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Single trial EEG signal analysis using outlier information Birch, Gary Edward 1988

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SINGLE TRIAL EEG SIGNAL ANALYSIS USING OUTLIER INFORMATION By GARY EDWARD BIRCH  .A.Sc., The U n i v e r s i t y o f B r i t i s h Columbia, 1983  A THESIS SUBMITTED  IN PARTIAL FULFILLMENT OF  THE REQUIREMENTS FOR THE DEGREE OF DOCTOR*OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department o f E l e c t r i c a l E n g i n e e r i n g )  We a c c e p t t h i s  t h e s i s as conforming  to the required  standard  THE UNIVERSITY OF BRITISH COLUMBIA June, 1988 © Gary Edward B i r c h ,  1988  In  presenting  degree  at  this  the  thesis in  University of  partial  fulfilment  of  of  department  this or  publication of  thesis for by  his  or  that the  her  representatives.  It  this thesis for financial gain shall not  Department The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3  for  an advanced  Library shall make it  agree that permission for extensive  scholarly purposes may be  permission.  DE-6(3/81)  requirements  British Columbia, I agree  freely available for reference and study. I further copying  the  is  granted  by the  understood  that  be allowed without  head of copying  my or  my written  ABSTRACT  The EEG an  goal of  s i g n a l and initial  t h e n , based on  investigation  event r e l a t e d to  s t u d y the  EEG  parameter  are  event  from  cleaning,  The into  EEG the  related  simple  additional  and  hence,  will  The  outlier  single  neurological  model  o f view o f  signal  content,  data  potentials  such as  as  the  they  four  viability  c o l l e c t e d under two  subjects of  this  Early  formed the single  conditions:  an  trial  Pursuing trial that  assumed  the  ongoing  By  EEG  into  content  the when  modeling  (GM-estimates) parameters and " c l e a n e d " EEG  i s extracted  basis  of  an  potentials,  "added"  outlier  robust  processing  related  are  additive  i s then p r o c e s s e d t o y i e l d event r e l a t e d  from  motor  estimator, a that  to  ongoing p r o c e s s .  estimated  robust  of  process  appear  models w i t h r o b u s t l y in a  a  potentials,  unique  point  useful  fundamental t o o l used  applications.  related  a  the  would e x t r a c t  The  the  pursue  a u t o r e g r e s s i v e modeling.  development  of  those models  obtained.  the  nature  w i t h AR  using  to  outlier  process  EEG.  estimation  a  considered EEG  signal  led  c h a r a c t e r i s t i c s of  need t o employ r o b u s t t e c h n i q u e s i n b o t h model  on  by  trial  c h a r a c t e r i s t i c s was  based  generated  ongoing  from s i n g l e  and  When event  study the  i n s i g h t s g a i n e d from these s t u d i e s ,  was  process.  to  a p r o c e s s i n g method t h a t  p o i n t e d t o the  ultimately  which  additive  signal  estimation  techniques  the  into  information  investigations  method  t h i s t h e s i s work was  from the  signal EEG  the by is  during  information.  the  processing  initial scheme.  investigation The  EEG  was  aictive t a s k i n which s u b j e c t s performed a  ii  skilled did  not  which  thumb movement and carry  out  provided  any  single  waveforms p o s s e s s e d events i n the not  contain  four  cost the  of  trial  consistent  of  activity. outlier  The  when  the  the  equal,  idle  to  decision 93%  trials  of  In the  the  the  the found  were i n c o r r e c t l y c l a s s i f i e d as  active.  to  case  be  these  related  of  active  to  trials  s t a t i s t i c r e s u l t i n g from  was  trials  s e t t o be  a c t i v e t r i a l s were c l a s s i f i e d c o r r e c t l y and  processed  active  waveforms. set  with  the  were c l a s s i f i e d  were i n c o r r e c t l y c l a s s i f i e d as  o f m i s c l a s s i f y i n g an i d l e t r i a l was  was  but  i d l e case the waveforms d i d  outlier  boundary active  content  classification  using a cost  time warping  In  which were  Bayesian  c a r r i e d out  outlier  waveforms.  features  features.  dynamic  subjects,  18%  motor  i n d i v i d u a l thumb movements.  misclassification and  i d l e t a s k i n which s u b j e c t s remained a l e r t  consistent  v e r s u s i d l e t r i a l s was application  an  active.  Across  the  cost  of  correctly When  f i v e times g r e a t e r ,  o n l y 1.7%  o f the  the  idle  80%  the of  trials  TABLE OF CONTENTS  Page ABSTRACT  i i  TABLE OF CONTENTS  iv  LIST OF TABLES  vi  LIST OF FIGURES  . .v i i  ACKNOWLEDGEMENTS  ix  CHAPTER 1: INTRODUCTION  1  1.1  Problem Statement  1  1.2  Motor P o t e n t i a l s  3  CHAPTER 2: MODELING THE EEG SIGNAL  7  2.1  Need f o r S t a t i s t i c a l  7  2.2  P r e v i o u s S t o c h a s t i c S t u d i e s on EEG  2.3  N e u r a l B a s i s f o r t h e G a u s s i a n Nature o f EEG  10  2.A  A p p l y i n g AR Modeling t o EEG  13  CHAPTER 3: THEORY OF AR MODELING  17  3.1  C o n v e n t i o n a l AR Parameter E s t i m a t e s  17  3.2  LSQ Parameter E s t i m a t i o n on S i m u l a t e d G a u s s i a n Data  24  3.3  D e v i a t i o n s from G a u s s i a n i t y  35  3.4  GM E s t i m a t i o n on S i m u l a t e d Contaminated G a u s s i a n Data  U n d e r s t a n d i n g o f EEG  iv  8  . . . .  36  CHAPTER A: OUTLIER PROCESSING OF SINGLE TRIAL EEG  Al  A.l  Experimental  Al  A.2  N e u r o l o g i c a l Premise .  A6  A.3  S i g n a l C l e a n i n g Process  A7  A.A  E x t r a c t i n g and P r o c e s s i n g O u t l i e r I n f o r m a t i o n  51  A.5  AR S p e c t r a l A n a l y s i s  60  A.6  A p p l y i n g O u t l i e r P r o c e s s i n g t o S i n g l e T r i a l EEG  67  A.7  S t a t i s t i c a l A n a l y s i s of Features  A.8  A p p l i c a t i o n o f Dynamic Time Warping t o O u t l i e r P a t t e r n s  Design  and EEG Data A c q u i s i t i o n  i n the O u t l i e r Patterns  . . .  78  . . .  87  CHAPTER 5: CONCLUSION  100  5.1  Summary o f Major R e s u l t s and R e l a t e d C o n c l u s i o n s  100  5.2  Areas f o r F u t u r e  10A  5.3  Significant Contributions  Investigation  106  REFERENCES  108  v  LIST OF TABLES  Page 4.1  F e a t u r e S t a t i s t i c s from S i n g l e T r i a l O u t l i e r P a t t e r n s  86  4.2  D i f f e r e n c e Between I d l e and A c t i v e Warping C o s t s  93  4.3  Group S t a t i s t i c s  95  4.4  Bayesian C l a s s i f i c a t i o n  96  4.5  Summary o f B a y e s i a n Cross-Matched  vi  Classification  99  LIST OF FIGURES Page 1.1  Primary Motor and  Somatosensory Areas o f the C e r e b r a l C o r t e x  3.1  L e a s t Squares AR  3.2  AR  Parameter E s t i m a t i o n on Gaussian  Data  3.3  AR  Parameter E s t i m a t i o n on Gaussian  Data w i t h  ...  Parameter E s t i m a t i o n  26 38 10%  AO  Contamination 3.4  AR  39  Parameter E s t i m a t i o n on Gaussian  Data w i t h 20%  AO  Contamination  4.1  Experimental  4  40  Apparatus and  Conventional  Grand Averages from  P r e v i o u s Motor P o t e n t i a l Study . . . . .  42  4.2  Robust AR  50  4.3  Process  4.4  Correlated Outlier Detection  4.5  AR  Parameter E s t i m a t i o n  to E x t r a c t O u t l i e r P o i n t s  S p e c t r a l Estimates  of One  52 54  Second EEG  Segments C o n s e c u t i v e l y  O f f s e t by a T h i r d of a Second 4.6  P r o g r e s s i o n of AR I d l e EEG  4.7  61  S p e c t r a l E s t i m a t e s w i t h I n c r e a s i n g Order u s i n g  Data  P r o g r e s s i o n of AR A c t i v e EEG  63 S p e c t r a l E s t i m a t e s w i t h I n c r e a s i n g Order u s i n g  Data  64  4.8  AR  S p e c t r a l E s t i m a t i o n of I d l e Task EEG  4.9  Average o f Segmented O r g i n a l and Cleaned  4.10  Example Waveforms  66 EEG  68 71  vii  A.11 Average Waveforms  75  A.12 Comparison o f N=15 Averaged O u t l i e r P a t t e r n s  79  A.13 S i n g l e T r i a l GM2 O u t l i e r P a t t e r n s w i t h C o r r e s p o n d i n g Thumb  A.lA  Movements  80  S i n g l e T r i a l Feature D e f i n i t i o n  8A  A. 15 Standard O u t l i e r P a t t e r n s  . . . .*  viii  89  ACKNOWLEDGEMENTS  I would l i k e t o thank everyone i n t h e P s y c h o p h y s i o l o g y L a b o r a t o r y a t UBC who have h e l p e d t o f a c i l i t a t e t h i s work. I n p a r t i c u l a r , I would l i k e t o thank A d e l l e F o r t h and Timothy Harpur f o r t h e i r i n v a l u a b l e a s s i s t a n c e i n g a t h e r i n g t h e EEG d a t a . I would l i k e t o express my deep g r a t i t u d e t o Dr. J . C. L i n d , who i n t r o d u c e d me t o r o b u s t s t a t i s t i c s and who reminded me t h a t i n some cases o u t l i e r s a r e l i k e diamonds i n a s e a o f s e m i - p r e c i o u s s t o n e s . I am g r a t e f u l t o my committee members, D r s . M. P. Beddoes, G. Dumont, and R. D. Hare, f o r t h e i r support and e x p e r t a d v i c e . I would l i k e t o g i v e s p e c i a l a d d i t i o n a l thanks t o Dr. Hare f o r making h i s l a b o r a t o r y f a c i l i t i e s a v a i l a b l e t o me and f o r h i s s t r o n g support o fraywork s i n c e t h e b e g i n n i n g . I extend my deepest a p p r e c i a t i o n t o my s u p e r v i s o r Dr. P.D. Lawrence f o r his g u i d a n c e , encouragement and d e d i c a t e d support throughout my graduate career. To my f a m i l y , I am t h a n k f u l for their encouragement throughout my education. I would l i k e t o e s p e c i a l l y thank Brenda f o r h e r p a t i e n t and n u r t u r i n g support.  ix  1  CHAPTER 1 INTRODUCTION  1.1  Problem Statement  Most  often  the  greatest  failing  of  technical aids  for severely  dis-  a b l e d persons i s the inadequacy o f the man/machine i n t e r f a c e .  With a u n i v e r -  sal  the  effective  of  providing  of  life  ideal  efficient  i n t e r f a c e , current technology  s u b s t a n t i a l independence and  i n t e r f a c e , researchers  important  robotic  have been s t u d y i n g  b r a i n p o t e n t i a l s to  arm  example  and  directly  the  capability  be  the  In p u r s u i t o f such  feasibility  communicate  a p p l i c a t i o n s would  the method of  has  hence, a g r e a t l y improved q u a l i t y  f o r even the most s e v e r e l y d i s a b l e d p e r s o n s .  electrical Two  and  utilizing  to p e r i p h e r a l  supervisory  i n p u t to a p e r s o n a l  of  an  devices.  control  computer system.  of  Such  a an  i n t e r f a c i n g c a p a b i l i t y would a l s o prove to be v e r y u s e f u l i n many man/machine i n t e r f a c e problems i n the a b l e - b o d i e d  population.  researcher  communication  is  to  provide  a  direct  The  ultimate goal of  system  between  this  man  and  external devices using e l e c t r i c a l b r a i n a c t i v i t y . Brain  potentials  are  a c t i v i t y which when r e c o r d e d signals.  Embedded  typically,  t h e r e i s a -6  EEG.  This  within  and  the  the  s c a l p , s i n c e the  are  i n the  order  of  the  db  EEG <are  t o a -9  current  5-50  of  continuous  random  electrical  are r e f e r r e d t o as e l e c t r o e n c e p h a l o g r a p h i c  activity  electrode implantation.  comprised  The  db  event  related potentials  r i s k versus  using  and  surface electrodes  benefit situation  does not  s i g n a l s as measured from the  micro-volts  (ERP)  s i g n a l - t o - n o i s e r a t i o between the  i s u s u a l l y recorded  EEG  (EEG)  and  are  easily  scalp  ERP on  justify surface  contaminated by  other  2  bio-electrical (EMG)  such  as  specific  averaged.  This  to the a v e r a g i n g  The  muscle  To date, the  o f EEG  recorded  study  during  With  to  the  recognizable  as  many drawbacks.  g r e a t e s t drawbacks are f i r s t l y ,  square  the  root  of  background  In the  the  number  of  there  However,  is a  i t is a  context  random s i g n a l  trials  of t h i s work, two  significant useful  i n c l o s e d loop l o s s o f unique  tool  for  obtaining  and  decreases.  t h a t i t i s not amenable t o r e a l - t i m e  of event r e l a t e d p o t e n t i a l s f o r use secondly,  a  averaging,  s i g n a l r e l a t e d t o the event i s assumed t o remain c o n s t a n t  becomes more  tion.  face.  and  r e l a t e d t o the event tends t o average out to z e r o a t a  proportional  approach has  and,  potentials  event, such as a f l a s h o f l i g h t , over many t r i a l s .  generally  hence  (EOG)  of the s c a l p and  m o s t l y been c o n f i n e d  random s i g n a l not  rate  ing  electroocular  p o t e n t i a l s , e s p e c i a l l y those  of ERP's has  the  signals  of  the  process-  control applications single t r i a l  a  general  informa-  idea  of  the  u n d e r l y i n g waveform of an event r e l a t e d response. One pursuing  of  the  most  significant  obstacles  the u l t i m a t e g o a l i s the e s t a b l i s h m e n t  that  must  be  overcome  o f a s i g n a l p r o c e s s i n g method  t h a t can e x t r a c t event r e l a t e d i n f o r m a t i o n from s i n g l e t r i a l EEG. been  some s i n g l e t r i a l  (see  f o r example  that  are  results partly  [1,2,3]),  related has  processing  been  to  specific  generally  dependent on  schemes, proposed by  t h a t were designed external  limited  have  The  their  been  critically  various  There have researchers  f e a t u r e s i n the usefulness  schemes  fundamental assumptions about the  i s t i c s of the EEG which at the p r e s e n t importantly,  to d e t e c t  events.  because  have  statistical  on  complex  EEG  of  their  often  been  character-  time are not w e l l u n d e r s t o o d and, dependent  in  external  more  visual  stimuli. The  goal  of t h i s  t h e s i s work was  t o study  the  c h a r a c t e r i s t i c s of  the  3  EEG an  s i g n a l and initial  event to  then, based on the  i n v e s t i g a t i o n into a processing  related information  study  i n s i g h t s gained  the  EEG  from these  method t h a t would e x t r a c t u s e f u l  from s i n g l e t r i a l  EEG.  s i g n a l c h a r a c t e r i s t i c s was  The  techniques  estimation  The  outlier EEG  signal  u l t i m a t e l y lead  method which was tive  and  to  based on  nature  estimation  the  a simple  c o n d i t i o n s : an  thumb movement and  an  c a r r y out any motor  1.2  applications.  development  of  a  Pursuing  robust  processing  n e u r o l o g i c a l model t h a t assumed an  processing  a c t i v e task  idle  Early  i n both model  single t r i a l  ongoing EEG  from f o u r s u b j e c t s formed the b a s i s o f the i n i t i a l  two  modeling.  techniques  o f event r e l a t e d p o t e n t i a l s to the  the v i a b i l i t y of t h i s s i n g l e t r i a l under  fundamental t o o l used  autoregressive  i n v e s t i g a t i o n s p o i n t e d t o the need 1so employ robust parameter  s t u d i e s , pursue  scheme.  addi-  process.  investigation into  The  EEG  was  collected  i n which s u b j e c t s performed a  skilled  t a s k i n which s u b j e c t s remained a l e r t but d i d not  activity.  Motor P o t e n t i a l s  B r a i n s i g n a l s t h a t are r e l a t e d t o movement are a type o f event r e l a t e d potential tials  are produced by the  movements cortex  and were f i r s t  of  the  which  body.  r e p o r t e d by Kornhuber and  Deecke  sensory-motor c o r t e x p r i o r Figure  illustrates  how  1.1  [5]  various  shows a parts  of  [4].  t o and  These p o t e n -  during  c r o s s - s e c t i o n of the  body  are  voluntary the  motor  functionally  mapped onto the motor c o r t e x . The  use  of ERP's r e l a t e d t o movement have many advantages over  ERP's i n the  context  as  of  a  result  a  of an  i n t e r f a c e system.  self-initiated  cognitive  other  Motor p o t e n t i a l s are produced process.  This  is  unlike  the  4  Primary Motor and Somatosensory Areas of the Cerebral Cortex  Primary motor  P  Figure 1.1  r  i  m  a  r  y  somatosensory  5  perhaps  better  external  known  stimulus  somatosensory  subset  i s required  or  auditory  associated with parts cognitive  control,  peripheral  device  spatial  to  electrodes. Brunia  and  be  Van  elicit  a  evoked  p o t e n t i a l s , where  response,  potentials.)  utilized  thereby  discern  As  what  to  ( f o r example:  Motor  obtain  minimizing  the  potentials  part  of  the  the  the  recent  work i n t h i s  Bosch  [6]  in  motor  which  control  l i k e l i h o o d of  body  across  den  unique  suggested by F i g u r e  from  Some of  and  to  called  1.1,  is  being  cortex  that  the  has  of  a  are have given  unintentional  i s i n f a c t some  moved  with  area  they  there  an  visual,  o f the body over which a d i s a b l e d p e r s o n does not  could  information  ERP's,  evoked  a c t i v a t i o n of t h a t d e v i c e . capability  of  by  use  utilizing of  surface  been c a r r i e d out  exploit  the  by  ipsilateral  and  c o n t r a l a t e r a l p r o p e r t i e s o f the motor p o t e n t i a l s t o demonstrate an a b i l i t y discriminate  between hand  body movements.  The  r e f i n e d needs t o be diversity  in  the  potentials.  throughout This  property  requires  a  robotic  arm. A  terms  of  afferent  control  there a  is  could  a  be  continuous  interface  work of Vaughan  t o which  this  control  that systems  i n the  [8]  could  requiring  i t was  will  need  f o r the  generation  such  be  in as  an in  from  i n the order  left  of  one  i s the  responses.  some motor  task  system  of  i n future role Due  second)  involvement.  interface the  the  be  averaged motor  cognitive  addressed  disabled,  o f motor  thought t h a t  the  substantial  to  r i g h t versus  derived  that,  (in  exploited  function  as  l e a s t i t can p r o v i d e  be  [7]  response  particularly  well  d i s c r i m i n a t i o n c a p a b i l i t y can  been demonstrated  task  as  a t the v e r y  that  sustained  issue  feedback  movements  functions  also  prolonged  specific an  extent  foot  pursed f u r t h e r but  I t has  potentials,  and  to  of to  that  guiding  a  work, i n peripheral the  early  the p o s i t i v e components a f t e r onset  6  of the movement were due to a f f e r e n t feedback.  However, Vaughan et a l . [9]  four years  p o s i t i v i t y i n deafferented  later  unexpectedly found a s i m i l a r  monkeys while they where carrying out a s e l f paced task.  Papakostopoulos  [10] postulated that as long as there are elements of s k i l l required i n the task, the p o s i t i v i t y w i l l be developed despite the absence of a f f e r e n t feedback.  The uncertainty surrounding  t h i s issue remains i n r e l a t i v e l y recent  work (see Grunewald et a l . [7]) and to resolve i t completely w i l l require the a b i l i t y t o analyze  the motor p o t e n t i a l s on a s i n g l e t r i a l basis because a  disabled person can not provide a movement t r i g g e r f o r conventional techniques.  