 200.0 , (X10 1 250.0 ';• PROBLEM C3DR:--F81'"1D2 ~i-8i u-u i -uj cj cr UJ° 3.0 40 3 RELATIVE fiO.O POWER r 120.0 S i UJ • CD? • CC t — ^r. UJ CJ s ° Cup. PROBLEM CQDE=F8T484 o.o 20.0 RELATIVE 40.0 POWER 60.0 ^A t — UJ C J S» °-?-PROBLEM C0DE=F7T3B2 3.3 80.0 RELATIVE ico.o POWER I 240.0 16, 28 . 302. 1 70, 14, 15. 17. 33. 36. 48, 63. 35, 74. 40, 1 59. 75. 61 . 41 . 59. 4 0 . .32. 27. 3437 019 1 6335 7136 5639 6755 0002 2140 0429 9 29 6 3923 219 6 7 739 2200 0798 008 7 7713 048 7 3566 61 65 9 61 0 9 OH 0 I 163.3 20:.o 11.629 2 19.5700 12 . 1 49 5 2 2 . 0 3 5 2 4 . 8 7 5 5 13-0033 10.2484 7.6756 6.8348 5-79 32 12.6312 1 4 . 0 8 5 6 10. 5900 10.1177 25.9771 5-5 620 9 . 8 0 9 7 5.6257 13.72 55 19 .9 545 17.65H4 80.0 100.0 17.2616 2 1 . 2 8 5 4 2 6 4 . 9 9 19 163.5333 14.3370 .34.09 78 13.3858 3 0 . 4 4 3 7 £6.998 7" 19.7081 8 7 . 9 0 3 8 34-8744 4 9 . 8 0 2 2 55-4155 140.1560 3 3 . 8 3 8 5 . 38.9 772 9 3 . 4 9 1 7 9 5 . 4 6 8 2 3 6 . 1 0 3 7 32.8582 74.5378 I 320.0 400.3 Figure 3.1 (cont'd) PROBLEM C0DE-F7T3B3 co3-J cr. uj CJ Cf. UJo , CL. -_| : 4 9 12 48 110 13 31 24 58 37 18 65 120. 110. 80. 183. 17. 42. 44. 185. 129. 6 4 . 4 6 . • 1334 48 4 7 3551 • 0023 • 1591 .9248 .7876 . 4845 • 0336 • 6412 • 9260 69 63. 9 548 017 7 3734 1213 6744 8357 208 3 6318 1 530 1 725 , , , . _^ 0.0 40.0 83.0 120.0 160 0 RELATIVE POWER 200.0 8-PROBLEM C0DE=F7T3B4 UJ - . CJ Cr" 8 . 3 3 5 4 13.8974 6.3247 18.2044 4 . 7 58 3 5 .7323 10.5183 9 . 0 3 2 9 1 0 . 4 4 6 5 6 . 8 3 9 5 7.4118 12.3929 13-259 5 7.2470 2 5 . 0 3 8 2 5 . 0 1 8 0 7.8868 7.1959 2 0 . 8 7 2 9 8 .69 32 3 6 . 2 2 1 0 18.5151 1 1 ~T 1 : T ' 1 3.0 40.0 S3.3 120.3 160.0 203.0 RELATIVE POWER F i g u r e 3.1 ( c o n t ' d ) PROBLEM CODE=POCOHl cc or UJo 0.2 0.4 T 0.6 0.8 COHERENCE PROBLEM C0DE=P0C0H3 cn CJ cc UJo n 0.2 0.4 0.6 o.a COHERENCE °- PROBLEM C0DE=CTCQH1 <^><a~\ UJ <_) CC U J o a-* 0.2 — 1 — 0.3 0.4 0.S COHERENCE o . 4859 0 . 5 2 0 0 0 . 7 8 6 2 0.614 7 .0.38 59 0.6659 .4264 .6355 .4468 . 2 8 5 5 .3300 0 . 3580 0 . 7 6 9 9 0 . 2 6 0 0 0 . 3 2 4 0 . 3 0 9 0 . 29 68 • 3052 .3822 . 7 58 1 . 6 0 6 5 .3313 1.0 1.0 1.2 ~1 1.2 5061 3860 4299 4604 5007 5504 0 . 3 0 2 0 0.2271 0 . 2 6 0 9 0.1913 0 . 6 6 3 4 4957 3394 501 5 6572 28 58 2229 0 0 0 0 0 0 0 . 6 1 5 5 0 . 2 6 4 7 0 . 6 7 3 6 0-6831 0 . 2 2 1 7 I o.s -1 0.7 °- PROBLEM C0DE=P0C0H2 20 0.69 22 •o 0.5019 CM 0.6615 0 . 4 2 4 4 0.3271 0-639 1 0 . 3 1 4 5 O 0 . 6 9 3 0 i n -0 . 4 5 5 7 0 . 3 8 0 6 0.2668 0 . 2 9 0 2 0.6808 CO CDS-0-2446 0.3317 • <X 0 . 3 2 6 0 i — ~ y 0.2919 UJ 0.2729 CJ CC U J o 0.3627 0 . 7 0 5 7 CL, - -0.6444 0 . 2 6 2 4 CD ti-er i— z. UJ CJ cc U J o 0.2 PROBLEM C0DEFP0C0H4 r 1 1— 0.24 0.34 0.44 0.54 COHERENCE PROBLEM C0DE=CTC0H2 i— UJ CJ £° Q.O. j in 0 . 4 8 30 0.6137 0-7418 0 . 4761 0 . 3 3 6 2 0 . 6 4 6 4 0-4708 0- 7857 0 . 599 7 0 . 2 8 6 5 0 . 2 79 0 0-2858 0 . 7 8 0 0 0 . 3 1 8 7 0 . 4 0 3 3 0 . 3 64 5 .4201 .4344 .3781 . 7 738 . 6208 .3156 0-0-0 . 0-0 . 0-_) 1 , : — p — 0 4 0.6 0.6 1.0 COHERENCE 5800 34 48 6030 3335 2618 4 48 3 3261 5726 3 570 2828 2 672 28 2 3 569 5 29 38 .3201 .3420 .3451 .3181 .3293 . o i K .4937 .3210 0.64 0.74 0.538 3 0.2710 0.4701 0 . 2 8 6 4 0.5409 0.489 5 0 . 2 6 8 6 0.4449 0.2658 0 . 2 8 3 4 0.6827 0.568 5 • 333 7 .5872 .3905 .3232 .3629 0.6551 0.2909 0.5477 0 . 7 4 9 3 0 . 3 6 2 0 - i - r - r— 0.2 0.4 0.6 0.8 1.0 COHERENCE 1.2 Figure 3.2 PROBLEM C0DE=CTC0H3 fe-cc UJ CJ cc U J o 0.2 • 0.4301 0.8853 0.4J21 0 . 8 4 0 0 0.5018 0.5058 0-3188 0.3316 0 . 8 6 7 6 0 . 3 8 4 8 0 . 6 0 3 4 0 . 6 5 6 5 0 . 8 78 1 0.5579 0.4829 0 . 2 574 0 . 2 78 5 0 . 5 4 6 7 0 . 8 9 77 0.5125 0.6257 0 . 3257 " T 0.4 OJS O.B COHERENCE — r ~ 1.0 1.2 PROBLEM-C0DE=CTC0H4 Ujc C D o j -CC UJ CJ ' cr. LUa , 0.25 0.3 0.35 0 .4 COHERENCE 0-I 0. i o-j - 0 0 I o ! o I 0 ! o '. o 0 0 0 3621 8473 4882 3975 3232 3353 37 68 £8 79 3736 3355 4951 5365 30 79 0 . 4 0 6 6 0 . 4 0 1 0 3001 3055 4300 3561 4376 4025 0 . 3 2 4 5 0.45 21 ~ l 0.5 PROBLEM C0DE=CPC0H2 fe-cc UJ CJ ce U J o O-Oj-0.2 0.4 0.6 O.B COHERENCE PROBLEM C0DE=CPC0H3 fe-cc UJ CJ oe U J Q ca 0.2 0.4 0.6769 0 . 688 5 0.7250 0.7163 0 . 74 69 0.7136 0 . 7 7 0 3 0 . 6 6 9 8 ' - 0 . 2 9 5 1 0 . 8 3 7 5 0 . 7 3 8 4 0.6918 0.5556 0.6638 0 . 6 5 0 0 0 . 6 5 5 3 0 . 7 6 2 5 0 . 6 4 8 3 0.7237 0-7184 0.6974 0.7125 "1 0.6 o.e COHERENCE ~ i — 1.0 1..Z UJ , cr UJ CJ Cd U J o PROBLEM C0DE=CPCQH4 0.2 I — 0.4 r 0.6 0 . COHERENCE 0 . 6 6 5 7 0 . 6 9 8 2 0-6669 0 . 6 2 0 0 0.6368 . 69 69 • 749 3 .589 5 • 2503 " . 7439 • 5606 0 . 5 0 5 7 0 . 6 3 0 3 0 . 6 4 5 3 0 . 5 0 3 2 0.7510 0.7833 0 . 7 3 3 6 0.7156 O. ( O i l 0-6527 • 0-7459 —\ 1.0 1.2 0.6931 0.6341 0 . 5 5 7 0 0 . 6 4 5 6 0 . 5 3 4 0 0 . 4 9 66 0.542K 0 . 5 8 3 9 0 . 2 5 8 3 0.7277 0.5249 0 . 5822 0-6049 0 . 5 8 4 8 0 . 4 5 6 6 0 . 5 3 3 6 0 . 5 7 0 6 0 . 6 0 0 5 0 . 5 5 9 0 0 . 6 5 8 5 0 . 5 0 8 5 0 . 5 1 3 5 " 1 1.0 " I 1.2 1. F i g u r e 3.2 ( c o n t ' d ) 22 PROBLEM CODE^FTCQHl caS-CC t— •z. UJ CJ ce UJo 0.16 0.26 r 0.36 0 COHERENCE 45 0.3758 0 - 5 0 6 2 0 - 6 5 3 5 0 . 5 0 3 2 . 0.3/116 0 . 6 70 5 0 . 5 0 E 6 0 . 4 5 4 3 0 . 2 2 4 7 0.4579 0 . 3 2 3 5 0 . 3 4 4 2 0 . 3045 0 . 6 2 0 0 0 . 4 5 3 2 0 . 2 9 40 0.2819 0.6012 0 . 4 5 1 7 0 . 38 69 0 . 2 9 0 4 0 . 4 2 2 4 ""I 0.56 0.66 PROBLEM C0DE=FTC0H2 Bs-cc t— • UJ CJ cc U J o 0.24 0.32 , COHERENCE 0.48 .4409 .4153 . 5502 . 6134 . 4 0 9 0 . 4035 .2612 .4534 0 . 3 7 9 3 0 . 34 79 0 . 2 8 8 4 0 . 2 79 6 0 . 3947 0 . 4 3 1 2 0.4152 0.3361 0 . 2 3 0 3 0 . 3 4 2 7 0.4149 0 . 3 3 3 0 0 . 2 5 3 4 0.3122 I 0.S6 0.64 PROBLEM C0DE=FTC0H3 cc UJ CJ cc U J o , r" :—1— 0.16 0.24 0.32 0.4 COHERENCE 0.4379 0 . 4 0 0 7 0 . 3 8 4 4 0 . 6880 0 . 4 7 8 6 0 . 3 3 6 S 0 . 4 3 0 0 0.4521 0 . 3 4 2 2 0.358 6 0 . 3 3 4 7 • 2851 • 38 14 • 54 1 S .3323 •3553 2875 458 1 58 44 5027 2356 2693 0.40 0.56 PROBLEM C0DE=FTC0H4 ac i — z: UJ CJ £° Q.O. -i 1— 0.16 0.24 0.32 0.4 COHERENCE 0 .3399 0 .3000 0 . 3 0 7 6 0 .4180 0 .3832 0-3518 0-3246 0-3145 0 -3485 0 .3916 0 .3106 0 .288 4 0 .3239 0.4128 0.3 748 0 .3410 0 . 2 6 9 6 0 .3689 0 .3816 0 . 3 8 8 0 0 . 3 3 5 6 0 .3319 0.48 0.56 F i g u r e 3.2 ( c o n t ' d ) C 3 -CM PROBLEM C0DE=P402B1 0 i n . o CD2-(X U J C J Cd L U o 3 1 0.0 80.0 160.0 240.0 RELATIVE POWER —i 320.0 400.0 C M . cn CM PROBLEM C0DE=P301B1 tx t— UJ o ct 74 3 2> ~T— 200.0 0.0 50.0 100.0 RELATIVE POWER 150.0 250.0 Fig. 3.3 Relative delta band spectral power in occipital region for four EEG records having excessive occipital slow wave activity 24 In order to evaluate the e f f i c a c y of using the normative data i n a v i s u a l EEG a n a l y s i s , feature values for some abnormal EEG records were located with respect to the histograms from the normal EEG records. In p a r t i c u l a r , the 22 abnormal records of the NASA data base were ana-lyzed to produce feature values, and these are displayed beside each histogram i n Figs. 3.1 and 3.2. Several comments are i n order regarding the abnormal data. F i r s t , the f i r s t 11 records constitute a "miscellan-eous" group of EEG abnormalities of which numbers 1, 6, and 9 are f o c a l abnormalities, numbers 2, 4, 5, and 8 are psychomotor variant discharge abnormalities, numbers 3 and 10 are paroxysmal slow dysrhythmia abnormal-i t i e s and number 11 having bursts of high voltage alpha-like a c t i v i t y i n cent r a l and temporal areas. The next 4 records had been c l a s s i f i e d as having excessive o c c i p i t a l slow wave a c t i v i t y and the f i n a l 7 records as containing excessive theta a c t i v i t y , c h i e f l y i n temporal and anterior leads. 3.5 Discussion As an example of how the system j u s t described might be used c l i n i c a l l y , consider the 4 EEG records i n the group of 22 abnormal EEG records which had been c l a s s i f i e d as having excessive o c c i p i t a l slow wave a c t i v i t y . The histograms of the r e l a t i v e s p e c t r a l power i n the de l t a band (corresponding to slow waves) for channels P402 and P301 ( o c c i p i t a l derivations) are displayed i n F i g . 