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Estimating the intensity function of the nonstationary poisson process Flynn, David Wilson 1976

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ESTIMATING OF  THE I N T E N S I T Y FUNCTION  THE NONSTATIONARY  POISSON PROCESS  by D A V I D WILSON .Sc.,  University  A THESIS THE  of Science  FYNN  and Technology  SUBMITTED IN PARTIAL REQUIREMENTS MASTER  Ghana,  FULFILLMENT  19  OF  FOR T H E D E G R E E OF OF  SCIENCE  in THE (INSTITUTE  We  OF A P P L I E D  accept to  THE  DEPARTMENT  this  OF  MATHEMATICS  thesis  the required  UNIVERSITY  MATHEMATICS  as  STATISTICS)  conforming  standard  OF B R I T I S H  August,  AND  1976  (c) D a v i d W i l s o n Fynn  COLUMBIA  In p r e s e n t i n g t h i s t h e s i s  in p a r t i a l f u 1 f i l m e n t o f the requirements f o r  an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and I f u r t h e r agree t h a t p e r m i s s i o n  for e x t e n s i v e copying of t h i s  study. thesis  f o r s c h o l a r l y purposes may be g r a n t e d by the Head of my Department or by h i s  representatives.  It  i s understood that copying o r p u b l i c a t i o n  o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d without my written  permission.  Department o f The U n i v e r s i t y o f B r i t i s h Columbia 2075 W e s b r o o k P l a c e V a n c o u v e r , Canada V6T 1W5  i  ABSTRACT  Let'{N(t), intensity  -T<t<T} be a n o n s t a t i o n a r y P o i s s o n p r o c e s s X ( t ) > 0 , assumed i n t e g r a b l e  function,  o n [-T,T] .  optimal l i n e a r estimator, X , of the i n t e n s i t y function Li  sidered  with The  i s con-  i n this thesis.  Chapter 1 d i s c u s s e s X as a f u n c t i o n L  of h ( t ; s ) , which i s  the unique s o l u t i o n o f the Fredholm i n t e g r a l equation second k i n d ,  •>  m ( s ) h . (s) + / K ( s ; u ) h . ( u ) d u = K ( t ; s ) , t a t  a<s<b.  b  Chapters  of the  2 and 3 a r e r e s p e c t i v e l y devoted  to a  discussion  o f some o f t h e e x a c t a n d a p p r o x i m a t e m e t h o d s f o r s o l v i n g t h e above i n t e g r a l e q u a t i o n .  To  illustrate  numerical oilwell  the use o f t h e techniques  examples a r e t r e a t e d .  discoveries  t o d a t a on t r a f f i c  Finally, counts  also  Britain.  presented.  i n C h a p t e r 5,  on t h e L i o n s  Gate B r i d g e , V a n c o u v e r , and t o d a t a on c o a l - m i n i n g i n Great  three  C h a p t e r 4 d e a l s w i t h d a t a on  i n A l b e r t a , Canada.  the model i s a p p l i e d  devised,  disasters  C o m p u t e r p r o g r a m s a n d numerous d i a g r a m s a r e  i i  TABLE  CHAPTER  CHAPTER  CONTENTS  1  ON  T H E E S T I M A T I O N OF  1.0  INTRODUCTION  1.1  OPTIMAL LINEAR ESTIMATORS THE I N T E N S I T Y FUNCTION  THE  INTENSITY  FUNCTION 1  1.2  THE B E S T L I N E A R E S T I M A T O R  2  SURVEY  OF  OF E X A C T METHODS FOR  FREDHOLM  INTEGRAL EQUATIONS  2 5  SOLVING (OF T Y P E I I )  2.0  SUMMARY  2.1  FINITE  2.2.1  Hadamard's I n e q u a l i t y  14  2.2.2  Hilbert  14  2.3  THE C L A S S I C A L  2.4  HILBERT-SCHMIDT, AND  CHAPTER  OF  9 D I F F E R E N C E APPROXIMATIONS  Spaces  GRANDELL  FREDHOLM  TECHNIQUES  12  16  KARHUNEN-LOEVE  TYPE SOLUTIONS  2.5  GRANDELL  2.6  SOME OTHER  3  A  3.1  INTRODUCTION  46  3.2  QUADRATURE  46  3.3  G E N E R A L I Z E D QUADRATURE  52  3.4  C O L L O C A T I O N METHOD  54  3.5  ERROR A N A L Y S I S  58  3.6  SUMMARY  61  SURVEY  TYPE SOLUTION  26  METHODS  OF A P P R O X I M A T E  AND  36  RULES  CONCLUSION  41 METHODS  i i i  CHAPTER  4  COMPARISON: APPROXIMATE  4.1  CASE  O U T L I N E OF W H I T T L E ' S D E R I V A T I O N OF  CHAPTER  AN E X A C T V E R S U S AN SOLUTION IN A S P E C I A L  (1.2.6)  .  63  4.2  THE EXACT SOLUTION  65  4.3  AN  69  4.4  COMPUTATION:  5  APPLICATIONS  APPROXIMATE  O I L WELLS  5.1  INTRODUCTION  5.2  E S T I M A T I O N OF AT  SOLUTION  TRAFFIC  DISCOVERY DATA  71  *  73  DENSITIES  THE L I O N S G A T E B R I D G E  5.3  COAL-MINING  5.4  CONCLUDING  DISASTERS  79  REMARKS  82  BIBLIOGRAPHY  APPENDIX  75  93  COMPUTER PROGRAMS AND  OUTPUT  1  THE OILWELL DISCOVERY PROCESS  95  2  THE. L I O N S G A T E B R I D G E P R O C E S S  104  3  THE COAL-MINING  112  D I S A S T E R PROCESS  iv  ACKNOWLEDGEMENT  I  would  like  t o express  for  g u i d a n c e and numerous  the  manuscript.  Dr.  S. W.  For  expert  Hardy  Nash  events.  comments  acknowledge  like  grateful  my  Zidek  t o W.  of  indebtedness to  f o rhelpful  t o thank Aseefa  and encouragement  t o D r . J i m V.  throughout the writing  of d i f f i c u l t material,  I am a l s o  I would  understanding  gratitude  a n d D r . J . M. V a r a h  typing  Bunn.  Finally,  I also  my  communication.  I am m o s t  Samaradasa Merali  i n the face  grateful to  a n d R.  Brun.  f o r patience, o f many  series of  CHAPTER 'ON  1  THE E S T I M A T I O N OF THE OF THE N O N S T A T I O N A R Y INTRODUCTION,  INTENSITY  FUNCTION  POISSON PROCESS: PRELIMINARIES  INTRODUCTION  In sity  this  chapter  we  consider  function of a nonstationary  formulation  given  by' C l e v e n s o n  linear Poisson  and Zidek  estimators process. ( 7 ) , we  of the intenUsing  consider  the a  point  process  { N ( t ) , - T $ t $ T } , 0<T^°°  with  independent  increments  and w i t h  p[N(b)-N(a)=n]=^[A(a,b)] exp[-A R  (a,b) ]  where  A(a,b)=/ A(t)dt, a b  and  the intensity  integrable form.  on  [-T,TJ.  For details  consider  function  linear  we  -T$a<b$T,  A(t)>0,  i s assumed  t o be  Riemann  The unknown A ( t ) d o e s n o t h a v e refer  estimators  to Grandell  parametric  ( 1 0 ) . I n s e c t i o n 1 . 1 , we  of the intensity  function, A(t), i n  2  general, and  the  and  two  natural  moving-average.  imposed  and  A ( t ) , the  particular  form.  a  integral  Fredholm  solution  of  estimator  1.1  OPTIMUM  Let  LINEAR  as  of  square  integrable  tation  with  respect of  an  A(t), i s determined  in  of  Clevenson  (7),  second  i n order  OF  THE  assumed  Clevenson  the  is  of  to  kind  and  Zidek  i s obtained;  obtain  INTENSITY  nonstationary  the  best  a  the  linear  process  i n t e g r a b l e on  [-T,T].  the  and  (7  Zidek  [-T,TJ.  A,  FUNCTION  Poisson  Following  joint  estimator,  approach ),  A(t)  Denoting  distribution i s measured  by  of  N  of  with We  seek  Grandell  i s assumed E  the  and  to  expec-  A,  the  by  2  the the  present estimate  record  leads  estimators  to with  .  however,  and  work  as  well  as  i s constrained The  numerical  histogram  tage,  a  A(t)>0,  to  criterion  form.  f u n c t i o n on  counting  the  given  the  Grandell's  histogram  T  In  tice  of  ESTIMATORS  by  work  the  E/^ [A(t)-A(t)J dt  (1.1.1)  above,  the  {A(t);-T$t$T}.  extended  performance  estimator  i s required  i n the  in particular,  s e c t i o n 1.2,  equation  function,  estimate  (10),  best  {N ( t ) ,-T^t$T.} b e  intensity an  (t)  In  Following  which  A  estimators,  motivation  that  are  i s that  moving  to for  be  Also,  average,  f o r many  a  this  relatively  algorithms.  the  i n the  references linear  are  function of  the  constraint i s that i t  easy two  cited  to  well  compute known  linear.  a p p l i c a t i o n s of  A  in  prac-  estimators, disadvan-  interest,  linear  be  3  estimates consider Here,  the  the  In of  may  major  must  (1.1.2)  be  sufficiently  estimation problem difficulty  general,  A(t)  for  not  by  have  the  the  (1.1.3)  For  example,  a(t)=0  this  is analytic  wish  linearity  to  constraint.  intractability.  the  estimate  A(t)  function a(t)  let  and  h(t,s)=r' 0,  t  otherwise. r  <  s  <  t  choice,  This  moving-average,  )/r.  r  is a  process.  Such  advantage  of  an  histogram  estimate  requiring  beyond  that  needed  common  practice  to  evaluate  times;  for  values.  to  no  f i t an  used  the  of  about  assumed  the  the  intensity  time  estimate  m u l t i p l e s of time  intensity  i t has  averaging  moving-average  integral  the  because  knowledge  select  the  instance, at  least-squares curve  estimate  i s widely  almost  process  sampled  may  h(t,s)d(N(s))  x(t)=pf£_ dN(s)= (N(t)-N(t-r)  to  the  constraint,  (1.1.3)  crete  One  form  some d e t e r m i n i s t i c  an  without  linearity  A(t)=a(t)+  As  accurate.  r,  function to  r.  It is  at  dis-  and  then  the  4  Clevenson is  representable  sense  by  the  function  DEFINITION: wide-sense a  Zidek  (7)  assumed  A  prior  condition that  stationary stochastic process  covariance  is  and  with  not  In  (1.1.4)  where The  s t o c h a s t i c process  independent  (7)  s  The  Clevenson  {x ( t ) ;-°°<t <°°}  of  t  and  i s said  y  and  to  be  f u n c t i o n m ( t ) =E-{x ( t ) }  i t s covariance  and  Zidek  function  function only  define  X (t)=[N(t )-N(t _ )] [ 2 A ] H  of  (t-s)  i  i  =T  i  1  i s any  estimator,  -  1  the  ,  estimator  t _ <t<t , ±  partition A  histogram  1  of  , i s defined  i  [-T,T] a n d  2A^=t^-t^_  by  A ( t ) = [ N ( t + A ) - N ( t - A ) ] [2 A ] " , 1  M  -T+ A <t  optimal  optimal  <T-A.  class  here.  width  window w i d t h  also obtained.  details  mean  wide-  separately.  moving-average  provided  were  and  -T=tp<t^<•..<t  (1.1.5)  the  t  constant  A(t)  K.  K ( t , s ) = E { x ( t ) x ( s ) } - E { x ( t ) }E{x ( s ) } i s a and  about  {A(t);-T<t<T} i s a  s t a t i o n a r y i f i t s mean-value  constant  knowledge  We  do  f o r the  f o r the not,  histogram  moving-average  however,  intend  estimator  and  estimator to  discuss  the  THE  1.2  It  i s at  performance mators the  can  least  intuitively  superior be  X  to  the  designed  intensity  Let  BEST LINEAR  process  denote  (1.1.2) when b o t h  error  is  the  consider  of  Grandell's  and  histogram  about  the  esti-  statistics  of  available.  linear and  estimate  h(s,t)  E[(A (t)- A  (t))  i s given  follows  the  technique.  L  H e r e , we  X(t)  of  are  that  selected  to  results  minimize  in the  by  ].  This  linear  Grandell  (10).  Clevenson-Zidek  T h u s we  seek  an  minimum The  (7)  mean-  method  generalization  estimator  of  the  form  T  drop  the  assumptions  s t a t i o n a r y and  The  problem  now  that  in section  m(t)=EX(t)Ey.  i s to  determine  the  1.1 We  to minimize  take  resulting these w i l l  optimal i n turn  choices  are,  minimize  Zidek,  say,  show, a f t e r  wide-  M(s) ^g (t)dt. =  functions  (t)-X (t)]  a (t) 0  (1.1.1).  is m  a(t)  and  2  f u n c t i o n a l E [x^  the  X(t)  that  ~ •h(t;.)  having  X (t)=a(t)+/^ h(t,s)d(N(s)-M(s)).  (1.2.1)  sense  here  estimators  " 2  estimate  we  that  moving-average  r  square  evident  i f more k n o w l e d g e  a(t)  mean-square e r r o r  ESTIMATOR  and  .  If  h°(t;.),  Grandell,  and  then  subsequently  Clevenson  and  (1.2.2)  E [ X ( t ) - X ( t ) ] = ( a ( t ) - m ( t ) ) + ( h ( t ; .) , h ( t ; .) ) - 2 L ( h ( t ; . ) ) 2  L  some m a n i p u l a t i o n ,  the  that  2  t  6  where  (.,.)  (1.2.3)  for  i s defined  ( x ( . ) ,y  a l lfunctions  nonnegative  for  T  (1.2.4)  Assume that  T b y L x (. ) =/_ f c  (x,x)<«>.  product.  K(.,.) i s The  linear  x (s) K ( t ; s ) d s ,  I t i s clear  1  from  (1.2.2)  that the  f o ra(t)i s  a°(t)=m(t)  that  T  (x,x)<»; a n d b e c a u s e  (x,x)<°° i s a n i n n e r  i s defined  choice  T  x,y f o r which  a l lx such t h a t  optimal  of  (.))=/^ x(s)y(s)m(s)ds+/^ /^ x(s)y(s)K(s,u)dsdu  definite,  functional  by  fora l l t.  0<infm(s) s  and that  i s a continuous linear  m and K a r e bounded, functional  i t follows  on t h e H i l b e r t  space  functions  H={x:(x,x)<co}.  It  i n turn  follows  by t h e R i e s z  L x ( . ) = (x(.) t  for  a l lxeH where  (1.2.5)  g  representation  theorem  that  ,g (.)) t  (.)eH.  h°(t;s)=g. ( s ) ,  Thus  the optimal  choice  f o r h,  7  is  the  unique  solution  i s the w e l l  kind  which occurs such  The  equation  equation,  work and this  the  linear  are  given  covariance  (1.2.6). and  numerical  and  of  the  second  a p p l i e d mathetheory.  studied extensively  problem  for  f o r e x a m p l e , H.  (1.2.6)  methods as  well  as  observations  Van  Trees  approximate  to design  main concern  approximation  emphasizing  functions are  thus  i s our  i n subsequent  I t i s perhaps worth  equation  been  filtering  (see,  of  Information  (1.2.6) has  to equation  exact  problem  mean and  equation  (19),  6).  solution  REMARK: the  i n C o m m u n i c a t i o n and  with  4 and  integral  f r e q u e n t l y i n many a r e a s  contain additive noise  The this  known F r e d h o l m  integral  chapters  for  as  connection  that  integral  T  This  in  the  x(s)m(s)+/2 x(u)K(s,u)du=K(t;s).  (1.2.6)  matics  i n H of  s o l u t i o n s the  techniques  chapters.  that theoretically  needed  the  in  to  solve the  estimator. data  only integral  However,  i n hand  is  for  very  useful.  