The s i n g l e t r i a l  a n a l y s i s method described  averaging  i n this  thesis  provides t h i s required a b i l i t y and therefore, w i l l u l t i m a t e l y f a c i l i t a t e the study of motor p o t e n t i a l s from persons who lack p e r i p h e r a l a f f e r e n t feedback.  7  CHAPTER 2 MODELING THE EEG SIGNAL  2.1 Need f o r S t a t i s t i c a l Understanding o f EEG  A prerequisite  t o t h e mathematical modeling  of a given  adequate u n d e r s t a n d i n g o f i t s fundamental c h a r a c t e r i s t i c s . gations random EEG be  (see P e r s s o n character  signals. based  [11] o r McEwen and Anderson  signal  i s an  Previous i n v e s t i -  [12]) have n o t e d t h a t t h e  o f EEG makes the t h e o r y o f random p r o c e s s e s a p p l i c a b l e t o  T h e r e f o r e , i f t h e approach t o t h e s i g n a l a n a l y s i s o f EEG i s t o  on random p r o c e s s s i g n a l  t h e o r y then b a s i c  statistical  character-  i s t i c s o f the EEG s i g n a l s h o u l d be w e l l u n d e r s t o o d . Statistical are  often  properties,  key f a c t o r s  processing  methods  particularly  i n the r e s u l t i n g  that  have  i n s t a n c e , a p p l i c a t i o n s o f Wiener  been  al.  [13]).  characteristics required  This were  could at  about  Gaussianity,  o f many o f t h e s i g n a l applied  t o EEG.  For  f i l t e r i n g , Kalman f i l t e r i n g and AR parameter and i n c o n s i s t e n t  very well  different  G a u s s i a n assumptions.  performance  conventionally  e s t i m a t i o n t o EEG have met w i t h mixed et  assumptions  be e x p l a i n e d  times  ranging  results  (see M c G i l l e m  i f i n fact  the s i g n a l  approximations  t o the  On o c c a s i o n s when t h e s i g n a l p r o c e s s i n g was  c a r r i e d out w h i l e the EEG was r e l a t i v e l y c l o s e t o G a u s s i a n , t h e r e s u l t  would  have been c l o s e t o o p t i m a l and r e l a t i v e l y good performance would be e x p e c t e d . However, carried  as w i l l  be  demonstrated  out on EEG t h a t  i n Section  was r e l a t i v e l y  3.4,  non-Gaussian  i f estimation  had been  t h e performance  would  have l i k e l y been v e r y poor. Hence, t o u t i l i z e  s i g n a l p r o c e s s i n g methods which make c e r t a i n  statis-  8  tical  assumptions,  signal  as w e l l  when those tical  both  about  the target  assumptions a r e n o t t o some e x t e n t met.  In t h e type o f s t a t i s -  employed  sections,  a  i n this  The f o l l o w i n g  Previous Stochastic  There  have  characteristics  been  section  details  understanding  begins  this  o f which  are provided i n  o f these  process  issues  i s very  of understanding  by  i n t o the s t a t i s t i c a l c h a r a c t e r o f EEG.  S t u d i e s on EEG  r e l a t i v e l y few i n v e s t i g a t i o n s  o f spontaneous EEG a c t i v i t y .  these few i n v e s t i g a t i o n s most  study,  satisfactory  reviewing previous i n v e s t i g a t i o n s  general,  knowledge  o f u s i n g a g i v e n method  important.  2.2  a statistical  as knowledge about t h e r a m i f i c a t i o n s  modeling  subsequent  requires  into  The r e s u l t i n g  have been l a r g e l y c o n t r a d i c t o r y  investigators  were attempting  t o measure  the s t a t i s t i c a l conclusions  from  and i n d e c i s i v e . the degree  In  o f wide-  sense s t a t i o n a r i t y and t o e s t i m a t e the amplitude p r o b a b i l i t y d i s t r i b u t i o n . McEwen and Anderson To  [12] d i d some e a r l y  e x t e n s i v e work i n t h i s  t e s t f o r wide-sense s t a t i o n a r i t y they d i v i d e d  halves both  and then  the sample  required could  that  with  out a two-sample  amplitude  both  and s p e c t r a l  the amplitude  as wide-sense s t a t i o n a r y .  a g i v e n EEG segment i n t o two  Kolmogorov-Smirnov  distribution  and s p e c t r a l  n o t be s i g n i f i c a n t l y d i f f e r e n t  sidered EEG  carried  area.  (K-S) t e s t  functions.  distributions  from  on  This  test  each  half  f o r the whole EEG segment t o be con-  They t e s t e d  f o r the Gaussianity o f a given  segment by u s i n g i t s amplitude d i s t r i b u t i o n i n a K-S goodness o f f i t t e s t unknown mean  rejected closed  and v a r i a n c e  using  a 0.05  level  of s i g n i f i c a n c e .  t h e i r n u l l h y p o t h e s i s t h a t EEG from awake r e s t i n g  was G a u s s i a n and wide-sense s t a t i o n a r y  They  s u b j e c t s w i t h eyes  approximately  15% o f the time  9  over  two  second  epochs.  Persson  statistical but  the  and  [14]  tests  that  In f a c t ,  cause  they  the  and  approximately  i n commenting they  digitization  correlated. would  epochs  rates  used  efficacy  consequently  of  on t h e i r  assume  used  McEwen and  resulted  the  results,  in  time  over  8  second  p o i n t e d out t h a t  samples  sampling  tests at  a  the  samples ( o b s e r v a t i o n s ) that  were  highly  noted t h a t too h i g h a sample  statistical  recommended  of  independent  Anderson  the  60%  t o be rate  as  rate  adversely affected little  above  the  N y q u i s t r a t e as p o s s i b l e . Persson coefficient  [14] went on t o argue  between  adjacent  that  samples  the maximum t o l e r a b l e  i s about  0.5  and  correlation  i n p r e v i o u s work  showed, based on an e s t i m a t e d a u t o c o r r e l a t i o n f u n c t i o n from r e a l EEG,  that to  meet t h i s requirement the sample r a t e s s h o u l d not be much g r e a t e r than 10 The  obvious  resulting  epochs can be  conundrum  i s that  c o n s i d e r e d s t a t i o n a r y and  i f only  approximately  he  two  Hz.  second  a sample r a t e i n the o r d e r o f 10  Hz  i s used then the r e s u l t i n g number o f samples would be so s m a l l t h a t a r e a s o n a b l e i n f e r e n c e cannot be made about the amplitude Weiss based  on  [15] approached  the second  and  t h i s problem  fourth  distribution.  by d e v e l o p i n g a c o r r e c t i o n  factor,  s p e c t r a l moments, f o r the Kolmogorov-Smirnov  goodness o f f i t t e s t which i s d e s i g n e d t o compensate f o r c o r r e l a t i o n data two data  which  Although still  he  was  generated  by  a  r e p o r t s good r e s u l t s  He  sample p o i n t s .  will  moments.  be In  further  t e s t e d t h i s method on s i m u l a t e d EEG  second on  this  g e n e r a l l y l i m i t e d by i t s a b i l i t y  o n l y two EEG,  sample p o i n t s back i n time.  i n the  order  autoregressive  processes.  simulated data, i t s usefulness i s  t o compensate f o r the c o r r e l a t i o n over  In a d d i t i o n , i t s e f f e c t i v e n e s s , i f a p p l i e d t o a c t u a l limited  S e c t i o n 2.4  by  the  a different  accuracy approach  of  the  estimated  spectral  i s d i s c u s s e d which would  be  10  potentially EEG  signal  more  flexible  and u l t i m a t e l y p r o v i d e more i n f o r m a t i o n about  the  characteristics.  2.3 N e u r a l B a s i s f o r the G a u s s i a n Nature o f EEG  R. E l u l was aspects  o f EEG  r e s p o n s i b l e f o r the i n i t i a l work devoted t o the s t o c h a s t i c  based  on  neuronal a c t i v i t y .  He  first  suggested  [16,17,18]  t h a t each i n d i v i d u a l neuron g e n e r a t o r was  independent o f the summed c o n t r i b u -  tions  Therefore, t h i s  EEG,  from  a l l the  c o u l d be  independent  neuron  thought  neuronal  generators.  o f as  the sum  of s t a t i s t i c a l l y  contributions  and  since  the  resultant  independent  sum, or  contribution  the  nearly  from  each  neuron i s v e r y s m a l l r e l a t i v e t o the r e s u l t i n g EEG t h e r e must be a v e r y l a r g e number o f neurons the  application  the  sum  later  (TTX)  into  carried  out an  Limit will  have caused  about  a  g i v e n time. Theorem  tend  experiment  the b r a i n o f c a t s .  a c t i v i t y was his  a t any  o f the C e n t r a l  of neuronal a c t i v i t y  [19]  should  contributing  (CLT)  toward  i n which  i n neural  is justified;  Gaussianity. he  The amount o f TTX  10% drop  Based on t h e s e arguments  that  Siegel  groups,  existence  Elul  t h a t was  activity.  g i v e n t o the c a t s The  resulting  EEG  reduced way below the l e v e l t h a t c o u l d be accounted f o r based on  [20]  a substantial  These  However,  administered tetrodotoxin  concept o f independent or n e a r l y independent n e u r a l A.  that i s ,  followed  up  on E l u l ' s  p r o p o r t i o n o f the neurons  however, would  o f many competing  be  necessarily  inputs  work and  activity. he  proposed  the  idea  b e l o n g t o s y n c h r o n i z e d groups. restricted  i n size  t o a g i v e n neuron which would  due  t o the  result i n  a t t e n u a t i o n o f the s y n c h r o n i z i n g e f f e c t  as one moves a l o n g a c h a i n o f i n t e r -  acting  t h a t because  neurons.  He  further postulated  of t h i s r e s t r i c t e d  size  11  there  would  still  be  a  mutually  unsynchronized  activity  was  occurred  i n the TTX  summed  able  to  activity  generators.  l a r g e number  groups predict  of the  experiment.  being  Elul  very  "internally  neurons". dramatic  This  of  the  synchronized  explanation  reduction  I t a l s o allows  independent  [18]  of  o f EEG  of  alluded to e a r l i e r  neuronal  activity  which  f o r E l u l ' s b a s i c idea of activity  of  the  when the  the  individual  a l s o suggested t h a t d i f f e r e n t degrees o f  between neurons, as was  but  independence  terms "independent"  and  " n e a r l y independent" were used, would be the major i n f l u e n c e on the degree o f Gaussianity:  as  the  dependence  becomes  greater  the  resulting  distribution  becomes l e s s G a u s s i a n . Siegel  [20]  elaborates  mechanism  that  theory  of  applying  states  that  separation long  as  would  as  then  the  the  produce  the  long  on  CLT  as CLT  this  to  the can  dependency o f  this  concept  result.  dependent  two  between  applied. neuronal  He  i n so  utilizes  variables.  dependence be  and  also follows that, at periods  Roughly  Therefore, generators  levels a  Elul  [18]  degree  of  it with  that  as  q u i c k l y enough  can s t i l l  be  justified.  of time when dependency i s l e s s , the  effec-  i s more c l o s e l y  suggested the a p p l i c a t i o n of t h i s concept to v a r i o u s  of mental a c t i v a t i o n :  greater  decreases  decreases  a  [21]  speaking,  S i e g e l argues  t i v e number o f independent c o n t r i b u t o r s i n c r e a s e s and the CLT approximated.  suggests  Bernstein's  variables  w i t h i n c r e a s e d s e p a r a t i o n , the a p p l i c a t i o n of the CLT It  doing  performing  interneuronal  an a c t i v e mental t a s k would r e q u i r e  coupling  than would  a mental i d l e  state.  Hence, the degree o f G a u s s i a n i t y would decrease d u r i n g performance o f mental tasks. support  He  carried  for this  dependency  out  some  i d e a , but  problem  that  empirical  the  was  work  statistical  described  in  with  EEG,  which  showed  a n a l y s i s s u f f e r e d from the the  previous  s e c t i o n as  some  sample  well  as  12  other methodological As  problems.  attractive  as  the  above  concept  at  first  c o n t r a d i c t o r y t o a common assumption about EEG. tage, slow  fast  activity  activity  resolves  implies  'desynchronized  i s indicative  this  apparent  of  As  with  the  Elul  (idle  following  i t i s however,  states:  ( a c t i v e ) EEG,  1  'synchronization'"  paradox  seems,  and  high-voltage  EEG).  Siegel  argument.  In  s t a t e t h e r e a r e , as s t a t e d p r e v i o u s l y , groups of neurons which a r e synchronized that  is  During  relatively a  mental  becomes more states: tions  but m u t u a l l y  electrical  neuronal  than  this  correspond  appears  simple  neuronal  the  activity,  relative  "within  i s because neurons must be  i n complexity  some r e c e n t a c t i v i t y and  net  model.  the  summed n e u r a l  Their  interneuronal  tions  in  to  although  that  of  the  activity  synchrony.  group" As  work by  neurons  Siegel  task  hence t h e r e w i l l be  the r e s u l t i n g EEG  principle  itself."  So  the  does not  have  the  a g r e a t e r amount o f  connectivity:  distribution addition,  of  they  the  membrane  discovered  model, when e x t e r n a l  i n p u t was  signals  to  were  applied  the  for  a  they  studied  in  given  level  individual of  net,  the  causing  greater  independent o f the the  neural  i n f a c t the  c o n n e c t i v i t y caused  a p p l i e d , as would be  neural  [22],  t h a t the main f a c t o r i n  occurred  potential  that  Elul  from G a u s s i a n i t y was  greater  This r e s u l t  and  EEG.  based on a r i g o r o u s a r t i f i c i a l  c o n c l u s i o n was  a c t i v i t y to deviate  from G a u s s i a n i t y .  Zenone  [20]  related i n configura-  more interdependent  Anninos,  idle  internally  a c t i v i t y c a n c e l i n g each o t h e r out r e s u l t i n g i n lower v o l t a g e  In  of  be  " i n - s t e p " synchrony.  same appearance o f s y n c h r o n i c i t y and neuronal  to  r e l a t i o n s h i p between  complicated  "Essentially,  which  the  and  [20]  the  independent, which r e s u l t s i n summation o f  high-voltage  task  "low-vol-  resulting  devia-  probability  elements.  connectivity i n the  level  In their  case when a f f e r e n t distribution  became  13  l e s s Gaussian. differences  T h i s f i n d i n g may add<some i n t e r e s t i n g i n s i g h t t o the p o s s i b l e  between  t h e motor  related  potentials  i n normal  and  disabled  persons. To  date, i t appears t h a t  activation or  to various  rejected  t h e i d e a o f r e l a t i n g v a r i o u s l e v e l s o f mental  l e v e l s o f G a u s s i a n i t y has n o t been c a r e f u l l y c o n f i r m e d  by e m p i r i c a l  measurements.  n o t e d i n S e c t i o n 2.2, t h a t  T h i s i s p r o b a b l y due t o t h e f a c t , as  a s a t i s f a c t o r y t o o l t o measure the EEG s t a t i s t i c s  does n o t seem t o be a v a i l a b l e .  2.4 A p p l y i n g AR Modeling t o EEG  A very general  l i n e a r model f o r t h e modeling  of stochastic  discrete-  time p r o c e s s e s i s the a u t o r e g r e s s i v e moving average (ARMA) model. I t i s g i v e n by  I  P  X  i  =  E k i-k k=l a  X  +  E j i-j j=0 b  where x_^ i s t h e d i s c r e t e residual parameters  error  has  '  1  s i g n a l sequence o f l e n g t h n, i = l , 2 . . . n ,  values  of the s i g n a l  (AR) model i s  the following x. i  2  e_^ i s t h e  sequence and a, , k=l,2...p and b., j=0,l,2....1 a r e w e i g h t i n g  on p a s t  autoregressive  e  and r e s i d u a l s  the " a l l pole" v e r s i o n  respectively.  The  o f the ARMA model and i t  form  = a.x. , + a„x. „ + ... + a x. + e. 1 l-l 2 i-2 p l-p i  2.2  where a g a i n x^ i s t h e s i g n a l sequence, a^ a r e w e i g h t i n g parameters, e^ i s t h e white signal  residual  error  x^ a t a g i v e n  sequence time  and  p  i s the order  o f t h e AR model.  The  i i s assumed t o be a l i n e a r l y weighted sum o f p  14  past  v a l u e s o f x. p l u s a random (white) e r r o r term e.. 1  1  Often t h i s l a s t  term  i s r e f e r r e d t o as t h e p r e d i c t i v e e r r o r s i n c e i t i s t h e d i f f e r e n c e between t h e measured v a l u e and t h e p r e d i c t e d v a l u e . The  AR model has some s i g n i f i c a n t model.  Firstly,  a  practical  closed  form  advantages over  solution  t h e more  general  ARMA  t o the minimization  problem  f o r t h e e s t i m a t i o n o f t h e ARMA model parameters does n o t e x i s t and  hence i t e r a t i v e n u m e r i c a l o p t i m i z a t i o n approaches must be u t i l i z e d . in  t h e case  o f t h e AR model, t h e c l o s e d form  solution  o f the minimization  problem does e x i s t and c o m p u t a t i o n a l l y e f f i c i e n t methods have been to  estimate  t h e AR model parameters.  [23] , t h e Wold process  decomposition  (in fact,  is,  utilizing in  model,  a  t h a t any s t a t i o n a r y  ARMA  as w e l l ) o f f i n i t e v a r i a n c e can be r e p r e -  that for a given signal  reasonable  developed  by Kay and Marple  AR model which may be o f i n f i n i t e o r d e r .  even i f i t i s argued  appropriate  as noted  theorem demonstrates  any MA p r o c e s s e s  sented by a unique  Secondly,  Whereas,  approximation  The i m p l i c a t i o n  an ARMA model i s t h e most can s t i l l  be  achieved  an AR model w i t h an a p p r o p r i a t e l y chosen model o r d e r .  a study by Beamish and P r i e s t l y  [24] t h e y note t h a t t h e time  by  Similarly, s e r i e s does  not have t o e x a c t l y conform t o a f i n i t e AR model b u t r a t h e r assumes i t can be modeled by an i n f i n i t e AR model. will  provide  i n some sense  can be w e l l r e p r e s e n t e d . important  the  the optimal  f i t with  a finite  model, t h e s i g n a l  S e l e c t i o n o f t h i s a p p r o p r i a t e model o r d e r i s a v e r y  i s s u e and i t i s d e a l t w i t h i n d e t a i l i n S e c t i o n 4.4.  Previous useful  Then by choosing an a p p r o p r i a t e o r d e r which  work  has i n d i c a t e d  that  AR modeling  t o o l i n t h e i n v e s t i g a t i o n o f t h e EEG s i g n a l .  statistical  definition  of regression i s :  between two o r more c o r r e l a t e d v a r i a b l e s used  would  prove  t o be a  Jansen e t a l . [25]  "a f u n c t i o n a l  note  relationship  t o p r e d i c t v a l u e s o f one v a r i -  15  a b l e when g i v e n v a l u e s o f t h e o t h e r s . " as  would be t h e case w i t h  [26].  at a given  purposes  discussed  of  model this  i n terms  order  d e f i n i t i v e l y a s s e s s whether an AR  i s adequately  study,  r e p r e s e n t i n g a segment  the appropriateness  of i t s r e l a t i v e  related,  an a u t o r e g r e s s i v e model can be a p p l i e d  In g e n e r a l , i t i s n o t c l e a r how t o  model For  EEG, then  I f these v a r i a b l e s a r e time  o f AR  performance when  modeling  o f EEG. will  be  i t i s a p p l i e d t o the  s p e c t r a l e s t i m a t i o n o f EEG (see S e c t i o n 4.4). The estimated  AR model as a p p l i e d t o EEG can be u t i l i z e d parameters  c o u l d p o s s i b l y be used  t i o n / d i s c r i m i n a t i o n problem. in  spectrum e s t i m a t i o n .  