3.3. The r e l a t i v e s p e c t r a l power i n the d e l t a band for the four abnormal records are located as points on these two histograms. We can see that a l l these points are to the r i g h t of the d i s t r i b u t i o n mean, with record #2 beyond the range of the graph f or both channels. S i m i l a r l y , those 7 abnormal records c l a s s -25 i f i e d as having excessive theta activity in the temporal and anterior leads (e.g. channels C4T4 and C3T3) can be compared to the corresponding normative histograms of theta band -spectral energies and in doing so i t can be seen that many of the feature values calculated are much greater than the corresponding means of the normative distributions. Unfortunately, no significant deviations of the coherence fea-tures of the abnormal records relative to the normative data were obtained. While many of the 22 abnormal records seemed to have coherence values slightly in excess of what would be expected for the normal group, there are no obvious differences between the groups. Also, unlike spectral i n -formation, coherence information i n the EEG cannot be readily given a c l i n i c a l interpretation. A serious d i f f i c u l t y with this system is that i t i s entirely pos-sible that combinations of feature values need be examined to determine ab-normality. This combination, although possibly subjectively accounted for by EEGers, i f in fact i t exists, might not manifest i t s e l f as an excessive deviation, of any one feature value relative to what might be construed as a normal range of values. This type of problem arising in a visual analysis scheme together with human var i a b i l i t y due to psychophysical and psycholo-gical factors presents a powerful argument for a s t a t i s t i c a l pattern re-cognition approach to EEG analysis. This approach i s the subject of the next chapter. 26 CHAPTER 4 The Pattern Recognition Approach to EEG Analysis 4.1 Introduction The use of automatic pattern recognition techniques for c l a s s i -f y i n g EEG patterns i s not new. Walter et. a l . [24] used pattern recogni-t i o n techniques to discriminate among states of consciousness by EEG measurements, Sklar et. a l . [8, 25] were able to discriminate between normal and dyslexic children using s p e c t r a l signatures i n a discriminant analysis program, and McEwen et . a l . [9] report success i n using pattern recognition techniques to monitor the l e v e l of anesthesia by automatic analysis of spontaneous EEG a c t i v i t y . However, the use of these techniques for the c l a s s i f i c a t i o n of c l i n i c a l EEGs has not been reported. A block diagram of an EEG pattern recognition system i s shown i n F i g . 4.1. A pre-processor transforms an analog EEG segment into a pattern sample from which features are extracted.- The.feature.extractor analyzes the pattern sample produced by the pre-processor and quantita-t i v e l y evaluates i t i n terms of a s p e c i f i e d set of features. F i n a l l y , the set of feature values extracted from a preprocessed pattern sample i s used i n conjunction with stored data to c l a s s i f y the pattern as e i t h e r normal or abnormal. The pattern recognition algorithm may be regarded as a procedure for c l a s s i f y i n g EEG records i n t o predetermined classes. Such an algorithm i s based on what i s known as the test-space representation of the data. That i s , each EEG record to be c l a s s i f i e d i s represented by an ordered set of numbers: ANALOG EEG DATA PRE-PROCESSOR FEATURE EXTRACTOR CLASSIFIER Fig.4.1 Schematic Diagram of Pattern Recognition System 28 where n i s the number of features. For instance, Lubin e t . a l . [26] used f i v e features, :these being the t o t a l s p e c t r a l energy i n each of 5 EEG f r e -quency bands. By assigning one axis i n a 5-dimensional space to each of the 5 bands, the ordered set of numbers represents the coordinate values, l o c a t i n g each EEG record as a point i n the 5-dimensional test space. Such a point represents a pattern to be c l a s s i f i e d . To each point i s also assigned a l a b e l i n d i c a t i n g the class to which the EEG record i s known to belong. The set of a l l points having a p a r t i c u l a r l a b e l , c a l l e d a group, occupies some region i n te s t space. The d i s c r i m i n a t i o n problem, then, i s to construct boundaries separating these groups i n such a way as to minimize the p r o b a b i l i t y of points being m i s c l a s s i f i e d . I f the groups do not overlap i n the te s t space then such boundaries are not d i f f i c u l t to construct, but freauently the groups do overlap and a s t a t i s t i c a l d e c i s i o n process i s necessary to determine the best boundaries. There i s no optimum procedure f o r s e l e c t i n g the best features to be used i n di s c r i m i n a t i n g between patterns corresponding to different-classes of EEGs. However, i n s e l e c t i n g features f o r a s p e c i f i c pattern recognition problem, experience has shown that a few well-chosen, heur-i s t i c a l l y - d e r i v e d features are usually b e t t e r than a larger number chosen more randomly. This i s because processing many features requires more computing time, more storage and more data f o r t r a i n i n g the c l a s s i f i e r [27]. Consequently, t h i s study was r e s t r i c t e d to considering only fea-tures which had previously been described as meaningful i n the l i t e r a t u r e on computer-based EEG analysis. Review papers have been published which describe the state of 29 the art in pattern recognition and current progress i n the f i e l d of EEG pattern "recognition [28, 3]. 4.2 Stepwise Discriminant Analysis A variety of algorithms have been developed to perform discrim-inant analysis. In this study a stepwise discriminant analysis algorithm was employed [65]. At each step in the algorithm the set of discrimin-ating variables i s augmented by that feature amongst the yet unused fea-tures for which the ratio of between-group variance to within-group variance (F ratio) is greatest. Equivalently, the selection pro-cedure may be regarded as choosing that feature with the smallest F-probability. The.F ratio indicates, by i t s magnitude the degree to which i t i s desirable to enter that feature into the discriminant func-tion. Once a particular variable has been entered into the discriminant function the algorithm readjusts the F values for each feature based on it s correlation with the selected feature. If a variable which had pre-viously been chosen for entry into the discriminant function has an F ratio which f a l l s below a chosen threshold lev e l , the algorithm w i l l e l -iminate that variable from the set of discriminating features. The s e l -ection algorithm continues un t i l 1) perfect classification of the data i s obtained or 2) there are no more features having F ratios larger than a specified minimum. A detailed description i s given i n appendix E. 4.3 Feature Selection The process of feature selection i s normally suboptimal since the exhaustive testing of a l l possible combinations of features i s usu-ally impossible [28]. Consequently, heuristically selected features were employed, these being selected on the basis of their use i n various EEG 30 analysis schemes previously reported i n the l i t e r a t u r e . An examination of the l i t e r a t u r e suggested that four general types of EEG features seemed to be good candidates for use i n a pattern recognition scheme: (i) Spectral features based on multichannel power s p e c t r a l analysis of the EEG ( i i ) Coherence features based on the coherence spectra be-tween homologous channels ( i i i ) Amplitude features based on the amplitude histogram of the EEG (iv) Zerocrossing features based on the zero crossing rate of each EEG channel and the zero crossing rate of each d i f f e r e n t i a t e d s i g n a l . For the f i r s t two types of features a t o t a l of f i v e features per EEG channel (or channel p a i r f o r the coherence features) were i n i t i a l l y chosen and these were the r e l a t i v e power and average coherence i n the S, 6, a, 3, and g„ bands. We w i l l r e f e r to these features as E., E_ , E , i Z o o a E„ , E D and C„, C A, C , C. , C„ for the s p e c t r a l and coherence features • p]_ P2 o fc) a P2 respectively. For the purposes of t h i s work the above bands were defined as follows: 6 band: 0.5 TO 4.2 Hz 6 band: 4.2 TO 8.0 I.,Hz a band: 8.0 TO 12.5 Hz band: 12.5 TO 20.0 Hz ^2 band: 20.0 TO 30.0 Hz The s p e c t r a l and coherence features i n each data set were