In the special  (1.2.7)  case  sequel  (e.g.,  i n chapter  where  m(t)=u,  and  K(t;s)= K(t-s).  4 ) , we  shall  consider  the  8  Thus  (1.2.6)  (1.2.8)  It  becomes  , y x ( s ) + / j X ( u ) K ( s - u ) d u = K ( t - s ) , -T<s<T.  f o l l o w s t h a t o u r l i n e a r e s t i m a t o r now  takes the s p e c i a l form  i  (1.2.9)  A (t)=y+/^ H(t;s)d(N(s)-ys), L  o b t a i n e d from  T  (1.2.1) by s e t t i n g a ( t ) = m ( t ) = y .  i n which our model w i l l and  5.  0  This i s the  form  be u s e d i n t h e a p p l i c a t i o n s i n c h a p t e r s  4  9  CHAPTER 2 SURVEY OF E X A C T METHODS FOR S O L V I N G FREDHOLM  I N T E G R A L EQUATIONS  (OF T Y P E I I )  SUMMARY  This methods kind),  chapter  f o r solving  theory  some t h e o r e t i c a l  and methodology  ( o f t h e second  background,  The n e c e s s a r y  background  a n d some a s p e c t s  sented.  Subsequently, on s u i t a b l e  of Hilbert  a l l integral  Hilbert  1.  be g i v e n  underlying these  sketched  a n d a number o f  By means o f  f o rthedifferences i n methods and t h e i r  i n linear space  algebra w i l l  theory w i l l  operators w i l l  invesbe  be p r e -  be viewed  as  spaces.  INTRODUCTION  An function  integral appears  Integral a  Equations  a r e a l s o d i s c u s s e d and compared.  some m o t i v a t i o n w i l l  tigation.  acting  Integral  T h e b a s i c m e t h o d s a r e t r e a t e d i n some d e t a i l , a n d  developments  examples,  t o a d i s c u s s i o n o f some o f t h e e x a c t  Fredholm  together with  applications. recent  i s devoted  equation under  equations  number o f y e a r s ,  The  actual  i s an e q u a t i o n  the integral  originally  t h e unknown  sign.  have been encountered i n the theory  development o f t h e theory  however, o n l y  i n which  i n mathematics f o r  of Fourier  of integral  Integrals.  equations  a t t h e end o f t h e n i n e t e e n t h century  began,  due t o t h e  10  works in  of the Italian  the year  published let  1900, when  have  From  been  V. V o l t e r r a ,  t h e Swedish  h i s famous work  problem*.  tions  mathematician  then  mathematician  o n a new m e t h o d  of research  principally  Ivar  of solving  o n , up t o t h e p r e s e n t ,  the subject  and  Fredholm the Dirich-  integral  f o r numerous  equa-  mathema-  ticians.  The many  theory of integral  different  areas  applied  mathematics  tions.  T o make  impossible. applied  equations  It  o f mathematics. c a n be s t a t e d  a list  Suffice  mathematics  linear  integral  vector  spaces,  o f such  applications  and m a t h e m a t i c a l a  Indeed,  i n t h e form  i t to say that  do n o t p l a y  i s worth  e q u a t i o n s has c l o s e  there  contacts with  many  problems  of integral  would  be  i s almost  physics  where  of equa-  almost no a r e a o f  integral  role.  mentioning, equations  at this  stage,  the fundamental  that  i n dealing  concepts  e i g e n v a l u e s and e i g e n f u n c t i o n s  play  of a  with  linear significant  role.  *"'Sur u n e n o u v e l l e m e t h o d e p o u r l a r e s o l u t i o n d u p r o b l e m e d e Dirichlet". O f v e r s , a f Kungl. Vetensk. Akad. FOrh., Stockholm, "57, n r . 1 (10 J a n . 1 9 0 0 ) , 39-46. * " S u r une c l a s s e d ' e q u a t i o n s f o n c t i o n n e l l e s " . S t o c k h o l m , 27 ( 1 9 0 3 ) , 3 6 5 - 3 9 0 .  Acta  Mathematica,  11  The m o s t f r e q u e n t l y  studied  i n t e g r a l equations arethe  following:  / K(t,s)f(s)ds  (2.1.1)  b  3.  f ( t ) + X / K ( t , s ) f ( s ) d s =g(t)  (2.1.2)  b  (2.1.3)  m(t)f(t)  + X / K ( t / s ) f (s)ds b  The a b o v e e q u a t i o n s a r e g e n e r a l l y tions of the f i r s t , interval  second,  known a s F r e d h o l m e q u a -  (a,b) may i n g e n e r a l be a f i n i t e 00  may d i v i d e  00  The f u n c t i o n K ( . , . ) , w h i c h  We n o t e t h a t  equations  i s generally  known as t h e k e r n e l ,  b  1  stated  1  2  f ( . ) enters the  earlier,  (s)] d s = C / K ( t , s ) f ( s ) d s + C f K ( t , s ) f b  2  and o f t e n  b  1  the equations  many s i t u a t i o n s ; i n s t a t i s t i c a l symmetric  the function  A l l t h e above  i n a l i n e a r manner s o t h a t  / K(t,s) [c f (s)+C f  As  we  i t t o (2.1.2).  g ( . ) a r e assumed known a n d f ( . ) i s s o u g h t . that.is,  The  i n t e r v a l o r (-°°,b],  are f i n i t e .  (2.1.3) by m ( t ) t o r e d u c e  equations are l i n e a r ;  -g(t)  and t h i r d k i n d , r e s p e c t i v e l y .  [a,°°), o r (- , ) , w h e r e a a n d b  and  =g(t)  2 J  (2.1.1)-(2.1.3)  problems t h e k e r n e l s  a l s o nonnegative  definite.  2  (s)ds.  arise i n are usually  12  In  the next  results, Hilbert cal  theory,  Section  We  including Slepian  shall  techniques  shall  consider  (7) t y p e  discuss  will  (18),  Although appropriate  and  mathematical  background o f The  i n Section  Hilbert-Schmidt,  Karhunen-  s o l u t i o n s and t h e r e s u l t i n g  some o t h e r  methods  we  not only  are dealing with  stage  consider  because  of great  insight  2.1  space  into  finite  finite  practical the nature  exact  m e t h o d s , we difference  difference  utility,  of integral  find i t  approxima-  approximations  but also  provide  equations.  theory.  (2.2.1)  i n the  (19) ,  TOOLS  F I N I T E DIFFERENCE APPROXIMATIONS  If,  expan-  i n the literature  s t a t e H a d a m a r d ' s i n e q u a l i t y a s w e l l a s some a s p e c t s  Hilbert  classi-  others.  to briefly  at this  certain  some  be c o n s i d e r e d  SOME U S E F U L M A T H E M A T I C A L  tions  state  t h e d e t e c t i o n t h e o r e t i c approach by Vantrees  2.  also  4 we  and G r a n d e l l  sions.  briefly  as u s e f u l t o o l s i n t h e sequel.  Fredholm expansion  Loeve,  shall  i n c l u d i n g a d i s c u s s i o n of the necessary space  In  are  s e c t i o n we  equation  f ( t ) - X / j K ( t , s ) f (s)ds = g ( t )  We of  shall  3.  13  we  r e p l a c e t h e i n t e g r a l by a s u i t a b l e n , f (t)-A L _ ^ K ( t ,£)f (i) i=l  (2.2.2)  for  l a r g e n and a c o n t i n u o u s  the  sum  at  n discrete  (2.2.3)  we  i n (2.2.1).  (2.2.4)  we  evaluate  =g(l), ^ n  K  (2.2.3) w h i c h may  (I-AH)F  where t h e i - j t h  equation  J  be r e w r i t t e n i n m a t r i x  form  = G  element i n the matrix  H i s (—)K(i,—), n n n  f (^-) and G has components  To s o l v e  (2.2.4) we  invert  the matrix  and  and F i s  g {^) .  find  F=(I-AH) G. - 1  The above i n v e r s e w i l l most  only  (2.2.1) by t h e a l g e b r a i c  components  at  (2.2.2)  j=l,2,...,n,  a vector with  (2.2.5)  to the  points  n , . . f(i)-A^ i (i,-)f(i) n 4-—= n n n n i=l  J  and c o n t i n u o u s f ( t ) ,  a c l o s e approximation  I f , furthermore,  have r e p l a c e d t h e i n t e g r a l  system  = g(t)  kernel K(t,s)  i n (2.2.2) r e p r e s e n t s  integral  sum:  n values.  determinantal  exist  f o r a l l A, w i t h  These a r e the r o o t s o f the  equation  j  the exception of characteristic  14  In which  (2.2.5)  no  0.  we  solution  eigenvalues. (2.2.3)  =  I-AH  (2.2.6)  We  see  exists. also  note  n e c e s s a r i l y - has  equation  (2.2.1)  need  2.2.1  there  THEOREM  state  this  2.2.1  upper  Let  estimate  H  be  values  that  every  eigenvalues,  not  as  may  Such  have  special  are  commonly  finite even  values  of X for  known  algebraic  though  the  as  system integral  eigenvalues.  Hadairiard's  We  An  that  Inequality  a  theorem.  be  a matrix  with  for i t s determinant  the  general  is given  element  h... ID  by  (2.2.7)  2.2.2  Hilbert  So  f a r we  question  of  Basically, the  it  what of  equation  additional should  purposes Hilbert  have we  not  mean by  course, to  an  a  i t will spaces.  to  a  a  prove  addressed solution  solution  identity.  restrictions belong  really  Spaces  on  But  the  be  of  any  often,  class  convenient  an  of to  the  as  equation.  must  however, such  to  integral  equation  solution,  particular to  of  ourselves  we  reduce may  demanding  functions. work  impose  i n the  For  that these  so-called  15  DEFINITION product inner  2.3.1  space  A linear  i f an i n n e r p r o d u c t  product  assigns  number d e n o t e d by  (f,g) w i t h  (af+0g,h) =  a(f,h)+3(g,h)  3.  (f,f)£0 a n d  ( f , f ) = 0 i f and o n l y  We  l e t (f,f)  DEFINITION Cauchy  an i n t e g r a l  to  an element  If  (2.2.1)  Then H i s s a i d  sequence converges  of  an  complex  of integral  operator  equations,  i s fundamental.  satisfies  (1) ( f , f ) =  space  (f,f).  a n d i^-^  t o be a H i l b e r t  t o an element  f i n H a new e l e m e n t ,  the operator  0.  i t t h e Norm o f f .  L e t H be an i n n e r p r o d u c t  s e q u e n c e i n H.  In the study  i ff =  s i n c e by p r o p e r t y  | | f | | and c a l l  2.3.2  Cauchy  Such  = (g,f)  (f,f) i s real, =  inner  the following properties:  2.  that  on i t .  p a i r f and g i n X a  (f,g)  note  we  t o every  t o be an  i s defined  1.  We  every  space X i s s a i d  we  a  space i f  i n H.  find  that  the notion  Such an o p e r a t o r  assigns  s a y K f i n H.  the condition  K ( a f + 6 g ) = ctKf+BKg  say that K i s a l i n e a r  operator i s  operator.  An example o f a  linear  16  Kf  where may  =  /jK(t,s)f(s)ds,  f eL2[0,l]  o r may  Consider  L [ a , b ] , where 2  Such  space  an  operator  H.  the interval  [ a , b ] may  be  If  / / |K(t,s)| dtds a a' b  b  2  1  then  i s continuous.  n o t be d e f i n e d on t h e whole  THEOREM 2 . 3 . 1 infinite.  and K ( t , s )  the operator  Kf  =  f  =  M <~ 2  K(t,s)f(s)ds  i s bounded.  a  The  stage  i s now s e t f o r o u r m a i n  concern  i n the sections  ahead.  2.3  THE C L A S S I C A L FREDHOLM  To the  begin  Fredholm  (2.3.1)  with  with,  i n order  meter in  consider  the solution  o f equa-  b  (2.3.1)  three  shall  f(t). = g ( t ) + A / K ( t , s ) f ( s ) d s a  a Riemann  A.  t o f i x o u r i d e a s , we  equation  integral  i n a given  F r e d h o l m was t h e f i r s t tion  TECHNIQUES  i n the general  The r e s u l t s  person form  to give  (a,b).  f o r a l l values  o f Fredholm's  theorems which  interval  of the para-  investigations  are contained  a r e among t h e m o s t i m p o r t a n t  and b e a u t i f u l  17  mathematical in  discoveries.  replacing the integral  this  equation  t o a system  The method u s e d  accordance  interval  c i — t i / tn r  1 and  with  of linear  —ID f  •••/ t  h. 1 1  parts  infinity.  by t h e p o i n t s  -™t•  l + l l  equation  to  and l e t t i n g t h e  1  n  replace  equations  of  F r e d h o l m s m e t h o d , we p a r t i t i o n t h e  (a,b), i n t o n e q u a l  2  consisted  i n ( 2 . 3 . 1 ) b y a sum, t h e r e d u c t i o n  n u m b e r o f t e r m s o f t h e sum t e n d  In  by Fredholm  n  (2.3.1) by  n  (2.3.2)  f (t)=g(t)+Ah5 K ( t , t . ) f (t. ) . i=l 1  Let  f ( t . ) = f . , K(t.,t.)=K..; I  I  '  J  I  J  t h e n we may  I  1  equations  as  (2.3.3)  (1-AhK, , ) f,-AhK, _ f 11 1 12 2 -AhK  The form  teristic  rewrite  the system  of  . .-AhK. f ' = g ( t , ) i n n ^ 1  , - A h K „-...+ ( l - A h K ) f =g ( t .) . nlf1 n2 nn n ^ n 1 4 =  solutions f  , 1  the  1  of ratios  , 2  f  f o f (2.3.3) c a n be e x p r e s s e d i n n  of c e r t a i n determinants  determinant  b y t h e common  charac-  18  l-xhK (2.3.4)  -AhK  D (A) = n  -  1 1  An  that  this  expansion  1  2  1- AhK nn  i s not equal t o zero.  (-Ah) K  i j L  +  YZ1 i»j=l  21  K . . K .. Di 33 K. 1  (-Ah) n n l  K. 1  i  l  f  i  , • • - i  2  n  t h e sake  K(t ,S ) 1  1  . 1 1  K. . 1 2  • 2 1  K. . 2 2 "  1  1  X  1  n  . l , 1  K. l i„ n 2  K(t ,S ) 1  2  ...  The and  K(t,s)  K(t ,S )  determinant  t h e above also  . K, in  1  n,  K(t ,S ) 1  n  K 1  i~i | 2 n  of simplicity, l e t  (2.3.6) n  K  1  :  K(t ,S )  K. . 1 n  '  X  1 K. l  For  i n t h e form  K. . K. .  n  (2.3.5)  +...+  In  - AhK„ 2n  22  o f ( 2 . 3 , 4 ) may b e e x p r e s s e d  = 1-Ah> i=l  n  -AhK  K  -AhK ~ n2  determinant  n D (A)  n  1-Ahk  2 1  •AhK , nl  provided  A  symbol  n  2  ...  (2.3.6) i s taken  l  #  t  2 * ' " n\ t  1 2  n  K(t ,S ) n  i s called  n  Fredholm's  determinant,  t o be d e f i n e d f o r e v e r y  i n a m u l t i - d i m e n s i o n a l domain.  kernel  19  The if  any  fundamental property  pair  of  transposed,  Using the  arguments  the value  the  the  lower  (2.3.6),  we  may  write  that,  sequence  the  is  sign.  the expansion i n  form  (2.3.7)  (-Ah)  D ( A ) = l - A h Yi~= l K ( t " , t ") + n  i  ,(-Ah)  Now sum  suppose  ,  (2.3.7)  Thus  arises  (2.3.8)  h+0  and  2  shown by  this  \ l  1/  rJ  of  the terms  double, t r i p l e  integral,  of etc.  a  b  -' 1  \s  a  ds,ds  2  s /  1 #  2  2  /S-^,S2,S2i /  a (s ,S ,S ) K  Fredholm,  f o r every value  vein  b  2T  f f  converge  Vt. , t ./  ^  n->-°°, t h e n e a c h  b  ~JT l l  d  s  l  d  S  2  on  the basis  of  2  d  s  3  +  o f Hadamard's  theorem,  A.  1  series  3  i  the Fredholm s  a c o n v e r g e n t power  form  y i ' 3  D(A)=l-A/ K(s,s)ds+A / / K  was  K  series  3  In  Ii , j = l  +  single,  a  which  1  1  K  t e n d s t o some the  It. , t  ft • , t . , t \ ^ r\  3  that  n  2  21  ±  i,D,r=l  in  or  the d e t e r m i n a n t changes  n  to  determinant i s  1  i n the upper  of  symbol  of Fredholm s  function  (Fredholm's  D(A) first  may  be  series)  expanded of  the  20  (2.3.9)  D(A)=l+5 i=l  ^ 1' ^ 2  (-A)  K ,  i !  .S  l f  S  ' " " * ' ^ 1  S . l ds ds ...ds  2  1  2  i  i n a n a r b i t r a r y d o m a i n Q.  We  (2.3.10)  now s e e k a s o l u t i o n o f t h e  form  f (t)=g(t)+A j N ( t , s , X ) g ( s ) d s  where t h e r e s o l v e n t k e r n e l N ( t , s , X ) i s t h e p r o d u c t  (2.3.11)  N(t,s,X)=D(t,s,X D(X) '  D ( t , s , X ) i s t h e sum  of a c e r t a i n j_  CO  (2.3.12)  sequence,  D(t,s,X)=C (t,s)+') i=l  (  Q  ~*  C (t,s)  }  i  and  (2.3.13)  C (t,s)=  *'s S  ±  f  i f  S  This l e a d s to the Fredholm  (2.3.14)  i  series.  >  (  •,  r  w h i c h has t h e same c o n v e r g e n c e  d  S l  ds ...d 2  S i  series:  -  D(t,s,X)=K(t s)+5 i=l  S.J  2  x  '  I  ... )  / t , S , . . . ,S  K J  n  1  \S,S, , • • . ,S,  p r o p e r t i e s as Fredholm's  I ds  ...ds  1  first  21  We  a r e now  theorems. THEOREM  2.3.1  Fredholm  Fredholm's  that  equation o f the second  the functions  kind,  under t h e  g ( t ) and K ( t , s ) a r e i n t e g r a b l e ,  D(A)=|=0 a u n i q u e  solution,  which  i s of the  (2.3.10).  THEOREM the  the three  x  i n the case  form  to state  -  assumption has  i n a position  2.3.2-  If A  homogeneous  (2.3.15)  Q  i s a zero  q of D(x),  of multiplicity  then  equation  f(t)=A /K(t,s)f(s)ds 0  n possesses  at least  one, and a t most  q, l i n e a r l y  independent  .solutions j "^2. ^2' " " * ' ^ j — 1 f  ** " ' ^ i  f  '.  for  not  j=l , 2 , . . . , i ;  identically  nation  zero,  of these  l^i$q,  and any o t h e r  solutions.  S _.  solution  D. d e n o t e s  i s a linear  a Fredholm  minor  combiof  I  order  i relative  THEOREM tion the  2.3.3  solutions equation  K(t,s).  F o r t h e nonhomogeneous  i n the case given  to the kernel  D(AQ)=0,  function ^^(t)  equation to possess  i t i s n e c e s s a r y and s u f f i c i e n t  a  that  g ( t ) be o r t h o g o n a l t o a l l t h e c h a r a c t e r i s t i c  (j =l , 2 , . . . , i )  of the associated  homogeneous  c o r r e s p o n d i n g t o t h e e i g e n v a l u e AQ, and f o r m i n g t h e  fundamental  system.  solu-  The g e n e r a l  solution  then  has t h e form  22  (2.3.16)  f(t)=g(t)  +  A  „ I  1  D  +)  g(s)ds  r  \s ,...,s  p  1  p p  C.(j)..(t), where $•(t)= J J 3  L  products  S  r^.  f  ^ r  q  Fredholm's Equation  V7e c a l l  - l'  t  D pl'"-' i-l' i' i+l'  j=i 2.3.2  q V  with  the kernel K(t,s)  r  ,s.  .t_ , \  r  \^*1' '  Degenerate  Kernel  d e g e n e r a t e i f i t i s t h e sum o f  o f f u n c t i o n s o f one v a r i a b l e n  (2.3.17)  K(t,s) = >  N (t)L (S) ±  i  i+1 Substituting  (2.3.17)  into  f(t)=g(t)+AJK(t,s)f(s)ds  we n o t i c e g(t)  immediately  that  and o f a c e r t a i n  a solution  linear  i s t h e sum o f t h e f u n c t i o n  combination  o f t h e f u n c t i o n s IsL ( t ) :  n  f(t)=g(t)+2  (2.3.18)  A N (t), i  i  A  i  being  constant  i=l  In order  to determine  expression  (2.3.18)  this  example:  by an  EXAMPLE  2.3.1  the constants  i n the integral  S u p p o s e we  are given  f ( t ) = g ( t ) + X / J ( t + s ) f (s)-ds  A^, we s u b s t i t u t e  equation.  We  the integral  illustrate  equation  The  solution  should  have t h e form  •f ( t ) = g ( t ) + A t + A , 1  2  w h e n c e we h a v e t h e i d e n t i t y  g (t) + A t + A = g ( t ) + A / J (t+s) [g (s) + A 1  and  2  the system o f  A  x  S+A ]ds 2  equations  ( 1 - ^ - A ) - A A= A / J g ( s ) d s , 2  —jA-^ A+A ( l — ^ A ) 2  H e n c e we o b t a i n A equation  ; L  and A  1  = A / J s g (s) d s  2  and t h e s o l u t i o n  of the given  i n t h e form  f (t)=g(t) 2A/l 6AtS 3(2-A)(t l) 2A +  +  U  provided roots  12-12A-A  +  of the equation  2  hence  X =-6 + 4 / 3 ,  A =-6-4/3 2  +  g  (  s  )  d  g  Z  A i s n o t one o f t h e e i g e n v a l u e s .  12-12A-A =0  and  .integral  The e i g e n v a l u e s a r e  The  corresponding  characteristic  •f (t)=c(x t+i-|x ), 1  where  C i s an a r b i t r a r y  REMARK  The Fredholm  important  role  tions,  since  f (t)=c(x t+i-|x )  1  1  solutions are  2  2  2  constant. equation  i n the theory  i tcan e a s i l y  with  degenerate  and a p p l i c a t i o n s  kernel  p l a y s an  of integral  be s o l v e d by a f i n i t e  equa-  number o f  integrations.  2.3.3  Existence  Equations Suppose the  that  Fredholm  of Solutions.  o f t h e second  the range  kind  have  of integration  Alternative  Either:  The F r e d h o l m  Alternative  an e x i s t e n c e i s finite,  theory:  then  we  have  as f o l l o w s :  I f X i s a regular value,  then  f ( t ) =g ( t ) + X / K (t', s) f ( s ) d s h a s a u n i q u e a b  the equation  solution  f o r any  arbitrary  g(t); Or:  If X i s a characteristic  value,  then  t h e homogeneous  equation  (2.3.19)  has <j>i  f (t)=X/ K(t,s)f(s)ds a  a finite  b  number  ( t ) , . . . , <t>^ ( t ) .  (p, say) o f l i n e a r l y  In t h i s  case  independent  the transposed  solutions  homogeneous  equation  25  (2.3.20)  ¥(t)=A/ K(t,s)¥(s)ds b  cl  also  has p s o l u t i o n s - ^ ( t )  V ( t ) ; -and t h e n o n h o m o g e n e o u s  P equation theV^,;  has a s o l u t i o n i f and o n l y that  i fg(t) i s orthogonal  to a l l  i s , i f and o n l y i f  • f k g ( t ) V. ( t ) d t = 0 , a  i=l,...,p.'  1  This any  solution i s clearly linear  combinations  Stated the  i n another  homogeneous  not unique,  s i n c e we  o f t h e ^ ' s , and o b t a i n  form,  equation  (2.3.19)  has the unique  t h e nonhomogeneous e q u a t i o n  (and  t h e s o l u t i o n must e v i d e n t l y be u n i q u e ) . implies  Before  we  observation REMARK  and so  f a r found  integral  turn  t o the next  regarding  wearisome  numerical,  i s solvable  solvability  analysis.  apart  s o l u t i o n f=0,  forarbitrary In other  g  words,  t h e above.  of eigenvalues,  only  I f  s e c t i o n we m a k e t h e f o l l o w i n g  The F r e d h o l m c l a s s i c a l  distribution  solution.  existence.  concerning  information  another  the Fredholm A l t e r n a t i v e says:  then  uniqueness  can add t oi t  techniques  yield  of equations  but often  Consequently  require  rather  and e x i s t e n c e rather  Fredholm's  equations.  and  tedious  formulae  a few a p p l i c a t i o n s , e i t h e r a n a l y t i c a l from p r o v i d i n g a foundation  precise  have  or  f o r the theory  of  26  2.4  H I L B E R T - S C H M I D T , KARHUNEN-LOEVE, AND GRANDELL T Y P E S O L U T I O N S  2.4.1  Hilbert-Schmidt  In called  the f i r s t  selfadjoint  compact  regarding  their  expansion  theorems.  integral  operators.  As the  part  of this  before,  we  shall  s e c t i o n we  operators  eigenvalues, These  Theory  and o b t a i n  eigenfunctions  results  define  with  information  and t h e a s s o c i a t e d  a r e then  be c o n c e r n e d  some  the so-  a p p l i e d t o compact  integral  equations  of  type  f(t)=g(t)+X/ K(t,s)f(s)ds  (2.4.1)  b  in  the Hilbert  to  belong  to L |a,b] 2  integrable,  (2.4.2)  space  The  and t h e k e r n e l  function g(t) will will  be assumed  be  assumed  t o be  square-  / / |K(t,s)I dtds<«. a a b  b  2  2.4.1 H.  L e t K be a bounded,  Hilbert  space  if  the sequence  that  2  so t h a t  DEFINITION  from  L [a,bJ.  i s a Cauchy  {f„}, i n ' H .  Then  K will  { K f } we n  sequence,  be  said  linear  operator  t o be a compact  can extract a  f o r any u n i f o r m l y  on a operator  subsequence  {Kf  } n  bounded  R  sequence,  27  Now l e t {<j>^}'be a n o r t h o n o r m a l  s e t i n H, a n d l e t  n K  where  n f  =  K  f /  i  y  (  f  ,  <  >  i ' * i  n-1,2,...  )  {y^} i s t h e sequence o f e i g e n v a l u e s o f K o r d e r e d  p  y  have t h e f o l l o w i n g THEOREM 2 . 4 . 1  }  i s  a  eigenvalues, these  convergent  sequence  so  - ' 4  We  thus  results: i n H.  Then f c a n be r e p -  Q  i  ±  i s a suitable, element  i n the nullspace of K  L e t {<j>^} b e t h e c o r r e s p o n d i n g  w i t h K, a n d s u p p o s e H i s L ~ [ a , b ] ,  4)  l i m ,b ,-b  Pa al ;  K ( t  2  ' -/_ s ,  W *.(t)*.(s:  dtds=0  i  . i = l  that  K(t,s)=^)  y ( | ) ( t ) (j) (s) ±  i  i  i=l converges  i n t h e mean  ( i n t h e sense  of  (i.e.,  eigenvectors  then  n  (2  i n H.  accumulate  (f , * ) * +f o  THEOREM 2.4.2 ciated  n f  L e t f be an element  f=>  f  K  o f nonzero  i n t h e form  (2.4.3)  where  that  1>  t h e o r i g i n , and (  resented  such  l t> I 2 "  Then i f K h a s an i n f i n i t y at  t  (2.4.4))  asso-  28  It  then  (2.4.5)  follows  / / |K(t,s)| dtds=\ b  We  b  note  that  2  oo sum <•  L_  i n t h e above  y.  i s vital,  need  1  n o t be  i=l  We kernel state  v  2  square-integrable the  that  2 ±  results  since  the fact  for arbitrary  K(t,s)  compact  i s  operators  finite.  .  also  recall  that  operators  f o r which  a r e r e f e r r e d t o as H i l b e r t - S c h m i d t the  that  K(t,s)  i s an  operators.  We  now  important  HILBERT-SCHMIDT  THEOREM  2.4.3  Every  function  f(t) of the  form  (2.4.6)  f(t)=  is  almost  everywhere  to  the orthonormal  metric  kernel  The  (2.4.7)  above  /K(t,s)h(s)ds  t h e sum  system  o f i t sF o u r i e r  series with  <j>^(t) o f e i g e n f u n c t i o n s  respect  o f t h e sym-  K.  theorem  implies  f(t) = \f.<j>.(t) /  1  that  converges,  1  i=l where  the c o e f f i c i e n t s  function  f(t) with  f ^ are the Fourier  respect  t o the system  coefficients (<|>^(t)},  of the  that i s  (2.4.8)  f , = / f ( t ) <j>. ( t ) d fi 1  and  t ^ i  1  the h^ are the Fourier  with  respect  (2.4.9)  • to  coefficients  t o the system  of the given  function  h  (<|>^(t) }:  h = / h ( t ) <t> (t) d t i  i  Consequently,  t h e sum  the function  of i t s absolutely  f ( t ) i s almost  and u n i f o r m l y  everywhere  convergent  equal  Fourier  series:  oo (2.4.10) 2.4.2  f (t)= >  Application of the Hilbert-Schmidt  Using a  series  the Hilbert-Schmidt  expansion  (2.4.11)  with  Theorem,  of the solution  Theorem  i t i s possible  to obtain  f ( t ) of the i n t e g r a l  equation  f (t)=g(t) + A/K(t,s) f (s)ds  respect  metric the  ^ij). (t) .  t o the system  kernel  K(t,s).  eigenvalues,  of eigenfunctions  Assuming  A=|=A^, t h e r e  that  exists  (<j>^(t) ) o f a  A i s not equal a unique  sym-  t o any o f  solution  f(t) in  L (n). 2  Specifically  equation  (2.4.11)  oo (2.4.12)  f  (  t  )  '  g  A  (  t  l  - \ ~ C L /  <!>• ( t ) 1  1  may  be expanded  i n the form  30  w h i c h may term  be  to  substituted  i n (2.4.11)  and  integrated  term  by  obtain oo  X C. (1-X ) (j> . ( t ) = T K ( t , s ) g ( s ) d s  )  (2.4.13)  x=l But,  (2.4.14)  according  to the Hilbert-Schmidt  J"K(t,s)g(s)ds=\  t h e o r e m , we  have  again  T^-jtt),  i=l where From  the  are the Fourier  (2.4.13)  and  X  (2.4.15)  i  g  i  i t follows  that  ' (t)=0. ±  i  i=l Multiplying because  (2.4.14)  c o e f f i c i e n t s of the function g(t)  both  i  sides  i n turn  of the orthogonality  (2.4.16)  we  €. (1-r—)-Y^=0 1  by  < | > , <J> , . . . a n d i n t e g r a t i n g , 1  2  obtain  f o r every i .  1  i C.= i X . -X l g  Consequently  Substituting the  required  an  absolutely  these  expansion and  values  i n the series  ( 2 . 4 . 1 2 ) , we  of the s o l u t i o n of equation  uniformly  convergent  functions  of the  (2.4.17)  f (t)=g(t)-xy~ -^-<() (t)  series  kernel:  x  i=l  i  (X=|=X ) i  obtain  (2.4.11)  i n terms of  as eigen-  31  REMARK  I f A were e q u a l t o one o f t h e e i g e n v a l u e s  A . « = < = A-. . ~ n w i t h r a n k pp=t=2 r - ' p +p q + q==il * A  i4 S  +  = x  w  i  t  h  r  q, then e q u a t i o n  n }  * * p =  =  (2.4.16) w o u l d  A p  +  1  be  § a t i § f i § S i f §nd o n l y i f  = / g C t ) <j,  (2.4.18)  g  Condition  (2.4.18)  p + i  (t) dt=0  p+i  ( i = 0 , 1 , 2 , . . . , q-1)  i s i n accordance  Theorem and i th a s t o be e x t e n d e d corresponding series the  to the value  a s many t i m e s  solution  A  p  w i t h the Fredholm  Third  t o a l l thee i g e n f u n c t i o n s  which  i s r e p e a t e d i n t h e above  a s t h e number o f i t s r a n k .  In that  case  takes t h e form oo *  f (t)=g(t)-A^) X^Y7 i=l  (2.4.19)  ( | >  i  ( t )  +  C  l  , f ,  p  ( t )  +  x  + C  2*p+l  ( t )  +  •••  +  C  q *  P  +  q - l  (  t  )  * where the  >  denotes  that  v a l u e s o f i e q u a l t o p, p+1,  X  p  - X  p+l~ p+2 X  where q i s t h e rank  By vent  i n t h e s u m m a t i o n we h a v e e x c l u d e d a l l  _ X  p+q-l'  of that eigenvalue.  