i n s e v e r a l ways.  as f e a t u r e s i n a s i g n a l  The  detec-  As i n d i c a t e d above, the parameters can be used  Many r e s e a r c h e r s  [23,25,27] have demonstrated t h a t  t h e r e a r e some d i s t i n c t advantages o f t h i s approach over t h e c o n v e n t i o n a l FFT methods o f spectrum e s t i m a t i o n . EEG  Another b e n e f i t o f a p p l y i n g AR modeling t o  i s the f a c t t h a t r e s i d u a l s , i d e a l l y , have t h e c o r r e l a t i o n o f t h e p r o c e s s  removed  (whitened) and s i n c e t h e AR p r o c e s s  acteristics Although will [29]  o f the o r i g i n a l  process  are s t i l l  it  the s t a t i s t i c a l  contained  char-  i n the r e s i d u a l s .  Andrews [28] demonstrates t h a t r e s i d u a l s from a non-Gaussian p r o c e s s  tend  t o mask  have developed  the evidence  of non-Gaussianity,  residual overcomes  Chambers  and Heathcote  a method o f c h a r a c t e r i z i n g t h e G a u s s i a n i t y o f a p r o c e s s  based on a s c a l e f a c t o r which i s determined the  i s linear,  error distribution. the problems  by the c h a r a c t e r i s t i c f u n c t i o n o f  The main b e n e f i t o f t h i s approach i s t h a t  of correlated  data  samples,  as was d i s c u s s e d i n  S e c t i o n 2.2. The thesis  g r e a t e s t b e n e f i t o f a p p l y i n g t h e AR model t o EEG i n terms o f t h i s  work  lies  representation,  i n the f a c t  which  allows  that  i t has a  f o r the s t r a i g h t  very  convenient  forward  use o f  state-space state-space  16  techniques  such as Kalman type  filtering  [30].  These techniques  p l a y a major  r o l e i n the EEG s i n g l e t r i a l p r o c e s s i n g scheme t h a t i s d i s c u s s e d i n d e t a i l i n Chapter A.  17  CHAPTER 3 THEORY OF AR MODELING  3.1  Conventional  EEG  Estimates  s i g n a l c h a r a c t e r i s t i c s a r e changing over time and hence, a s i n g l e  time-invariant estimate  AR Parameter  model  can n o t be  model parameters  applied.  This  results  i n t h e need t o  from t h e EEG s i g n a l i n a manner t h a t w i l l  t o account f o r time v a r y i n g c h a r a c t e r i s t i c s .  attempt  The e s t i m a t i o n o f t h e AR model  parameters can g e n e r a l l y be c a r r i e d out e i t h e r by b l o c k mode e s t i m a t i o n o r by recursive  estimation.  Recursive  data p o i n t by data p o i n t .  methods s e q u e n t i a l l y update t h e parameters  They have t h e p o t e n t i a l advantage o f b e i n g  such t h a t t h e e s t i m a t i o n o f parameters adapts t o time v a r y i n g of  a  weight  signal  [31].  This  i s essentially  t o newer i n f o r m a t i o n  accomplished  s t r a t e d f o r EEG s i g n a l s , which a r e s l o w l y t i m e - v a r y i n g adaptive Jansen is  recursive  estimation  [25,32] p r o v i d e s  not r a p i d  enough  evidence  f o r short  expected t o be t i m e - v a r y i n g of  1 t o 2 seconds  schemes  over l o n g epochs, t h a t  can be e f f e c t i v e  segment  analysis  quickly.  [32,33].  However,  f o r t h e work  o f EEG s i g n a l s  Short carried  greater  I t has been demon-  which i n d i c a t e s t h a t t h e a d a p t a t i o n  relatively  are t y p i c a l  characteristics  by a s s i g n i n g  than t o o l d e r i n f o r m a t i o n .  s e t up  process that are  segments i n t h e o r d e r out i n t h i s  thesis.  Hence, a b l o c k mode approach, where t h e AR model parameters a r e e s t i m a t e d on the b a s i s o f a s h o r t data segment, i s employed throughout t h i s work. Various model  block  parameters  describes  from  mode methods have a sample  signal  t h e most common c o n v e n t i o n a l  been used  t o estimate  segment.  The f o l l o w i n g  methods f o r AR parameter  a s e t o f AR discussion estimation.  18  In the p a s t , the s t a n d a r d method t o e s t i m a t e AR parameters was the  Yule-Walker  the  autocorrelation  Bishop  [34].  equations.  The  These e q u a t i o n s  function  and  the  R  (0) R X  X  X  X  X  X  (1) R  X  X  X  and  X  (p-2) . . . R X  X  r  i s the  R  (2-p)  a  X  R  2  ap  (0)  X  X  X  X  X  by  solving  the  t i o n f u n c t i o n at s p e c i f i c  Note the  . (m) = xx n  also  that  (p) X  f u n c t i o n f o r l a g k, p i s the AR model T h e r e f o r e , by  system  of  Equations  3.1.  Kay  I  . i=0  for a  where 6*  augmented w i t h 3.3 equations  [23]  I  . * k=l  i s the  Marple  autocorrela-  x.^ x. l+m I  3.2  stationary  process  function R  (m)=R X  the  conjugate  (-m) X  symmetry p r o p e r t y  6  an e x p r e s s i o n t h a t  allows  [23]  a. R (k) + 6* k xx e  v a r i a n c e of the  of  can a l s o be u t i l i z e d i n s o l v i n g  X  the v a r i a n c e o f the r e s i d u a l s t o be c a l c u l a t e d (0) =  and  can  l a g s w i t h the f o l l o w i n g e x p r e s s i o n  In a d d i t i o n , the Yule-Walker e q u a t i o n s y i e l d  xx  obtaining  n-m-1  autocorrelation  R  and Marple  f u n c t i o n , e s t i m a t e s o f the AR parameters  X  3.1.  and  3.1  recommend, t o a c h i e v e low mean-squared e r r o r , e s t i m a t i o n of the  R  Ulrych  (2)  a ^ k=l,2...p are the AR model parameters.  obtained  (see  (1)  R* X  autocorrelation  e s t i m a t e s of the a u t o c o r r e l a t i o n be  r e l a t i o n s h i p between  parameters  X  . . R  (0)  on  i n m a t r i x form i s  U-p)  . . R X  X  r  where R ^ t k ) order,  (-1) X  (p-1) R  R  model  a  d e r i v a t i o n o f these e q u a t i o n s i s reviewed by Kay  [23] and the f i n a l r e s u l t expressed R  AR  provide  o f t e n based  3.3  residual  sequence.  Equation  3.1  can  be  t o y i e l d the f o l l o w i n g a l t e r n a t i v e form o f the Yule-Walker  19  R (0) xx R (1) xx  6* e  R (-1) xx R (0) xx  R x x  (-(p-l)) 3.A  R (p) xx Least  R  R (p-1) xx  square  common method.  X  (LSQ)  (Oj  X  0  a LPJ  e s t i m a t i o n o f the AR  parameters  i s another  very  In f a c t , as w i l l be shown l a t e r , a l l the c o n v e n t i o n a l methods  d i s c u s s e d i n t h i s s e c t i o n can be shown t o be based on l e a s t squares m i n i m i z a tion  criteria.  method  The method  can be d e r i v e d  that  i s most  commonly  i n the f o l l o w i n g manner  referred  [23].  t o as the "LSQ"  From E q u a t i o n 2.2 the  p r e d i c t i o n e r r o r can be w r i t t e n as  i  e  =  x  i"  a  3.5  k i-k x  k=l The sum o f squared p r e d i c t i o n e r r o r s i s then n SSE  1=  i=l To  find  the AR  X i=l  [  parameters  x  i * that  X k i-k k=l a  x  minimize  3.6  ] z  3.6  the p a r t i a l  r e s p e c t t o each a ^ are taken and s e t e q u a l t o z e r o . 3(SSE) = 0 3 a  derivatives  with  That i s 3.7  q=l,2.  The r e s u l t o f a p p l y i n g 3.7 t o 3.6 i s P y a. k=l, k  n y x. ,x. i.= l, l - k l - c  n  I  x  i=l  ,-l l - q x  q= 1,2... p  3.8  Then by s u b s t i t u t i n g 3.8 i n t o 3.6, the minimum SSE can be shown t o be [23] n  SSEmm.  =  Now by expanding  .Y. .  i=l  x ?  p  + iE . I  a  k=l  n  ik  E .  x  - i I l-k x  ,  i=l  3.8 i n t o m a t r i x form f o r model o r d e r p r e s u l t s i n  3.9  20  £ i-i i-i x  x  I_ _ x  x  i  1 i  Ex.  £ i-2 i-r ^ i-2 i-2  x  x  x  2  ^  X  X  .  £ i i-l X  .  1-p 1-J  X  E i i-2 X  X  3.10  _  i - i i-p  Z x  i-p  z x  a i - p j LPJ  Zx.x. _  1  l-p_  By c a l c u l a t i n g the summations i n 3.10, from a data sequence o f n p o i n t s , system  o f e q u a t i o n s can be s o l v e d  parameters. except  Note  t o determine  this  t h e LSQ e s t i m a t e o f t h e AR  t h a t i f the a u t o c o r r e l a t i o n e s t i m a t e g i v e n i n 3.2 i s used,  f o r the s c a l i n g  the AR parameters,  factor,  1/n, which  does n o t e f f e c t  the above e q u a t i o n s reduce  parameter e s t i m a t i o n .  the s o l u t i o n f o r  t o t h e Yule-Walker method o f  T h e r e f o r e , the Yule-Walker e s t i m a t e s a r e e q u i v a l e n t t o  the LSQ e s t i m a t e s f o r s u f f i c i e n t l y l a r g e n. An ment  implicit  assumption, which i s p a r t i c u l a r l y e v i d e n t i n t h e d e v e l o p -  o f t h e Yule-Walker  solution,  i s that  the a u t o c o r r e l a t i o n  assumed t o be z e r o o u t s i d e t h e d a t a segment o f i n t e r e s t . relatively  short  correlation [25,23],  function Burg  lengths  can  result  [35] addressed  autocorrelation lated  segment  (MEM).  ance  over  relatively in  this  utilized, method.  method  short  thesis MEM  problem  poor  by u s i n g  selected  parameter  estimates  extrapolation  o f the  e n t r o p y and he  formu-  Many a u t h o r s have noted the s u p e r i o r p e r f o r m -  d a t a segments  was  of the auto-  e s t i m a t i o n , known as t h e Burg method o r t h e  t h e Yule-Walker  work, because  I n p r a c t i c e , when  truncation  f u n c t i o n based on concepts o f maximum  maximum e n t r o p y method this  this  i n relatively  this  a method o f AR parameter  of  a r e used,  function i s  type  methods  when  ( f o r example see [23,25,36,37]). typically  as  relatively  the c o n v e n t i o n a l  short AR  applied  to  Therefore,  epochs  o f EEG a r e  parameter  estimation  21  The  fundamental  extrapolation and  idea  behind  Burg's method  of the^autocorrelation  i n c l u d i n g the p  should  impose  l a g : see E q u a t i o n 3.1)  t h e fewest  possible  would  To a c h i e v e t h i s , he r e q u i r e d be r e p r e s e n t e d  have maximum e n t r o p y . series,  given  produces entropy  on t h e  t h a t the h y p o t h e t i c a l  extrapolated  about t h e known  time s e r i e s , which  autocorrelation  function,  should  T h i s requirement maximizes t h e randomness o f t h a t time  the constraints  a minimum b i a s estimate  zero  Burg argued t h a t t h e e x t r a p o -  constraints  by the e x t r a p o l a t e d  (up t o  as opposed t o t h e i m p l i e d  a u t o c o r r e l a t i o n f u n c t i o n w i t h o u t compromising any i n f o r m a t i o n lags.  a non-zero  f u n c t i o n beyond t h e known l a g s  e x t r a p o l a t i o n as i n t h e Yule-Walker e q u a t i o n s . lation  i s to provide  o f the estimate  solution.  i s based  on  of the function,  From a s p e c t r a l p o i n t  choosing  a spectral  and hence  view, t h e maximum  estimate  such  t h a t the  e n t r o p y (E) p e r sample  l/2f E =  J"  -l/2f  3.11  In F ( f ) d f  S  X  s  where F ( f ) i s the s p e c t r a l e s t i m a t e o f the d a t a segment and f X  s  i s the sample  frequency, i s maximized s u b j e c t t o  l/2f J -l/2f It  can be  subject  F ( f ) exp-(j2TTk—) d f = f  X  s shown  [38]  that  R X  k=l,2,  its first  estimate p  which  the estimate of the s p e c t r a l density  In a d d i t i o n ,  i t i s shown t h a t  3.12  maximizes  Fourier coefficients  e x a c t l y t o t h e sample a u t o c o r r e l a t i o n f u n c t i o n e v a l u a t e d is  . . .p  s  the s p e c t r a l  t o the c o n s t r a i n t that  (k)  X  entropy  correspond  a t the f i r s t  p lags  f u n c t i o n o f an AR model o f order p.  the e s t i m a t e s o f t h e parameters and t h e power  22  i n the r e s i d u a l sequence, P , can be w r i t t e n as R  (0) . . . R  X  xx  X  (p) r  .1 a  R  X  X  (p)  . R X  X  11  p  0  —  a  (0)  _  Lp_  p  0  3.13  _  Note t h a t  these e q u a t i o n s  a r e o f the same form as t h e augmented  Equations  3.4 where the v a r i a n c e o f t h e r e s i d u a l s  Yule-Walker  f o r t h e z e r o mean r e s i d u a l  sequence i s equal t o P . The  a l g o r i t h m t h a t was u t i l i z e d  parameter e s t i m a t e s system  i s based  of Equations  3.13  i n this  on a procedure  t h e s i s work t o c a l c u l a t e  outlined  by Anderson  i s s o l v e d i n a s e q u e n t i a l manner.  MEM  [36].  Beginning  The with  p=0, P , i s e s t i m a t e d by 1  P  n  = X x. o . , 1 n i=l  2  <  Then t h e model f o r p=l i s determined forward  prediction  error  sequence  backward p r e d i c t i o n e r r o r sequence. 1 2 For  1 n-1  3.14  as t h a t which minimizes averaged  together  with  t h e power i n t h e t h e power  i n the  T h i s average power i s g i v e n by  n  I  t h e g e n e r a l case  [(x.-a (l)x.. ) 1  i=l+l  2  1  o f p r o g r e s s i n g from  +  (x _ -a (l)x ) ] 2  i  1  1  i  3.15  o r d e r p-1 t o o r d e r p, i t i s shown  that 1  1  2 n-p i=p+l  k=l  +  ( x  i-p"  J ^ k ^ i - p + k ^  3.16  23  and  i t can  be m i n i m i z e d  with  r e s p e c t t o the s i n g l e parameter a ^ ( p ) .  t h a t the arguments on the parameters at  a g i v e n stage  model parameters a (p)  i n this  i n d i c a t e the c u r r e n t maximum model o r d e r  s e q u e n t i a l procedure.  The  dependence o f the o t h e r  a t the p t h stage are g i v e n by the L e v i n s o n r e c u r s i o n  = a (p-l) - a (p)a _ (p-l)  k  Note  k  p  p  3.17  k  3TT  Applying  Equation  3.17  t o 3.16,  taking a p  ^—j P  = 0 and s o l v i n g f o r a  P  (p)  results i n _ I  2 a  p  (p) =  P  1 =  n  r  b  (i-l)e  (i) 3.18  + 1  I (|b . ( i - D I ' + le ,(i)l ) • ii P~l P~l i=p+l 2  P where b  (i)= P  forward a  i  0  0  errors.  In  the  Hence, the v a l u e o f p ( p ) a  remaining  backward then  Y a ( i ) x ( i - p + k ) and e ( i ) = Y a ( i ) x ( i - k ) a r e the backward and P P i= P  prediction  (0)=1.0.  the  =  P  and  parameters forward  being  prediction  i n c r e a s e d t o p+1  and  above  assumed  that  i s c a l c u l a t e d v i a E q u a t i o n 3.18,  with  calculated error  notation  it  v i a Equation  is  3.17  sequences are updated.  the procedure  i s repeated u n t i l  and The  then  the  order i s  the d e s i r e d model  o r d e r i s reached. van fitting  den  Bos  [38]  o f an AR model.  can be viewed  noted  MEM  i s equivalent to  More s p e c i f i c a l l y , Kay and Marple  a  least  squares  [23] note t h a t  as a c o n s t r a i n e d l e a s t squares m i n i m i z a t i o n problem because,  i s i n d i c a t e d by E q u a t i o n 3.15, energies  that  the sum  MEM as  o f b o t h the forward and backward e r r o r  (squared e r r o r t e r m s ) , as g i v e n below  24  n  n  I  E =  le ( i ) 1  i=p+l  I  +  2  i s the m i n i m i z a t i o n c r i t e r i a  3.2  LSQ  lb Ci) 1  i=p+l  P  3.19  2  p  for  MEM,  Parameter E s t i m a t i o n on Simulated Gaussian  LSQ methods, as i s shown i n S e c t i o n 3.3, Gaussian  random  provides  the  process  best  in  linear  terms  of  unbiased  Data  are o p t i m a l when a p p l i e d t o a  maximum  likelihood  estimates.  estimation  Simulation studies using  parameter e s t i m a t i o n were c a r r i e d out on computer generated Gaussian AR The  which LSQ  data.  s i m u l a t e d d a t a were generated u s i n g a Gaussian sequence, e^ w i t h v a r i a n c e  1.0  and  mean  0.0,  driving  an  8th  order  AR  model  with  the  following  parameters: a, = 0.838 1 2  a. = -0.471  a. = 0.638 3  a, = -0.429 4  a  a, = -0.304  a.  a_ = -0.243  = 0.518  c  T h i s s e t o f parameters was as  being  typical  sequence  e^ was  g i v e n by Box based T.G.  for  an  calculated 8th  generated  and M u l l e r  order  from  [39].  excellent  and  W.H.  random  power, where Fifty  j  Payne  from EEG  AR  model  data and the s e t was of  actual  a u n i f o r m white  EEG.  shift  [40] .  The  Gaussian  sequence v i a the  procedure  a  sequence  that t h i s  p e r i o d of  2  generator  raised  i s the i n t e g e r word l e n g t h which i n t h i s case was of  estimations  using  lengths  n=1000, n=500, n=100, n=64, and  squared  error  (MSE)  between the  actual  was  r e g i s t e r a l g o r i t h m " which i s g i v e n by  They demonstrate  p r o p e r t i e s and  sets  selected  The u n i f o r m pseudo-random number g e n e r a t o r  on a " g e n e r a l i z e d feedback Lewis  = 0.182  n  different n=32 were and  random  data  carried  estimated  out.  to  has  the j t h  31. segments The  of  mean  parameter v a l u e s were  25  calculated results  for  each  show t h a t  length  was  set.  the  MSE  decreased  l e n g t h was  to  is  order  EEG  (see  data  that for  at  was  sampled  second  EEG  data.  at  since  Deviations  Smith and  cations based optimal  Lager  Gaussian  performance of these break  case o f a c t u a l EEG,  64  Section  Hz  (see  data p o i n t s .  i t was  significantly  be  the  absolute of  as  4.1),  Since less  small  and  the MSE  as p o s s i b l e .  and  seconds  u t i l i z e d i n the s i n g l e t r i a l <  one  SE became so  a t n=64, i t was  lower bound on 1.5  there  therefore, a  decided  segment  length  (n=96), m o d e r a t e l y  a n a l y s i s method  d i s c u s s i o n on segment l e n g t h r e l a t i n g  researchers,  estimation in  segment  i n Section  (see  specifically  to  4.5.  from G a u s s i a n i t y  down under  such as M c G i l l e m e t a l . [13], Jansen e t a l . [25]  [41], employed AR  i n v o l v i n g EEG.  on  segment  l a r g e when the  segment s i z e  (n=64) should  Further  These  I d e a l l y , segment s i z e s i n  i n the  data used i n t h i s t h e s i s work i s p r o v i d e d  Several and  became v e r y  U l t i m a t e l y , a segment s i z e  S e c t i o n 4.4).  3.3  they  3.1.  s i g n i f i c a n t l y when the  to make the  o n l y 64  above t h i s lower bound, was  the EEG  increase  i n Figure  T h i s p o i n t s t o , i n terms o f parameter e s t i m a t i o n  S e c t i o n A.5)  n=32 and  one  summarized  lower bound on segment s i z e .  second segment c o n t a i n s large  n=100 and  are  o f n=500 would be p r e f e r a b l e , but  a need  The  results  began t o  reduced to n=32.  efficacy, a practical the  The  parameter e s t i m a t i o n i n v a r i o u s  In t h i s work, the methods  involving  processes.  even s l i g h t  least  squares,  i t has  been  which  are  shown  [42]  under c e r t a i n c i r c u m s t a n c e s ,  deviations  d i s c u s s e d e a r l i e r i n S e c t i o n 2.2  r e q u i r e d parameter e s t i m a t i o n  However,  methods can,  and  2.3  appli-  from G a u s s i a n i t y . t h a t t h e r e was  was  generally that  the  significantly  Although i t  considerable  was  evidence  LEAST SQUARES AR PARAMETER ESTIMATION Gaussian N=50 0.023 0.021 0.019 0.017 0.015 0.013 0.012 0.010 o Z3 0 . 0 0 8 Z> 0 . 0 0 6 c o  0.004  OJ  0.002  ^  0.000  3  4  5  7  8  Parameter Number  . . LSQ NPTS=100  LSQ NPTS=500 F i g u r e 3.1a  LSQ NPTS=1000  LEAST SQUARES AR PARAMETER ESTIMATION Gaussian N=50 0.0904  /  0.0829 0.0753  (- 0 . 0 6 7 8 o 0.0603 c < _ L x J  - T J  /  -  —  /  /  \  /  -  / /  0.0527 0.0452  C U  c_  0.0377  a a  0.0301  co  0.0226  c o  0.0151  O J  —  0.0075 0.0000  3  4  5  7  8  Parameter Number  LSQ NPTS=32  LSQ NPTS=64 Figure 3.1b  .SQ NPTS=100  tSJ  28  to expect EEG t o be sample  segments  questions  about  particular  g e n e r a l l y Gaussian,  o f EEG w i l l both  relatively  always  the G a u s s i a n short  t h e r e i s c e r t a i n l y no guarantee t h a t  be  strictly  nature  segments,  model parameter e s t i m a t i o n t e c h n i q u e s  Gaussian.  