calculated using the Direct Method, i . e . by d i r e c t Fourier transformation of the data. Sets of 2048 samples per EEG channel, corresponding to 20.48 seconds of EEG were used for these c a l c u l a t i o n s and t h i s resulted i n s p e c t r a l e s t i -mates with 16.0 degress of freedom. Inherent i n the s p e c t r a l calculations 31 i s the assumption of s t a t i s t i c a l s t a t i o n a r i t y . The v a l i d i t y of t h i s as-sumption has been investigated by McEwen [30]. A d d i t i o n a l d e t a i l s con-cerning the s p e c t r a l and coherence calculations are dealt with i n appen-dix D. Besides the raw s p e c t r a l features this study included as fea-tures r a t i o s of s p e c t r a l powers f o r several EEG frequency bands. These were: E, + E, 6 R l " E + E Q where E„, E„, E , and E. are the r e l a t i v e s p e c t r a l 6 6 a powers i n the 6, 9, a and &1 bands respectively. ( i i ) R 2 = where E, i s the r e l a t i v e 6-band s p e c t r a l power for a °r r i g h t - s i d e EEG channel and E g. i s the r e l a t i v e 6-band sp e c t r a l power for the c o n t r a l a t e r a l homologous channel. ( i i i ) R 3 = E a r E a t where E i s the r e l a t i v e a-band s p e c t r a l power for a a r r i g h t - s i d e EEG channel and E i s the corresponding power for the c o n t r a l a t e r a l side. The motivation f o r including features of this sort i s based on reports concerning success i n detecting f o c a l abnormalities or slow wave pathol-ogies with s i m i l a r features [51, 52]. The amplitude s t a t i s t i c s of the EEG were used to generate two more features, the skewness and ku r t o s i s of the EEG amplitude. These features were included with the hope that they might have been useful i n v 32 the detection of epileptogenic abnormalities. We w i l l refer to these features as and respectively. Finally, the zero-crossing features chosen for each EEG channel were based on the sta t i s t i c s of the interval between zero crossings. These fea-tures were: (i) Mean interval between zero-crossings ( i i ) Variance of zero-crossing interval ( i i i ) Skewness of zero-crossing interval (iv) Kurtosis of zero-crossing interval (v) Mean interval between zero-crossings of f i r s t derivative of signal and are referred to as to respectively. These features were chosen be-cause of their usefulness i n other studies [2, 3, 25]. In a l l , a total of eleven different sets of features were used for the evaluation of the pattern recognition systems. These are given in Table IV. The f i r s t seven feature sets were chosen so that the same features were used for ture sets. System #8 was chosen by calculating the F-ratios for a l l possible features without regard to their correlation with the other features and those 4 0 features with the largest F-ratios (smallest F-probabilities) were chosen to form this feature set. In system #9 those 1 6 features of system #1 which had been employed least over the 4 9 training sessions were deleted and replaced by the features and over a l l channels. System #10 was formed by deleting from system #9 those 1 6 features used least over the 4 9 training sessions and replacing them with the features R^ , R^* and R^  over a l l channels. Finally, system # 1 1 was constructed by removing those two features from system #1 used least when trained on the second data base (NASA + hospital data) and adding the features T 1 and T 0 over a l l channels. SYSTEM NO. FEATURES USED 1 v v E u - v a n d % a l l channels 2 V V <V V a n d C 6 2 a l l homologous channel pairs 3 Z l ' Z2' Z 3' V a n d Z5 a l l channels 4 V V V C a - Z2> a a d Z5 a l l channels or homolgous O V W O i J channel pairs 5 V z2' zy EJ> Ee a l l channels 6 V V r2> V Es . a l l channels 7 ' R2> R3- r i - r2* V Ee a l l channels 8 W r r r2 channel P402 Ee, Zj , z4, z5, r 1, r 2 channel P301 V V z i - V r2 channel C4T4 zv z 2 . r 2 channel C3T3 E 6 . Z 2 . ZV Z 5 channel CAP4 V E 6 ' Z3 channel C3P3 E6. Ee2. Z 5 channel F8T4 V EB2 channel F7T3 c o, c B l , c 8 j . channel pair P402/P301 V channel pair C4T4/C3T3 channel pair C4P4/C3P3 V R3 ... channel pair F8T4/F7T3 9 V V V r i - F2 channel P402 V V V r i - r2 channel P301 V Z2' r i - F2 channel C4T4 V V 22- V r i - r2 channel C3T3 «,. z 2. * s . r r r 2 channel C4P4 7- r r V 1' 2 Channel C3P3 z_. z_. r . . r. r h w n o l P R T / i *«• TV F2 . channel F7T3 channel pair P402/P301 c. . ' channel pair C4T4/C3T3 channel pair C4P4/C3P3 channel pair F8T4/F7T3 10 i , . E , . rlt r 2 . R3 channel P402 Z2' rl> r2- R3 channel P301 Z2, TV R3 channel C4T4 E«> V z2 ' V r r V R 3 channel C3T3 E E , Z2, R, channel C4P4 R3 channel C3P3 C e > K2 channel pair P402/P301 Ce, R p R2 channel pair C4T4/C3T3 h-h • channel pair C4P4/C3P3 V C a - V «2 channel pair F8T4/F7T3 . 11 • , . z 2 . r r r 2 channel P402 V V z2 r i - r2 channel P301 V V W r i - r2 channel C4T4 V V z2- zs- r i - r2 channel C3T3 V V z2- V r i - r2 channel F8T4 E 4 . E b . z 2 . z s > r l t r 2 channel F7T3 V c . channel pair P402/P301 Channel pair C4T4/C3T3 channel pair F8T4/F7T3 Table IV Diffe rent feature sets used in pattern recognition systems 34 4.4 Evaluation of Pattern Recognition Systems Performance re s u l t s are presented for the various pattern re-cognition systems using d i f f e r e n t feature sets and two d i f f e r e n t data bases. The c r i t e r i o n usually adopted for assessing the o v e r a l l perform-ance of a pattern recognition system, as previously mentioned, i s i t s pro-b a b i l i t y of m i s c l a s s i f i c a t i o n e r r o r , denoted P g, which can be estimated by c e r t a i n techniques on the basis of a f i n i t e number of data sets [29]. One technique, known as the U Method, estimates P g by averaging the re-s u l t s of N experiments i n each of which the system i s tested on a s i n g l e pattern sample a f t e r being trained on the other N-1 pattern samples. Be-cause t h i s estimation method can be shown to give the most r e l i a b l e es-timate of P g [29], i t was chosen r f o r - t h i s study. The two d i f f e r e n t data bases employed i n this study were: (i) NASA data (27 normals/22 abnormals) ( i i ) NASA data + h o s p i t a l data (2 7 normals/30 abnormals) The f i r s t data base consisted of 8 channels of EEG data per subject and the second data base consisted of 6 channels of EEG data per subject. This difference i n the number of EEG channels arose from constraints encountered i n recording the h o s p i t a l data at Vancouver General H o s p i t a l . Two ser i e s of pattern recognition systems were developed and evaluated, one on each data base. The estimated m i s c l a s s i f i c a t i o n p r o b a b i l i t i e s are given i n Tables V and VI, respectively, together with descriptions of the systems i n terms of the features used. No. of Average No. ?est System FPROBI Features of Features P e imber Number Inputted Used 1 1 0.20 40 8.4 0.367 2 1 0.35 40 10.2 0.408 3 1 0.50 40 18.1 0.367 4 2 0.20 20 4.4 0.449 5 2 0.35 20 9.1 0.429 6 2 0.50 20 11.7 0.449 7 3 0.20 40 11.1 0.429 8 3 0.35 40 15.6 0.469 9 3 0.50 40 27.7 0.449 10 . 4 0.17: 40 10.1 0.327 11 4 0.20 40 10.8 0.265 12 4 0.23 40 11.0 0.204 13 4 0.26 40 11.2 0.225 14 4 0.30 40 11.4 0.225 15 4 0.35 40 11.8 0.225 16 4 0.50 30 18.0 0.327 17 4 0.80 40 32.2 0.449 18 ,' 5 0.20 40 12.1 0.388 19 5 0.35 40 20.8 0.347 20 5 0.50 40 25.7 0.347 21 6 0.20 40 5.2 0.347 22 i 6 0.35 40 11.1 0.469 23 6 0.50 40 19.3 0.510 24 7 0.20 40 5.5 0.327 25 7 0.35 40 7.2 0.306 26 7 0.50 40 12.2 0.490 27 8 0.20 40 8.8 0.388 28 8 0.35 40 12.5 0.347 29 8 0.50 40 17.9 0.306 30 9 0.20 40 13.2 0.225 31 9 0.30 40 15.8 0.204 32 9 0.35 40 16.1 0.204 33 9 0.40 40 16.7 0.184 34 9 0.50 40 21.0 0.225 35 10 0.20 40 14.9 0.306 36 10 0.35 40 20.9 0.306 37' 10 0.40 40 22.7 0.306 38 •> 10 0.50 40 25.7 0.347 Table V Performance of c l a s s i f i e r using NASA data base (49 subjects) 36 No. of Average No. Test System FPROBI Features of Features p Number Number Inputted Used e 1 4 0.20 30 5.0 0.351 2 4 0.35 30 6.2 0.386 3 4 0.50 30 12.7 0.368 4 7 0.20 30 4.7 0.439 5 7 0.35 30 6.8 0.421 6 7 0.50 30 9.7 0.456 7 11 0.20 40 6.1 0.367 8 11 0.35 40 10.4 0.387 9 11 0.50 40 18.3 0.551 Table VI Performance of c l a s s i f i e r using NASA data plus hospital data (57 subjects) 37 4.5 Discussion Several comments concerning the re s u l t s of the previous section are i n order. F i r s t , r e f e r r i n g to Table V, we see that for a given set of features and pattern samples that the success rate of the c l a s s i f i e r may depend strongly on the parameter FPROBI, the F- p r o b a b i l i t y required of a p a r t i c u l a r feature f o r i t s entrance i n t o the discriminant function produced by the stepwise discriminant analysis algorithm (see appendix E). This i s i l l u s t r a t e d i n F i g . 