a s i m i l a r m e t h o d we may  kernel  (2.4.20)  '••  p+q-1, f o r w h i c h  find  the expansion  o f ther e s  N u s i n g t h e i n t e g r a l e q u a t i o n s a t i s f i e d b y N:  N(t,s,  A)=K(t,s)+A/K(t>y)N(y,s A)dy. /  We  obtain  in this  case  (2.4.21) 4>  (t)=0,  i  i=l whence, it  as  follows  before,  account  of  the  orthogonality  (<j>^(t)},  of  that  (s) b.(s)= i X . ( A.-X)  (2.4.22)  v o /  1  The  on  required  obtained  1  expansion  i n the  of  the  resolvent  kernel  is  therefore  form  oo  (X=A ) , i  —<i> (t)<j).(s: N ( t , s , X ) = K ( t , s ) - X )i = l _ _ _ x  (2.4.23) the  x  convergence  absolute  and  of  which  uniform  REMARK  The  seem more  convenient  functional titative  ticular  apply  results provided  approach kind  of  }  by  series.  be  to This  the  and  than  the  series  may  not  the  similar  \  <t> . ( t ) t y . (s) i l x.l  are  clearer  i^T  (2.4.23)  classical  expand  of  Fredholm's  often  KARHUNEN-LOEVE  will  i n view  of  (2.4.17)  a n a l y t i c techniques  THE  The  convergence  to  x  i s evident  expansions  2.4'  (A  formulae, lead  to  the  but  and the  quan-  techniques.  EXPANSION  the  function  i s analogous  to  a  f(t)  in  Fourier  a  par-  series  expansion  i n terms  of  sine  weighting  coefficients;  and  a  good  cosine  functions with  reference  appropriate  i s Helstrom  (126  pp.124-  133) .  To set  of  be  more  orthonormal  coefficients  the  we  d e s i r e an  functions  expansion  <f>^ ( t ) w i t h  i n terms  uncorrelated  of  a  weighting  r^.  Recall over  precise,  that  interval  a  set  of  (0,T)  functions  {<j>  i s orthonormal  m  f l ,  ( t ) : i = l , 2 , . •. . ) d e f i n e d  i  on  this  interval  i f  i=j  / *. (t) ** . (t)dt=. 0  If  the  function  (2.4.1)'  The of  1  lo,  3  orthonormal  f ( t ) ,defined  f  set on  i s complete,  (0,T)  i  (2.4.1)' by  zero  r ^ may  4>*j ( t )  and  functions are  except  (2.4.2)'  square-integrable  represented  as  i  be  determined  integrating  / J f (t) <fr*j ( t ) d t = )  the  be  a  (t)=2_r * (t)  coefficients  Since  may  then  f o r j = i , and  r =/Qf ±  (t) * * ( t ) d t . k  multiplying  over  the  each  interval  side  (0,T),  /Q r «|) (t)+* (t)dt. i  orthonormal, so  by  i  j  the  right-hand  side  is  34  For  illustrative  integral  where  purposes  i s the kernel  e i g e n f u n c t i o n , . and X^ a n d  values  know  that  t h e homogeneous  we  assume  that  4>^(t)  i s an  eigenvalue.  positive.  the eigenfunctions  set of functions  function  and where  for a positive definite  are s t r i c t l y  definite, the  consider  equation  K(t-s)  We  we  kernel,  Furthermore, form  <}>^(t) i s s a i d  the  i f K(t,s)  a complete  set.  t o be c o m p l e t e  By  eigeni s positive definition,  i f the only  g(t) satisfying  T  / - g ( t ) cf, (t)dt=0 i  for a  a l l i i s the function  function  of  tions the  g(t) not identically  functions  The  <j>^(t), t h e n  In essence  zero,  g(t) i s also  to f ( t ) .  n  i=l  means  that i f  to theset  an e i g e n f u n c t i o n .  the series expansion Convergence  this  i s orthogonal  s i g n i f i c a n c e o f the completeness  <f>^(t) i s t h a t  mean  g(t)=0.  property (2.4.1)  i n t h e mean  means  of the  func-  converges i n  35  The cj)^(t)  series  chosen  expansion  as  ( 2 . 4 . 3 )' i s t h e  it  can  Karhunen-Loeve  facts  theorem  be  equation  the eigenf unctions  Some o t h e r Mercer's  of  will  which  expanded  be  states  i n terms  of  of  (2.4.1)' w i t h t h e the  integral  functions  equation  expansion.  of  eventual interest.  i f K(t,s) i s positive  One  is  semidefinite  e i g e n v a l u e s and e i g e n f u n c t i o n s  as  oo ( 2 . 4 . 4 )'  K ( t , s )= > L  1  i=l.  •  1  I n some c i r c u m s t a n c e s  kernel  K  (2.4.5/  The  X <j) (t)<f>* . ( s ) . 3  "*"(u,v)  , 0<t,  1  kernel  has  t o use  the  inverse  by  /j!JK~ (t,u)K(u,v)du=6 (t-v)  inverse  given  defined  i t i s convenient  an  expansion  v<T.  i n terms  of  <!>^(t) a n d  X^  by 00  (2.4.6)'  K  _  1  4» (t) <|)* (u)-  ( t , u ) = ^  i  I=T The  usefulness of  analytical.  In of  the  a random p r o c e s s  series  1  inverse  In p r a c t i c e ,  s u m m a r y , we  may  kernel  I t may  be  represent a  over  of orthonormal  i  a  finite  functions.  i s , however, difficult  to  mainly determine.  square-integrable function  observation interval  in a  The  be  coefficients  may  made  36  to  have  the  useful  property  REMARK  Homogeneous  tant  i n communication  are a  roi© used  is  that  Fredholm  i n determining  random  process.  A  of  being  integral  theory.  the  uncorrelated.  As  equations a  theoretical  Karhunen-Loeve  difficult  i t i s often difficult  aspect to  of  find  play  to  they  theory  theory,  solutions  impor-  tool,  expansion  this  an  of  .  however,  the  equations  involved.  2.5  GRANDELL  In chapter we  (10),  G r a n d e l l used  1,  obtain  briefly  to  review  the  TYPE  SOLUTION  a method  best  linear  G r a n d e l l s method 1  similar  to  that  estimator. and  make  In  some  used this  in section  comments  thereon.  With Grandell the  the  assumption  adopts  as  q u a d r a t i c mean.  (2.5.1)  where  the  determined  only  criterion Thus  he  the  f o r the  seeks  covariance choice  estimates  of of  i s known, the  the  estimate, type  X*-(t)=a(t)+/Je (s)d(N(s)-s) . t  functions a(t) so  minimized.  and  that  E{A*(t)  is  a  that  -  X(t) }  2  3  (s)  (as  i n chapter  1)  are  37  By  virtue  theorem can  o f t h e Karhunen-Loeve  of the preceding  be r e p r e s e n t e d  section,  K(s,t) =  and M e r c e r ' s  the covariance  kernel  K(s,t)  by  (t) (2.52)  expansion  . (s)  x  i=l the  a n d <j>^ b e i n g  eigenvalues  and e i g e n f u n c t i o n s ,  respectively,  2 By  assumption  exists,  a ( t ) a n d 3 (.s) t  are finite  a n d E{ X * ( t ) - X ( t ) }  so 3 ( s ) has t h e form t  (2.5.3)  3 (s)=^) t  B <j) (s)+b (s) i  i  t  i=l where  b ( s ) i s orthogonal  t oty^ , cf> ^ t • • • •  t  We  now  (2.5.4)  proceed  t o minimize  the expression  E(A*(t)-X(t)} =E{(a(t)-l)+/g 2  B  f c  (s)dN(s)-s)-(Nt)-1)}  oo  oo  + \  —  Since  (2.5.4)  i  t  i=l.  _ 3 . + .(t)  i  A - " i  "  i=l  i s being  minimized,  a(t)=l  00  EU*  1  -2 y  i=l  ^  2  oo  —*. (t) 2  2  i=l  O  O  e +/Qb (s)ds+^>  =d(a(t)-l) +^) 2  oo  g  2  ( t ) - X (t) } =^ 2  (B,-*, ( t ) ) {3. +-i—2  2 Z  }  and b ( s ) = 0 . t  Thus  38  so  that  &E{A»(t)-Ut) >  - ,«  2  38.  "  2  i  3  2(@ -^ (t)) 1  1  V-  +  x setting  this  x  equal  i  t o zero  we  obtain  . 1+y.  1  which  corresponds  t o a minimum  3 E { A * ( t ) - A (t) } 2  •30.  It  follows  and  w  (2.5.1)  2+  2  U  that  8 t^ ( s ) = )/  (2.5.5)  /  i n turn  2_  2 =  since  > Q  i  (substitution i n  <i>, (t)4> . (s) - V r1+y. r-^ ,  becomes  -<f>, (t)<j>, ( s ) (2.5.6)  (2.5.3)),  A*(t)=l+/g^>  d(N(s)-s);  1  X 1  +  y  and 2 2 \ T ^ i E { A * (t) - A ( t ) } = > - i 1+y i=l 00  (  (2.5.7)  If  we  integrate  now over  multiply  t  )  i  expressions  the i n t e r v a l  (0,T),  (2.5.2) we  obtain  and  (2.5.5),  and  39  00  CO rpy  < .. (u) vJ p) .. ( V t W) < v| p> V " /  v  <f) . (u) |>. ( s ) v p -<  vp.  /j3 (u)K(u,s)du=/J^_ >  d  t  i=i <t> ( t )  D=l  1  u  3  (s)  i  (l+y )P i  ±  i=l (JK  (t)  <p  i  (s)  ^-—  y.  Thus  6 (s) i s a  i  (s)  1+y.  t  B (S)+/QK(U,S)B (u)du  REMARK  Although  T  =  T  equation  o f the second  K(t,s)  (2.5.8)  kind,  equation  i s a Fredholm  Grandell  fails  integral  to note  this  fact  (10) .  Grandell in  p  (  K(s,t)-6 (s)  (2.5.8)  in  (t)  solution, of the integral  t  equation  i  / i=l  i=l =  <p  the case  unbiased  estimate  of  interested  To EXAMPLE  that  3 ( t ) i s a unique  of  X(t).  i s infinite,  X(t) which reader  may  illustrate (2.5).  K(s,t) '  refer  Suppose  =  p  covariance On  kernel,  the other  hand,  i t i s impossible  i s useful  this  solution.  t  of a degenerate  eigenfunctions estimate  shows  f o r every  to Grandell's  theory,  t  there  Further, exists  i f t h e number  to find  an  i n (0,T).  paper.  we  consider  an  the kernel  i s given  as  example.  an of  unbiased The  40  As  was  noted  i n the previous  (t)  are  positive.  (2.5.9)  the  only  =  solution  <|> ( t ) =  The  T  so  i n this  example  2  get the  equation  being  C,  =  that  * (t) =  we  yi/Jcj) ( s ) d s ,  a  constant.  o r t h o g o n a l i t y requirement  / J c T ( t )  a l leigenvalues,  y / g K ( t , s ) (f)(s)ds  Thus  • (t) =  section,  —  Tp  2  =  1,  of the  <J>^(t)  implies  y  Substitution  i n equation  /T so  (2.5.9)  gives  P  that  Now  to get the best  linear  estimator,  A * , we  use equation  (2.5.6) ,  thus  X * ( t ) = .1 +  1/T 1+(P/T)  0  from  w h i c h we  obtain  (2.5.10)  X* -  REMARK mate  We  +  (2.5.10),  the given  N  (  T  Helstrom  (  s  , , ) ~ s  }  estimate as:  example  on t h e covariance  i s the best  to other  the engineering  linear  i n t h e above  covariance  now t u r n  N  )  SOME  We  {  P+T  that  2 .6  in  P  i s dependent only  estimate, with  note  the best  d  linear  esti-  f u n c t i o n and so t h e  estimate  f o revery  process  function.  OTHER METHODS  solution  literature  (12b) a n d o t h e r s .  linear  the best  techniques  as d e s c r i b e d  which  appear  mainly  by Van Trees ( 1 9 ) ,  Integral of  signal  tered  equations  are frequently encountered  d e t e c t i o n and e s t i m a t i o n .  i n connection  with  Two  such  i n the  equations  the detection of signals  theory  encoun-  i n nonwhite  noise are:  (2.6.1)  where  f ( t ) and  (2.6,2)  where  Af(t)  /j!JF(t-s)f (s)ds  i s t h e sum  noise,  0<t<T  A a r e t o be d e t e r m i n e d ,  and  = g(t)  of parts  By  assumption  corresponding  the covariance  t o white  and non-  so  N  R(t-s)  and  =  f ( t ) i s t o be d e t e r m i n e d .  function white  /jF(t-s)f(s)ds  substituting  0  =  (t-s)+K(t-s)  this  into  equation  (2,6.2)  produces  the  integral  equation  N  0  (2.6.3)  which of  we  immediately  t h e second  As (2.6.3)  T (t)+/jK(t-s) f (s)ds = g ( t )  as a Fredholm  integral  equation  kind.  mentioned will  recognize  0<t<T  i n sections  generally exist  (2.3) a n d unless  (2.4),  (-N /2) n  a  i s an  solution  to  eigenvalue  4.3  of a  t h e homogeneous  integral  positive-definite  negative of  equation  kernel,  eigenvalue,  and  (2.6.1).  the integral  there  i s no  Since  equation  trouble  K(t-s)  cannot  about  the  i s  have  existence  a solution.  2.6.1  Applications  In  numerous  of Fourier  Transforms  applications, integral  equations  of the  type  oo (2.6.4)  are If  /_  O O  K(t-s) f (s)ds  encountered.  The  = g(t)  integral  K ( T ) , g ( x ) e I>2 I" / ] ' -0 0  sides  to  00  w  e  c  a  n  on u  s  e  the l e f t  i s a  convolution.  the F o u r i e r transform  of  both  obtain  (2.6.5)  /2nG(K)G(f)  (2.6.6)  G(f) =  =  G(g),  and  so  ,  ./2JIG(K) provided  If finally  ( 2  . .7) 6  G(K) does  the right  not  side,  vanish.  as  a  obtain  f .  -i-GMljlf)  •  function  o f u,  i s i n L2[-  O o  a  ,  c o  ]  we  EXAMPLE  Consider  2.6.2.  (2.6.0)  f(t)-A/  By a d i r e c t  G  £  M  I  1  S  lf(s)ds  integration  (  e  - j t |  {VA  =  l+u so  =g ( t )  2  that  G(f)-A/2n  and  '-^<3(f) l+u  so ,  (2.6.9):  G(f)=  _  2  l + u -2A  w h e r e we r e q u i r e  ^ \ K<  for  ; > i <  w  e  -2A  1,°° -  p-iut  ,  2 l + u -2A 0 1  obtain the solution  (2.6.10)  ' f = ^  REMARK  Techniques  Mellin  .  /  +  'j'  G(g)  Then  %  _ ,v 2/n l+u  and  = G(g)  transforms  w h e r e y = /1-2A  and p r o p e r t i e s o f L a p l a c e , Hankel, and  (Helstrom  (12b))  c a n be d e r i v e d by  relating  t h e m t o F o u r i e r t r a n s f o r m s ; we d o n o t , h o w e v e r , i n t e n d t o d i s cuss  these  transforms  i n the present  work.  Equations  2.6.3  In  this  case  we  with  may  Separable  expand  the  Kernels  kernel  i n the  form  n (2.6.11)  where K  A  and  (t,s).  (2.5)  to  REMARK  worthy  T h u s we solve  of  methods  i s the  leads  to  are  can  the  omission  cesses,  y Aj ^ ( t ) ^ ( s ) , j= l  =  <f>j ( t )  Lack  numerous  which  K(t,s)  the  use  given space  which for  eigenvalues  the  prevents  finite  found  complete  eigenf unctions  of  sections  of  (2.4)  and  equation. us  from  i n the  observation  state-variable a  techniques  integral  are  and  d i s c u s s i n g a l l the literature.  and  formulation  solution.  A  nonstationary of  Kalman  and  noteproBucy  (6b)  46  CHAPTER A FOR  THE  SURVEY  OF  APPROXIMATE  SOLUTION  OF  FREDHOLM  OF  THE  3.1  SECOND  METHODS  INTEGRAL  EQUATIONS  KIND  INTRODUCTION  Since usually here,  exact  s o l u t i o n s of  difficult  the  private of  3  to  obtain,  possibility  of  communication,  quadrature  methods  the  i t is natural  finding  Professor of  Fredholm  the  integral to  equation  consider,  as  approximate  solutions.  