o f sample  This  segments  and t h e r e s u l t i n g  prompts  o f EEG, i n  performance  o f AR  under c o n d i t i o n s o f v a r y i n g degrees o f  d e v i a t i o n from G a u s s i a n i t y . A c o n s i d e r a b l e amount o f work has been c a r r i e d tion  of location  case  o f independent  basic  goal  provide order  parameters  r e g r e s s i o n model  and i d e n t i c a l l y d i s t r i b u t e d  o f a robust  good e s t i m a t e s  procedure,  (i.i.d.)  f o r the purposes  f u n c t i o n t o be contaminated.  should provide e s t i m a t i o n r e s u l t s  thesis,  i n the case  In a d d i t i o n , the robust  with  The i s to  ( i n the o f EEG, procedure  which a r e n o t s i g n i f i c a n t l y d i f f e r e n t  the c o n v e n t i o n a l LSQ methods when t h e data i s n o t contaminated Location  i n the  observations.  of this  c a u s i n g t h e assumed, Gaussian  estima-  parameters  when t h e data has a s m a l l number o f o u t l i e r s  o f 5 t o 20 p e r c e n t )  distribution  and l i n e a r  out on r o b u s t  from  outliers.  Case  There a r e a number o f r o b u s t methods b u t t h e most s a t i s f y i n g appear t o be those  g i v e n by m o d i f i c a t i o n s t o maximum l i k e l i h o o d .  t o date  Hogg [43]  p r o v i d e s a good background t u t o r i a l on r o b u s t methods and t h e f o l l o w i n g b r i e f description 11  x^  f(x-O)  t  ^2 *  o f m o d i f i e d maximum l i k e l i h o o d methods i s based on t h a t **••»*  x ^ aire a random sample from a p r o b a b i l i t y  where 0 i s a l o c a t i o n  tutorial.  density function  parameter, then t h e l o g a r i t h m o f t h e l i k e l i h o o d  f u n c t i o n i s g i v e n by JnL(0) =  n n I inf(x.-e) = I p(x.-0) i=l i=l 1  3.20  1  The maximum l i k e l i h o o d method maximizes I n L(0) o r i n terms o f t h e p f u n c t i o n  29  minimizes n I p(x.-G) = K(0) i=l  .  3.21  1  Assume t h a t the m i n i m i z a t i o n can be a c h i e v e d by d i f f e r e n t i a t i n g K'(9)  = 0.  and s o l v i n g  In other words, f i n d the v a l u e o f G t h a t s a t i s f i e s  n  I  V> (x.-9) = 0  i=l  3.22  1  where \|>(t) = P (t) = - f ( t ) / f (t) 1  The  value  o f 0 t h a t minimizes  estimate  o f 0 and i s d e n o t e d  Equation  3.22  except  that  o f M-estimate. will  The  K(0) i s termed the maximum  as 0.  likelihood  R o b u s t M - e s t i m a t e s a r e generated v i a  different  d e s c r i b e d i n E q u a t i o n 3.23. type  3.23  1  psi  functions  are  used  than  that  Each d i f f e r e n t p s i f u n c t i o n d e s c r i b e s a s p e c i f i c basis  protect  of robust  functions  that  against  influence  on the e s t i m a t i o n r e s u l t . *  M-estimates  outlier  data  i s to f i n d  points  An example o f such  that  p s i (\Ji)  cause  undue  a p s i function i s  t h a t due t o Huber [ 4 4 ] . I t i s designed t o d e a l w i t h d a t a t h a t i s d i s t r i b u t e d normally  in  the  distribution).  c  with  double  exponential  tails  ("heavy-tailed"  The p s i f u n c t i o n i s g i v e n by -c t c  ij)(t) = {  where  middle  t < -c | t | <. c t > c  i s a tuning constant.  3.24  A s c a l e i n v a r i a n t v e r s i o n o f the M-estimator  i s g i v e n as n  x.-0  i=l  s  I  where  s  = 0  i s a r o b u s t e s t i m a t e o f the p r o c e s s  3.25 scale.  In t h i s  case, c = 1.5  30  is  often  selected  uninfluenced protection  by  this  from  data  points.  (IWLS)  should  Gaussian  a truly while  Gaussian  still  The s o l u t i o n  distribution  providing  t o be  the  desired  t o Eqn. 3.25 i s t y p i c a l l y  mode i t e r a t i o n scheme, such as t h e i t e r a t e d - w e i g h t e d procedure  [45].  Hampel's t h r e e p a r t r e d e s c e n d i n g It  from  p s i function  outlying  found by a b l o c k squares  to allow  be  noted  that,  distribution  l i k e l i h o o d estimator.  Other  commonly  used  least  p s i functions  are  and Tukey's B i - w e i g h t .  the c a l c u l a t i o n  demonstrates  that  of the p s i function  t h e LSQ  estimate  for a  i s t h e maximum  The G a u s s i a n d i s t r i b u t i o n i s g i v e n by [46] -(x-0)* 2o» — — V2TTO r  e  x  p  [  f(x-O) =  n  ]  3.26  J  and  hence p(x-G) = \ fin(2Tro*) +  (  *~f  3.27  } 2  and \b(x-0) = Applying value  3.28  Eqn. 3.22 i t f o l l o w s  of 0 that 1  (x-O) that  t h e maximum  likelihood  estimator  i s the  satisfies  n  I (x.-6) = 0 . , I i=l  ±7 a 2  yielding  3.29  s  t h e w e l l known r e s u l t  0 = x  3.30  which i s e x a c t l y t h e LSQ e s t i m a t e o f 0  [43].  R e g r e s s i o n Case The case  above methods can be extended t o t h e b l o c k  (see Hogg  [43]).  G i v e n t h e l i n e a r model  mode l i n e a r  regression  31  Y = Xa + e where:  It  3.31  X  i s n x p data matrix  a  i s a parameter v e c t o r o f o r d e r p  e  i s a r e s i d u a l vector of order n  p  i s the o r d e r o f the model  n  i s the number o f data p o i n t s  then f o l l o w s t h a t the e x p r e s s i o n t o be minimized i s  now  T y. - x. a P( ) e  n  I  1  3.32  1  a  i=l  where:  y_^  i s the i t h d a t a p o i n t  x. -i  i s the i t h row o f the m a t r i x X  C o n s i d e r i n g the p f i r s t p a r t i a l d e r i v a t i v e s , the f o l l o w i n g s e t o f p e q u a t i o n s must be s o l v e d T n  I  x. *  i=l where s Again,  y -x (  e  1  a _ ) ~  1  e i s a robust  this  3.33  0  estimate  set of equations  of the  can be  s t a n d a r d d e v i a t i o n o f the  residuals,  s o l v e d u s i n g a IWLS procedure and  Hogg  [43] recommends the f o l l o w i n g r o b u s t e s t i m a t e  s  e  =  median |e. - median(e.)I — 0.6745  3 34 *  Autoregression Case The behind [47], series  application  the a p p l i c a t i o n  of  robust  methods  t o the i . i . d .  to  t i m e - s e r i e s data  case, p r o b a b l y ,  as  residuals  s e t s as w e l l  due  to  data  as  the p o s s i b l e  outliers  that  occur  dependency t h a t i n patches  lagged  suggested by M a r t i n  because o f the c o n s i d e r a b l e d i v e r s i t y i n q u a l i t a t i v e data  has  f e a t u r e s o f timemay  arise  (correlated).  i n the Martin  32  [47] a p p l i e s r o b u s t e s t i m a t i o n t o t i m e - s e r i e s d a t a and shows t h a t many o f the concepts  from  the  i.i.d.  observations  case  can  be  applied directly.  g r e a t e s t d i f f e r e n c e seems t o occur i n the d e f i n i t i o n s o f the r o b u s t of  the  g i v e n methods due  Martin it  mostly  t o the  qualities  mentioned  above.  [47] a l s o p o i n t s out t h a t f o r d a t a t o q u a l i f y as a t i m e - s e r i e s o u t l i e r  only  has  to  process  scale.  smaller  than  be  "different"  S i n c e the the  on  the  Martin  innovations  innovations scale  process  scale  the  v i s u a l l y d e t e c t i n a p l o t o f the raw  series  added d i f f i c u l t i e s  The  (residual)  is typically  outliers  will  scale  10 t o  o f t e n be  not  the  10,000 times  impossible  to  data.  [47] i d e n t i f i e s t h r e e types o f o u t l i e r s t h a t may  occur i n time-  data:  1) independent i s o l a t e d g r o s s - e r r o r o u t l i e r s which may r e c o r d i n g (measurement) e r r o r s  be  caused  by  various  2) p a t c h y type o u t l i e r s whose b e h a v i o r seems t o be u n c o r r e l a t e d w i t h the b e h a v i o r o f the r e s t of the d a t a - t h i s may be caused by b r i e f malfunct i o n s i n the d a t a c o l l e c t i o n system, i n h e r e n t b e h a v i o r o f the p r o c e s s or maybe o t h e r unaccountable e f f e c t s 3) p a t c h y o u t l i e r s whose b e h a v i o r does appear t o be r e l a t e d t o the r e s t the d a t a w i t h the p o s s i b l e e x c e p t i o n o f an i n i t i a l jump - t h i s type o u t l i e r may be caused by unusual events w i t h i n the p r o c e s s  He a l s o suggests  t h a t two  above types o f o u t l i e r types  types o f o u t l i e r models can r e a s o n a b l y s i m u l a t e the  activity.  The  a d d i t i v e o u t l i e r model would a p p l y f o r  1 and 2 w h i l e the i n n o v a t i o n s o u t l i e r model would a p p l y f o r type The  of of  Innovations O u t l i e r  (10) model i s d e s c r i b e d i n [47]  3.  as  P x. = £ l i . k=l where t h e  a, x. . + e. k l-k l  i n n o v a t i o n sequence e^ i s  and the o b s e r v a t i o n s are g i v e n by  3.35  i . i . d . w i t h a symmetric d i s t r i b u t i o n G  33  y. = x. J  l  1  Innovation o u t l i e r s have  found  that  .  3.36  occur when G  this  type  of  i s heavy-tailed. deviation  except perhaps i n extreme cases, estimation  of parameters from a  from  Martin  and  Gaussianity  s e r i o u s problems f o r the  Thompson  does  not  [45]  cause,  conventional  time-series.  In the case of the A d d i t i v e O u t l i e r  (AO)  model, the o b s e r v a t i o n s  are  given  by y. = x. + v. i l l  3.37  J  where x^ v^  has  i s d e f i n e d as i n 3.35  a symmetric  i.i.d.  case  which has  is  the  distribution.  the  f o l l o w i n g form 2  y  is  represents With t h i s y^  = x^  a  the  series  (l-r)N(0,0) of  (Gaussian)  distribution,  which equals  1-/  and  10 case, i t has  A l t h o u g h , v^ has  has  [45]  that  with  the  also  deal  best  +  f o r v^  estimator A  are  central  3.38  2  contamination distribution  and  with  the  mean  u  notation and  patchy  robust  2  variance  in typical  a p p l i c a t i o n s 0.01  a . 2  that  < y < 0.25.  been shown [45,47,48] t h a t c o n v e n t i o n a l  been r e s t r i c t e d above to the i . i . d .  type  of  outliers.  irrelevant  type of  In time-  the d e t a i l s of the o u t l i e r  because  i t w o u l d be  a poor  M-estimate  (GM-estimate)  of  case i t outlier  Furthermore, M a r t i n  t h a t depended s i g n i f i c a n t l y on a g i v e n d i s t r i b u t i o n f o r generalized  N(u,o )  t h a t v^ = 0 i s the p r o b a b i l i t y  schemes which d e a l w e l l w i t h t h i s  largely  the  component  rN(0,o )  Thompson [45] p o i n t out t h a t , i n p r a c t i c e , bution  degenerate  for v^for  i s h i g h l y non-robust under t h i s a d d i t i v e type  contamination. found  with  the p r o b a b i l i t y  parameter e s t i m a t i o n  been  suitable distribution  and  [45]  proportion  normal  c o n t r a s t to the  A  contaminated-normal  CND(r,o ) = where  w i t h G G a u s s i a n , v^ i s independent o f x^,  and  distrirobust  v^.  [47] , a member of  the  34  class  of  bounded-influence  autoregression  have r o b u s t q u a l i t i e s f o r the AO  case and  (BIFAR)  estimates,  i s g i v e n by  was  shown  to  [42]  T I i=p+l  5  W(x  N o t e t h a t except  > «  role  of  the  3.39  P  f o r the w e i g h t i n g  same form as the e q u a t i o n s [47] , the  " " ") = 0 e i  s  f a c t o r s W(x^), these e q u a t i o n s  g i v e n f o r the M-estimate.  additional  weighting  factors  are o f the  As d e s c r i b e d by M a r t i n i s to  down-weight  the  T summands o f E q u a t i o n  3.39  f o r which x^a  i s a poor p r e d i c t o r because one  more o f t h e v a l u e s i n x^ are too l a r g e .  He  l a t i o n o f these weights can be a c h i e v e d by  shows t h a t an a p p r o p r i a t e c a l c u -  letting  WCx^) = wtcL) where d. i s d e f i n e d as l T -1 x. C x.  d»(x.) = — - i and  w(.)  3.40 3.41  P  C i s the pxp  and  or  c o v a r i a n c e m a t r i x f o r the p ^  i s a non-negative  order AR model o f the  decreasing weighting  process  function, t y p i c a l l y of  the  form w(t)  = c \|)(t/c)/t  where c i s a t u n i n g c o n s t a n t . r e q u i r e d : one  in  3.41  3.42 Hence, t h e r e are a t l e a s t two  f o r the p s i f u n c t i o n and T -1  i s p r o p o r t i o n a l to  x^C  one  x^ ;  Mahalanobis d i s t a n c e , p r o v i d e s , as noted  [45]  The  GM-estimates can be  f o r the w f u n c t i o n .  this  by M a r t i n  which the r e l a t i v e " l a r g e n e s s " o f x^ can be  tuning  constants  The v a l u e  e x p r e s s i o n , known as [47], a n a t u r a l m e t r i c  d  2  the by  determined.  s o l v e d u s i n g an IWLS procedure  as g i v e n below  35  I  x.  i=p+l  W(x.  )  r  B^  (y. - xT  r  a  J + 1  )  =0  3.43 j = 1,2,  r  ...  NIT  where T y. - x.  *(  1  ~  - 1  'j a J  " )  P  e = s  l  J  e  *  .  3.44 T "j y. - x. a l —i-p — J  J  The p r o c e s s t o o b t a i n GM sequentially  increasing  model order i n t h i s 3.43  e s t i m a t e s r e q u i r e s s t a r t i n g w i t h model o r d e r p=l p  until  the  sequence, a s e t o f  are s o l v e d f o r each of the NIT  corresponding iterations  d e s i r e d order  robust  estimate  of Equation  3.43.  equations r e s u l t i n g  iterations.  of s The  p  are used  g  reason  i s reached.  MEM as  from  At  and each  Equation  e s t i m a t e s f o r a and starting  v a l u e s f o r the  t h a t the model parameters must  e s t i m a t e d s e q u e n t i a l l y i s due t o the manner i n which a r o b u s t e s t i m a t e o f is calculated.  I t has been found  robust  o f C"  estimate  C" where  A  i s based  1  1  the  be C"  1  [47] t h a t a s u c c e s s f u l approach t o o b t a i n a on the  factorization  = A A  3.45  T  i s upper t r i a n g u l a r and i s g i v e n by -a(p-k) X  % > k  K  s (p-k) e  s (p-k) g  I  0 where k = residual value  of  1,2  standard s  2 e  (0)  p-1  and  < k  a ( p - k ) , s ( p - k ) are the parameter e s t i m a t e s e  d e v i a t i o n f o r model order i s set equal  p-k  and  and  the r e q u i r e d s t a r t i n g  t o the v a r i a n c e o f the o r i g i n a l d a t a sequence  36  y\ .  Hence, t h e  and up  pxp m a t r i x C"  the corresponding t o p-1.  parameter  residual  Therefore,  by  i s r e p r e s e n t e d i n terms o f t h e AR parameters  1  s t a n d a r d d e v i a t i o n s d e r i v e d f o r model  fitting  AR  e s t i m a t e s a t model o r d e r  p-1  the A m a t r i x a t t h e p ^ i t e r a t i o n , via  Equation  3.45 t o be used  models  orders  i n s u c c e s s i o n , the p r i o r  GM  w i l l enable the c o n s t r u c t i o n o f A , P  which p r o v i d e s t h e r o b u s t e s t i m a t e o f C"  1  f o r the c u r r e n t GM parameter e s t i m a t e a t o r d e r  P-  3.A  GM E s t i m a t i o n on Simulated Contaminated Gaussian  Simulation  studies  were  carried  out  to  Data  determine  the  relative  performance o f r o b u s t GM e s t i m a t i o n methods as compared t o t h e c o n v e n t i o n a l LSQ  (MEM)  with  0%,  studied type  e s t i m a t i o n method when 10% and 20% l e v e l s  because  method  concepts. model  variance  contamination.  (see S e c t i o n  i n Equation  1.0 and mean 0.0.  4.2) i s l a r g e l y s  f o r these  3.37  with  AR G a u s s i a n  on  additive  s t u d i e s was produced  the d i s t r i b u t i o n  3.38 w i t h  carried  out on d a t a  reflects Chapter  t h e segment  based  G being  this trial  outlier on t h e AO  Gaussian  with  The d i s t r i b u t i o n f o r v. was o f t h e form g i v e n i n l  Equation  under  because, t h e s i n g l e  based  data  Additive o u t l i e r s are  e s t i m a t i o n i s not r o b u s t  (see S e c t i o n 3.3) and a l s o  The c o n t a m i n a t i o n  given  o f AO  c o n v e n t i o n a l parameter  of contamination  analysis  a p p l i e d t o e i g h t h order  o =2.0 and 2  T=0,  0.1, and 0.2.  &  The s i m u l a t i o n s were  segments w i t h a l e n g t h equal t o 100 p o i n t s because l e n g t h used  on t h e a c t u a l EEG s i g n a l s  this  as d e s c r i b e d i n  4. Figures  3.2, 3.3, and 3.4, show the MSE performance o f the LSQ, GM,  GM1, and GM2 e s t i m a t i o n methods on f i f t y  (N=50) random s i m u l a t e d 8 t h o r d e r AR  37  Gaussian  processes with  0%,  10%, and 20% l e v e l s  tively.  The GM1 and GM2 methods a r e e x t e n s i o n s t o the GM e s t i m a t e which w i l l  be d e s c r i b e d i n S e c t i o n A.3. Thompson used  In these  studies,  [A5] , the Huber p s i f u n c t i o n ,  f o r the f i r s t  two i t e r a t i o n s  o f AO c o n t a m i n a t i o n r e s p e c -  as suggested  by M a r t i n and  as d e s c r i b e d i n E q u a t i o n  3.2A,  was  (j = l,2) o f E q u a t i o n 3.A3 and t h e n Tukey's  bi-weight M  t  )  m  V  {t>  r  t ( l - (t/c)»)»  was used  |t|< c  0  1  It| * c .  f o r the l a s t i t e r a t i o n  Through t r i a l  J  ,  4  /  (j=3=NIT).  and e r r o r , the v a r i o u s t u n i n g c o n s t a n t s were s e l e c t e d t o  p r o v i d e the b e s t performance  i n term o f MSE and a r e summarized below c  Huber p s i  1.0  Tukey p s i  3.0  w based on Huber p s i  1.3  These r e s u l t s  show t h a t  t h e r o b u s t methods perform,  almost as w e l l as the LSQ method on the uncontaminated o f AO  contamination  level  The  r o b u s t methods perform much b e t t e r w i t h the GM2 method p e r f o r m i n g  performance  LSQ  method  o f f dramatically. almost  significantly.  However, the  c l e a r l y the b e s t performer w i t h a MSE g e n e r a l l y l e s s  i n the  10%  contamination  expectations of robust e s t i m a t i o n i n that as w e l l  falls  With a  At the 20% l e v e l o f c o n t a m i n a t i o n the  a c r o s s a l l the methods has dropped  GM2 method i s s t i l l the  case.  performance  Gaussian d a t a .  10%  as w e l l as i n the uncontaminated  the LSQ  i n terms o f MSE,  case.  results  support  the  the r o b u s t methods p e r f o r m n e a r l y  as the LSQ method on uncontaminated  b e t t e r than the LSQ method on contaminated  The  than  d a t a but perform  data.  significantly  RR P A R A M E T E R GAUSSIAN  ESTIMATION N=50  0.1200 0.1100 0.1000 0.0900 0.0800 0.0700 0.0600 0.0500 0.0400 0.0300 0.0200 0.0100 0.0000 L_  3  4  5  Parameter Number  LSQ  GM Figure 3.2  GM1  AR P A R A M E T E R E S T I M A T I O N A D D I T I V E O U T L I E R 10'/. V A R . = 2 , 0  N=50  0.122 0.112  —  0.102 0.092 o c _ c _  UJ  0.082 0.071  -a 0.061 a) <_ 0.051 o r> 0.041 cr co 0.031 tr 0.020 o 0.010  —  —  .  0.000 3  4  8  5  Parameter Number  ._. LSQ  . GM1  GM Figure  3.3  GM2  RR P A R A M E T E R E S T I M R T I O N A D D I T I V E O U T L I E R 20% V A R . = 2 . 0  . . . LSQ  GM Figure 3.4  GM1  N=50  GM2  41  CHAPTER 4  OUTLIER PROCESSING OF SINGLE TRIAL EEG  4.1  Experimental The  signals  Design and EEG Data A c q u i s i t i o n  o b j e c t i v e of the experimental  from  subjects  during  data  a controlled  a c q u i s i t i o n was t o o b t a i n EEG  voluntary s k i l l e d  motor  activity  ( a c t i v e task) and d u r i n g a c o n t r o l l e d s t a t e i n which t h e s u b j e c t s were a l e r t but n o t i n v o l v e d i n any motor a c t i v i t y  (idle  task).  