4.2 where the dependence on FPROBI of the percentage of c o r r e c t l y c l a s s i f i e d pattern samples for a t y p i c a l EEG data base ( i n this case the NASA data using the set of features corresponding to system #4 of Table IV) i s shown. Although increasing the value of FPROBI generally increases the number of variables i n the discriminant function, merely having a large number of discriminating variables does not n e c e s s a r i l y insure good c l a s s i f i e r performance, and the "best" choice of FPROBI must be determined e m p i r i c a l l y . Next, we should note that choosing features s o l e l y on the basis of t h e i r i n d i v i d u a l F - p r o b a b i l i t i e s without regard to t h e i r c o r r e l a t i o n with the other features does not generally produce a good feature set, as evidenced by the performance of system #8 shown i n Table V. As an example of how features may be correlated consider the power spectrum of a s i g n a l and i t s r e l a t i o n to the zero-crossings of that s i g n a l . I f X i s the den-s i t y of zero crossings for a Gaussian s i g n a l with power spectrum S(f) then we have [66] 12 = A — J°° f 2 S(f) df * S(f) df so c l e a r l y we should expect zero-crossing and s p e c t r a l information to be 38 50* i l i i 0.2 OA 0.6 0.8 FPROBI Rg. 4.2 Typical effect of FPROBI on classification accuracy 39 correlated. The fact that systems #9 and #10 gave the best performance indicates that the alternative feature selection procedure, that of choosing those features most frequently used by the stepwise discriminant analysis algorithm over the course of the 49 training sessions, is to be preferred. Again referring to Table V, we observe that the best performance obtained achieved 81.63% correct classification. Misclassification of pattern samples was probably due to four sources: (i) Some EEG segments used to train the c l a s s i f i e r may not have been representative of the abnormality they supposedly displayed. Specifically, the NASA data was i n i t i a l l y examined for abnormalities by a panel of e l -ectroencephalographers who observed over half an hour of paper tracings. Most c l i n i c a l EEG records contain a similar amount of data. On the other hand, this study dealt only with 20.48 second segments, and i t is entirely possible that c l i n i c a l l y significant EEG patterns might have appeared i n -termittently in such a way that they were obvious to those examining the longer paper tracings but did not appear in the segment used for the training and testing of the c l a s s i f i e r s . This statement i s particularly true for EEG abnormalities such as paroxysmal slow dysrhythmia, which by definition occurs intermittently. ( i i ) The features used in the pattern recognition system may not have been sufficient for the discrimination of a l l the EEG abnormalities. While i t is reasonable to assume that the spectral and zero-crossing features might be useful in the detection of spectrally related EEG ab-normalities such as excessive occipital slow wave activity, i t is by no 40 means clear that features of t h i s sort are useful i n the detection of wave form abnormalities such as epileptogenic a c t i v i t y or psychomotor variant discharges, which may not manifest themselves s p e c t r a l l y . In t h i s regard Kellaway [42] reports that "the evaluation of wave form to determine abnormality i s probably the most complex problem encountered i n the automatic analysis of the EEG, perhaps because i t i s most d i f f i -c u l t f o r the electroencephalographer to specify what steps he takes i n the v i s u a l analysis of a waveform." Indeed, computer-based techniques of (time-domain) waveform evaluation are woefully lacking i n electroen-cephalography, and t h i s may be the foremost block to the eventual auto-mation of EEG analysis. ( i i i ) The c l a s s i f i c a t i o n algorithm used may not have possessed suf-f i c i e n t power to properly c l a s s i f y a l l the pattern samples. The stepwise discriminant algorithm used produced a l i n e a r surface (hyperplane) i n N-dimensional space, where N i s the number of features used i n the d i s -criminant a n a l y s i s , but other c l a s s i f i c a t i o n algorithms might perhaps y i e l d superior r e s u l t s . For example, the varying types of pattern samples i n the class of abnormals might order themselves i n clusters i n feature space thereby requiring a more complex surface than a hyperplane for d i s c r i m i n a t i o n from normal EEGs. Several possible a l t e r n a t i v e algorithms suggest themselves. F i r s t , c l a s s i f i c a t i o n algorithms based on nonlinear boundaries, p a r t i c u -l a r l y quadratic boundaries, appear promising. Cooper [31] has shown that for multivariate Gaussian d i s t r i b u t i o n s the optimal separating surface i s a hyperquadratic and that t h i s surface i s also optimal for some other p r o b a b i l i t y d i s t r i b u t i o n s . Another nonlinear algorithm i s the "nearest 41. neighbor" algorithm [32] whereby membership of an unknown pattern sample is decided by a majority vote of the K nearest neighbors. Secondly, the Bayes c l a s s i f i e r , based on the Bayes decision rule [33] appears to be particularly promising i n so far as i t is relatively simple to implement and has the property that the probability of a misclassification error is minimized when the features are s t a t i s t i c a l l y independent and the required probability estimates are either known or obtained using Bayes estimation. [34]. (iv) A f i n a l possibility is that differences i n error performance for the two data bases may be related to different experimental condi-tions in the recording of the analog EEG data. If the typical spectrums in the two data bases are colored even slightly differently, this could reflect i t s e l f i n a reduction i n performance since many features relate to the distribution of energy with frequency. Conceivably, such di f f e r -ences could arise from differences in the recording instrumentation used. Finally, a comment concerning the differences between the results based on the NASA data (Table V) and the results based on the NASA data plus the hospital data (Table VI) i s in order. The best resultsobtained from the latter data base corresponds to a misclassification rate of over 30%, which is more than 10% greater than the minimum error rate for the NASA data base. There are two probable reasons for this difference in error rate. First, as already mentioned, the hospital data base was composed of only 6 EEG channels as opposed to the 8 channels of the f i r s t data base and i t is reasonable to assume that this reduction i n training data could adversely affect c l a s s i f i e r performance. Secondly, the epileptogenic EEG records which were added to the NASA data to construct the second data base possibly are not capable of being discriminated by the feature sets used in this study. CHAPTER 5 CONCLUSIONS 5.1 Conclusions This thesis sought to explore two str a t e g i e s of automatic or semi-automatic analysis of the electroencephalogram as a step toward the eventual goal of r e l i a b l e automatic analysis of EEGs for e i t h e r c l i n i c a l or mass screening purposes. The f i r s t strategy, which to some degree mimicked the approach of c l i n i c a l electroencephalographers, was to examine the v a r i a b i l i t y of various EEG parameters f or a population of normal sub-j e c t s so as to e s t a b l i s h "normal l i m i t s " to these parameters; EEGs to be examined were then processed to see i f they f e l l w ithin normal l i m i t s . This approach to EEG analysis was the subject of chapter 3. The second strategy involves thp HRP nf nat-fprn ropnftm'Hnn t 0chnirtzec tc dcvclcp " c l a s s i f i e r which, on the basis of a set of t r a i n i n g data c o n s i s t i n g of EEG pattern samples known to be normal or abnormal, c l a s s i f i e d unknown EEGs in t o normal or abnormal categories. This approach, which was d i s -cussed i n chapter 4, used a stepwise discriminant analysis algorithm to construct a hyperplane d i v i d i n g the feature space. The best r e s u l t ob-tained, based on t r a i n i n g data c o n s i s t i n g of 27 normal EEG pattern samples and 22 abnormal EEG pattern samples, was 81.6% . correct c l a s s i f i c a t i o n (P = 0.184) as estimated by the U method. This and other encouraging re-s u l t s i n d i c a t e that further work i n t h i s area i s warranted, p a r t i c u l a r l y i n view of the many p o t e n t i a l payoffs which could r e s u l t from r e l i a b l e automatic EEG an a l y s i s . 