James  suggested  Varah  we  In the  are do  a use  form  n  (3.11) i=l i.e., the we  the  integral  integrand explore  at  this  a  methods  for  we  In in  particular  consider then  can  section  suggested  other  are  the  to  we  (3.5)  the  the  we  Trapezium  tackle  improvements  of the  on  of  a weighted points  deal  and  proof  with  sum  of  In  this  values  of  their  (6),  simple  (12) ,  We  the  validity,  quadrature  rules.  shall  (14) .  rules; briefly  rules  in  section  (3.3),  and  collocation  in  section  (3.4).  In  question  errors  the  methods  of  discussed  and  the  thus  of  chapter  in general,  literature  Simpson's  these  t^.  methods;  without  cited  shall  method  by  approximate  explained  refer  (3.2)  number  generalizations to  consider  section  finite  and  numerical which  i s represented  subsequent  far.  Finally,  47  in  section  ( 3 . 6 ) we  shall  discussed  i n detail  here.  a  a n d some  summary  mention T h e n we  concluding  3.2  other  their  summarize  error  the p r i n c i p a l  terms.  shall  are not  end t h e c h a p t e r  with  o f type  with  remarks.  QUADRATURE  We  methods which  RULES  formulae  L e t us w r i t e  (3.1.1)  I(f) f o r the integral  /  f(t)dt. cl  For n is  the repeated  equal  steps  applied  order that mulae  the interval  o f l e n g t h h, so t h a t  over  sub-Intervals.  derivative this  forms,  i s continuous.  use e q u a l l y spaced  h =  (b-a)/n,  The e r r o r  of f(t) at a point  derivative  [a,bj i s d i v i d e d  points,  and the  depends  The f o l l o w i n g l e t  The M i d - P o i n t  t^=a+ih.  Rule  n I (f) = h ^  f (t _j )-+R i  s  1  where R-^ =  (3.2.2)  1 2 II 2-^(b-a)h f "  (£), i s the remainder  The T r a p e z i u m  Rule  n-1 1(f)  = |h{f(a)+2^> i=l  f ( t ) + f (b) } + R i  2  we  three  i ~ 1^ 2/• ••/ n •  (3.2.1)  formula  o n some  £ i n [a,bj , where  s o we  into  term  high assume for-  48  where R  - ™ ( b - a ) h  2  (3.2.3)  2  f "  (5)  Simpson's 1 2 1(f)  Rule 1  -i  2 ^ }  n  = |h{f (a)+4^~f (t  2 i  _ )+2^> 1  i=l  f(t  2 i  ) + f (b) } + R  3  i=l  where R  REMARK all  3  - y i o  In formula  (  "  b  type  important  may  a  h  4  f  l  V  (  5  n must  points,  be d e r i v e d .  )  be even.  The above  and h i g h e r - o r d e r  However,  t h e above  formulae  formulae  three  of the  a r e t h e most  i n practice.  (3.2.4)  Formulation  Fredholm  equations  case  i s now  of Discrete  o f the second  a s t r a i g h t f o r w a r d way  general  )  (3.2.3),  use e q u a l l y spaced  same  in  =  b y means  clear.  We  Equations  kind  c a n be  of quadrature  choose  approximated  formulae.  any q u a d r a t i v e  The  formula  n / f(t)dt b  =  ^>  w f(t ; i  i  I=T involving general  the n points  t ^ and t h e c o r r e s p o n d i n g  Fredholm'integral equation  o f the second  weights kind  w^.  The  4.9  (3.3.1)  is  / K(t,s)f(s)ds+g(t)  then  the  replaced  unknowns  tion  to  the  by  f(t^)  i n matrix  (3.3.2)  (I-KD)f  the  a  matrix  system  (to  integral  Written  where  =  b  cL  K  of  n  indicate  equation  form,  =  f(t)  linear  that  this  f ( t ) has  this  algebraic  system  i s only  been  of  resents  elements the  We  equations  EXAMPLE  the  the  =  elements  =  t^  ^2  Then  K  0, =  K j ±  t =  by  f (t)) .  becomes  K..=K(t.,t.)  s o l u t i o n of  values  above  of  these  f(t) at  methods  and  D  has  the  1 3  by  equations,  the  two  points  simple  f^,  rep-  t=t^.  examples  t + / j K ( t , s ) f (s)ds  K(t,s)  Take  i.e.,  the  i s of  (13)  ±±  The  kernel  (Hilderbrand =  approxima-  Consider  f(t)  where  w^.  approximate  3.2.1  for  g,  has  illustrate  an  replaced  13 diagonal  equations  =  s ( 1  notes \'  t ( l - t j ) , i  form  t(l-s) { _  (1-t^ , K  ±  the  i f  t )  i  f  that  =  i<j,  t  ~ t  >  s  this  ^5 ±2  t<s  1  ^  a  kernel n  d  n  (l-t ) ,  i , j =  2  ~  i s weakly \  (trapezium  ..., K  1,...,5.  3 5  =  singular.) rule)  t j(l-t ) 5  ...  50  so  that K  0  0  0  0  0  0  3/id  1/8  i/16  0  0  1/8  1/4  1/8  0  0  1/1-6  1/8  3/16  0  0  0  0  .0  0  and D = diag  Hence the  from  h(l/2,  (I-KD)f=g  system  of  with  1,  1,  g=  [o  1  1,  1/2)  1/4  1/2  3/4  l ] , we  obtain,  equations  = 0 61/6 4 f - l / 3 2 f - l / 6 4 f 2  3  = 1/4  4  -l/32f +15/16f -l/32f  4  =  -l/64f -l/32f +61/64f  4  = 3/4  2  3  2  3  V 2  = 1  The  solution  to this  f  =  set of equation i s  K f  0.0000 0.2943  2  0.5702  *3  0.8104  *4 f  EXAMPLE solution  3.2.2  1.0000  5  Suppose  -  we  of the integral  are required equation  to find  an  approximate  51  /JK(+.,S) f ( s ) d s + g ( t )  We  = f (t) .  u s e Simpson's r u l e t o a p p r o x i m a t e  /Ju(t)dt  where E this  =  represents  2  remainder  the i n t e g r a l  l/6{u(o)+4u(l/2)+u(l)}+E  the e r r o r or remainder  f o r t h e m o m e n t , we  obtain  Neglecting  relation  l / 6 ( K ( t , o ) f ( o ) + 4 K ( t , | ) f ( | ) + K ( t , l ) f (1) } + g ( t )  In  this  equation  t  we  form  2  term.  the  i n the  = f (t) .  write  = 0, j a n d 1 s u c c e s s f u l l y , a n d  obtain  l / 6 { K ( o , o ) f ( o ) + 4 K ( o , | ) f ( | ) + K ( o , l ) f (1) } + g ( o ) = f (o) l / 6 { K ( | , o ) f (o)+4K(|,|)f  ( | ) + K ( | ) , l ) f (1) } + g ( l / 2 )  l / 6 { K ( l , o ) f ( o ) + 4 K ( l , | ) f ( l / 2 ) + K ( l , l ) f (1) }+g(D  w h i c h may  be w r i t t e n (I-KD)f  with  We  =  f(l/2)  = f(l)  as g  D = d i a g ( l / 6 , 4/6,  can, therefore,  =  solve  1/6).  f o r t h e unknown  values  f , , f_ and f_  52  3.3  GENERALIZED  Assume Consider [a,bj.  to develop  Simpson's  i s Lebesque  here  n  the nodal  (3.3.1)  fa,bj.  i n t e g r a t i n g f ( t ) <\> ( t )  over  polynomial  inter-  (b-a)/n  and d e f i n e  t  t^ =  linear  , t-^,  Q  Rule  section, l e t  the piecewise points  Trapezoidal  a+ih,  i =  0,  1,  interpolation function t  ±  ±  off ( t )  ; i.e.,  f ( t ) = i { ( t - t ) f ( t _ ) + . ( t - t _ ) f ( t ) }, n  n.  1  i  1  t _ <t$t  ±  i  1  i=l,...,n.  Substituting  (3.3.1)  into  / f b  n  ( t ) <j> ( t ) d t ,  we  obtain  n (3.3.2)  / f ( t ) <j)(t)dt = y  ia f (t _ ) + 6 f (t ) }  b  n  ±  i  1  i  ±  i=l where  a. and x  (3.3.3)  rule  rule.  =  f ( t ) be  on  the generalizations of the trapezoidal  i n the preceding  h  integrable  i s to use piecewise  The G e n e r a l i z e d  As  at  <j> ( t )  of numerically  approach  3.3.1  Let  and  the problem The  polation and  fec[a,b]  QUADRATURE  3. a r e g i v e n 1  -  by:  3  i ^ i _  2  i  (  v  t  )  t  (  t  ,  a  t  i  6  . .  ( t  _  t  •  )  m  )  d  t  ±  53  REMARKS ate  the  Since as  a  For  this  form  i n t e g r a l s of  usually simple  the  of  <j> ( t )  quadrature, and  singularity  function,  these  i t i s necessary  tty ( t )  over  of  integrand  an  arbitrary can  to  evalu-  intervals. be  isolated  i n t e g r a t i o n s may,  in general,  o> ( t ) =1,  the  not  be  difficult.  We  also  trapezoidal  note rule  a. = 1  =  we  quadratic  when  in this  ordinary  case  Generalized  Simpson's  n>l,  interpolation function  quadrature  obtain  2  l e t h=(b-a)/2n,  quadratic  we  £  l  The  Here  The  since  B.  3.3.2  being  that  on  each  formula  and to  f  subinterval  Rule  define  f  on  t^,  t^,  ^2i-2'  t  as  the  piecewise t  2  ;  n  2 i ^ ' """ ^' =  ^n n  beomces: n  (3.3.4)  / f (t).<f) ( t ) d t =  ^ t a  b  n  i  f  (t _ )+B f i  2  i  (t  2 i  _ )+Y f 1  i=l where a  (3.3.5)  i  i  &  =  Z =  zVt , 2  2h  ^l f  h  Y.  =  —  2h  1  2i-2  ( t - ^ . X t - t ^ J M t J d t ,  2i  (t-t 2i-2  2  i  _ ) 2  /^2i (t-t . 2i-2  (t-t  2  i  )*(t)dt  ) (t-t .  ) <j> ( t ) d t  i  (t  2  i  ) }  *  We we  are  shall  see  dealing  the  with  error  The  above  concealing need  special  t o be  analysis  form  1  sider  methods  modified  Green  independent part  the  of  an  integral  (3.4.1)  f(t) =  assume  an  these  generalizations  i n section  In  have  a  sense,  i n order to  take  particular  the  (3.5).  are  advantage  e q u a t i o n may  $  2  on  disadvantage  they  ( 1 2 ) , l e t <j>^, t)> , functions  when  METHOD  often  errors.  features'which the  linearly <JK s  direct  anomalous  Following  and  of  COLLOCATION  3.4  and  usefulness  [a,b].  orthonormal basis  We  for L  2  n  form  too of  of  general, any  posses.  a  set of  assume  that  the  [a.,b] .  Now  con-  equation  g(t)+ A / K ( t , s ) f (s)ds, b  approximate  solution  of  the  form  n (3.4.2)  f  (n) i=l  where  the  C.'s x  Substituting  where  e  are  undetermined  (3.4.2)  into  (3.4.1)  n  n  i=l  i=l  denotes  the  error  constant  coefficients.  yields  involved  as  a  result  of  assuming  the  55  solution as  (  f  n  ) -  W  to minimize  In  this  e  a  i  the  vein  error  we  determine  the  vanish  each  of  b . =  g(t.)  at  to  m  choose e  coefficients  i n such  a  way  n  choose  coefficients these  the  a by  set of the  points  t^, t  requirement  2  ,  that  t e^  f i  (t)  ,  and  should  points.  Let  3  3  and a.. xj  =  substituting of  n  linear  4>. ( t . ) - A / K ( t . ,s) <!>. ( s ) d s , ± j a j x b  these  into  the  expression  ( 3 . 4 . 3 ) , we  obtain  a  system  equations: n  2 i j i=l  (3.4.4)  from  a  which  Now  to  numerical of  the  solve  (i)  The  (ii)  At  j  j =  are  t o be  ( 3 . 4 . 4 ) , we and  1,  2,  n  determined,  choose  the  the  numbers  points a..  are  t j from  the  o b t a i n e d by  use  formula. solvability:  determinant least  b  a t hand,  a quadrature for  =  C^'s  data  Conditions  i  c  one  of  of the  the  system  b.'s  must  (3.4.4) be  must  be  non-zero.  non-zero.  3  If f  these i  n  conditions the  form  of  are  satisfied,  the  solution  a polynomial approximation.'  of We  (3.4.4) now  gives  give  an  example. \  56  EXAMPLE  3.3.1  (3.4.5)  Consider  f(t)=  the  equation  t+/j!jK(t,s)f (s)ds  with s K  S u p p o s e we  (  '  t  ±  s  /  us assume f  with t-^=0,  )  =  have . • t  Let  s  (  3  s< t  t  {  s>t  the points ' _  n  i  1 ,  u,  4  = -  )  ( t )  t =l.  :  2  2  simplify  =  c  The  We  3  the  1  +  4  3 , , i .  of the c  2  t  +  c  3  t  form  2  the error  should  use the a n a l y t i c  a, . ID  a.. are given  l-/oJ 0  . - t.-/^j 2] D 0  n  by  sds-/^ t.ds t_. j  j  j  s ds-/?~ t .s d s t. ij 2  D  a j 3  2 t = t J~-/"QD  and b . = t. D  D  3 1 S ds-/  t  vanish  form  computation.  coefficients  a  1 ,  a solution  the c o n d i t i o n that t  2  ~> tjS'ds  at the points  of the kernel to  57  Thus  l  ll  l  12  W^sds-O/Jds = 1  , W  = Hence  C.j + 52  f  (  C  2  1,1 , i / 2 ;  d  =  s  2 +  assumes =  0  3  =  96  9 C  3  =  12  approximation:  (t)  1.988t-0.434t'  3  )  =  f^=0, f = 0 . 8 3 6 , 2  the points  best  f o r t h e chosen  3  f^=1.554,  = 0.470;  f  t  2  5/8  t h e form  to the  where  mate  _  C  i s automatically  at  S  (3.4.4)  which  f  , d  etc.  2  ±  leads  s  C +17  C+  6 which  u^,  t h e system  120  .1/2 0  and gives  p o i n t s t=0,  1,  the solutions  = 1.247  4  = ^- a n d t ^ = 3 / 4 ,  solution to equation  respectively.  (3.4.5)  Thus  the approxi-  i s :  0.000  (3.4.6)  0.470 0 . 836 1.247 1.554  REMARKS that not  The c o l l o c a t i o n  an exact control  matching  the size  suffers  from  solutions  all  points  the disadvantage  of the solution at certain points  o f t h e d e v i a t i o n between  approximating the given  method  a t other  which  may  points  the exact  (unless  and t h e  we w a n t  n o t be c o m p u t a t i o n a l l y  does  t o choose  feasible).  58  At because used)  any at  rate, least  methods  the  collocation  i t i s an  of  method  improvement  on  i s worth the  successive approximations  considering  direct  and  (and  the  widely  quadrature  methods.  3.5  ERROR  ANALYSIS  3.5.1  We the  now  result  consider of  discussed.  the  In  the  the  problem  calculation sequel  we  of  e s t i m a t i n g the  f o r any shall  of  denote  the  accuracy  quadrature  the  norm  of  methods  any  function  4- ( t ) b y | U|  | =  / |<j, ( t ) | d t b  d  Returning function  (3.5.1)  to  the  problem  f ( t ) i s r e q u i r e d to  J K ( t , s ) f (s)ds+g(t) b  of  e s t i m a t i o n we  satisfy  =  the  integral  f (t)  a. The  computed  (3.5.2)  where  function f(t) actually  / K ( t , s ) f (s)ds+g(t) b  cl  E (t) d e n o t e s  the  error  =  satisfies  f(t)-E(t)  term.  know  that  of  the  equation  5?  If  we  write e(t)  for  the  error  (3.5.3)  =  f(t)-f(t)  i n our  r e s u l t , we  / K(t,s)e(s)ds  =  b  find  by  subtraction  that  e(t)+E(t) .  ci  In  the  usual  way,  K(f) the  error  writing  =  e(t)  (3.5.5)  || e|| <||  in the  terms  of  =  Under  can  examining  It  (3.5.6)  _ 1  E (t) ,  estimate  remains  operator:  and  so  | | ||E||.  estimated.  or  perhaps  assumptions  finite  [6] , [9] , s t a t e s  be  case,  suitable  t e r m s we  (1-K)  - 1  derivative, either  nonsingular  integral  by  -(1-K)  can  a  the  b  i s given  ||E||  for  / K(t,s)f(s)ds,  (3.5.4)  Now  K  the  E(t)  will  usually  of  the  of  f ( t ) only  about the  complete  the  expressed  integrand  in a  smoothness of  magnitudes of  be  singular the  derivatives  in case.  various by  differences.  