For the a c t i v e t a s k , s u b j e c t s p l a c e d t h e i r which o r i e n t e d t h e i r hand i n a s t a n d a r d p o s i t i o n . aim  for a "target" position  with  their  against  right  a lever  thumb.  by p e r f o r m i n g  During  that provided  In the s t a r t i n g  position  a s m a l l opposing  the l e v e r  r e s t e d on a support  lever  position  thumb movement  extension  f o r c e t o t h e thumb movement.  to  encoded  (ramped)  t h e movement t h e t i p o f t h e i r thumb p r e s s e d  l o a d and t h e thumb was i n a r e l a x e d s t a t e . provided  apparatus  The t a s k r e q u i r e d them t o  a slow smooth  initial the  r i g h t hand i n an  signal.  information  so t h a t t h e r e was no  A potentiometer  which  was  used  A s k e t c h o f the apparatus  attached  to derive  an  i s given i n F i g .  4.1a. The long. the  duration  o f the ramped e x t e n s i o n <  was  approximately  one  second  A f t e r s u b j e c t s completed t h e e x t e n s i o n and had r e t u r n e d t h e i r thumb t o  starting  position,  visual  feedback,  v i a different  colored  lights,  was  p r e s e n t e d t o them i n d i c a t i n g whether t h e y had h i t , o v e r s h o t , o r undershot the target position. visual  During  feedback a r e a  subjects practice  from trials,  i n an attempt  looking they  thumb movements s u b j e c t s were asked t o f i x a t e on t h e  at  their  were asked  t o minimize eye movements and t o p r e v e n t thumb.  After  subjects  t o c a r r y out f i f t y  were  given  some  self-paced repetitions  Thumb bracket  ight  Linear potent iometer Lever,  Lever support for s t a r t i n g position  la  R i g h t hand g r i p s d o w e l w i t h thumb p l a c e d u n d e r t h e thumb b r a c k e t and p e r p e n d i c u l a r to the lever  43  with  their  second  right  before  thumb.  thumb  On  each  movement  trial,  onset  acquisition  (determined  by  thumb movement s i g n a l ) and then c o n t i n u e d f o r 4.5 the feedback was  presented.  presenting  feedback  seconds.  the The  potentials analysis  above  carried they  The a c q u i s i t i o n was  which  active  out  by  utilized  resulted  t a s k was  in  based  Grunewald  and  conventional  the  EEG  monitoring  more seconds  total  epoch  a previous  techniques  the  second  study  the  idle  task  the  subjects  were  kept  in  the  of  6.5  motor  In  their  [7] . and  after  into  their  averages a c r o s s seven s u b j e c t s , 35 t r i a l s each, are g i v e n i n F i g u r e For  grand  4.1b.  same  physical  s i t u a t i o n as i n the a c t i v e t a s k but i n t h i s case t h e y were not p e r f o r m i n g thumb movements.  Twenty epochs o f EEG,  from the s u b j e c t s under t h i s feedback  lights  flashed  6.5  condition.  to i n d i c a t e  seconds  onus  was  on  the  subject  to  their  p r e p a r a t i o n f o r the onset o f the next The chloride  EEG  signals  electrodes.  were  of  10  the  seconds  A f t e r t a k i n g a s h o r t pause the eyes  on  the  feedback  lights  in  epoch.  recorded  T h i s type  collected,  t o the s u b j e c t t h a t they had  fixate  any  i n l e n g t h , were c o l l e c t e d  A f t e r each epoch was  to r e l a x b e f o r e the onset o f the next epoch.  1  encoded  length  Grunewald-Zuberbier  averaging  started  a t which time  then h a l t e d one  a on  of  from  the  scalp  e l e c t r o d e possess  using the  silver/silver  most a p p r o p r i a t e  c h a r a c t e r i s t i c s f o r EEG r e c o r d i n g i n terms o f low p o t e n t i a l d i f f e r e n c e s , l o n g time  constants  and  low  resistance  s u r f a c e o f the e l e c t r o d e . hole  i n the  top  and  t h e y were  w i t h the use o f c o l l o d i o n . under the "cup" between  the  The  between  the  electrolyte  and  the  metal  e l e c t r o d e s were "cupped" shaped w i t h a s m a l l firmly  a t t a c h e d t o the  E l e c t r o l y t e j e l l y was  scalp  around  the r i m  i n j e c t e d i n t o the a i r space  o f the e l e c t r o d e which p r o v i d e d a good e l e c t r i c a l c o n n e c t i o n  scalp  and  the  electrode.  Typically  the  electrode  impedance  LEFT INDEX FIN6ER  A  B CD  E  RIGHT INDEX FINGER  F  A  B CD  E  F  Ramp p o s i t i o n i n g movements of the l e f t and r i g h t index f i n g e r . Grand averages across seven r i g h t handed s u b j e c t s . Recordings of f o r c e , r e c t i f i e d EMG, p o s i t i o n , v e r t i c a l EOG, and slow p o t e n t i a l s h i f t s i n l e f t and r i g h t p r e c e n t r a l ( C 3 ' C4') and p o s t c e n t r a l (C3'', C4**) EEG (1cm a n t e r i o r and 2 cm p o s t e r i o r t o C3, C4 p o s i t i o n s ) . f  Figure 4.1b  45  between the s c a l p The  s i g n a l was  and the r e f e r e n c e  initially  recorded  electrode from  system e l e c t r o d e s i t e s Cz, C3 and C4 of  these  electrodes  was  referenced  s i g n a l and t h e c o r r e s p o n d i n g  was  three  (See Jasper to linked  approximately  standard  2500 ohms.  i n t e r n a t i o n a l 10/20  [ 4 9 ] ) . The s i g n a l from each ear l o b e s .  A  bi-polar  EOG  encoded thumb movements were a l s o r e c o r d e d .  EOG e l e c t r o d e s were p l a c e d on t h e supra o r b i t a l r i d g e and t h e e x t e r n a l  The  canthi  of the r i g h t eye. The criterion that  EOG s i g n a l was u t i l i z e d  i n a very  which r e j e c t e d any EEG epoch t h a t  a t anytime d u r i n g  conservative  had a c o r r e s p o n d i n g  T h i s v a l u e was n o m i n a l l y  As  was  any EEG  contained  epoch  peak v a l u e s  that  s e t a t 17 m i c r o v o l t s .  n o t r e j e c t e d by the above c r i t e r i o n but  t h a t exceeded b a s e l i n e by 43 m i c r o v o l t s ,  t o f a c i a l EMG, were a l s o r e j e c t e d as a r t i f a c t - c o n t a m i n a t e d . the  a c t i v e case,  rejected.  t r i a l s not c o n t a i n i n g a reasonable  t y p i c a l l y due  In a d d i t i o n , f o r  thumb movement were a l s o  U l t i m a t e l y , based on t h e above s e l e c t i o n c r i t e r i a ,  both the a c t i v e and i d l e The  cases were u t i l i z e d  15 t r i a l s  from each o f t h e f o u r  from  subjects.  EEG s i g n a l t h a t was used f o r a l l t h e subsequent s i g n a l a n a l y s i s was taken  from the C3 e l e c t r o d e s i t e which was c o n t r a l a t e r a l t o t h e thumb In Beckman 100Hz  a l l cases  the EEG and EOG  711 p o l y g r a p h u s i n g  o f 14.7 seconds.  at a rate  Before  any  preprocessed cutoff  frequency  lowpass  a  filter  with  movement.  a m p l i f i e d by a a -3dB p o i n t a t with  a time  The s i g n a l s f o r each epoch were d i g i t i z e d i n r e a l  analysis  schemes  201-point  phaseless  o f 29Hz  initially  and a h i g h p a s s analogue f i l t e r  o f 1024 samples p e r second  signal by  s i g n a l s were  an analogue  (20dB per decade r o l l - o f f )  constant time  EOG s i g n a l  the epoch f l u c t u a t e d above or below b a s e l i n e by more  than a g i v e n t h r e s h o l d v a l u e . well,  artifact rejection  (-3dB  were  and were s t o r e d on a h a r d applied,  digital  the  lowpass  EEG  filter  signals which  p o i n t ) , a t r a n s i t i o n w i d t h o f 3Hz  disk. were had a  (-24dB a t  46  32Hz), and [50]  that  Hertz. the  a minimum stopband a t t e n u a t i o n almost  Therefore,  relatively  the  the  rate  of  there  above d i g i t a l 64  Hz  which  one  filter  of the h i g h e s t  to  make  the  best  possible  sample r a t e t h a t w i l l a l l o w  series brain  agreed  and  thirty  resampled  because  as  the  at  sample  the AR model  order  number of sample p o i n t s .  trade-off  between u s i n g  f o r the a c c u r a t e  a  representation  frequency o f i n t e r e s t and y e t w i t h i n t h a t framework keep i t as  Neurological  The  d a t a was  i s desirable  as p o s s i b l e so t h a t the r e q u i r e d model order  4.2  the  i s a c o r r e s p o n d i n g need t o i n c r e a s e  wants  s u f f i c i e n t l y high  It i s generally  i s between zero  time dependency i s spread over a g r e a t e r  Generally,  low  power i n the normal EEG  with  low  rate increases, since  a l l the  o f -27dB.  Premise  concept  outliers activity  i s minimized.  that  event  i s based at  a  ongoing e l e c t r i c a l  on  given  r e l a t e d information  the point  activity  that  following on  the  i s contained  model of  scalp.  sums, s p a t i a l l y  i n EEG  summation o f  Under and  idle  time  electrical  conditions  temporally,  at a  the given  p o i n t on the s c a l p can be modeled as an o v e r a l l ongoing p r o c e s s as  "observed"  from t h a t  When event  related unique  point  on  the  potentials, additional  scalp  such  as  process  during* a p a r t i c u l a r time i n t e r v a l . motor  they  are  related  potentials,  "added"  into  the  are  generated  pre-existing  p r o c e s s and would appear as a d d i t i v e o u t l i e r content when c o n s i d e r e d point  o f view of  distinguish trial provide  active  the  outlier EEG  information  ongoing p r e - e x i s t i n g p r o c e s s . points  time  from  series  pre-existing  then  these  Therefore,  process  outlier  points  points  could  from  the be  about event r e l a t e d p o t e n t i a l s on a s i n g l e t r i a l  a  ongoing  i f one in  by  the  could single  used basis.  to  47  The  underlying  generate  an  estimation  AR  p  rinciple  model  method  of  that  the  is  then used i n a r o b u s t  estimated  signal  between the to  be  Martin or  ongoing,  content  Thomson  [45],  underlying  (see  underlying  3.3).  This  (see S e c t i o n 4.3) EEG  the  Section  They used t h i s  contained  time  t h e r e i n , of  s e r i e s which  s p e c t r a l estimates.  estimate  estimated  Section  robust  4.4.1).  process  estimated  The  model an  difference  signal i s  The  by  which produces  process.  estimated  parameter  considered  outlier  content  that  is  provide  Process  Kalman s i g n a l e s t i m a t o r . an  (see  ongoing,  a  event r e l a t e d i n f o r m a t i o n .  information  robust  points  the  using  scheme i s t o  approach t o d e t e c t o u t l i e r p o i n t s i n a time s e r i e s was  and  "cleaned"  represent  signal  processing  (see S e c t i o n 4.4.2) to produce waveform p a t t e r n s  Signal Cleaning  An  EEG  measured s i g n a l and  additive outlier  single t r i a l  single t r i a l  s i g n a l estimator  the  original  then p r o c e s s e d  4.3  of  active  will  down-weighting unusual data  o f the  o f the pth  The  AR  the  used  as  t o study  outliers  but  part  a  of  This cleaning process  original  order  was  system not  model o f  the  (3.37) which f o r convenience i s r e p e a t e d  rather  process  the AO  process  x^  content. as  given  T h i s AR p r o c e s s  —I  =  $  X.  + u. —1  a  produced  "robustified" i s to  provide  It relies i n t h e AO  on  a  model  4.1  l  -i-1  produce  below  w r i t t e n i n s t a t e v a r i a b l e form i s g i v e n x.  to  by  character,  that  i s based on a  = x. + v. l  the  o b j e c t i v e of the c l e a n i n g p r o c e s s s i g n a l without  proposed  by 4.2  48  T where x. = (x. ,x. -i I i - i  1 0  2 0 1  0  0  a  Note  l  T u. = (e.,0,0 - i i '' '  x. i-p+1  0) , and ' •  a  that, given  0 0  . 1  4.3  0  t h e above d e f i n i t i o n , the s t a t e x^ i s equal t o the c u r r e n t  v a l u e o f x. and p a s t v a l u e s o f x. up<to x. ,,. l I i-p+1 r  "Robust" e s t i m a t e s o f the s t a t e x^ a r e c a l c u l a t e d r e c u r s i v e l y w i t h the f o l l o w i n g e x p r e s s i o n [45] m. y. - x i-1x. = <J>x. , + — s. \|) ( -i - i - l s? l s. l y  where  i s the f i r s t  column  )  o f t h e pxp m a t r i x  4.4 which  i s recursively  c a l c u l a t e d as f o l l o w s M  l+l  l  4.5  x  and y. - x . , a m.m. r> w / i ~ i - l \ - i - i P. = M. - w ( ) s. where Q i s a pxp m a t r i x w i t h a l l zero e n t r i e s except is  equal t o a r o b u s t e s t i m a t e  Q(l,l)= S  2 g  4.6  the f i r s t  element which  o f the v a r i a n c e o f the r e s i d u a l sequence, i . e .  and s^ i s a time v a r y i n g s c a l e d e f i n e d by s* ±  = M (l,l) i  4.7  49  The  cleaned data  a t time  i , will  then be  the f i r s t element of the  estimated  s t a t e x., which i s -i y. J  In  = x.  other words, y^  the  additive  that  there  i s an e s t i m a t e of the p r o c e s s  outliers,  i s no  4.8  1  1C  v^.  without  the i n f l u e n c e o f  Note t h a t w i t h the s c a l i n g g i v e n 4.7  i n f l u e n c e from  the p s i f u n c t i o n ,  as would be  and  given  the d e s i r e d  r e s u l t when t h e r e i s no o u t l i e r c o n t e n t , x. = y.. Hampel's t h r e e p a r t r e d e s c e n d i n g p s i f u n c t i o n [43] was E q u a t i o n 4.4  used i n  and i t i s g i v e n as f o l l o w s :  Itl  0 <. | t | < a a <. | t | < b  ii(t) = s i g n ( t )  { c-b 0  where a, b and of  w(t) The procedure based  on  w =  cleaning  process and  4.10 described GM2  AR  c l e a n e d time  provides  a  block  diagram  parameter e s t i m a t e s . it  was  found  estimate of  of the  above  was  utilized  parameter e s t i m a t e s . series  i t e r a t i o n where the parameters from GM1 improved  the  and  a GM2  are used cleaned  procedure  to  as  part  A GM1  estimate  of  performance  the  estimate i s is a  further  i n the c l e a n e r t o p r o v i d e a time  series.  obtain  GM,  Figure GM1,  and  Through the s i m u l a t i o n s t u d i e s d e s c r i b e d i n S e c t i o n  t h a t the b e s t  case  form  f(t)/t  estimated  theoretically  In a s i m i l a r f a s h i o n as i n the  f u n c t i o n i s o f the  t o o b t a i n GM1 the  c £ |t|  c are t u n i n g parameters.  GM-estimation the  4.9  Itl < c  b *  i n terms of m i n i m i z i n g  o b t a i n e d by s e t t i n g the t u n i n g parameters i n E q u a t i o n 4.9  the  MSE  as g i v e n below  4.2 GM2 3.4 was  50  ROBUST AR PARAMETER ESTIMATION y(k) 3/ GM ESTIMATION  y(k)  GM ESTIMATE a  CLEANER  \1/  9(k) c  GM ESTIMATION  y(k)  GM1 ESTIMATE .A  a CLEANER  GM ESTIMATION Figure 4.2  GM2 ESTIMATE  51  t u n i n g parameter  4.4  a  1.8  b  2.2  c  3.0  E x t r a c t i n g and P r o c e s s i n g O u t l i e r  4.4.1  Extracting Outlier  The  outlier  Information  extraction  t a k i n g the epoch, 6.5  process  the  expected  model  4.5).  to  The  parameters  f o r the  seconds l o n g , and  w i t h each segment o v e r l a p p e d by order  Information  for idle account  .75  t a s k EEG  for active  dividing  seconds.  data  i s accomplished  i t into  1.5  i n f o r m a t i o n i n the  i n the  c l e a n e r d e s c r i b e d i n S e c t i o n 4.3.  outlier  e x t r a c t i o n p r o c e s s was  expected would values  be  12th  additive  e x t r a c t i o n process the  (see  Section  The  outliers  and  are  cleaned  then  signals  *  simulated  (correlated)  from  EEG  of  s i g n a l from each segment i s then c l e a n e d u s i n g the e s t i m a t e d model  (see F i g u r e 4.3).  Gaussian  segments  hence, r e d u c i n g the a b i l i t y  c a l c u l a t e d by t a k i n g the d i f f e r e n c e between the o r i g i n a l  The  second  by  Each segment i s modeled a t the  (12-14) and task  EEG  outlier  had  simulated  order  AR  data  t h a t i n the  case  correlated.  As  ability  Patchy  of r e a l  by  This  by  contained test  "patchy"  confirmed  was  used  the a d d i t i v e event Martin  dividing  10%  that  t o r e c o v e r the o u t l i e r  contamination  EEG  suggested  f o r v. were g e n e r a t e d  which  contamination.  some d i s t i n c t  signal.  i n i t i a l l y t e s t e d by a p p l y i n g i t t o  and  the  Zeh  since  the  content it  was  related  potentials  [51] , the  correlated  segment i n t o  equal halves.  P R O C E S S TD E X T R A C T OUTLIER POINTS  A  yCk)  PRRRMETER  7>  ESTIMRTION  a  P Ck) t  CLERNER  yCkJ-y^k)  yCk)  OUTLIERS yCk)  Figure  4.3  53  Immediately  following  the f i r s t  non-zero  non-zero v^'s were grouped t o g e t h e r . c o r r e l a t e d v_^'s v i a the f o l l o w i n g v? l  =  These grouped v^'s were used t o produce  expression  0v? . + (1 - Q ) ' v . i - l l 2  where a v a l u e o f 0 = 0.6  was  results  outlier  in a  v ^ i n e a c h h a l f the r e s t o f the  correlated  1  4.11  2  used i n these s i m u l a t i o n t e s t s . series  which  has  This procedure  r o u g h l y the same v a r i a n c e  as i n the independent case [51]. It psi  was  found  function given  stronger Section  i n this  application  i n E q u a t i o n 4.9  needed  i n f l u e n c e than i n the parameter 4.3.  p r o c e s s was  By  trial  and  error  the  the t u n i n g parameters  f o r the  t o be s e t such t h a t i t p r o v i d e d a  estimation application discussed i n best  performance  of  the  extraction  o b t a i n e d by s e t t i n g the t u n i n g parameters as g i v e n below  t u n i n g parameter  Some example potential  that  *  a  1.0  b  1.2  c  1.8  results  performance  from of  these t e s t s , the  extraction  which  qualitatively  process, are  demonstrate  shown i n F i g u r e  the 4.4.  c Each  example c o n t a i n s the same randomly g e n e r a t e d v_^ c o n t a m i n a t i o n which  used  to contaminate  each p l o t outlier  d i f f e r e n t G a u s s i a n AR sequences x^.  was  The broken l i n e i n  r e p r e s e n t s the a c t u a l v^ v a l u e s and the s o l i d l i n e r e p r e s e n t s the  content that  F i g u r e s 4.4a  was  through 4.4c  extracted  v i a the the o u t l i e r  show t h r e e d i f f e r e n t  parameters i n the c l e a n i n g p r o c e s s .  extraction process.  examples o f u s i n g GM2  F i g u r e s 4.4d and 4.4e  and  LSQ  are the second and  CORRELATED OUTLIER DETECTION: flR-12 A D D I T I V E O U T L I E R 10% V O R . = 2 . 0 NPTS=100 O F F S E T - I 5.892 ••.983 4.073 3.163 2.234 1.344 0.433  •v  -0.475 —  1.384  -2.294 -3.203 —  4.113  OM2 Parameter Estimates h i  — S.022  io  20  3 0  4-0  70  50  eo  9 0  Data Points 4.831  j  4.0Z9  j  3.206  !  2.383  j  1.560  j  0.737  I  — 0.08S  j  —0.908  j  .  — 1.731 —2.354  '  -3.377  !  —  4.199  —  3.022  b -  LSQ Parameter Estimates • i  V  io  20  50  30  6.0  7 0  8 0 ~  9 0  Data Points £xtraot«d  Figure 4.4a  Outltor*  CORRELATED OUTLIER DETECTION: AR-12 A D D I T I V E O U T L I E R 107. V A R . = 2 . 0 NPTS=100 O F F S E T - 4 3.271 4,244  i  3.2X7 2.190 1.163 0.133 -0.892 — —  1,919 2.946  GH2 Parameter Estimates  -3.973 — 3.OOO -6.Q28  lO  20  30  A-O  I— 6.0  SO  Data Points  7"b  9fJ  80  A  4.851 4.029 3.20& 2.383 1.3&0 0.737  -o.oes -0.90B —  1.731  —  2.334  —  3.377  —  4.199  —  3.022  LSQ Parameter Estimates  - J _ - -  _i  lO  I  SO Data Points  20  _i_  6,o  > °  ""  8  0  9  0  Extraotad Outli«n Aotual OutlMr*  Figure 4.4b  CORRELATED OUTLIER DETECTION: AR-12 A D D I T I V E O U T L I E R 107. VAR.