5.2 Suggestions For Further Work There are a va r i e t y of areas where the work reported here could 43 be extended. The acquisition of a much larger data base on which to dev-elop strategies of automated analysis i s important. Unfortunately, few EEG laboratories are oriented in this direction in that their EEG recordings are in the form of paper tracings rather than being recorded on magnetic tape. Consequently a substantial change in c l i n i c a l EEG recording procedures w i l l have to be realized before progress in this direction can be expected. The only normative data library presently available is that produced by NASA [10, 11] based on 200 male subjects aged 24 to 35 years. Although this data does display some EEG abnormal-i t i e s the incidence of the various abnormalities does not reflect that which would be expected c l i n i c a l l y . This i s almost certainly due to the fact that subjects used in the study were neurologically normal. Con-sequently, there is a strong need for more normative data. A second area where extensions are possible concerns the detec-tion of transient waveform abnormalities. Of those c l i n i c a l EEG records which are visually classified as abnormal, about 60 to 70 percent display waveform abnormalities of an epileptogenic nature [10], which are decidedly transient. Consequently there is a strong motivation to develop strate-gies for detecting abnormalities of this sort, particularly spikes and spike-and-wave complexes. Similarly, the problem of automatic detection of focal abnormalities, be they focal slow waves or focal spike abnormal-i t i e s has been largely unsolved, and despite the work of Steadman [51] much work in this area remains to be done. A f i n a l suggestion for further work concerns the pattern re-cognition algorithm used in this study. It was mentioned in section 4.5 that alternate classification algorithms such as the Bayes c l a s s i f i e r \ 44 might produce better results. Although one cannot present analytical arguments to this effect, the general line of argument in this regard relates to the fact that a linear hyperplane probably would work best to disciminate pattern classes which form two specific clusters but since the sources of the EEG abnormalities are themselves quite varied, i t i s reasonable to expect that a number of clusters would be encountered in the test space, thus requiring a more powerful classification algorithm than one which simply constructed a simple hyperplane. APPENDIX A Glossary of Neurophysiological Terms axon - the central core which forms the essential conducting part of a nerve fiber. cortex - the convoluted layer of gray matter that covers each cerebral hemisphere. cortical - referring to the cortex. dendrite - a branched and tree-shaped protoplasmic process from a nerve c e l l which conducts impulses towards the c e l l body. excitory postsynaptic potential (EPSP) - an induced change in the mem-brane potential of a neuron caused by an excitory neural trans-mitter. inhibitory postsynaptic potential (IPSP) - an induced change in the mem-brane potential of a neuron caused by an inhibitory neural trans-mitter. . -1 1 ~ 1 J J «... « C «-,•„„,.„ — — " ^  " " J f — "wEJ-w— . * — - . neuron - a nerve c e l l . pyramidal c e l l - a pyramid like c e l l in the cerebral cortex believed to be responsible for the electroencephalogram. reticular formation - a large complex of nerve cells and fibers occupy-ing the central region of the brain stem which is believed to control the cardiovascular and respiratory systems, muscle tone, movement and the control of cortical e x c i t a b i l i t y . soma - the main body of a neuron. synapse - the region of contact between two adjacent neurons, forming a junction across which neural transmitters are passed. thalamus - the main relay center for sensory impulses to the cerebral cortex APPENDIX B Physiology of EEG Generation B.1 Introduction The electroencephalogram i s composed of o s c i l l a t i n g p o t e n t i a l s derived from the scalp surface and o r i g i n a t i n g from the c e l l s i n the cerebral cortex of the b r a i n . As yet no exact p h y s i o l o g i c a l mechanism regarding the generation of the electroencephalogram has been found, but an adequate explanation would encompass the following: (i) The i d e n t i t i e s of the c e l l s generating the e l e c t r i c a l p o t e n t i a l s . ( i i ) The type of e l e c t r i c a l p o t e n t i a l involved. ( i i i ) The neural network determining the time and frequency domain properties of the EEG. (iv) The neural mechanisms responsible for modifying the potentials when the e x c i t a b i l i t y of the b r a i n i s a l t e r e d . On a non-specific l e v e l there i s reasonably good agreement amongst researchers regarding the general nature of the neural mechan-isms involved i n the generation of the EEG. The pertinent points may be summarized as follows: (i) The e l e c t r i c a l p o t e n t i a l s recorded from the surface of the b r a i n or from the scalp are summated synaptic p o t e n t i a l s generated by the pyr-amidal c e l l s i n the cerebral cortex. ( i i ) The synaptic potentials are the responses of c o r t i c a l c e l l s to rhythmic discharges from thalamic n u c l e i . ( i i i ) The properties of the thalamic discharges (and hence the c o r t i -c a l p o t entials) are determined by the s p e c i a l arrangements of excitory and i n h i b i t o r y interconnections among thalamic c e l l s . (iv) During " a c t i v a t i o n " , inputs from the r e t i c u l a r formation abol-i s h rhythmic discharges i n the thalamic n u c l e i and cause the c o r t i c a l p o t e n t i a l s to become desynchronized. B.2 C e l l s Generating EEG Potentials H i s t o l o g i c a l studies of the cerebral cortex show that three types of c e l l s may be found to e x i s t , and they are classed as s t e l l a t e , spindle, and pyramidal c e l l s . Of these, only the pyramidal c e l l s are believed to be responsible f o r the EEG generation, since they are the only c e l l s which are uniformly oriented perpendicularly to the c o r t i c a l surface and with dendrites s u f f i c i e n t l y long to form e f f e c t i v e dipoles. Desser de Barenne and McCulloch [35] showed that the deeper pyramidal c e l l s were p a r t i c u l a r l y important by showing that the EEG p e r s i s t s even i f super-f i c i a l layers of the cortex are destroyed by heat. Also, d i r e c t e v i -dence of pyramidal c e l l involvement has been obtained from i n t r a c e l l u l a r recordings of c h a r a c t e r i s t i c synaptic p o t e n t i a l s . B.3 Generation of E x t r a c e l l u l a r P otentials The available experimental evidence suggests that the e l e c t r i c a l p o t e n t i a l s measured on the scalp r e s u l t from excitory and i n h i b i t o r y postsynaptic p o t e n t i a l s developed by the soma and the larger dendrites i n the neurons. Considerations i n v o l v i n g the i n t e r n a l resistance of the axon and smaller dendrites i n the neuron ind i c a t e that i t i s u n l i k e l y that they contribute s i g n i f i c a n t l y to any surface p o t e n t i a l s , while the low i n t e r n a l resistance of the soma and larger dendrites would, on the other hand, ensure large e x t r a c