to  that  estimate  || ( 1 - K )  || .  A  well  known  result  60  where  K i s any bounded  vided  that  interval  | |K | | <1.  [a,b],  operator  I f we c h o o s e  then  t h e norm  | | K | |=  (3.5.7)  linear  i n a Banach  t h e norm  space,  pro-  | | f | | =max | f | , o v e r t h e  of Ki s  / |K(t,s)|ds, b  and (3.5.6)  holds  provided  II KI We  therefore  have  a rigorous  3.5.2  bound  Generalized  Turning quadrature  to the corresponding  rules,  and u s i n g  for the error  i n the result.  Error  error  the notation  term o f t h e g e n e r a l i z e d of section  (3.3),  we  have  (3.5.8)  E  (f)= / { f ( t ) - f  (3.5.9)  | E ( f ) | < | U| | || f - f j | •  b  n  a.  has  that  f o r Lagrange  Atkinson  f " ^ i s continuous,  and  that  2  |f  1 ! L  | |  |U||.  the error  subinterval  the generalized  bound  | E ( f ) | <^h | n  and u s i n g  i n t e r p o l a t i o n on each  (3) h a s s h o w n  the error  (3.5.10)  (t)}4> ( t ) d t ,  n  Assuming mula  n  for-  [t_._^,tj] ,  trapezoidal  rule  61  REMARK that  We  note  of the ordinary  below,  this  quadrature  will  order of  convergence  trapezoidal rule,  n o t be t r u e f o r t h e  i s t h e same  b u t , a s we  shall  generalization  as  find  of a l l  Simpson's  rule,  the corresponding  bound f o r  error i s :  | E (f) I < ^ | h | | f n 27  (3.5.11)  As  the  rules.  Considering the  that  3  hinted  earlier,  we  1  find  1  1  ! I ' I I* II .  that,  unlike  the case  of the  trape3  zoidal of  rule,  the generalized  convergence  order  whereas  at least, the generalized on t h e o r d i n a r y  3.6  SUMMARY  We  have  rules  methods Fredholm  on  Simpson's  only  has an h  order  i s of a  higher  rule  4  improvement  the  the regular  rule  h .  Thus,  son's  Simpson's  considered plus  their  rule  provides  an  method.  AND  CONCLUSION  i n some  detail  the trapezium  and  generalizations  as convenient  quadrature  f o r the numerical i n t e g r a l equation  method  Simpson's  approximation o f the second  o f c o l l o c a t i o n may  the r e s u l t s o f the former  yield  to the solution of the kind.  We  a significant  methods.  Simp-  found  that  improvement  62  We the  do n o t c l a i m  t o have  a v a i l a b l e methods.  methods  go by v a r i o u s  Eighths"  rule,  mentioned  Indeed, names  Newton-Cotes  a l l , o r even  f o r quadrature  as:  Simple  rule,  Radau  Gauss  most, o f  rules, rule,  some  other  "Three-  quadrature,  Lobatto  rule, etc.  Although methods ties.  here,  found  conclusion,  are expansion  we  cases  wish  equations  do n o t f e e l ,  however,  way  too inferior  t o one g i v i n g may  bearing  considered  as a model  f o r some  place.  such  possibilias t h e  a l l of which  out that  methods  that  certainly  necessary,  an approximate  methods  s o l u t i o n s be  and n u m e r i c a l  absolutely  only  further  i s i n general  We  form  some  techniques,  to point  can such  approximate  i n closed  to discuss a l l  may  be  literature.  of integral  exceptional  tion  there  exploring  and t h e R a y l e i g h - R i t z  solutions  various  an o p p o r t u n i t y  y e t i t i s worth  i n the cited  In  any  do n o t have  F o r example,  Galerkin  in  we  found. have  an e x a c t  real  system,  used.  method  solution.  A  i si n solu-  but i s rarely  the integral i s almost  o f the system  Only  Generally,  t o be  be c o n v e n i e n t , that  explicit  difficult.  an approximate  i n mind  representation  finding  equation,  certainly  i n the  first  63  CHAPTER COMPARISON:  AN  EXACT IN  VERSUS A  4 AN  SPECIAL  APPROXIMATE  SOLUTION  CASE  INTRODUCTION  Whittle t i o n ' (1.2.6) a  method  (22) d o e s  obtain  for certain  special  techniques.  Whittle's  method  to obtain  equation  (1.2.6)  integral estimator  example of  this  i s treated.  linear  estimator  solution  and c o n s e q u e n t l y  equa-  suggests  using  use a s l i g h t l y modified  an e x a c t  practical  problem  Estimate  A  .  consists  by w i l d c a t  t o 1971.  estimator  A BRIEF  The d a t a  discovered  1953  approximate  4.1  and  W h i t t l e ' s work  linear form  the  linear  We  compare  of successive  a  numerical  30-day  totals  exploration i n Alberta for  Clevenson  and compute  devised,  and Zidek  (7) c o n s i d e r  the approximate  our exact  and  optimal  Clevenson-Zidek's  results.  O U T L I N E OF W H I T T L E ' S  A ( t ) by  a linear  DERIVATION  smoothing  OF  formulae  (1.2.6)  so t h a t  A (t)  L will  of  t o the Fredholm  obtain  i l l u s t r a t e the use o f the techniques  period  linear  shall  estimators  (1.2.1).  o i l wells  the  We  linear  cases.  of obtaining the optimal  smoothing  To  Bayes  be e s t i m a t e d  by  s t a t i s t i c s of the  type  64  (4.1.1)  for  X (t) =  / w (s)dN(s), b  L  which,  (4.1.2)  t  conditionally,  EX  (t) =  / w. a  X  (s)X(s)ds,  b  j_i  given  L.  and  (4.1.3)  varX  T  (t) =  / w (s)X(s)ds. a t b  J_i  Assume a  population of  priori will  then  where  such  be  of values.  obtained  C*[t;w]  =  If  by  The  the  optimum  X ( t ) ' s have linear  an  of a  estimator  minimizing  E[/w Xds+(/wXds) -2X(t)/wXds+X (t)], 2  2  2  respect  to prior  distribution  X(t) s. 1  EX(t)=p(t)  respect  (4.1.5)  f u n c t i o n , X ( t ) , i s a member  f u n c t i o n s , so t h a t  E denotes expectation with  the  with  the p a r t i c u l a r  distribution  (4.1.4)  of  that  2  and  E [x(t)X(s)]=y(t,s)  t o w^_(s) y i e l d s  y(s)w  f c  (s)+/p(s,u)w  the  integral  (u)du =  then  minimizing  equation  u(t,s).  (4.1.  65  By  a normalization  C (s) t  o f the form  w ( s ) /{y ( § ) / y ( t )  s  t  r(t,s)  =  (4.1.6)  y(t,s)  = r(t,s)/y(t)y(s),  (4.1.7)  w (s)  y ( t , s ) / / y (t) y (s)  we g e t  t  Substituting  =  these  C ( s ) /y ( t ) / y ( s ) ' t  into  (4.1.5)  we  obtain,  after  some  tion ,  (4.1.8)  which  C (s)+/ r(s;u)? (u)du b  t  t  i s obviously  THE  (4.2.1)  where  seek  the optimal  X (t) =  (4.2.2)  EXACT  linear  form.  SOLUTION  estimator  o f the form  y+/^ h(t;s)d(N(s)-ys)  L  h(t;s)  r(t;s),  of the required  4.2  We  =  T  i s the solution of the integral  yh (s)+/^ K(s;u)h (u)du t  T  t  = K(t;s),  equation  -T<s<T.  manipula-  66  Consider upon  the  ( t - s ) so  (4.2.3)  where, and  special that  case  where  (4.2.2)  may  yh(s)+/ K(s-u)h(u)du a b  f o r convenience,  we  the kernel be  written  depends  only  as  = K(t-s),  have  dropped  the  subscript,  t, of  h  [a,b]=[-T,T].  We  further  specialize  Cov(A(t),A(s))  our  =  results  by  assuming  EA(t)A(s)-EA(t)EA(s)  = y(t,s)-y 2 /. s a p(t-s)  =  so  K(t;s)  2  =  2 -a a e  1  t-s  \  that  /+.  \ = a 26 ^ , - a'l t - n+y s L 2-.  y(t,s) Since (t,s)  =  K(t,s)  =  K  we  U  ' , /y(t)y(s) (  t  S  )  obtain  (4.2.4)  Whittle  K(t-s)  treats  the  special  form  K ( t ) = a( +ee~ ' ' ) a  Y  H+y  = ^—e  t  case  where  the kernel  i s of  the  67  which  i n o u r example ay  Following be  =  (4.2.5)  we  find  by r e p e a t e d  h"  that  and ag =  y  Whittle  converted  means  (s)-0 h(s) 2  a  /  2  that  y  the integral  = -2^—  6(t-s)+ya f/ h(u)du-ll, 2  b  L  2 0  =  2  (4.2.3) c a n  differentiation to  y  with  equation  a.  2  2 ^ - a  +  a  *  y  The  6-function  derivative hold  the  of e ~  without  tives  We (4.2.6)  ( 4 . 2 . 7 )  t  -  s  l  at s-t.  a n d —6  magnitude.  Equation  (4.2.6)  l  a  here because  the 6-function 2 -  o f h(s)  same  arises  a  Whittle  f o rs ^ t , however ,  ~  l must  s  has a general  = ^-  A  into  e-®  +  a  2  /  Q  /3-a  e  step  ( s  ( £  s-t  + Q  l - a | s-u| e  +  - G ( b - u ) I  a  I  .  +  R  )  ]  d  u  e"  solution  o f t h e form  P, Q a n d R b y  u } u  ( b  e  "  s )  substituting  +R  - 0 | s-u|  +  p  e  must  discontinuities of  ^-C^^+^^KQ&^^-^+R.  +pe"® ~ >  br  (4.2.5)  f o rs=t t h e d e r i v a -  (.4.2.3) t o g e t  1 s _ t 1  +  have  that  of the  ^  determine the quantities back  asserts  y  (4.2.5)  h(s)  '  of the discontinuity  - 0 (u-a)  68  2 where  A  Recalling  that  involved,  (4.2.7)  (4.2.8)  h (s)  If  yields  =  t  (4.2.8)  (4.2.9)  L  x[e  the integration  the solution,  -0|s-t| -0(t-a)-8(s-a) -®(b-t)-0(b-s)-| +e  +e  i s substituted  ( t ) =  A  , [ a , b ] = [-T,T] , a n d p e r f o r m i n g  into  ( 4 . 2 . 1 ) we  >  obtain  y+i +i +i , 1  2  3  where  x  2  I  3  .  f  ,  r  with  The  =  /  the intensity  to  obtain  N  (  s  b -©(-t-s+2b) a e  _  )  d  (  N  y  (  )  S  s  )  _  u  S  )  (4.2.9)  (  N  (  s  )  _  y  S  )  i s the exact  optimal  linear  estimator  A ( t ) , and i t i s t h e form which i s used  solution  (b) d i s p l a y s  to the oilwell  the exact  f o r comparison.  i s given  d  2 „2 2aa . 2 Q = +a y  function,  the exact  graphically  (  p  expression  Figure  d  b -®(t+s-2a)  2 aa -; yG  of  (4.2.9)  p  a  x  A  b -©|s-t|  discovery  and t h e approximate  A Computer Program  i n Appendix ( 1 ) .  problem.  results  t o compute  69  4.3  AN  This by  presentation  h (t;s)  follows  Clevenson  yx(s)+/'^ x ( u ) K ( s - u ) d u  (4.3.1)  l e th  S O L U T I O N OF  1.2.6  and Zidek  (7).  Denote  the solution of  T  and  APPROXIMATE  = K(t-s),  T  O T  (t;s)  be o u r a p p r o x i m a t i o n  -T<s<T,  (t;s);  to h  that i s ,  ( t ; s ) i s the solution of  h  oo CO  (4.3.2)  yx(s)+/  co  with  / • x —  Using show  2  CO  Fourier  -co< <oo,  = K(t-s),  s  transform  techniques,  Clevenson  and Zidek  (7)  that  =  T  Trvco f o r e a c h  Their  x'(u)K(s-u)du  (s) ds<co.  e (t;s)  as  CO  —  argument  h (t;s)-h T  fixed also  (t,s)  gives  o o  (t;s)^0  under  a bound  suitable for e  T  regularity  i n t h e form, 1  (4.3.3)  |e ( t ; s ) | < A y " { / | | 1  T  u  > T  with A  =  Is Ik  {/°° —  CO  1  I  2  (s)ds}  2  h  2  (t;u)du} {T 2  D  -1  2  +By  _ 1  },  conditions.  70  and B  =  {/°° K  Corresponding  (4.3.4)  2  (s)ds}  2  to equations  X (t) =  (4.3.1)  and  (4.3.2), l e t  u+/^ h (t;s)d(N(s)- s)  T  T  T  M  and  X  (4.3.5)  (t) =  respectively  be  tion  Then  a  to  bound  A  .  y  +/  —  00  the optimal by  (t)- x  estimator  E|A|<4TAy~ {/ 1  1  when  t  and  i s not  Zidek  [-T,T].  X  the boundaries  near  lent  to our  Thus  remark  Further, approximation  Clevenson  that  this  2  2  the boundaries  X^ w i l l  after  approxima(4.3.3),  form:  }.  _ 1  oo  (7) o b s e r v e  of  _1 +By  2  > i  i n the  {T  Ji (t;u)du}  too near  period, T  ,  U  1  Clevenson only  | i  the  1  1 (4.3.6)  and  i n inequality  (t) I i s o b t a i n e d  oo  1  1  linear  a p p l y i n g t h e bound  f o r E I A I=EI X 1  h (t;s)d(N(s)-ys]  OO  n o t be  that  a  period.  bound  i s small  of the observation  good  approximation  Note  that  this  i s equiva-  (4.2.8).  and  of the optimal  Zidek  give  linear  the  large  estimator  X  T  Li  time, (t) as  to  T,  71  (4.3.7)  X j t )  U+e/^e- ^ 7  -  S  'd(N(s)-ys) ,  where  ^ 2  Q  and  To  in  - l  . , 2 -1 -1,"2 (l+2a y j )  n  3 =  a  Y =  ct(l+2a y  determine  given  y  "^a  2  the  accuracy  inequality  (4.3.8)  of  (4.3.6)  4.4  = '4AT{T  ~2  +By  -1  COMPUTATION:  From  the  data  the  Cov(X(t),A(s))  where  p(u) using  = e  the  mean,  process;  on  Also  [0,°°).  y=0.70,  the  as  Time  an  = a  l  a u  l ,  data.  regressive  108  evaluated,  (4.3.7)  and  found  the  to  bound  be  2 -1 -2vT }3 Y 6 •  OIL  2  data,  was  approximation  ^. P  citly  the  Pcosh(syt),  where  used  -T<t<T,  we  WELLS  y,  i  DISCOVERY  was  DATA  estimated  as  y-0.70.  i  p(|t-s|), with The  a=0.05,  form  of  believe that  p  was  chosen  i s that p  of  without a  i s symmetric  and  to  measured  observations  i n the  in  30-day  period  intervals  [-T,TJ.  For  and  there  this  expli-  one-step  express our u n c e r t a i n t y about the *2 value a =0.25 was chosen, with support 2 estimate of o .  was  We  auto-  decreasing  choice from  the  were  particular  72  case the  T=110.5, and following A  values  -  3 =  The  bound  0.01  son  the  the  linear  to  the  constants  0.0913,  y  =  /5/2, 0.195.  in inequality  ( 4 . 3 . 6 ) was  earlier,  we  estimator  and  the  The  graphs  i n Figure  (b)  support  i s , the  X  found  display i n Figure  linear  that  (4.3),  obtained:  exact  that  in section  t o be  less  than  jt|<87.  boundaries  cates  were  B =  mentioned  REMARK tions;  reference  5//2,  given  provided  As  with  m  of  the  will  estimator,  bound  not X  T  i s small only  observation be  a good  , near the  (B)  l a r g e time  the  when t  period,  boundaries  i s not  of  to that  compari-  approximation.  above  [-T,T].  approximation  for  calculatoo  This the  near indi-  optimal  period.  73  CHAPTER 5 APPLICATIONS  5.1  INTRODUCTION  The  main purpose o f t h i s  chapter i s t o apply the techniques  developed  thus  f a rto practical  different  from t h e s p e c i a l case c o n s i d e r e d i n c h a p t e r  t h a t we h a v e b e e n c o n c e r n e d A , of the intensity process.  