=2.0 NPTS=100 O F F S E T = 8 7,000 6.000 5.000 4.000 3.000 2.000 1.000 •° 0.000 ^ — 1 . 0 0 0 =—2.000 " - 3 . 0 0 0 =«-4,000 — 5.000 — 6.000 — 7.000 v  1— . GM2  Parameter  Estimates  I  o  10  .... - U .  2 0  3 0  4-0  t 50  so  7 0  s>o  Data P o i n t s 7.000 6.000 s.000 4.000 3.000 2.000 1.000 0.000 -1.000  - a .  -vr  -2. OOO -3.000 -4.000  LSQ  -s.000  Parameter  Estimates  -6.000 -7.000  lO  :o  3 0  4-0  SO  6.0  70  80  90  Data P o i n t s E x t r a c t e d Outlian ftotuol Outliers Figure  4.4c  CORRELATED OUTLIER DETECTION: A R - 1 2 A D D I T I V E O U T L I E R 107. V A R . = 2 . 0 NPTS=100 OFFSBT=4 6.134 5.132 4.10? 3.087 2.064 1.042 0.019 -1.003 -2.026 •3.048  GM1  -4.071  Parameter  Estimates  •3.093 -6.116  IO  _L  2 0  Data  SO  7 0  5 0  4-0  3 0  9 0  Points  4.286  A,  3.431 2.573 1.713  A  0.858 O.OOO -0.838 -1.713 -2.573 -3.431  GM  -4.288  Parameter  Estimates  -5.146 -6.003 . . .  . 1  IO  3 cF  "  4  o  s o  6,0  J  7 0  8 0  L.  Data P o i n t s Extraot«d Outlisrs Aotual Outliers Figure  4.4d  —i  CORRELATED OUTLIER DETECTION: A D D I T I V E O U T L I E R 107. VAR.=2.0 NPTS-100 7.OOO 6, O O O 5.OOO 4,000 3.OOO 2.OOO 1,000 O.OOO -l.OOO -2.OOO -3.OOO -4. OOO -5.OOO -6.OOO -7.OOO  OFFSET--8  .A.  tz L  GM1  lO  2 0  4-0  3 0  SO  6.0  Parameter  7 0  Estimates  8 0  9 0  Data P o i n t s 7.OOO 6.OOO 5.OOO 4.OOO 3.OOO 2.OOO l.OOO O.OOO -l.OOO -2.OOO -3.OOO -4.OOO —3 . O O O -6. OOO -7.OOO  GM Parameter E s t i m a t e s  lO  2 b  3 0  4-0  5 0  6 0  7 0  8 0  9»0  Data P o i n t s  F i g u r e 4.4e  E x t r a c t e d Outlier* Aotual Outliers  59  third  examples r e p e a t e d u s i n g GM  and  GM1  parameter e s t i m a t i o n .  from these examples t h a t the p r o c e s s performs w i t h GM of GM2 case  estimates and GM1  that  much b e t t e r than w i t h  LSQ  estimates.  estimates The  than  performance  are q u i t e s i m i l a r w i t h perhaps some s u b t l e improvements i n the  o f GM2.  performance  and  b e t t e r w i t h GM2  It i s clear  S i n c e these i n using  tests  LSQ,  GM,  r e v e a l e d t h r e e c l e a r l y d i s c e r n a b l e jumps i n and  GM2  subsequent s t u d i e s u s i n g o u t l i e r  parameter  estimates,  i t was  decided  d e t e c t i o n i n t h i s t h e s i s work would be  r e s t r i c t e d t o those t h r e e e s t i m a t i o n methods.  4.4.2  Processing O u t l i e r  Information  P r o c e s s i n g the e x t r a c t e d o u t l i e r i n f o r m a t i o n i s accomplished the o u t l i e r outlier  content  content  from each EEG  from  the c o r r e s p o n d i n g  an o u t l i e r p a t t e r n spanning is  then  smoothed by  which i s based  segment and  i t together with  o v e r l a p p i n g segment.  the whole 6.5  convolving  averaging  i t with  second a  epoch.  by t a k i n g the  This results i n  The o u t l i e r p a t t e r n  16 p o i n t t a p e r e d  on a minimum-bias s p e c t r a l window suggested  smoothing window by P a p o u l i s [52].  I t i s g i v e n by ,,,,, W(k)  1  =  , 16TT  1  C  +  COS(2TT  4.12  [16(2ir k / 1 6 ) where  k = 0, +/-1, The  k/16)  2  resulting  2  +/-2  -  IT ] 2  2  +/-16  smoothed p a t t e r n c o n s t i t u t e s the output waveform o f the  s i n g l e t r i a l p r o c e s s i n g method.  60  A. 5  AR S p e c t r a l A n a l y s i s  Preliminary providing signal. EEG  studies  some measure  i n v o l v i n g AR  of the a b i l i t y  As w e l l , these  spectral  analysis  were  useful i n  o f t h e AR model t o r e p r e s e n t  t h e EEG  s t u d i e s were i n s t r u m e n t a l i n e s t a b l i s h i n g a p p r o p r i a t e  segment l e n g t h s and a procedure f o r the s e l e c t i o n o f t h e AR model o r d e r . It  can  be  shown  [23]  that  the  AR  spectral  estimate  i s  S (f) S(f)  =  A.13  P II - I a k=l  k  e x p - ( ^ ) | s  2  where S ( f ) i s t h e power spectrum o f t h e r e s i d u a l sequence e. and f I  6  sample frequency. t h e o r y white, flat  i s the  S  S i n c e the term S ( f ) a p p l i e s t o the r e s i d u a l s which a r e i n e  t h e r e s u l t i n g power d e n s i t y f u n c t i o n o f t h e r e s i d u a l s s h o u l d be  and t h e r e f o r e S ( f ) e  will  be a c o n s t a n t  I d e a l l y , the v a l u e o f t h i s c o n s t a n t  independent  of  frequency.  ( n o t i n g t h a t the mean o f t h e r e s i d u a l s i s  zero) w i l l be p r o p o r t i o n a l t o t h e v a r i a n c e o f t h e r e s i d u a l s [A6]. Hence, t h e final  expression  f o r the conventional  r e p l a c i n g S ( f ) i n A.13 w i t h ° e  estimate  i s o b t a i n e d by  s / f , where s i s an e s t i m a t e e s e  o f the v a r i a n c e  2  r  of  The  EEG  signal  h i g h l y a c t i v e mental spectral  considerable two  2  t h e r e s i d u a l s and t h e 1/f term i s i n c l u d e d i n t h e numerator so t h a t t h e s  t r u e power o f t h e c o r r e s p o n d i n g  AR  AR s p e c t r a l  seconds.  consecutive  estimates  analogue s i g n a l w i l l be r e p r e s e n t e d [ 2 3 ] .  characteristics  from  subjects,  s t a t e s , a r e changing r e l a t i v e l y from  adjacent  change i n s i g n a l  one second  characteristics  particularly  quickly.  segments  c o u l d occur  An example o f t h i s i s p r o v i d e d i n F i g u r e A.5. AR  spectral  plots,  each  derived  from  during  Single  demonstrated over  this  that  span o f  I t contains  a one second  trial  four  segment o f  61  Figure 4.5 AR spectral estimates of one second segments of active trial EEG consecutively offset by a third of a second.  62  active was  EEG  and  each o f f s e t  discussed  by  0.333 seconds  i n S e c t i o n 3.2,  i t was  (total  found  span o f *  that a p r a c t i c a l  segment l e n g t h , from a parameter e s t i m a t i o n p o i n t o f view, one  second.  the s i n g l e t r i a l  need  for  short  segments the  because  as  short  was  found  Akaike's  segments  of  was  to  Information  the  of  relatively the  segment  (AIC)  rapid  orders  was  changing  signal lower  were s i m i l a r  to  work w e l l w i t h  Jansen  [32]  within a  results.  reasonable  range  these  i n that  I t was (for a  found u s e f u l  sample r a t e o f  f o l l o w i n g the t r e n d of the e s t i m a t e as  increased. Features  were i d e n t i f i e d  increased,  a c o n v e n t i o n a l FFT based e s t i m a t e .  expecting  peaks began t o o c c u r . two  t h a t seemed  22  12 t o  from  example AR for  The  during  the  was  o r d e r was s e q u e n t i a l l y  a p p r o p r i a t e model o r d e r was  then  spurious  s e l e c t e d to  be  T y p i c a l l y , model o r d e r s were s e l e c t e d i n the  14 from s u b j e c t s d u r i n g the i d l e  subjects  the  reasonable  f e a t u r e s t o become b e t t e r d e f i n e d , u n t i l  or t h r e e below t h a t v a l u e .  range o f to  the  The  the  and e r r o r and, i f  based on both the a - p r i o r i knowledge o f the c o n d i t i o n under which the EEG c o l l e c t e d and  in  seconds  l e n g t h above the  does not  knowledge o f expected  64Hz, somewhere between 8 to 25), order  as n o t e d  the model o r d e r v i a c o n v e n t i o n a l methods  Criteria  Conclusions  some a - p r i o r i  t r y a number  model  approximately  an attempt t o t r a d e o f f the  s e l e c t i o n of an a p p r o p r i a t e model o r d e r r e q u i r e s some t r i a l possible,  on  bound  the parameter e s t i m a t i o n e f f i c a c y .  that selecting  [53].  T h i s was  desire to r a i s e  bound f o r purposes o f improving  such  lower  second segment l e n g t h w i t h an o f f s e t o f 0.75  p r o c e s s i n g method.  c h a r a c t e r i s t i c s with  It  As  These f i n d i n g s u l t i m a t e l y l e a d t o the u t i l i z a t i o n ,  S e c t i o n 4.4.1, o f a 1.5 in  o seconds).  active  task.  t a s k and Figures  i n the range o f 4.6  and  4.7  s p e c t r a l p l o t s to demonstrate t h i s model order s e l e c t i o n  the i d l e and a c t i v e t a s k r e s p e c t i v e l y .  18  provide procedure  63  Figure 4.6 Progression of AR spectral estimates with increasing order using BEG data from an example idle trial, a) conventional FFT b) model order 8 c) model order 10 d) model order 12 e) model order 14 f) model order 16  64  LO  • as  •e a  *. a* e 0)04 0  ao  "12''  i«'" 'to'' '2C  s i r  24  12 16 20 r r a q u w i o y CHx) tf)  24  8  F r a q u a n o y (Hz) Co>  Fr«qu«noy Cd)  LO —  (Hz)  LO  • as _ 4.  a« >a4U a2  12 I * 2 0 F r « q u « n o y (Hs) (•)  24  ao  Figure 4.7 Progression of AR spectral estimates with Increasing order using BEG data from an example active task, a) conventional FFT b) model order 8 c) model order 10 d) model order 12 e) model order 14 f) model order 16  65  In  the  alternative  t o assuming  calculation FFT  based  signal  derivation  of  that  the  autoregressive  the r e s i d u a l s w i l l  o f S ( f ) would be t o e s t i m a t e g  estimate.  because  The r e s i d u a l  the information  signal  that  spectral  be p e r f e c t l y white  an  i n the  that quantity with a conventional can be thought  can be r e p r e s e n t e d  o f as a whitened by an AR model has  been s u b t r a c t e d r e s u l t i n g i n a s i g n a l w i t h a much f l a t t e r FFT  estimate  spectrum.  When t h e  i s a p p l i e d t o t h i s prewhitened s i g n a l t h e i n h e r e n t drawback o f leakage i s  greatly  reduced.  Application  of  conventional  leakage  control,  such  as  Blackman windowing, s e r v e s t o f u r t h e r reduce t h i s problem. The prewhitened AR e s t i m a t i o n method, t h e r e f o r e , combines t h e s p e c t r a l i n f o r m a t i o n from both t h e AR model and the r e s i d u a l FFT s p e c t r a l e s t i m a t e . S(f)  —  =  I  1 k=l  where S. Ne T  I t i s g i v e n by [45]  r  4.14  a, exp - ( l ^ f ) f s (f) i s a s p e c t r a l estimate of the r e s i d u a l  conventional represent  K  FFT method.  short  Some i n s i g h t  segments  of  EEG  was  prewhitened AR s p e c t r a l  estimates.  appropriate  was u t i l i z e d  model  order  into  s e q u e n c e e. u s i n g a l  the a b i l i t y  gained  by  o f t h e AR model t o  pursuing  using  These s t u d i e s demonstrated t h a t when an the conventional  AR s p e c t r a l  were r e a s o n a b l y  good compared t o t h e prewhitened AR e s t i m a t e  of  retained  information  studies  i n the r e s i d u a l s  (see B i r c h  estimates  which makes use  et a l .  [53]).  This  i n d i c a t e s t h a t the AR model, a l t h o u g h n o t p e r f e c t , does r e p r e s e n t much o f t h e information conventional EEG  contained  in a  short  and a prewhitened  i s g i v e n i n F i g u r e 4.8.  segment  12th o r d e r  o f EEG. AR s p e c t r a l  An  example  estimate  o f both  of idle  a  task  66  RR SPECTRAL ESTIMATION OF IDLE TASK EEG  Frequency Figure  4.6  67  A.6  Applying O u t l i e r Processing to Single T r i a l  A.6.1  Comparison of Idle  original  idle  evidence  of  motor  and  of idle  should  occur,  noting  that  plots  demonstrate  distinct positivity at  about  cases the  active  above n e u r o l o g i c a l  cleaned a c t i v e activity  N=6  motor  signals.  signal  whereas  the  and  Original  averaged  Motor  the  original  cleaned  that  the  the  three  in  conventional the  active  averaged s i g n a l d u r i n g the  seconds).  However,  a v e r a g i n g does not  reveal  in any  the  first  cleaned  due  seconds  technique  to motor p o t e n t i a l  set  the  case epoch,  epoch.  These  reveals  (raised  two  active,  active  of  no  level  some of  t h r e e seconds w i t h a peak active  and  original  idle  Hence,  the  s t r o n g n e g a t i v e peak i n  the  v i s u a l evoked response t o  the  The  t a s k p l o t at about 6 seconds i s the i s not  the  d i s t i n c t motor a c t i v i t y .  i s s u b s t a n t i a l l y borne out.  feedback l i g h t and  in  case  Each  original  second i n t o the  averaging  original  the  signal  provides  trials.  active,  activity  first  active  F i g u r e A.9  potential  during  premise, t h a t  s h o u l d have l i t t l e or  o t h e r w i t h N=15  thumb movement began one  activity  i n the  trials  conventionally  actual  above p r e d i c t i o n N=6  with  approximately,  the  two  the  potential  one  plots  original  and  g i v e n the  evidence o f motor p o t e n t i a l a c t i v i t y .  plots:  contains  predicted,  signal  should contain of  Original Active  Signals  I t would be  sets  Segmented Cleaned A c t i v e ,  EEG  activity.  AVERAGE OF SEGMENTED ORIGINAL AND CLEANED SUBJECT*! N=6 ORIGINAL.  ACTIVE  Tim* <8*oonds)  CLEANED  lO 0  ACTIVE  — C3M2  * 2 O —2 —*  -lO  O.O 0.5 l.O 1.5  Z.O  ~2.5 3.0 3.5 4..'o 4.3" S.O 5.5 ~6.C Tim* ( 8 « e o n d s )  DRIGINflL  —I  1  I  I  i  j  i  IDLE  i  I  i  i  .  ,  O.O O.S l.O 1.5 2.0 Z.5 3.0 5.5 4.0 4.5 S.O "5.S fe.O Tim* (Saoonds)  F i g u r e 4.9a  69  AVERAGE OF SEGMENTED ORIGINAL AND CLEANED EEG SUBJECTS! N=15 ORIGINAL  O.O  O.S  l.O  l.S  2.0 Tim*  CLEANED  O.O  O.S  l.O  1.5  ACTIVE  2.S  ACTIVE  2.0 Tuna  S.O  3.S  4.0  (Saoonda)  2.S  -  3.0  (8«aond«)  ORIGINAL.  Figure  IDLE  4.9b  C3M2  3.5  4.0  70  4.6.2  Examples o f S i n g l e T r i a l O u t l i e r P a t t e r n s and  To  qualitatively  processing u s i n g GM2 in  method,  four  Figure  4.10.  which  Results  provided  i n the  these  outlier  point  trial  can  results the  of  not  be  single t r i a l  amount of easily  the  in  patterns  EEG  provided  raw  the  i s r e l a t e d to  the  the  are  that  raw  average  waveforms  from  the  is in  EEG  the  signals.  information  thumb movements.  to note the many s i m i l a r i t i e s  grand  trial  outlier  information  seen  EEG  single  f o l l o w i n g s e c t i o n s demonstrate t h a t the  patterns  with  cited i n Section  However,  o f these  at  single  Grunewald  study  4.1.  Comparison o f Averaged A c t i v e O u t l i e r P a t t e r n s , Averaged I d l e O u t l i e r P a t t e r n s and  To case  significant  i t i s interesting  patterns  4.6.3  example p l o t s o f  Note the  patterns  this  the  model parameters p a i r e d w i t h the c o r r e s p o n d i n g  outlier  in  demonstrate  S i n g l e T r i a l Raw  demonstrate  outlier  patterns,  that  patterns  the  and  the C o n v e n t i o n a l  there very  p l o t s i n Figure  Average o f A c t i v e  i s some s t r o n g  little 4.11  consistency  consistency  i n the  have been p r o v i d e d .  the averaged o u t l i e r p a t t e r n s f o r N=6  and N=15  u s i n g GM2  parameter  a c t i v e case i s a l s o i n c l u d e d i n t h i s  figure.  that  shape s i m i l a r t o the information trial.  the  average  single t r i a l  r e l a t e d to  the  active  case  patterns  case  active outlier  These p l o t s c o n t a i n  p l o t of the c o n v e n t i o n a l  fact  i n the  idle  As w e l l , f o r comparison purposes, a  The  EEG  estimates.  average f o r the  maintain  a  general  patterns, strongly i n d i c a t e s that there i s  thumb movement t h a t  i s c o n s i s t e n t from t r i a l  to  On the o t h e r hand, the f a c t t h a t the average reduces i n magnitude  and  EXAMPLE VRVEEORMS Subject #1 - Trial #3  OUTLIER  PATTERN  0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 Time (Seconds)  RAW E E G  0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 Time (Seconds)  Figure 4.10a  EXAMPLE WflVEEORMS Subject #1 - Trial #6  OUTLIER  PATTERN  0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 Time (Seconds)  RAW E E G  0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 Time (Seconds) Figure  4.10b  EXAMPLE WAVEFORMS Subject #1 - Trial #15 OUTLIER  PATTERN  0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 Time (Seconds)  RAW E E G  0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 Time (Seconds)  Figure 4.10c  EXAMPLE VAVEEORMS Subject #1 - Trial #35  OUTLIER  PATTERN  0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 Time (Seconds)  RAW E E G  172 111  50 -11 -72 -133 -193 -254 -315 -376  _L  J_  _L  0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 Time (Seconds) F i g u r e 4.10d  75  AVERAGE WAVEFORMS N=6 CONVENTIONAL  — i  O.O  I O.S  1  l.O  1  1.5  i  2.0  i  2.5 Tim*  ACTIVE  — i  O.O  i  O.S  1 l.O  l  1.5  i  2.0  — i  O.O  1  O.S  1  l.O  1  l.S  i  3.S  1  i  2.5  1  3.0  Ttma  J S.O  i  5.5  i  6.0  —  i  3.5  i  4.0  GM2  i  4.5  i  S.O  i  i _  S.S  e>.0  I 5.S  I 6.0  CSakoondart  i  2.5  I 4.3  (Saaonda)  OUTLIER  2.0  i  4.0  OUTLIER  TBTMI  IDLE  I 3.0  AVERAGE  i  3.0  I 3.5  C8«)Oondaf)  Figure  4.11a  —  I 4.0  GM2  i  4.5  i.  5.0  76  AVERAGE WAVEEORMS N=15 CONVENTIONAL.  Tim*  O^O  5^  3LO  l £  AVERAGE  (Saoonds)  zfe zTg Tim* W«nond«)  37o  sts  4^3  77  subtle  features  become  less  pronounced  as  N  is  increased,  indicates that  t h e r e i s a s i g n i f i c a n t amount o f uniqueness i n i n d i v i d u a l t r i a l s t h a t i s l o s t as many t r i a l s due  are  averaged t o g e t h e r .  T h i s uniqueness i s c e r t a i n l y i n p a r t  to the v a r i a n c e i n thumb movements from t r i a l  clearly  i n S e c t i o n A.7.  uniqueness  i s due  to  I t i s a l s o expected cognitive  patterns  clearly  is  r e i n f o r c e d across  being  a c t i v e t r i a l s shows t h a t w i t h N's is is  q u i t e l i m i t e d and discussed  occurring  in  averaging study  the  which has  trials.  above  for  the  active  these  conventional  Section  A . l ) , the  significant The  o f event  case  outlier  averages.  information  mental  seen trial  intensity  conventional  with  outlier  information  15 the motor p o t e n t i a l  "smearing" e f f e c t  that  average  of  information  related information that patterns  Hence,  with  would  the  also  be  conventional  as i n the case o f the Grunewald  obtained  will  be  limited  to  that  trials.  A c t i v e O u t l i e r P a t t e r n s Degrading w i t h Higher Model Orders  I t would a l s o be expected, single t r i a l  based on the n e u r o l o g i c a l premise, t h a t  p r o c e s s i n g method would perform  to b e s t  model would be related  i s no  o f 6 and  the  remained r e l a t i v e l y c o n s i s t e n t a c r o s s the  LSQ  selected  task  as  by  average o f the i d l e t a s k  method, even w i t h much g r e a t e r N's  (see  A.6.A  The  demonstrates t h a t t h e r e idle  that a d d i t i o n a l t r i a l  f a c t o r s such  which the s u b j e c t c a r r i e d out the t a s k .  to t r i a l which can be  f i t the  expected  activity  in  idle  case.  As  b e s t when the AR model o r d e r  the model o r d e r  to g a i n some improved a b i l i t y  the  active  task  EEG.  