e l l u l a r p o t e n t i a l s . Consequently, the non-propogating somatic and d e n d r i t i c postsynaptic p o t e n t i a l s are fav-oured as being responsible f o r the generation of the observed p o t e n t i a l s . B.4 The EEG Pacemaker The q u e s t i o n a s to whether the repetive polysynaptic p o t e n t i a l s i n the pyramidal c e l l s are evoked by discharges from other c o r t i c a l c e l l s or from a cen t r a l "pacemaker" has been the subject of i n v e s t i g a -tion f o r some time. The work of Andersen and Anderson [36] suggests that the cortex may contain neural networks capable of generating e l e c -t r i c a l rhythms. On the other hand, two observations suggest that the c o r t i c a l networks are normally driven by an external pacemaker. F i r s t l y , experiments have been performed to show that spontaneous e l e c t r i c a l a c t i v i t y can s t i l l be recorded when c o r t i c a l impulse a c t i v i t y , and therefore i n t r a c o r t i c a l d r i v i n g , has been abolished by anaesthesia. Secondly, K r i s t i a n s e n and Courtosis [37] have shown that i f the th a l a -mus i s removed from one side of the br a i n that spontaneous spindles on that side are abolished f o r several hours. The general f e e l i n g among e l e c t r o p h y s i o l o g i s t s i s that some other part of the br a i n i s normally responsible f o r d r i v i n g the slow wave a c t i v i t y of the cortex. Dempsey and Morison [38, 39] i d e n t i f i e d the non-specific thalamic n u c l e i as the probable pacemaker region f o r the EEG. Their experimental r e s u l t s l e d Andersen and Sears [40] to sug-gest a neural model f or the generation of the thalamic waves within a thalamic nucleus. B.5 The Re t i c u l a r Formation If a relaxed subject opens h i s eyes or performs mental arithme-t i c the alpha rhythm commonly found over the v i s u a l cortex i s usually replaced by fast i r r e g u l a r a c t i v i t y of smaller amplitude. Similar changes occur i f strong sensory information i s given to a drowsy or l i g h t l y anaesthetised animal. In such cases the change i n EEG a c t i v i t y i s r e f e r r e d to as " a c t i v a t i o n " of the EEG. Work by Magoun and h i s colleagues [41] has i d e n t i f i e d the region of the b r a i n which controls a c t i v a t i o n to be the r e t i c u l a r formation. This was done by showing that a l e s i o n i n the r e t i c u l a r formation of an animal would cause i t to become somnolent and to develop rhythmic c o r t i c a l a c t i v i t y while l e s i o n s made outside the r e t i c u l a r formation located so as to i n t e r r u p t the som-atosensory pathways showed no retarding e f f e c t on the animals behaviour and caused no sychronization of the EEG. APPENDIX C Visual Analysis of the EEG The visual analysis of the EEG begins with the determination of the age of the patient and his state of alertness and general well-being. These two factors influence the EEG to a considerable extent, and must be taken into account in the analysis of the EEG. Once the age and state of the patient are known, the EEG i s exam-ined in terms of the following parameters [42, 43]. (1) Frequency. The range of normal frequency for the occipital rhythm (alpha rhythm) has been established on an emperical basis as 8 to 13 cycles per second for the adult. (2) Voltage. The voltage range varies from 5 to 100 uV and i s de-pendent both on the voltage present at the cortex and the thickness and character of the scalp and skull. The voltage/frequency ratio i s an im-portant factor i n c l i n i c a l evaluation. In general, whether a waveform is considered to be of high or low voltage is assessed i n terms of the frequency of the activity; the higher the frequency of the activity, the lower the expected voltage. (3) Locus. The topological location of the EEG waveform being ex-amined must be considered, as different scalp areas can have vastly d i f -ferent characteristic patterns. For instance, the frontal area typically has l i t t l e alpha activity while the occipital area usually shows consid-erable alpha activity in a normal awake adult with eyes closed. (4) Waveform. The evaluation of waveform to determine abnormality can be useful in diagnosing epilepsy where characteristic sharp waves (epileptiformic activity) can sometimes be detected. A general rule of thumb i s that i n the waking s t a t e a l l a c t i v i t y should be of r e l a t i v e l y simple waveform. The more complex the form and the sharper or more square-topped the c o n f i g u r a t i o n , the l e s s l i k e l y the a c t i v i t y i s to be abnormal b r a i n wave. (5) Interhemispheric Coherence. The degree of congruence of the various parameters on the two s i d e s of the b r a i n i s c a l l e d i n t e r h e m i -s p h e r i c coherence (not to be confused w i t h the power spectrum coherence). The o c c i p i t a l alpha rhythm should have i d e n t i c a l frequency c h a r a c t e r i s -t i c s on both s i d e s w i t h i n one c y c l e per second. However, some asymmetry of voltage i s a c h a r a c t e r i s t i c of the normal b r a i n , the v o l t a g e u s u a l l y being lower on the dominant hemisphere of the brain.1 Voltage asymmetry up to 50% i f the low s i d e i s dominant and up to 10% i f the nondominant si d e i s low i s considered to be w i t h i n normal l i m i t s . (6) Character of Wave Occurence. A wave t r a i n may be c h a r a c t e r i z e d i n terms of i t s p e r s i s t e n c e ; i f the waveform c o n s i s t e n t l y d i s p l a y s a c h a r a c t e r i s t i c s t r u c t u r e then i t may be considered to have constant per-s i s t e n c e , whereas i f the wave t r a i n s occur i n runs or i n b u r s t s the per-s i s t e n c e i s considered to be i n t e r m i t t e n t . (7) Regulation of Frequency and Voltage. The r e g u l a t i o n of f r e -quency and amplitude c h a r a c t e r i s t i c s of EEG-rhythms tends to decrease w i t h i n c r e a s i n g age and the corresponding development of c a r d i o v a s c u l a r i n s u f f i c i e n c y although otherscauses are known to e x i s t . The o c c i p i t a l alpha rhythm u s u a l l y shows a frequency v a r i a t i o n of l e s s than ij0.5 c y c l e per second w h i l e the voltage of the o c c i p i t a l alpha rhythm u s u a l l y waxes and wanes w i t h i n a u s u a l l y smooth envelope. Poor frequency regu-l a t i o n i n the EEG i s o f t e n c a l l e d dysrhythmia. (8) Reactivity. The occipital alpha rhythm is reactive to eye opening and eye closing, with eye opening or alerting stimuli blocking the rhythm (alpha blocking). (9) Artifacts. The artifacts caused by swallowing, eyeblinks, head movement and the electrocardiogram must be recognized and ignored by the electroencephalographer. In the visual analysis of the EEG a l l these factors must be taken into account in order to arrive at an evaluation of the EEG. APPENDIX D Power Spectral Analysis of the Electroencephalogram D.l Introduction Perhaps the e a r l i e s t method used to describe the c h a r a c t e r i s -t i c s of the EEG was to examine which "frequency bands" were present i n the EEG recording. I t was not long a f t e r Hans Berger [44] published the f i r s t report on the EEG i n man that the terms alpha waves, beta waves, d e l t a waves, and theta waves, became accepted as h e u r i s t i c d e s c r i p t o r s o for EEG an a l y s i s . In f a c t , the c l i n i c a l EEG i s s t i l l described (at l e a s t i n part) using these terms [45]. In these cases the EEG i s usu-a l l y analyzed by manually counting the number of waves occuring i n a one second i n t e r v a l . ger's EEG recordings to Fourier analysis and compared the r e s u l t s of normal and pa t h o l o g i c a l EEGs. This motivated the development of f r e -quency analyzers by Grass and Gibbs [47]. However, the d i f f i c u l t i e s inherent i n analog frequency analyzers made automatic frequency analysis i m p r a c t i c a l u n t i l the advent of d i g i t a l computers when D.O. Walter and hi s co-workers [48-50] showed d i g i t a l s p e c t r a l analysis to be a valuable q u a n t i f i c a t i o n and analysis technique f o r electroencephalography. In