s i t u a t i o n s which  may be  slightly  4.  Recall  with the optimal l i n e a r estimator,  function,  A  (t),  of a nonstationary poisson  I t h a s b e e n shown t h a t X ( t ) i s a f u n c t i o n o f h ( t ; s ) L  which  i s the solution of the integral  (5.1.1)  equation  m(s)h. ( s ) + / K ( s ; u ) h . (u)du = K ( t ; s ) L. a t  a<s<b,  b  where  m(s) = y i s a c o n s t a n t  I n many c a s e s  (in chapter 4).  the assumption  t h a t m(t) i s a c o n s t a n t  over  the e n t i r e p e r i o d o f o b s e r v a t i o n [a,b] i s u n r e a l i s t i c .  It is  t h e r e f o r e o f i n t e r e s t t o study other s p e c i a l  where  we d r o p t h a t a s s u m p t i o n .  To f a c i l i t a t e  g e n e r a l model t o such p r a c t i c a l equation scribed is  situations  the a p p l i c a t i o n of our  situations,  consider the i n t e g r a l  (5.1.1) w h e r e m ( t ) i s n o t a c o n s t a n t b u t i s a p r e function of t .  An i n t e g r a l e q u a t i o n w i t h t h i s  property  sometimes c a l l e d a Fredholm i n t e g r a l e q u a t i o n o f t h e t h i r d  type.  H o w e v e r , b y s u i t a b l y r e d e f i n i n g t h e unknown f u n c t i o n , h (s) ,  74  and/or the k e r n e l , K ( t ; s ) , i t i s always p o s s i b l e an e q u a t i o n i n t h e f o r m o f t h e s e c o n d  I n p a r t i c u l a r , when m ( t ) [a,b], Hilderbrand i n the  (5.1.2)  /mT^)h (s)+/  such  type.  i s p o s i t i v e throughout the i n t e r v a l  ( 1 3 ) , shows t h a t  rewritten  to rewrite  t h e e q u a t i o n (5.1.1)  can  be  form  b  K  (  S  ;  U  )  t  ^ M  h  t  i  u  )  d  U  / m (s) m (u)  =  K j t ^ /m (s)  f  or  (5.1.3)  x  (s)+/ r(s;u)x  (u)du - g ( s ) .  b  "c  Thus i n t h i s  •  a.  f o r m one must x  h  "c  (s) =  u  recompute  (s) r  /m(s)  a f t e r x ( s ) has been f o u n d . t  H a v i n g shown t h a t by a p p r o p r i a t e l y i n v o l v e d , we  can r e w r i t e  t y p e e q u a t i o n , we  give  (5.1.1)  r e d e f i n i n g the  functions  i n a s u i t a b l e f o r m as a  two p r a c t i c a l e x a m p l e s  second  of such s i t u a t i o n s .  75  5.2  E S T I M A T I O N OF T R A F F I C D E N S I T I E S AT THE L I O N S G A T E B R I D G E  Volumes of  cars  on  the  five-minute crossing of  the  this of  of  a  data  have  been  Lions  Gate  Bridge  counts  intensity  function  and  other  effects  are  useful  another  lane  look  that,  from  day  traffic  at  to  of  appears  traffic  Lions  be  over  i n an  counts  one  to  from  treat  the  realization observed. niques lying  at  Gate  an  an a  of  effects and  which  to  the  i n the process.  us  to  previous  weather,  time  about  these  should  detector are  convinces  highly  reproducible  one  cars which  pass  the  time  every  the  same  i s large time,  should  apply  least,  ones?  be  underlying schedule  enables  the  days  arrival  next  say  working  nearly  gives  estimator  example,  underlying schedule. time  An  of  (£)  (southbound)  knowledge  For  a nonstationary Poisson  developed  To  counts  on  Figure  i n 1974.  individual  Bridge,  individual  day  This  intensity  the  distribution  traffic  sought.  existing  composed  apparent of  the  detector at  uncertainty  day  variables  "In p a r t i c u l a r ,  to  counts  to  counts  mainly  the  X ( t ) was  total  i n decision-making.  the  day.  f o r the  reflects  f o r the  i n Vancouver.  "typical"  exogenous  added  the  t o be  location there  be  on  a  function  day,  one  traffic  d e t e c t o r on  intensity  A  of  collected  our  I f one  compared the quite  S p e c i f i c a l l y , we  particular day;  were with  to  thus  to  make  the  small.  In  mathematically,  model  expect  variance in  process  chapters  would  i s assumed and  study  seek  a  the the  order a to  techunder-  linear  be  76  estimator,  (5.2.1)  X , of the intensity  X ( t ) = y +/^ L  t  where h ( t ; s ) subscripted assumption chapter  y^ t o i n d i c a t e t h a t t h a t m(t)=y  X ( t ) , i n the form  h(t;s)d(N(s)-H(s) ) ;  i s a s o l u t i o n o f (5.1.1).  4 about  (5.2.2)  T  function,  i n this  i s constant.  t h e k e r n e l , we  2  a  that  h e r e we  write  e x a m p l e we d r o p t h e  With  require  C o v ( X ( t ) ,X ( ) ) = a (t ^  Note  the assumptions i n  that  S  S  and  .2 (5.2.3)  K(t;s)  = K(t-s)  -  °y(\  t - s | ) +y y, f c  U  We  a r e now  faced  with  t  the problem of f i n d i n g  estimates  2 for  y^'s.  This  spection  and a , and a l s o  i s where,  comes i n .  essentially any  o  the constants  following the Bayesian  The m o t i v a t i o n  the desire  t o base  available information,  information  a method o f o b t a i n x n g  o f some o t h e r  recipe,  the  intro-  f o r B a y e s i a n methods i s  c a l c u l a t i o n s and d e c i s i o n s  whether nature,  i t i s sample such  as t h a t  on  information based  on  or  past  experience. We Barnett values  shall  use e m p i r i c a l Bayes  (4, pp. 189-200), published  collected  from  1  methods as suggested  t h e 1974 d a t a  b y L e a , N.D.  a t hand, and a l s o  and A s s o c i a t e s  t h e same l o c a t i o n w a s  by  used.  (16b) where  some data  77  CALCULATIONS:  Consider t h e data i n T a b l e  5.1.  TABLE HOURLY V A R I A T I O N :  Time  u  t  appealing and  Day  10  11  1,240  11  - 12  1,140  12  - 13  950  13  - 14  1,400  14  - 15  1,150  15  - 16  16  - •17  1,300  17  - 18  1,300  • 1,175  Traffic  t o use t h i s data,  with  Unit,  traffic  have  of Vancouver  to obtain  i t i s convenient y^. v a r i e s  would  be  increased  counts.  to divide  expect 1966.  One  we  intuitively figures  But t h i s  that  mean  hourly,  12th t since  t h e 1966  as t h e ^ ' s .  since  the p r i o r  t o l e t y^ change  f o r every  vehicle  s a t i s f a c t o r y ; one would might  City  information  five-minute  possibility  1966  Volume  use the r e s u l t i n g values  entirely  BRIDGE,  1,550  i n our c a l c u l a t i o n s dealing  GATE  - 10  f o r t h e 1974  thus are  order  LIONS  9  Source:  In  of  5.1  cannot  t h e volume  Calculations  by  12  be  of show t h e  78  h o u r l y v o l u m e f o r 1974 o f 1966  was  t o s t e p up  was,  1,050; t h u s t h e 1966  on t h e a v e r a g e ,  i n t h e r a t i o o f 11:7.  was  f l u c t u a t i o n s and  o t h e r c o n s i d e r a t i o n s , the  R e c a l l t h a t o u r c h o i c e o f p(u)=<S autoregressive process.  in arriving  at a value  was  used  H o w e v e r , t o make room f o r  used i n the computer program p r e s e n t e d  one-step  that  Hence, i n o r d e r  v a l u e s , t h e c o r r e c t i o n f a c t o r 11/7  i n t h e c a l c u l a t i o n o f t h e y^_. sampling  1,650, w h i l e  f o r a, and  01  factor  i n Appendix  3/2  (B).  I I i s i n the form of u  T h i s k n o w l e d g e may a l s o i n choosing  a  be e x p l o i t e d  an e s t i m a t e  for  2 a .  Computer programs were r u n f o r the f o l l o w i n g v a l u e s o f a and  a:  The shown by REMARK  0 . 01  1.0  0.01  2.0  0.05  2.0  0.2  2.0  r e s u l t i n g estimates of the i n t e n s i t y the graph of F i g u r e s The  (D,  f u n c t i o n , X ( t ) , are  E).  g r a p h s i n d i c a t e t h a t , as i n t h e o i l w e l l  f i x e d a, as a i n c r e a s e s , t h e e s t i m a t o r X d a t a - s e n s i t i v e , and  hence  irregular.  becomes  example f o r  increasingly  79  We is  a  now  consider  prescribed  nonconstant  5.3,  The  disaster or  Cox  are  i s defined  and  preceding  the  this  Lewis  this  that sort  this of  s e c t i o n may  s e c t i o n where  and  as  Bayes  an  illustrate  discussed  empirical  period pp.  formal as  periods, 1875  to  2-6).  A  death  of  statistical alternative  model.  will  regarded  the ( 8,  here  our  400-day  i n v o l v i n g the  more  data;  apply  for  Lewis  accident  example  be  EA(t)=m(t)  t.  Britain  of  problem  mean  successive  discuss  set  a n a l y s i s , we  in  Cox  a mining  and  analyzing  i s hoped  ticular,  as  from  of  the  DISASTERS  numbers,  taken  Cox  Lewis  detail  the  where  function  d i s a s t e r s i n Great  data  for  It more  gives  m o r e men.  methods to  (F)  coal-mining  1951.  10  example  COAL-MINING  Figure of  another  in an  this  in  a  work.  extension  ideas  were  little In  to  cited  par-  the as  2 justification tions now  of  section  require Fig.  (5.3.1)  f o r our  m(t)  (F)  (5.2)  s t i l l  to  a  be  suggests  Cosht  =  choice  ^(e^e  of  a  and  hold  with  continuous  )  , CO  Suppose  the  exception  function  consideration  t  a.  of  <t<°°.  the  of  the  t.  function  assump-  that  we'  80  Since, of  as F i g .  disasters  the  function  m(t)  t o have  (5.3.2)  where  (F) i n d i c a t e s ,  the average  i s decreasing with (5.3.1)  time,  i s unsatisfactory.  side of  L e t us choose,  instead  t h e form  m(t) = | ( e  0<a,b$l  a  +e"  t  b  t  ),  -l$t<l  and b>a.  restriction  The  r e s t r i c t i o n s on a and b a r e imposed that  o f occurrence  the p o s i t i v e  The  Note  rate  with  on t i s t o prevent  these  m(t) from  for similar  r e s t r i c t i o n s m(t) takes  k  becoming  1  1\  l ~l ~ i -1  1  ~l ~ 1  0  1  reasons.  the following  1  l\  too large  t  ^  form  81  The  s a m p l e mean o f  knowledge,  and  3=0.5 and  also to  b-1.0.  the  given  simplify  Using  the  data  the  2;  calculations,  techniques  "2 o  i s about  of  with we  section  this  choose  (5.2),  2 =0.25 i s c h o s e n a s  a=0.05  i s found  an  t o be  estimate  a good  COMPUTATION:  As  has  and  of  calculation  the method  ceding with  section.  the  compute the Note  a l r e a d y been  on  t  in  function m(t),  that i n this  .  Also,  are  said,  the  form  essentially to note  (5.3.2),  each  value  time  i n the value  of  those  t  the of  that i n  te[-30,30j , w h i l e  example  the  choice.  It i s interesting  restriction  a  of  kernel  the  pre-  conformity  subprogram  to  i s d i v i d e d by f o r m(t)  we  30.  require  te[-i,l] .  Another curiously,  interesting  the  values  feature of  a=1.75 and  the  b=1.0  present  example  were used w i t h  i s that,  the  same  ~ 2 values  of  produced which of  are  a  a  and  values  of  as  above.  the  estimator of  p r e t t y much t h e  computation  as  given  same.  by  the  b=1.0  a = 0 . 7 5 ; b=1.0  Thus one  i s persuaded  and  i s w h a t was  this  to done  the  out  that both  intensity  Nevertheless, computer  Time  Values a=0.5;  I t turned  are  (sec.)  as  the  Cost 1. 34  2.791  1. 39  here.  to the  values  function, A (t), cost  and  time  follows:  2.784  stick  programs  ($)  a=0.5 and  b=1.0,  82  Figures and  so  a  again  (G)  that  i t is  estimator  and  one  can  seen  to  (H) see  that  become  display the  for  more  A_  effects a,  fixed  Fredholm for of  the a  integral optimal  linear  intended  are  as  available  occur  agree  Zidek  (7)  of  solving  with  Grandell  that  only  respect  is  g e n e r a l l y a p p l i c a b l e when  obtaining  the  numerical  solution  In chosen  the to  optimal  numerical  reflect  calculations  of  our  4  the  a  and  linear  of  applied  of  (22),  of  the  emphasis  estimator,  examples,  a  A  A  special  l  the i  n  chapters  some  of  equations  and  ,  are  the which  Clevenson  linear  order  A  ij  estimation,  placed  , which  involves  form  the  of  which  process.  was  covariance the  quest  function,A  estimator,  class  main  i n our  the  mathematics.  second  about  solving  These  A i s any  obtain  the  approximation  type  linear  (1.2.6).  required to  3.  Fredholm  equation  belief  Here  versa.  intensity  incomplete,  restricted  5  of  Several  optimal  Bayes  1,  to  the  (10), W h i t t l e  the  a.  a causes  encountered  i n chapter  areas  in  vice  methods  II  of  necessarily  i n many  in chapters  type  process.  is  Thus,  with  of  presented  survey,  methods  frequently  We  a  increase  exact  estimator  Poisson  also  an  o and  varying  a  of  REMARKS  various  equation  nonstationary  techniques  and  2 presents  of  d a t a - s e n s i t i v e and  CONCLUDING  Chapter  for various values  on the  kernel  structure.  numerical  example  was The where  83  the  assumption  of  constant  more e x t e n s i v e t h a n the  assumption  To here  the  had  chapter the  been  i n the  4,  exact  r e q u i r e d f o r the  knowledge, the  exact  applied to r e a l - l i f e  started,  suggested  i s dropped  m(t)=y,  simple  are  case  much where  upheld.  author's  not  w o r k was  is  those  mean,  even though  optimal  of  linear  situations  the methods had  literature.  