Hence,  i s i n c r e a s e d the  the was AR  t o r e p r e s e n t the motor the  performance  of  the  s i n g l e t r i a l method s h o u l d b e g i n t o degrade s i n c e the c l e a n i n g p r o c e s s , which utilizes  the  higher  order  AR  model, would l o s e some o f i t s e f f e c t i v e n e s s i n  78  d e t e c t i n g motor r e l a t e d o u t l i e r s . plots using  LSQ parameters f o r model order  i d l e case) and model order shown  i n Figure  degrade,  i n terms  averaged  outlier  active  4.7  A p a i r o f averaged a c t i v e o u t l i e r  4.12.  22 ( g e n e r a l l y a p p r o p r i a t e  These  plots  demonstrate  o f b o t h t h e amplitude pattern,  when  order  f o r the  f o r t h e a c t i v e case) a r e  that  t h e performance  and t h e d e t a i l  t h e model  of features  i s better  matched  does  i n the t o the  case.  S t a t i s t i c a l A n a l y s i s o f Features  The  s e t o f 15  corresponding 4.13.  12 ( g e n e r a l l y a p p r o p r i a t e  pattern  encoded  active  trial  i n the O u t l i e r Patterns  outlier  thumb movements  from  patterns Subject  The p a t t e r n s , a l t h o u g h unique from t r i a l  g e n e r a l l y c o n s i s t e n t waveform which c o n t a i n s events  i n t h e thumb  determine  movements.  i f features  i n the corresponding  movement  and  statistical  three  features  analysis.  The  analysis  patterns.  i n the o u t l i e r features  are given  the  i n Figure  f e a t u r e s t h a t appear r e l a t e d t o  Statistical  outlier  #1  with  t o t r i a l , do seem t o posses a  i n t h e i n d i v i d u a l thumb  features  superimposed  was  movements Two  pattern  are described  c a r r i e d out t o are r e l a t e d to  features  i n t h e thumb  were u t i l i z e d  below  i n the  and a r e shown i n  F i g u r e 4.14.  Feature  1:  Time from epoch onset t o t h e p o i n t when t h e thumb movement  f i r s t reaches t h e "on t a r g e t " p o s i t i o n .  Feature first  2:  Time from epoch onset t o t h e p o i n t when the thumb movement  l e a v e s the "on t a r g e t " p o s i t i o n .  COMPARISON OF N=15 AVERAGED OUTLIER PATTERNS  L  0.0  0.5  1.0  S  Q  1.5  O  R  D  E  2.0  R  12  2.5  3.0  3.5  4.0  3.5  4.0  Time (Seconds)  L  0.0  0.5  1.0  S  Q  1.5  O  R  2.0  D  E  R  22  2.5  3.0  Time (Seconds) Figure 4.12  S I N G L E T R I A L GM2 O U T L I E R WITH CORRESPONDING THUMB  PATTERNS MOVEMENT TRIAL #3  0.0 0.5 1.0 145 2.0 245 3.0 345 4.0 4.5 Tins (Stoonds)  0.0 045  LO 145 2.0 245 3.0 34J 4.0 445 TMM Oaoonda)  TRIAL #6  TRIAL #8  0J0 0J5 1.0 1.5 2.0 245 3.0 345 4.0 445  OX) 045  TIM OMMMS)  Figure 4.13a  145 2.0 24) 3X1 345 AX) 445  oo o  R.lotlv.  T8  daonltud.  R.lotlv.  N a g n 11 u d •  Ralativ*  Z8  Magnitude  B«lotlv«  Nagnltuda  \?T £9 £«I OUTLIER PATTERNS WITH CORRESPONDING THUMB MOVEMENT r  L  RIftL  G  T R I A L #41  0.0  0.5 1.0 1.3 2.0 245 3.0 345 44) TfeM tttootxfc)  M  2  T R I A L #44  4J  00) 041 1.0 14$ 24) 245 34) 34S 4.0 T I M WtooBdrt  445  84  SINGLE TRIAL FEATURE DEFINITION SAMPLE THUMB MOVEMENT WITH STANDARD GM2 OUTLIER PATTERN FOR SUBJECT #1  0,5  1,0  1,5  2.0  2.5  Tftne (Seconds)  Figure 4.14  3.0  3,5 4.0  85  Feature  3:  Time  from  epoch  onset  t o the f i r s t  amplitude) p o s i t i v e peakn t h e o u t l i e r  Feature  A:  Time  outlier  pattern  from epoch onset after  feature  (greatest  pattern.  t o the f i r s t  3,  dominant  that  has  negative  a  peak i n t h e  minimum  of  5  units  magnitude p e a k - t o - t r o u g h d i f f e r e n c e .  Feature  5:  Time  outlier  pattern  from epoch after  onset  feature  t o t h e next  A,  that  has  p o s i t i v e peak i n t h e  a minimum  o f 20  units  magnitude p e a k - t o - t r o u g h d i f f e r e n c e on b o t h s i d e s o f t h e peak.  There  was  an e x p e c t a t i o n  r e s u l t i n g from  Grunewald and Grunewald-Zuberbier A.13  the e a r l i e r  conventional  [7] and from o b s e r v a t i o n s  study  taken from  by  Figure  t h a t f e a t u r e 1 would be p a r t i c u l a r l y r e l a t e d t o f e a t u r e s 3 and A whereas  f e a t u r e 2 would be p a r t i c u l a r l y r e l a t e d t o f e a t u r e 5.  The sample c o r r e l a t i o n  c o e f f i c i e n t s between a l l o f t h e f e a t u r e s  #1 were c a l c u l a t e d and  are  summarized  coefficients related Hence,  between  this  outlier  pattern.  outlier  pattern  A.l.  the  demonstrates  features  difference  Steiger  A.l.  are the strongest,  between  the  i n Table  [5A]).  These  features with that  i n the thumb  from S u b j e c t results  that  were  coefficient there  is a  movement  In p a r t i c u l a r ,  show  that  expected  the c o r r e l a t i o n  t o be  values  a l l greater  strong  consistent  and f e a t u r e s  t h e r e l a t i o n s h i p between  correlations  The r e s u l t s from  calculated  these  on  They show t h a t f e a t u r e s 3 and A c o r r e l a t e d w i t h  0.77.  relationship  features  dependent  t e s t s are a l s o  than  i n the s i n g l e  and i n t h e thumb movement was examined u s i n g between  particularly  trial i n the  the z - t e s t f o r samples  summarized  (see  i n Table  feature 1 s i g n i f i c a n t l y  86  TABLE 4.1 SINGLE TRIAL FEATURE STATISTICS Feature C o r r e l a t i o n Matrix feature feature feature feature feature  1 2 3 4 5  feature 1 1 .0 0.76 0.78 0.88' 0.60  z-test  feature 2 1.0 0.51 0.69 0.80  3  1.0 0.71 0.41  feature 4  feature  1.0 0-.57  1.0  on t h e D i f f e r e n c e Between C o r r e l a t i o n s C a l c u l a t e d on Dependent S a m p l e s  C o r r e l a t i o n Coef f i c i e n t s feature 1 feature 2 feature 3  feature  feature 4 0.78 0.88 0.60  feature 5 0.51 0.69 0.80  z  p (one - s i d e d )  1.91 1 .67 1 . 55  0 .029 0 .048 0.060  87  more s t r o n g l y  than w i t h  between  feature  feature  1, but  5  and  feature  feature  2  (p  2 was  < 0.05).  As  expected, the c o r r e l a t i o n  l a r g e r than t h a t  between f e a t u r e  t h i s d i f f e r e n c e achieved only a marginal l e v e l  5  and  of  significance.  4.8  A p p l i c a t i o n of Dynamic Time Warping t o O u t l i e r  All one  the  initial  subject,  revealed  work w i t h a c t u a l EEG  referred  that  quantitative  the  use  measure  compared to the  to of  of  dynamic  subjects  following  as  (template) subject.  described  by  representative The  waveform B by  where  warping the  (DTW)  Hence, DTW  on the d a t a from  initial  investigations,  provided  single t r i a l  the  processing  a n a l y s i s was  i n t h i s study.  using  Roberts single  et  a l . [55]  trial  was  active  s h i f t i n g , expanding or c o n t r a c t i n g  waveform A,  is  used t o  outlier  to  best  Specific results  obtain  patterns  Q i s the  warped  standard for  match waveform A  The  cost  to  o f warping  [55] 4  time  each  the time s c a l e of waveform A  " c o s t " o f warping.  following cost function  the  method  ultimately  Q(A,B,W) - X P(w)  w  best  Dynamic Time Warping  time warping procedure attempts  i s based on the =  for  These  included  such a manner t h a t minimizes the  C  time  c a r r i e d out  subsections.  Standard O u t l i e r P a t t e r n s  DTW  #1.  other previous a n a l y s i s .  are summarized i n the  in  Subject  performance  a p p l i e d to a l l f o u r of the  4.8.1  as  was  Patterns  function  (warped  time  axis)  used  *  to  1 5  warp  c o r r e l a t i o n between warped waveform A and waveform B,  P  88  is is  a p e n a l t y f u n c t i o n and X i s h e p e n a l t y c o e f f i c i e n t .  nonlinear  such  proportionately contractions.  that  much  the penalty higher  than  The X c o e f f i c i e n t  how expensive i t i s t o warp.  f o r l a r g e expansions o r c o n t r a c t i o n s i s the  penalty  f o r small  reasonable  expansions  i s a t u n i n g parameter t h a t d i r e c t l y  warped waveforms;  warpings whereas l a r g e r v a l u e s  smaller  and e r r o r t h a t t h i s  values  o f produced  or  effects  In a l l t h e DTW a p p l i c a t i o n s used i n t h i s  a X=75.0 was u t i l i z e d because i t was found by t r i a l produced  The p e n a l t y f u n c t i o n  study value  X extreme  o f X produced warpings t h a t were o n l y  shifted  i n time and c o n t a i n e d v e r y l i m i t e d expansions o r c o n t r a c t i o n s . A s t a n d a r d p a t t e r n f o r each o f t h e f o u r s u b j e c t s was a c h i e v e d by u s i n g the  procedure recommended by Roberts e t a l . [ 5 5 ] . The s e t o f a c t i v e s i n g l e  t r i a l p a t t e r n s was warped a g a i n s t each p a t t e r n i n t h a t s e t . produced the lowest as  the best  then  for  representative  constructed  being  mean c o s t  and v a r i a n c e  pattern  by a v e r a g i n g  of that  together  across set.  t h e s e t was then s e l e c t e d The s t a n d a r d  a l l the patterns  warped t o the above s e l e c t e d p a t t e r n .  The p a t t e r n t h a t  p a t t e r n was  i n the s e t a f t e r  P l o t s o f the s t a n d a r d  patterns  a l l f o u r s u b j e c t s u s i n g LSQ, GM and GM2 parameter e s t i m a t i o n a r e p r o v i d e d  i n F i g u r e A.15.  A.8.2  DTW Cost  S t a t i s t i c s on I n d i v i d u a l S u b j e c t s  Once a s t a n d a r d  a c t i v e case p a t t e r n was o b t a i n e d  f o r each s u b j e c t , i t  was warped a g a i n s t t h e 15 t r i a l s o f a c t i v e o u t l i e r p a t t e r n s and t h e 15 t r i a l s of i d l e  o u t l i e r patterns  the  standard  the  single t r i a l  f o r each s u b j e c t .  pattern a cost value  Each time a p a t t e r n i s warped t o  i s produced.  p a t t e r n " f i t " the standard  T h i s c o s t r e f l e c t s how w e l l  pattern.  The lower t h e c o s t t h e  S  T  A  N  D  A  R  GM2  1.5  2.0  M  O O  D  E  U L  T  L  I P  E  R A  2,5  SUBJECT  •  30  C CS  20  • >  A  M  E  T T  T E  E R  3.0  R  N  S  S  3.5  4.0  4.5  0.0  0.5  1.0  1.5  2.0  #2  2.5  3.0  3.5  4.0 4.5  T I M (Saoonds)  #3  S U B J E C T #4  40  •*»  o  R  A  SUBJECT  Tn* (Saoonds) 50  P  #1  SUBJECT  1.0  D  to 0 -10  — -20  •  cc -30  -40  1_ 0.0  0.5  1.0  1.5  _l_  2.0  2.5  TBM (Saoonds)  _1_  3.0  _l_  3.5  4.0 4.5  1.5  Figure 4.15a  2.0  2.5  T I M (Saoonds)  oo  06  S  T  A  N L  D S  A Q  R  D M  O  D  O  U  E  L  T  L  I P  E  R A  SUBJECT #1  P R  A  M  A E  T T  T E  E R  R  N  S  S  SUBJECT #2  SO  •  40 9. 4» 30 C 20 a o 10  •  >  0  4* -10 —o -20  O• S -30  -40  T I M Oaoonda>  TIM  SUBJECT #3  £0  _I_  2s  3JO  _l_  53 <ui its  CSaoonds)  SUBJECT #4  0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5  1.5 2.0 2.5 TBM  -L  &6 o5 Co 15  0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 +.0 4.5  Ocoonds)  TIM  Figure 4.15c  (Seconds)  92  b e t t e r the f i t . cases would be test  Therefore,  i t would be expected t h a t the c o s t s i n the a c t i v e  s m a l l e r than the c o s t s i n the i d l e case.  the d i f f e r e n c e between two  t  means, g i v e n by  and  1  N i s the  +  s  2  N  x  are  2  parameters shows t h a t  i d l e cases. for  a l l four  i n the  GM2  statistically significant case  are  except  also  for  #4  subjects  These r e s u l t s  LSQ  and  s  a r e the sample v a r i a n c e s ,  2 2  a p p l i e d t o the mean c o s t s  r e s u l t s of t h i s are  test  summarized  f o r GM2,  GM,  and  A.2.  i n Table  LSQ This  d i f f e r e n c e between the means i s h i g h l y  (p < .001).  The mean d i f f e r e n c e s i n the GM (ranging  parameters  from p  where  s t r o n g l y support  from the a c t i v e case would be turn, this  The  case the  with  2  i d l e c a s e s , was  statistically significant  Subject  significant.  28  the sample means, S j  number o f a c t i v e and  f o r the a c t i v e and  test  [56]  2  d f = 2N - 2 = where x  to  A.16  /  model  designed  = /  and  A t-test  the  s m a l l e r than the  and  LSQ  <  .01)  d i f f e r e n c e was  not  <  the  .001  expectation  to p  t h a t the  costs  c o s t s from the i d l e case.  i m p l i e s t h a t on average the a c t i v e case p a t t e r n s  f i t the  In  standard  p a t t e r n s much b e t t e r than the i d l e case p a t t e r n s .  4.8.3  Grouped DTW  The considered lation  average together  (for  a c t i v e cases. (over  the  trials)  15  was  Cost  Statistics  a c t i v e and to provide  a l l possible  idle  cost values  inferential  the  four  s t a t i s t i c s about the  For each s u b j e c t , the d i f f e r e n c e between the average i d l e  cost  evaluated.  and  the  average  The  differences  differences  a c t u a l popuand  trials)  mean  s u b j e c t s were  idle  idle  s u b j e c t s ^ of  from  a c t i v e cost appear  in  between  (over Table  the  15  4.3  and  active form  93 T A B L E 4.2 D I F F E R E N C E BETWEEN I D L E AND A C T I V E WARPING COSTS t-Test Results f o r the Difference Between Means: df=28  SUBJECT GM2 1 2 3 4 GM 1 2 3 4 LSQ 1 2 3 4  MODEL  MODEL  MODEL  DIFFERENCE BETWEEN THE MEANS  TWO-SIDED t-VALUE  P <  13.3 16.4 1 1.6 11.5  6.05 5.04 4.66 6.25  0.001 0.001 0.001 0.001  8.7 12.7 6.1 4.9  4.18 3.74 2.93 3.35  0.001 0.001 0.01 0.01  12.7 17.2 12.9 3.5  4.08 3.65 2.96 0.65  0.001 0.002 0.01 X  9A  the  basis  carried the  o f the i n f e r e n c e  t o be made f o r a l l p o s s i b l e  out by c o n s i d e r i n g  actual population  the n u l l  hypothesis  i s z e r o and then a p p l y i n g  that  subjects.  This  was  t h e mean d i f f e r e n c e o f  a statistical  t e s t to deter-  mine whether t h a t h y p o t h e s i s s h o u l d be r e j e c t e d .  The s t a t i s t i c a l  t e s t was a  t-test  two means w i t h  correlated  designed  (paired)  to test  samples.  the d i f f e r e n c e  I t i s g i v e n by [50]  MD SE  df = N - 1 = 3  4.17  MD  where MD i s the sample mean d i f f e r e n c e is  between  (mean o f the p a i r e d d i f f e r e n c e s ) , S E ^  t h e s t a n d a r d e r r o r o f t h i s sample mean d i f f e r e n c e and N e q u a l s t h e number  o f mean d i f f e r e n c e s  (number o f s u b j e c t s ) .  The s t a n d a r d e r r o r o f t h e sample  mean d i f f e r e n c e i s g i v e n by  V  N  where s^ i s t h e sample s t a n d a r d d e v i a t i o n o f t h e mean d i f f e r e n c e .  With  three  degrees o f freedom a 95% c o n f i d e n c e i n t e r v a l f o r the mean d i f f e r e n c e i s g i v e n by [50] CI  g 5  The  MD ± 3 . 1 8 ( S E )  group  summarized given  =  this  statistics  i n Table small  A.19  MD  A.3.  sample  f o r GM2, In t h e GM2  o f four  d i f f e r e n c e o f the a c t u a l population rejected found  (p < .002).  that  with  GM,  and LSQ parameter  case t h e s t a t i s t i c s  subjects,  imply  the h y p o t h e s i s  o f mean d i f f e r e n c e s  estimation that,  that  even  t h e mean  i s zero i s s t r o n g l y  A l t e r n a t i v e l y , i n terms o f c o n f i d e n c e i n t e r v a l s , i t was  95% c o n f i d e n c e  the i n t e r v a l o f 9.7  t o 16.8 c o n t a i n s  a c t u a l mean d i f f e r e n c e v a l u e .  The i m p l i c a t i o n s a r e s i m i l a r but l e s s  cant i n t h e GM and LSQ c a s e s .  These r e s u l t s imply t h a t t h e mean c o s t  ences between a c t i v e and i d l e t r i a l s based on a l l p o s s i b l e s u b j e c t s u n l i k e l y t o be z e r o .  are  the  signifidiffer-  i s highly  95 T A B L E 4.3 GROUP S T A T I S T I C S Mean C o s t D i f f e r e n c e B e t w e e n A c t i v e and I d l e Cases GM2 MODELING Subject 1 2 3 4  #  Mean C o s t S t d . Dev. Std. Error  GM MODELING  LSQ MODELING  8.7 12.7 6.1 4.9  13.3 16.4 11.6 11.5 13.2 2.28 1.14  8.1 3.45 1 .73  Summary f o r GM2  Modeling  Mean D i f f e r e n c e = 13.2 two s i d e d t - t e s t t = 11.58 : H r e j e c t e d p < 0.002 9 5 % C o n f i d e n c e I n t e r v a l : 9.6 t o 16.8 0  Summary f o r GM  Modeling  Mean D i f f e r e n c e = 8.1 two s i d e d t - t e s t t = 4.68 : H r e j e c t e d p < 0.05 9 5 % C o n f i d e n c e I n t e r v a l : 2.6 t o 13.6 Q  Summary f o r LSQ M o d e l i n g Mean D i f f e r e n c e = 11.6 two s i d e d t - t e s t . t = 4.03 : Hjj r e j e c t e d p < 0.05 9 5 % C o n f i d e n c e I n t e r v a l : 2.4 t o 20.8  12.7 17.2 12.9 3.5 11.6 .5.77 2.88  96  TABLE 4.4 BAYESIAN CLASSIFICATION A s s u m i n g P r o b a b i l i t i e s o f E i t h e r Case O c c u r r i n g a r e E q u a l a)  Cost  SUBJECT  of M i s c l a s s i f i c a t i o n s e t Equal ACTIVE CLASSIFIED CORRECTLY  FALSE POSITIVE  1 1 .35 19.88 19.10 17.05  14/15 15/15 13/15 14/15 56/60=93%  2/15 1/1 5 5/15 3/15 11/60=18%  15. 28 19. 10 16. 93 13. 26  12/15 15/15 12/15 13/15 52/60=87%  4/15 4/1 5 8/15 5/15 21/60=35%  22.58 41 .06 34.08 20.52  13/15 14/15 13/15 5/15 45/60=75%  4/15 5/15 7/15 2/15 18/60=30%  BOUNDARY VALUE  GM2 MODEL  1  2 3 4 GM MODEL  1  2 3 4 LSQ MODEL  1  2 3 4 b)  C o s t o f M i s c l a s s i f y i n g an I d l e Greater  GM2 MODEL 1 2 3 4  19.88 19.10 17.05  GM MODEL 1 2 3 4 LSQ MODEL 1 2 3 4  9.18  Case a s A c t i v e S e t t o  be 5 T i m e s  14/15 12/15 10/15 12/15 48/60=80%  0/1 5 1/1 5 0/15 0/15 1/60=1.7%  8.57 15.58 7.53 6.38  9/15 11/15 2/1 5 0/15 22/60=37%  0/15 2/1 5 0/15 0/15 2/60=3.3%  12.22 28.68 21 .35 2.92  4/15 ' 5/15 7/15 0/15 16/60=26%  1/15 2/15 3/1 5 0/1 5 6/60=10%  97  A.8.4  B a y e s i a n C l a s s i f i c a t i o n o f A c t i v e Cases v e r s u s I d l e  A active  Bayesian  cases v e r s e s i d l e  Gaussian  This  conditions:  misclassifying  an  was  cost  idle  applied  cases.  distribution.  different  In  classifier  I t was  to  the  assumed t h a t  classification of  was  results  subjects  of  the  f o r GM2,  condition  fairly tion.  93%  classification  GM  and  LSQ  o f the GM2  under  are  active  h i g h i f the a c t i v e Hence,  the  both  out  classify  equal  under  and  a  two  cost  of  f i v e times g r e a t e r .  provided  cost  of a f a l s e  correctly  i n table  was  The  classified  results  classifying  were f a i r l y  good  very  h i g h percentage  reduced  condition  u t i l i z i n g GM2  was  i n a control carried  case the percentage  to a  first  out  applicawere  the  still  o f GM2  active  v e r y r e s p e c t a b l e 80% 1.7%  ( o n l y one  but  cases i n so  i d l e case out  incorrectly). and  LSQ  outlier  in classifying of  false  T h i s i s perhaps parameter  p a t t e r n s under the f i r s t  active  positives.  e v e r , the c l a s s i f i c a t i o n performance dramatically.  the  t o the a c t i v e case mean by i n c r e a s i n g the  In t h i s  u s i n g GM  tion  Under  four  The number o f f a l s e p o s i t i v e s i s  doing reduced the f a l s e p o s i t i v e s t o a v e r y low s i x t y was  4.4.  the  cases were c o r r e c t l y c l a s s i f i e d w i t h 18% o f  moved c l o s e r  positive.  classified  conditions across  cases are going t o be used  second  d e c i s i o n boundary was  of  set  case s e t t o be  the i d l e cases b e i n g c l a s s i f i e d as a c t i v e .  