f a c t , d i g i t a l s p e c t r a l analysis of EEGs has now been applied c l i n i c a l l y with some success [51, 52]. D.2 D e f i n i t i o n s We begin by considering a given EEG channel to be a time s e r i e s x ( t ) . The power spectrum of x(t) (also known as the autospectrum or variance .spectrum) i s then defined as S x x ( f ) = E[X(f) X*(f)] = E[|x(f)| 2] (D.l) where X(f) = F[x(t)], F denotes Fourier transformation, denotes complex conjugation and E denotes expected value. This definition of the power spectrum of a time series i s based on the "direct" method of power spectral estimation and i s contrasted to the "indirect" method based on the Wiener-Khint-chine Theorem [53] where the power spectrum is defined as the Fourier transform of the autocorrelation function of x(.t) . Our motivation for choosing the direct method l i e s in i t s computational efficiency. Heuristically, we can look at the power spectrum as a measure of the average intensity of the time series x(t) as a function of frequency; i t decomposes the variance of the time series into contributions from individual frequency bands. Next, we introduce another EEG channel represented as a second time series, y(t). To obtain information concerning s t a t i s t i c a l inter-relationships between the time series x(t) and y(t) we define the cross-spectrum S (f) = E[X(f) Y*(f)] (0.2) x y where X(f) = F[x(t)], Y(f) = F[y(t)]. It should be noted that while the power spectrum S x x ( f ) i s necessarily real valued, S Xy i s generally complex valued with a phase spectrum given by i> (f) = arg S (f) rxy . 6 xy The phase spectrum may be considered to be a measure of the average phase difference between the various frequency components of x(t) and 5 5 y ( t ) . A n o t h e r i m p o r t a n t d e f i n i t i o n i s t h a t o f t h e c o h e r e n c e s p e c -t r u m 1 : , , lSxy<£)l C ( f ) = y X y V S x x ( f ) S y y ( f ) ( D . 3 ) T l f x ( t ) a n d y ( t ) a r e l i n e a r l y r e l a t e d , i . e . y ( t ) = h ( t ) * x ( t ) f o r some h ( t ) , t h e n C ^ C f ) = 1. T h u s t h e c o h e r e n c e s p e c t r u m c a n b e c o n -s i d e r e d t o b e a m e a s u r e o f t h e d e g r e e o f l i n e a r r e l a t i o n b e t w e e n two t i m e s e r i e s [ 5 4 ] . F i n a l l y we s h o u l d o b s e r v e t h a t a l l c o h e r e n c e m e a s u r e s h a v e t h e p r o p e r t y t h a t 0 < C x y ( f ) < 1 D . 3 T h e F a s t F o u r i e r T r a n s f o r m I n 1 9 6 5 C o o l e y a n d T u k e y [ 5 5 ] p u b l i s h e d a f i r s t v e r s i o n o f a f a s t a l g o r i t h m f o r t h e c a l c u l a t i o n o f t h e F o u r i e r c o e f f i c i e n t s o f d i s -c r e t e p e r i o d i c f u n c t i o n s . I t h a s s i n c e come t o b e c a l l e d t h e f a s t F o u r -i e r t r a n s f o r m ( F F T ) . T h e b a s i s f o r t h e s u p e r i o r c o m p u t a t i o n a l p o w e r o f t h e F F T l i e s i n t h e f a c t t h a t i n t h e c l a s s i c a l a p p r o a c h t o t h e c a l c u l a -t i o n o f t h e d i s c r e t e F o u r i e r t r a n s f o r m ( D F T ) a l a r g e p a r t o f t h e sums a n d p r o d u c t s a r e c a l c u l a t e d m o r e t h a n o n c e . T he F F T a v o i d s t h i s com-p u t a t i o n a l r e d u n d a n c y . S p e c i f i c a l l y , l e t N b e t h e n u m b e r o f d a t a p o i n t s i n o u r E E G r e c o r d . T h e n b y u s i n g t h e c l a s s i c a l DFT we f i n d t h a t t h e Some a u t h o r s u s e t h e d e f i n i t i o n C = l Sxy< f )| 2 x y ' • S x x ( f > S y y ( f ) H o w e v e r , o u r d e f i n i t i o n i s t h e m o r e common. 56 number of operations needed i s about N 2. However, the FFT requires a number of operations given by N times the sum of the factors of N where N i s r e s t r i c t e d to be a composite number [56]. I f N i s a power of two (and almost a l l FFT subroutines require this) the number of operations required i s about N log2 N. We can e a s i l y see that f or a reasonably large number of data points, the d i f f e r e n c e between N 2 and N log2 N be-comes huge. For example, with 1024 data points N 2 i s approximately 10 6 while N log2 N i s approximately 10^. Thus for 1024 points the FFT re-quires 1/100th as many operations and consequently about l/100th as much time. An extensive amount of work has been done concerning the FFT and i t s myriad a p p l i c a t i o n s . Some p a r t i c u l a r l y good treatments of the FFT are a v a i l a b l e [5 7-60] and the i n t e r e s t e d reader i s r e f e r r e d e l s e -where for the computational d e t a i l s . D.4 A l i a s i n g Errors The FFT must be used properly to avoid serious errors i n i t s a p p l i c a t i o n . The problems i n the use of the FFT are not i n i t s r e l a t i o n to the DFT; i n fact the smaller number of operations required by the FFT mean reduced round-off e r r o r . Rather the errors usually a r i s e from an incomplete understanding of the r e l a t i o n of the DFT to the continuous Fourier transform (CFT). The most common source of e r r o r i s c a l l e d a l i a s i n g . This err o r occurs when the time function contains frequency components higher than the f o l d i n g frequency i A t , where At i s the time i n t e r v a l between samples. In the development of the r e l a t i o n s h i p of the DFT to the CFT i t was assumed that the assumptions of the sampling theorem had been met. I f , however, there are frequency components to the s i g n a l higher than the f o l d i n g frequency, they w i l l be "folded" 57 into the frequencies below the f o l d i n g frequency. For example, a f r e -quency component 10 Hz higher than the f o l d i n g frequency w i l l appear to be a component 10 Hz lower than the f o l d i n g frequency. A l i a s i n g errors can be e a s i l y avoided by low-pass f i l t e r i n g the EEG records at a cut-off frequency lower than h a l f the sampling frequency. In t h i s study the EEG records were low-pass f i l t e r e d at 45 Hz and a samp-l i n g rate of 100 Hz was used. D.5 Windows and Leakage As with a l l s t a t i s t i c a l estimation tasks, the process of s p e c t r a l estimation bases i t s estimates on a f i n i t e number of samples. For a record of length T the e f f e c t of the fi n i t e n e s s of the data may be considered to be that of m u l t i p l i c a t i o n of an i n f i n i t e time ser i e s x(t) with a rectangu-l a r gate function g(f) of amplitude 1 between -T/2 5 0 - T/2 and zero out-side. This gate function may be regarded as a time window through which the observer looks at x ( t ) . Now, m u l t i p l i c a t i o n of x(t) with the gate function g(t) i s equivalent to the convolution of the Fourier transform of x(t) with the Fourier transform of g ( t ) . The l a t t e r i s c a l l e d the sp e c t r a l window G(f) and takes the well-known form of the sampling function s i n One unfortunate property of the s p e c t r a l window G(f) i s the slow decay of i t s sidelobes. The consequence of this i s considerable overlapping of adjacent frequency bands [61] which introduces spurious c o r r e l a t i o n between the r e s u l t i n g s p e c t r a l q u a n t i t i e s . This e f f e c t i s known as "leakage". To reduce leakage, window modifications are necessary; _ The implementation of..such modifications can be c a r r i e d out e i t h e r i n the.frequency domain or i n the time domain. One popular time domain procedure (and the one used i n t h i s study) i s that of apply-ing Tukey's "interim" data window to the input s e r i e s . That i s , the data sample x^ i s m u l t i p l i e d by the weight w where To.5 (1 - cos( i f(n-l/2)/r) - n = l , . . . , r w = < 1 n = r+1 N-r n |,0.5 (1 - cos(7r(N-n + 1/2)/r) n = N-r+1,...,N This e f f e c t i v e l y amounts to m u l t i p l i c a t i o n of the input data by a p a i r of cosine b e l l s , the period of which i s determined by the para-meter r. In this study r was chosen to be the greatest integer function of N/10. The Fourier transform of the sequence iw^} i s then the spec-t r a l window used. F i n a l l y , we should note that the problem of leakage reduction i s a subject on which much has been w r i t t e n . The "best" choice of technique