off  to  the c o s t v a l u e s had  carried  misclassification  case as an a c t i v e  values  b o t h c o n d i t i o n s the p r o b a b i l i t i e s o f e i t h e r case o c c u r r i n g were s e t e q u a l .  The  of  cost  Cases  cases c o r r e c t l y but b o t h had  Under the second  u s i n g GM  condi-  and  LSQ  a  c o n d i t i o n , how-  outlier  patterns  the b e s t d e m o n s t r a t i o n o f the  fell  superiority  e s t i m a t e s i n the s i n g l e t r i a l p r o c e s s i n g method.  98  4.8.5  Cross V a l i d a t i o n and  I n t e r s u b j e c t R e l i a b i l i t y Using  A combined measure o f the unrelated  set  of  outlier  s t a n d a r d p a t t e r n s was one  subject  to  standard  patterns  of  of  each  active means  the  and was  a  similar  cost  patterns  subjects.  costs. in  The a  t-test  similar  as  in  g r e a t e r was  carried  the  correct using  percentage and  and  of  out  4  false  cross-matched  classified from  the  using other  positives. cross  idle  and  patterns  and  reliability.  subject.  p o s i t i v e s was  standard  was  results  on that  data  overall  these  Subsection  4.8.2,  patterns  67%  with  was  the  same  fell  only  of  subject a  3%  patterns  used  in  provide  active on  the  overall  indicate that not  with five On  the the  the  times  average  o f f by  about  patterns for  subject  2.  same f o r b o t h the  average  the  In a d d i t i o n , i n  4.5.  for  of  sample and  classification  i n Table  exactly  from one  strongly  that  exactly  patterns  two  a c t i v e case s e t t o be  standard  almost  patterns  subjects  an  GM2  a d d i t i o n sets  c o r r e c t u s i n g matched s t a n d a r d  i t stayed  The  in  r e s u l t s are g i v e n  cross-matched  while  as  the  The  outlier  d i f f e r e n c e between  4.8.4, Bayesian  case as  the  cases.  These  validate  different  (p < 0.001) i n e v e r y case.  16% when compared to the p e r c e n t 1,3  a  the  of  w i t h the s t a n d a r d p a t t e r n from  r e s u l t e d i n twelve  manner  p a t t e r n on a  intersubject r e l i a b i l i t y  from  for  subsection  o f m i s c l a s s i f y i n g an  subjects  the  This  highly significant  manner  percent  and  from each s u b j e c t were a p p l i e d to the GM2  applied,  d i f f e r e n c e was  patterns  outlier  other  idle  c r o s s v a l i d a t i o n of a s t a n d a r d  o b t a i n e d by a p p l y i n g DTW  the  DTW  The  matched  cases  correctly  outlier  patterns  average  standard  false  patterns  construction  substantial  of  of  do  these  intersubject  99  SUMMARY OF BAYESIAN Cost  T A B L E 4.5 CROSS-MATCHED  CLASSIFICATION  o f C l a s s i f y i n g an I d l e C a s e a s an A c t i v e S e t t o be F i v e T i m e s G r e a t e r  Standard Pattern #  Data  1 C  F  C  from S u b j e c t # 2 F C  F  Case  F  C  1  14/15  0/15  12/15  1/15  9/15  0/15  8/15  0/15  2  11/15  0/15  12/15  1/15  8/15  0/15  1 1/15  0/15  3  12/15  0/15  12/15  1/15  10/15  0/15  9/15  0/15  4  1 1/15  1/1 5  12/15  1/15  6/1 5  0/15  12/1 5  0/1 5  Active Trial  Classified  AVERAGED With Matched Standard Pattern Subject Number  Correctly  F = False Positive  RESULTS With Cross-Matched Standard Pattern  Active Correct  False Positive  Active Correct  False Positive  1  93%  0%  75%  2%  2  80%  7%  80%  7%  3  66%  0%  51%  0%  4  80%  0%  62%  0%  80%  2%  67%"  3%  OVERALL AVERAGE  100  CHAPTER 5 CONCLUSION  5.1  Summary o f Major R e s u l t s and R e l a t e d  Conclusions  AR Parameter E s t i m a t i o n Simulation robust  methods the GM2  (AO)  contaminated  Gaussian  estimates  finding  equation  almost  t h a t the  as w e l l as  processes.  from  3.37)  these  Amongst  the GM  However, i t s h o u l d be  a r e a l s o the most c o m p u t a t i o n a l l y expensive  In terms o f the s i n g l e t r i a l  important (see  methods performed  method p r o v i d e d the b e s t performance.  t h a t the GM2  calculate.  (GM)  e s t i m a t i o n demonstrated  (LSQ) method on Gaussian p r o c e s s e s and s i g n i f i c a n t l y b e t t e r  additive outlier  noted  parameter  g e n e r a l maximum l i k e l i h o o d  the l e a s t squares on  s t u d i e s on AR  o u t l i e r p r o c e s s i n g method, t h e most  s i m u l a t i o n s t u d i e s i s t h a t g i v e n the AO model  the r o b u s t  e s t i m a t i o n methods,  i n particular  the  method, d e m o n s t r a t e d a s t r o n g a b i l i t y t o model the p r o c e s s x^ without unduly  to  GM2  being  i n f l u e n c e d by the a d d i t i v e o u t l i e r s v^.  N e u r o l o g i c a l Premise and O u t l i e r E x t r a c t i o n The b a s i c n e u r o l o g i c a l premise f o r the s i n g l e t r i a l is  that  ongoing  event EEG  resulting  related  process.  overall  distinct  which  ability  I f the o u t l i e r  combined p r o c e s s ,  t i o n c o u l d be o b t a i n e d . estimator,  p o t e n t i a l s have  an  p r o c e s s i n g method  additive outlier  content  c o u l d be  then s i n g l e t r i a l  effect  extracted  event  on the from the  related  informa-  S i m u l a t i o n s t u d i e s demonstrated t h a t a r o b u s t  utilizes  robustly  to extract a  estimated  significant  AR  amount  model  parameters,  o f the a d d i t i v e  signal has  a  outlier  101  content most  from a contaminated p r o c e s s .  effective  should  be  estimated  when  expected  GM2  parameter  since  T h i s e x t r a c t i o n p r o c e s s was found t o be estimates  t h e GM2  were  parameter  utilized.  estimation  This  i s based  result on  an  c l e a n e d s i g n a l x^ and hence, i t has t h e b e s t o p p o r t u n i t y t o p r o v i d e  a good e s t i m a t e d  m o d e l r e p r e s e n t i n g t h e a c t u a l p r o c e s s x^.  m o d e l o f x^ t h e b e t t e r t h e e x p e c t e d  The b e t t e r t h e  performance o f the o u t l i e r e x t r a c t i o n  process.  S p e c t r a l A n a l y s i s o f EEG AR s p e c t r a l EEG  signal i t s e l f  EEG  data  involving Spectral signals  used  analysis provided  deal  of insight  i n t o both t h e  and i n t o t h e a p p l i c a t i o n o f AR modeling t o t h e EEG s i g n a l .  i n this  motor  a great  thesis  activity  work  and an  was  idle  collected  task  changing  an a c t i v e  n o t i n v o l v i n g motor  a n a l y s i s demonstrated t h a t t h e s i g n a l were t y p i c a l l y  during  characteristics  at a r e l a t i v e l y  rapid rate.  task  activity.  o f these EEG Hence, an EEG  segment s i z e as s m a l l as p r a c t i c a l parameter e s t i m a t i o n c o n s i d e r a t i o n s would a l l o w was u t i l i z e d . EEG AR  U l t i m a t e l y , i n the s i n g l e  epochs were broken down i n t o spectral  a n a l y s i s was  model o r d e r s . able  for idle  also  I t was found t a s k EEG w h i l e  f o r a c t i v e t a s k EEG.  information  p r o c e s s i n g method, t h e  1.5 second segments o f f s e t by 0.75 seconds. t h e key t o o l  that orders orders  i n t h e EEG s i g n a l .  i n determining  appropriate  i n the range o f 12 t o 14 were  AR  suit-  i n t h e range o f 20 t o 24 were s u i t a b l e  F i n a l l y , t h e study  was u s e f u l i n p r o v i d i n g some i n s i g h t the  trial  o f prewhitened AR s p e c t r a l a n a l y s i s  i n t o how w e l l t h e AR model This  study  indicated that,  represented given the  s e l e c t i o n o f an a p p r o p r i a t e model o r d e r , t h e AR model does r e p r e s e n t much o f the i n f o r m a t i o n c o n t a i n e d i n a s h o r t segment o f EEG.  102  Single T r i a l O u t l i e r Processing I n i t i a l i n v e s t i g a t i o n i n t o the s i n g l e t r i a l o u t l i e r processing method was c a r r i e d out on the EEG data from one subject. conventional  averaging  a n a l y s i s , that the cleaned  I t was shown, through a c t i v e task EEG d i d not  contain any s i g n i f i c a n t motor r e l a t e d p o t e n t i a l s . This r e s u l t i n d i c a t e d that much of the motor r e l a t e d a c t i v i t y had been extracted from the a c t i v e task EEG s i g n a l by the a p p l i c a t i o n of the cleaning process.  By using the o u t l i e r  information extracted from the a c t i v e EEG, s i n g l e t r i a l o u t l i e r patterns were produced.  These patterns had strong s i m i l a r i t i e s to previous r e s u l t s using  conventional averaging  averaging  active t r i a l  techniques  over  o u t l i e r patterns  much of the information was  consistent  many t r i a l s together across  of a c t i v e  EEG.  By  i t was demonstrated that active t r i a l s  whereas, i n  contrast, the average of i d l e case patterns showed that there was no s i g n i f i cant information that was consistent across i d l e t r i a l s . averaging  In a d d i t i o n , t h i s  also demonstrated that there was a s i g n i f i c a n t amount of informa-  t i o n i n the a c t i v e task patterns that was unique to i n d i v i d u a l t r i a l s which was  lost  when the patterns  were  averaged together.  I t i s , therefore,  expected that t h i s same loss of information i s occurring i n the conventional averaging method of EEG a n a l y s i s . I t was shown that consistent features i n the a c t i v e o u t l i e r were s t r o n g l y c o r r e l a t e d with features i n the thumb movement.  patterns  This a n a l y s i s  demonstrated that there was a strong r e l a t i o n s h i p between the information i n the o u t l i e r patterns and events i n the thumb movements on a t r i a l by t r i a l basis. I t was found i n the i n i t i a l i n v e s t i g a t i o n s that dynamic time warping  103  (DTW) a n a l y s i s p r o v i d e d of  the s i n g l e t r i a l  all  the best  processing  f o u r o f the s u b j e c t s used  trial  processing  sentative  active  obtained. trials  associated  warped cost  i n this  trial  was  with  costs  across  obtained  together  i n v e s t i g a t i o n i n t o the s i n g l e  patterns  both  standard  f o r each  patterns.  With  reflected  These c o s t v a l u e s  how  repre-  subject  the a c t i v e t r i a l s  which  and t h e i d l e  each well  were  warping the  an  outlier  r e v e a l e d t h a t t h e r e was  (p < .001) d i f f e r e n c e between t h e i d l e and  a l l four  GM2 parameter e s t i m a t e s .  a n a l y s i s was a p p l i e d t o  the a p p l i c a t i o n o f DTW,  the standard  f i t the standard p a t t e r n .  mean  initial  from  a highly s t a t i s t i c a l l y significant active  Hence, DTW  outlier  patterns  against  value  method.  Through  single  The o u t l i e r  were  patterns  method.  q u a n t i t a t i v e i n f o r m a t i o n on t h e performance  subjects  with  The c o s t v a l u e s  outlier  patterns  derived  from the f o u r s u b j e c t s  pooled  i n a group, demonstrated t h a t t h e mean o f t h e a c t u a l p o p u l a t i o n o f  mean d i f f e r e n c e s between a c t i v e and i d l e cases was h i g h l y (p < .002) u n l i k e l y to be equal  zero.  Bayesian classify cost  classification  active patterns  versus  o f m i s c l a s s i f y i n g an i d l e  greater,  applied  idle  t o the warping  patterns.  cost  values  demonstrated  because w i t h  the  case as an a c t i v e case s e t t o be f i v e  were i n c o r r e c t l y c l a s s i f i e d as a c t i v e . superiority  of  to  I t was found t h a t w i t h t h e  80% o f t h e GM2 a c t i v e p a t t e r n s were c l a s s i f i e d c o r r e c t l y w h i l e  1.7% o f the i d l e cases also  was  utilizing  GM2  only  This analysis  parameter  the u t i l i z a t i o n o f GM and LSQ parameter e s t i m a t e s  times  estimates  the c l a s s i f i -  c a t i o n performance f e l l o f f d r a m a t i c a l l y . Bayesian  classification  from u s i n g  standard  the  subjects.  other  patterns These  was a l s o a p p l i e d t o the c o s t v a l u e s from  one s u b j e c t  results  strongly  on t h e o u t l i e r indicated  that  obtained  patterns the  from  standard  104  patterns these  do c r o s s v a l i d a t e on data  patterns  and  that  these  t h a t was n o t used i n t h e c o n s t r u c t i o n o f  patterns  provide  substantial intersubject  reliability.  In ranging  conclusion,  Gaussian  single  trial  validity active  the pursuance  properties  methods  to deal  with  o f t h e EEG s i g n a l l e d t o t h e development  method based on u t i l i z i n g  of a The  o f t h i s p r o c e s s i n g method t o e x t r a c t event r e l a t e d i n f o r m a t i o n  from  EEG has been e s t a b l i s h e d through  outlier  the  information.  task  processing  of robust  the r e s u l t s  obtained  from t h e  i n v e s t i g a t i o n s undertaken i n t h i s t h e s i s work.  5.2  Areas f o r Future  Investigation  There a r e many areas the  single t r i a l  gations.  models  processing  involve  accuracy  better  Regardless  on AR  t h e parameter  parameters models,  estimates.  based  types  to  provide  the  investi-  of  the underlying on these  t h e type  models.  o f model  i s an important further  work  One p a r t i c u l a r  starting  signal since This  o f models such as those  o f methods t h a t a r e more r o b u s t  methods,  procedure.  represent  i s fundamentally  functions.  estimation  utilization based  method  will  o f the e s t i m a t e d  GM  improve  that  be pursued i n f u t u r e  recommended areas a r e :  the a p p l i c a t i o n o f d i f f e r e n t  orthonormal  using  p r o c e s s i n g method t h a t s h o u l d  Some o f t h e most important  1) Pursue the  i n v o l v e d w i t h the p o s s i b l e f u r t h e r improvement o f  could aspect  be  based on  employed,  issue.  may  the  I n terms o f  carried  to consider  out t o i s the  than MEM e s t i m a t i o n , such as  estimates  i n t h e GM  iteration  105  2) Study the that For  i s best  effects  suited  for  i n s t a n c e , i t may  lying  process  segments  are  would  of v a r y i n g  be  the  segment s i z e  representation  found t h a t the  changing  allow  the  slowly  for  an  of  the  t o determine the underlying  EEG  length  process.  s i g n a l c h a r a c t e r i s t i c s o f the  enough  improved  such  that  the  modeling  representation  of  under-  of  the  longer  underlying  process. 3) F u r t h e r which the  i n v e s t i g a t e the o u t l i e r  performance c o u l d be  suggested by M a r t i n b o t h forward  The  specific  area to consider,  a cleaning process  the  current  method  of  smoothing o f the e x t r a c t e d o u t l i e r  i n t o other  outlier  One  t h a t makes use  as of  processing  the  outlier  c u r r e n t method i s r e l a t i v e l y u n s o p h i s t i c a t e d i n t h a t i t i s  a careful  beneficial  [45], i s t o u t i l i z e  a l t e r n a t i v e s to  information.  tion  improved.  t o determine ways i n  and backward p r e d i c t i o n i n the e s t i m a t i o n o f the s i g n a l x^.  4) Peruse  simply  d e t e c t i o n process  approaches  of  processing  this  i n revealing additional information  information.  information t h a t may  may  be  Investigaprove  contained  to  be  in  the  on  the  data.  Future single  trial  should  be  empirical processing  EEG  experimentation  method.  The  should  be  goals  of  initial  t o f u r t h e r v a l i d a t e the method on a new  mended paradigm would be  to  collect  a training  carried  out  these i n v e s t i g a t i o n s  s e t o f EEG  data.  A recom-  set of a c t i v e t r i a l s  f o r the  c o n s t r u c t i o n of a standard  p a t t e r n . * Then c o l l e c t a s e t o f i n t e r m i x e d a c t i v e  and  at  idle  trials,  classification evaluated.  perhaps  performance  Later  goals  the of  of these  discretion the  of  processing  the  subject,  method  i n v e s t i g a t i o n s should  could be  on  which be  the  further  oriented to using  the p r o c e s s i n g method t o l e a r n more about motor p o t e n t i a l s , p a r t i c u l a r l y  the  106  motor p o t e n t i a l s  rom d i s a b l e d p e r s o n s ,  application  p o t e n t i a l s can be taken  these  so t h a t when u t i l i z e d advantage  i n a control  o f i n t h e most  appro-  p r i a t e manner. Finally, work  i n real  i n v e s t i g a t i o n s on making  time  computationally  must  be undertaken.  intensive.  made i n t h e e f f i c i e n c y  Although  o f these  the s i n g l e t r i a l As i t stands,  processing  method  t h e method  i s very  some improvements c o u l d undoubtedly be  computations, t h e most s i g n i f i c a n t  advances  towards t h i s g o a l would l i k e l y be i n t h e implementation o f some o r a l l o f t h e component p r o c e s s e s  5.3  Significant  The  i n s p e c i a l i z e d hardware.  Contributions  signal  processing  significant contribution. the  outlier  deriving ongoing  content  very  process.  method, a l s o the  robust  model  that  orders,  that  i n the future  Also,  i n this  thesis  signal  short  is a  unique  event type  work  is a  serious  that ranging  of this  levels  from an  processing  of Gaussianity i n  c o n s i d e r a t i o n be g i v e n  a p p l i c a t i o n of various  approach t o  information  l e d t o t h e development  l e d t o the understanding  methods  processing.  the underlying  and r e l a t i v e l y  The path  EEG s i g n a l r e q u i r e s  developed  Modeling the u n d e r l y i n g s i g n a l and then e x t r a c t i n g  from  low l e v e l  method  types  t o t h e use o f o f EEG s i g n a l  a s u c c e s s f u l approach t o t h e s e l e c t i o n o f a p p r o p r i a t e  the s e l e c t i o n o f  assessment o f t h e r e l a t i v e  appropriate  ability  EEG  segment  lengths  o f AR models t o r e p r e s e n t  AR  and t h e  t h e EEG s i g n a l  were e s t a b l i s h e d through the s t u d i e s on AR s p e c t r a l a n a l y s i s . The single  ability  trial  to c o n s i s t e n t l y acquire  i s an  important  event r e l a t e d i n f o r m a t i o n  contribution  to  the  field  of  EEG  from a signal  107  analysis. event  In a d d i t i o n ,  related  the  information  l i s h e d as a v i a b l e model. understand  event  related  neurological  i s contained  premise i n v o l v i n g i n the  T h i s model s h o u l d be potentials  and  the  o v e r a l l EEG  way  i n which  s i g n a l i s estab-  c o n s i d e r e d when a t t e m p t i n g  their  relationship  to  ongoing  to EEG  processes. Finally, important  the  work  contribution  providing  the  single t r i a l  EEG.  this  towards  for control applications. by  in  thesis,  the  ultimate  taken  as  goal  of  I t overcomes perhaps one  framework  for  the  extraction  a  of  whole,  represents  harnessing  o f the useful  EEG  greatest  an  signals obstacles  information  from  108  REFERENCES  1. 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