depends both on the nature of the data and the ob-[61, 62], D.6 Spectral S t a b i l i t y Consider the Fourier transform {X r) of a d i s c r e t e time sequence {Xn}. The periodogram of the sequence {Xn} i s defined as the sequence {Pr}: P r = |x r| 2 i (D.4) We observe, then, that the power s p e c t r a l density of {X^Hs j u s t the expected value of the periodogram. Jenkins and Watts [63] have shown, however, that one cannot use the periodogram as a consistent s p e c t r a l estimator simply by using a s u f f i c i e n t l y large number of data points i n (D.4). This i s because the variance of the periodogram does not de-crease with increasing N. Rather, we must accomplish variance reduc-ticm by breaking up our time sequence {X n} into K shorter sequences {Xn} 1 by using the f i r s t N/K points of {Xn> for ( X ^ 1 , the second N/K points for { X ^ 2 e t c . From these K sequences we can obtain K periodo-grams P r i , i = 1,2,...,K and by averaging them we then get p; -= k X (D-5) 1=1 To see how this averaging accomplishes variance reduction and therefore y i e l d s more stable s p e c t r a l estimates, we f i r s t note that the variance of a periodogram estimate i s dominated by a constant term 0 , where a 2 i s the variance of the time s e r i e s . This term can be shown to be x independent of the record length used i n the periodogram [63]. However, with a straightforward a p p l i c a t i o n of the c e n t r a l l i m i t theorem [64] we — 1 °% var{P r} = - var{P r} i - f - ( D _ 6 ) Thus the variance of the estimate decreases with i n c r e a s i n g K and the s p e c t r a l estimate has been "smoothed". This method of s p e c t r a l smooth-ing i s c a l l e d B a r t l e t t smoothing [63] and r e s u l t s i n more stable spec-t r a l estimates. We should note, however, that t h i s increase i n s t a b i l -i t y i s obtained at the expense of s p e c t r a l r e s o l u t i o n . In s p e c t r a l an-a l y s i s of actual EEG data, t h i s compromise between s t a b i l i t y and reso-l u t i o n has to be evaluated. One measure of the degree of s t a b i l i t y of a s p e c t r a l estimate i s i t s "degrees of freedom" defined by the r e l a t i o n df = £ • (D.7) where N i s the number of samples used and b i s the number of frequency 60 bands i n the estimate. C l e a r l y i t i s desirable to have s p e c t r a l e s t i -mates with large degrees of freedom. The s p e c t r a l estimates used i n t h i s study were based on 2048 data points and 128 frequency bands r e -s u l t i n g i n estimates with 16 degrees of freedom. D.7 The Spectral Estimation Program The s p e c t r a l analysis program used i n t h i s study was an exten-s i v e l y modified version of the UCLA "Jet Speed Spectrum Estimation" pro-gram c a l l e d BMDX92 [65] (now known as BMD03T). Modifications were made to the input subroutine to allow the use of unformatted magnetic tapes (to allow f o r comp a t i b i l i t y with our tapedrive), to the output sub-routine to allow the s p e c t r a l and coherence estimates to be w r i t t e n on magnetic tape for subsequent analysis and p l o t t i n g f a c i l i t i e s were i n -COi 'J /O j . " t x t c J . L O c * l l 0 w Ccil<_v_imp p l o L. Lo.iig, u i L I I C a p c C L i-ctj. w i i i i i u w , Ltifc c iu i_u — s p e c t r a l estimates, the cr o s s s p e c t r a l estimates and the coherence e s t i -mates. The program employed the d i r e c t method of s p e c t r a l estimation and used the fast Fourier transform for c a l c u l a t i o n of the required Fourier c o e f f i c i e n t s . The window used i n order to reduce s p e c t r a l leakage was Tukey's "interim" window [61], and s t a t i s t i c a l s t a b i l i t y of the spec-t r a l estimates was insured by use of B a r l e t t smoothing. [63]. APPENDIX E Computational Procedure f o r Stepwise Discriminant Analysis Notation: p = number of variables g = number of groups n m = number of cases i n group m n = t o t a l number of cases ^mki = v a-'- u e °f v a r i a b l e i for case k of group m FPROBI = F ~ p r o b a b i l i t y for i n c l u s i o n of v a r i a b l e s i n t o dis criminant function FPROBD = F- p r o b a b i l i t y for de l e t i o n of va r i a b l e s from d i s -criminant function Step 1: The data are read and the following are formed: Means: p n~ x, — E E x . . i = : l « 2 « « . . , P r n T , n mkx m=l k=l Group Means: 1 r > ^ l l , 2 , . . . , P x . = — E x , , mi n T O . T mki .. „ m k=l m = 1,2,... ,g Group Standard Deviations: I nm l J _ 1 9 p ± v t „ \ 2 2 x , i . , . . . , r - . -i T, (x . . - x .)mi n -1 , , mki mi m k=l Within and Total Covariance Matrices: m = 1,2 ,... ,g 1 g nm _ _ W = (w. .) ; w. . = E E (x . . - x .) (x . . - x .) i j i j n-g ^ mki mi mki nrj -, g 'n _ _ T = (t..) ; t . . = — E E m (x ,. - x.)(x . . - x.) i j IT, n-g m = 1 k = 1 mki i ' mki j ' i = 1,2 ,. . .',p j = 1,2, ... ,p Within Groups Co r r e l a t i o n Matrix w. . . _ R = ( r . .) ; r. . = , — 1 - J . , p j = 1,2 ,... ,p Step 2: At each step i n the procedure the v a r i a b l e s are divided i n t o two d i s j o i n t sets: those included i n the discriminant func-t i o n s , and those not included. Assume f o r s i m p l i c i t y that the f i r s t r are included. l e t W ./ " U " 1 2 ) I - N W21 W22 / \ T 2 1 T22 where W^ and T^^ are r x r matrices. W _ 1 W T . A I 11 11 L e t A = 'w w-1 w „ 1 W12 V ' -w91 w.71 w10 ) ( a i j } m — 1 T i - 1 rn I T T - 1 T - T T - 1 T / ( b i i } 21 11 22 l 2 1 11 12 ' 2 The c o e f f i c i e n t s and constant terms of the discriminant functions are computed Sci = * *kj a i j i = l , 2 , . . . , r k = 1,2,...,g r _ Ck0 = 1 / 2 .\ Sci \ i k " 1.2....,g i = l Step 3: At each step i n the procedure the F values f o r each v a r i a b l e are computed: ( i ) I f v a r i a b l e j-has entered i n t o the discriminant function 63 a. . - b . . , i . . F = _JJ 11 . n ~ r ~ S+l with degrees of freedom g - 1 and n - r - g + 1 ( i i ) . I f variable j has not been entered b. . - a . F = -22 11. " - * - 8 j a.. g - 1 with degrees of freedom g - 1 and n - r - g To move from one step to the next, one variable i s added or removed from the discriminating set according to the following rul e s : (I) I f there are one or more variables which are entered with an F value with associated F pr o b a b i l i t y larger than FPROBD, the one with the smallest F value i s deleted from the set of d i s -criminating variables. (II) I f there are one or more variables which have not been entered with an F probability less than FPROBI, the one with the l a r g -est F value i s entered. (III) I f neither of the above i s s a t i s f i e d , the process i s ter-minated. Step 4: After the l a s t step i n the discriminant function construction procedure the following are computed for Jl = 1,2,...,g ; m = 1, 2>...»g>k. = l,2,...,n^: (I) The square of the Mahalonobis distance of case k i n group m from group I: r r _ _ D2 = E E (x . . - x ) a., (x - x ) £mk . , . 1 mlcx £x x i mkx £j 1=1 3=1 ( I I ) The classification x 2 value for case k in group m as having come from group £: 2 ("-I)(n-g-1) n 1 2 Dmmk X£mk (n-1) (n-g) ( n - 1 ) 2 n 3 1 2 D2 (n_.-l) (n-g) mmk for m = £ z = (n-1) (n-g-l)_ J 2 £j . x£mk (n-1) (n-g) J £mk 4 (n-1) (n-g) n n _ _ _ _ D2 , + D2 - E E (x .- x..) a., (x .- x..) I 2 mmk £mk ._. ._, mi £i i j mi £j •*• 1 n. 1 1 D 2 (n^-l)(n-g) mmk for m ^  £ ( I I I ) The posterior probability of case k in group m as having come from group £: £mk g E e«p(-iXJ ) 1=1 Pattern samples are then assigned to that class £ for which P. , is maximized. £mk Finally, the relationship between a given F value and i t s asso-ciated F-probability can be expressed as oo FPROB(F) = / p(£)d£ F where p ( Q is the F-distribution with N-^  and ^  degrees of freedom: Ni N 2 N N -ni 1 2 ( N i 2 r 12 , r 65 REFERENCES [I] B.F. Farley, L.S. Frishkopf, W.A. Clark, and J.T. Gilmore, "Com-puter Techniques for the Study of Patterns i n the Electroencepha-logram", IRE Transactions on Bio-Medical E l e c t r o n i c s , V ol. 9, P. 4 (1962) [2] P. Kellaway and I. Petersen, E d i t o r s , Automation of C l i n i c a l E l e c -troencephalography, Raven Press, New York, 1973 [3] J.R. Cox, J r . , F.M. Nolle, and R.M. 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