a comparison  techniques  approximate  estimator,  A  before  this  f r e q u e n t l y been  It is gratifying  the  considered  to note  that  estimator,  in  and  shows t h a t t h e  bound,  i given case  by of  Clevenson the  and  oilwell  Zidek  ( 7 , p.  discovery  Another  interesting  ,  the  21),  is satisfied  in  the  example.  observation i s that,  in  choosing  '2 y,  a  a and  Bayesian may  use It  other  approach empirical i s worth  constants which  i s a very  our  i t appears  is  of  i n order  is  careful usually  safely  At  least  done  t o assume t h a t t h e t h a t such  here  about  choice  examination helpful.  maintain  the  the  an  of  the  likely data,  I n many assumption  at  assumption  i t i s much  special  to y i e l d  cases,  t h a t m(t)=y,  one  may  more  since, cause  A word  of  t h e mean f u n c t i o n ,  least  the  i t is i f  mean i s c o n s t a n t  choice of  i s very  model,  here.  A t o become more d a t a - s e n s i t i v e .  "off-the-mark" A  i n our  n o t i n g t h a t i n some s i t u a t i o n s  not  estimators  tool.  B a y e s m e t h o d s a s was  satisfactory results,  useful  occur  from  the caution  m(t);  intangible  an  results.  in graphical  form,  h o w e v e r , one  may  a constant.  Also  one  84  may  use  the  optimal  yield  the  fairly  function,  large linear good  A(t).  time  approximate  estimator, results  for  estimator,  A ,  since  the  estimation  T  these  A  , instead  methods of  the  of  often  intensity  CO  A. Histogram estimators of the intensity of wildcat oil well discoveries using class widths of (a)30days ,(b)90days and (c) 360 days .  00  B. A comparison of the exact optimal linear estimator and the approximate solution of the intensity function of the oilwell discovery process.  minute v e h i c l e counts — — ro co ro o~> O  5 o  •£» o  o  o  o  o  ro o  CO CO  70  V 60  -o o  \  °- I 50 c £  a = 0.01 , cr = 2 . 0  \  140  X  CL  X  in  °  130  \  a = 0 . 0 1 , a- = l . o  E  \  120  I 0  00&  \ 1  -54-5© 9AM  10AM 1 I 1  -40  HAM !  -30  1PM  12N00N I -20  1  -10  1.30PM 1  0 Time  of  PP^ I 10 day  3Pk 20  4 PM 1  30  i  5  ,  1  40  D. The optimal linear estimator of the intensity function of the Lions Gate bridge process using various prior parameters.  P  M  i l  50  54 6PM  89  190 180  h  I 70 160  -a  \  " \  \ \  \  150  a = 0.05, cr = 2.0  o  S. 140 a>  | in  v_  \  130  /—•  \  .•1 1  /  /  \ /: \  120  <D CL  \ 0  V  a a o  a;  %  2  1  0  \  0  v  90  a = 0 . 0 2 , cr = 2.0  80 70 10AM HAM 60 54-50-40 -30 9AM  1  1  I2N00N IPM 1.30PM 2PM 3,PM  4PM  6  30  -20  -10  10  20  T i m e of day  5PM 40  50 54 6PM  E. The optimal linear estimator of the intensity function of the Lions Gate bridge process using various prior parameters.  o  •o o 9 V—  <D  CL  >»  O  TD  o o  8 7 6  cu  CL  (/> l. co to  o  to  TO  5 4 3  ^_  o 2 co  E 2  I 0 0  10  20  40  30 T i m e (in 4 0 0 d a y  50  units)  F. Cool- mining disasters in Great Britain for the.period 1875 -1951 Numbers in successive 400-day periods.  60  4.5  4.0  po  3.5  CU D.  >»  3.0  o  X3  O o  2.5  I—  c u CL in  2.0  cu  i sa  CO  o cu E. 3  2  1 .5  1.0  0.5  0 -30 1875  -20  - 10  0 Year  10  20  30 1951  G. The optimal linear estimator of the intensity function of the mining disaster process using various prior parameters.  CN  4.Of T3  O \_ CD CL  a  X3  O O  cu  3.5  a = 0.05 ,<r = 0.5  /  \  3.0  \  \  a = 0.01, cr = 0.5  2.5  .\-  CL  £  V)  2.0  D W  o  1.5  CD  E  1.01—  0.5  -30 1875  20  10  0 Year  10  20  H. The optima! linear estimator of the intensity function of the mining disaster process using various prior parameters.  30 1951  93  BIBLIOGRAPHY  A n d e r s o n , B.D.O., " K a r h u n e n - L o e v e e x p a n s i o n f o r a c l a s s o f c o v a r i a n c e s " ; P r o c . I n t e r n a t i o n a l C o n f e r e n c e on Systems S c i e n c e , H o n o l u l u (1969), 779-782. A n s e l o n e , P.M. a n d R.H. M o o r e , " A p p r o x and o p e r a t o r e q u a t i o n s " ; J . Math. 268-277.  solutions of integral A n a l . 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Probability, 444-504.  smoothing of p r o b a b i l i t y d e n s i t y 2 0 , No. 2 ( 1 9 5 8 ) , 3 3 4 - 4 3 .  functions";  95  APPENDIX COMPUTER PROGRAM: OF THE  1  THE O P T I M A L L I N E A R E S T I M A T O R  I N T E N S I T Y F U N C T I O N OF THE O I L W E L L D I S C O V E R Y  C  ***  OILWELL DISCOVERY  C  ***  EXACT REAL  ***  SOLUTION USING WHITTLE'S  DS(120),  EXACT(225)  DIMENSION T M ( 2 2 5 ) , C E X ( 2 2 5 ) , R E A D 2 , MU, 2  SIGMA, A L F A  FORMAT(3F4.2) N=108 READ4,  4  (P(I),  DS(I),  = A L F A * S I G M A * SIGMA/MU  T T A = SQRT(2.*WK  + ALFA*ALFA)  VK = WK/TTA PRINT6, 6  FORMAT('  WK,TTA,VK ',3F10.4)  UT1 = 1 1 0 . 5 UT =  221.  T= - 1 1 0 . K=0 8  IF(T.GT.110.) SUM= 0. J  10  1=1,N)  FORMAT(F5.0, F3.0) WK  SUGGESTION  MU,SIGMA,ALFA,EXACT  DIMENSION P ( 1 2 0 ) ,  = 0  J = J+l  GOTO 20  PROCESS  CAT(225)  ***  96  IF(J.GT.N)  GOTO 12  R = ABS(T-P(J)) S=T+P(J) PA1=-TTA*R I F ( P A 1 . L T . - 1 0 0 . ) A=0. A=EXP(PA1) PA2=-TTA*(UT+S) I F ( P A 2 . L T . - 1 0 0 . ) B=0. B=EXP(PA2) PA3=-TTA*(UT-S) IF(PA3.LT.-100.) C=  C=0  EXP(PA3)  ABC = A+ B + C IF(ABC.EQ.O.)  GOTO 10  SUM=SUM+D'S ( J ) *ABC GOTO 10 12  CVA =  VALU(TTA,UT1,T)  ADVA = M U * V K * ( 1 .  - CVA)  K = K + 1 EXACT(K)  = VK*SUM  + ADVA  P R I N T 1 4 , T, E X A C T ( K ) 14  FORMAT(  1  T .= T + 1 GOTO 8 20  NT=221 M=NT - 1 N=NT  ', F 6 . 0 , F 1 0 . 2 )  97  TM(1)=1. DO  22  1=1,M  •TM(I+1)=TM(I)+1. CONTINUE DO  2 4 1=1,NT  CAT(I)=TM(I) CEX(I)=EXACT(I) CONTINUE *** NOW CALL  CALL SUBROUTINE AJOA  TO  DO  THE  PLOTTING  AJOA(CAT,CEX,NT)  TERMINATE PLOTTING CALL  AND" STOP  **  PLOTND  STOP END  .  S U B R O U T I N E A J O A ( X , Y, DIMENSION  N)  X ( N ) , Y(N)  CALL  SCALE(X,N,10.0,XMIN,DX,1)  CALL  SCALE(Y,N,10.0,YMIN,DY,1)  CALL A X I S ( 0 . , 0 . , CALL A X I S ( 0 . ,  ' T I M E * , - 4 , 1 0 . , 0.,  0.,  'EXACT',  CALL LINE(X,Y,N,1) C A L L P L O T ( 1 2 . 0 , 0., RETURN END  -3)  XMIN,  DX)  5, 1 0 . , 9 0 . , Y M I N ,  DY)  98  FUNCTION  VALU(X,Y,Z)  REAL X,Y,Z Al=  -X*ABS(-Y-Z)  Bl=  -X*ABS(Y-Z)  I F ( A 1 . L T . - 1 0 0 . . A N D . B 1 . L T . - 1 0 0 . ) P F = 0. I F ( A 1 . G E . - 1 0 0 . . A N D . B 1 . L T . - 1 0 0.) P F = E X P ( A l ) IF(Al.LT.-lOO..AND.Bl.GE.-lOO.)  PF=-EXP(Bl)  IF(Al.GE.-lOO..AND.B1.GE.-100.)  PF=EXP(Al)-EXP(Bl)  A2=  -X*(Z+Y)  B2 = - X * ( 3 * Y +Z) I F ( A 2 . L T . - 1 0 0 . . A N D . B 2 . L T . - 1 0 0 . ) QF=0. IF(A2.GE.-100..AND.B2.LT.-100.)  QF=EXP(A2)  I F ( A 2 . L T . - 1 0 0. .AND.B 2 . G E . - 1 0 0 . ) QF= - E X P ( B 2 ) IF(A2.GE.-100..AND.B2.GE.-100.)  QF=EXP(A2)  - EXP(B2)  A3 - - X * ( Y - Z ) B3 = - X * ( 3 * Y - Z ) I F ( A 3 . L T . - 1 0 0 . . A N D . B 3 . L T . - 1 0 0 . ) RF = 0 . IF(A3.GE.-100..AND.B3.LT.-100.)  RF=  EXP(A3)  IF(A3.LT.-100..AND.B3.GE.-100.)  RF=  -EXP(B3)  I F ( A 3 . G E . - 1 0 0 . .AND.B 3 . G E . - 1 0 0.) R F = E X P ( A 3 ) VALU  = ,(PF +.QF  RETURN END  + RF)/X + X  - EXP(B3)  99  COMPUTER OUTPUT: THE A P P R O X I M A T E  AND  THE  OILWELL DISCOVERIES;  EXACT L I N E A R ESTIMATORS,  X  AND X  oo R E S U L T S FOR T  .  L  o=0.5, a=0.05.  A (T)  T  L  T  A (T L  -110.  •1.21  0. 50  -89.  0 .88  0.88  -109 .  1.16  0.58  -88 .  0 . 91  0 . 91  -108.  1. 08  0.60  -87.  0.94  0 .94  -107.  1.03  0. 64  -86.  0 .93  0.93  -106 .  0.96  0. 63  -85.  0 .92  0.92  -105.  0. 88  0. 61  -84 .  0.91  0.91  -104 .  0.83  0.62  -83.  0. 86  0.86  -103.  0. 81  0. 64  -82 .  0 .84  0.84  -102.  0.83  0. 68  -81.  0 . 81  0 . 82  -101. .  0. 83  0. 72  -80.  0.82  0. 82  -100.  0.87  0. 78  -79.  0.81  0 .82  -99.  0.83  0.76  -78.  0.77  0.77  -98.  0.79  0.73  -77.  0. 75  0.76  -97.  0. 77  0.72  -76.  0.73  0.73  -96.  0.78  0. 74  -75.  0.69  0.69  -95.  0. 82  0 . 79  -74 .  0.68  0.68  -94.  0. 79  0. 76  -73.  0. 69  0.70  -93.  0. 78  0.76-  -72.  0 .73  0.73  -92.  0.80  0. 78  -71.  0.79  0. 80  -91.  0.81  0.80  -70 .  0. 89  0.8 9  -90.  0. 85  0.84  -69 .  0.94  0 . 94  100  T  A  (T)  A  (T)  CO  T  A  (T)  A  T  (T)  -65.  0.93  0.94  -40.  0.60  0.61  -64.  0.81  0.81  -39.'  0.67  0.68  -63.  0.71  0.71  -38.  0.77  0.77  -62.  0.64  0.64  -37.  0.86  0.86  -61.  0.59  0.59  -36.  0.94  0.94  -60.  0.56  0.56  -35.  0.98  0.99  -59.  0.55  0.55  -34.  0.96  0.96  -5-8.  0.56  0.56  -33.  0.93  0.93  -57.  0.59  0.59  -32.  0.90  0.90  -56.  0.60  0.60  -31.  0.86  0.87  -55.  0.53  0.53  -30.  0.86  0.87  -54.  0.47  0.48  -29.  0.86  0.86  -53.  0.43  0.44  -28.  0.85  0.85  -52.  0.41  0.42  -27.  0.86  0.87  0.37  0.37  -26.  0.88  0.88  -50.  0.33  0.34  -25.  0.93  0.93  -49.  0.31  0.32  -24.  0.90  "0.90  -48.  0.30  0.31  -23.  0.87  0.87  -47.  0.30  0.30  -22.  0.87  0.87  -46.  0.31  0.31  -21.  0.87  0.87  -45.  0.32  0.33  -20.  0.89  0.90  -44.  0.35  0.36  -19.  0.88  0.89  -43.  0.39  0.40  -18.  0.90  0.91  -42.  0.44  0.45  -17.  0.96  0.96  -41.  0.51  0.52  -16.  1.01  1.01  -51.  .  101  T  A  (T)  °o  (T)  A  T  A  (T)  a>  L  (T)  A  L  -15.  0.99  0.99  10.  0.77  0.78  -14.  0.97  0.98  11.  0.83  0.83  -13.  0.99  0.99  12.  0.88  0.88  -12.  0.93  0.94  13.  0.92  0.93  -11.  0.88  0.88  14.  0.89  0.90  -10.  0.85  0.86  15.  0.90  0.90  -9.  0.86  0.86  16.  0.9 3  0.94  -8.  0.86  0.86  17.  0.93  0.94  -7.  0.85  0.86  18.  0.89  0.90  -6.  0.85  0.85  19.  0.89  0.89  -5.  0.87  0.87  20.  0.87  0.88  -4.  0.92  0.93  21.  0.90  0.90  -3.  0.94  0.94  22.  0.91  0.92  -2.  0.95  0.96  23.  0.93  0.93  -1.  0.93  0.93  24.  0.91  0.91  0.  0.86  0.87  25.  0.84  0.85  1.  0.83  0.84  26.  0.81  0.82  2.  0.80  0.80  27.  0.81  0.82  3.  0.76  0.76  28.  0.84  0.84  4.  0.70  0.71  29.  0.89  0.90  5.  0.68  0.68  30.  0.98  0.99  6.  0.67  0.68  31.  1.04  1.04  7.  0.7 0  0.70  32.  1.06  1.07  8.  0.74  0.75  33.  1.08  1.09  9.  0.74  0.75  34.  1.15  1.15  102  T  A  (T)  CO  A (T) J_,  T  A  (T)  CO  A (T) J_,  35.  1.18  1.19  60.  0.42  0.42  36.  1.18  1.19  61.  0.42  0.43  37.  1.23  1.24  62.  0.40  0.41  38.  1.26  1.26  63.  0.37  0.37  39.  1.15  1.15  64.  0.34  0.34  40.  1.04  1.05  65.  0.32  0.33  41.  0.94  0.94  66.  0.32  0.32  42.  0.87  0.88  67.  0.32  0.33  43.  0.83  0.84  68.  0.34  0.34  44.  0.79  0.80  69.  0.37  0.37  45.  0.78  0.78  70.  0.41  0.41  46.  0.79  0.80  71.  0.42  0.43  47.  0.80  0.80  72.  0.42  0.42  48.  0.76  0.77  73.  0.43  0.44  49.  0.71  0.72  74.  0.45  0.46  50.  0.70  0.70  75.  0.46  0.46  51.  .0.67  0.67  76.  0.48  0.49  52.  0.62  0, 63  77.  0.52  0.52  53.  0.57  0.57  78.  0.57  0.58  54.  0.53  0.53  79.  0.65  0.65  55.  0.51  0.52  80.  0.67  0.68  56.  0.48  0.48  81.  0.69  0.69  57.  0.46  0.46  82.  0.66  0.66  58.  0.45  0.46  83.  0.61  0.62  59.  0.43  0.43  84.  0.59  0.59  103  T  X  (T)  X  T  (T)  T  X  (T)  X  T  (T)  85.  0.55  0.55  98.  0.34  0.31  86.  0.53  0.54  99.  0.33  0.29  87.  0.53  0.53  100.  0.34  0.29  88.  0.55  0.55  101.  0.35  0.29  89.  0.59  0.59  102.  0.38  0.30  90.  0.61  0.61  103.  0.42  0.32  91.  0.62  0.62  104.  0.47  0.36  92.  0.58  0.57  105.  0.54  0.40  93.  0.52  0.52  106.  0.63  0.46  94.  0.4 9  0.4 7  107.  0.71  0.50  95.  0.43  0.41  108.  0.78  0.52  96.  0.39  0.37  109.  0.83  0.52  97.  0.36  0.33  110.  0.89  0.50  104  APPENDIX COMPUTER PROGRAM: OF THE  2  THE O P T I M A L L I N E A R I N T E N S I T Y FUNCTION  OF THE L I O N S GATE  BRIDGE  PROCESS  ** L I O N S GATE B R I D G E E X A C T S O L U T I O N DATA FROM 9A.M. REAL  ESTIMATOR  TO  6  **  P.M.  MU,SIGMA,ALFA,FACT  F A C T I S A M U L T I P L Y I N G FACTOR  TO S C A L E UP  M(T)  DIMENSION  P(120),DS(120),SK(10),EXACT(120)  DIMENSION  TM(120),  READ 2,  CEX(109), CAT(109)  SIGMA,ALFA,FACT  FORMAT(3F6.3) N=109 M=9 T=-54 . UT=109. UT1=54.5 READ 4 , ( P ( I ) ,  DS(I),  FORMAT(F6.0,F6.0) READ5,  (SK(I),  FORMAT(F6.01) L=0 K=0 KT=0 L=L+1 Y=SK(L)  1=1,M)  1 = 1,N)  105  C A L L F U N C T I O N UTVAR TO COMPUTE NEW M ( T ) MU=UTVAR(FACT,Y) RMT=SQRT(MU) WK=ALFA*SIGMA*SIGMA/MU TTA=SQRT(2.*WK + A L F A * A L F A ) VK=WK/TTA 8  IF(T.GT.54)  GOTO 20  ** CHANGE M ( T ) A F T E R IF(KT.GT.12) SUM  EVERY 12TH INTERVAL  GOTO 7  =0.  J=0 10  J=J+1 IF(J.GT.N)  GOTO 1 2 -  R=ABH;(T-P ( J ) )  S=T+P(J) PA1=-TTA*R IF(PA1.LT.-100.)  A=0  A=EXP(PA1) PA2=-TTA*(UT+S) IF(PA2.LT.-100.)  B=0.  B=EXP(PA2) PA3p-TTA*(UT-S) I F ( P A 3 . L T . - 1 0 0 . ) C=0. C=EXP(PA3) ABC= A+B+C IF(ABC.EQ.O-) SUM  = SUM +  GOTO 10  DS(J)*ABC  106  GOTO 10 ** 12 **  TO COMPUTE X T ( S ) C V A = V A L U ( T T A , U T 1 , T) TO COMPUTE H T ( S ) A F T E R C A L C U L A T I N G X T ( S ) .  **  HAVE = CVA/RMT ADVA = M U * V K * ( 1 . - HAVE) K=K+1 E X A C T ( K ) = V K * S U M + ADVA + MU P R I N T 1 4 , T, E X A C T ( K ) 14  FORMAT('  ' , F 6 . 0 , F 1 0 , 2)  KT=KT+1 T=T+1 GOTO 8 20  NT=109 TM(1)=1. DO 2 1 1=1,N TM(I+1)=TM(I)+1.  21  CONTINUE DO 22 1=1,NT CAT(I)=TM(I) CEX(I)=EXACT(I)  22 **  CONTINUE NOW C A L L S U B R O U T I N E CALL  *  AJOA(CAT,CEX,NT)  TERMINATE CALL STOP  A J O A TO DO THE P L O T T I N G **  P L O T T I N G AND STOP **  PLOTND  107  END C S U B R O U T I N E A J O A ( X , Y, DIMENSION  N)  X ( N ) , Y(N)  CALL  SCALE(X,N,10.0,XMIN,DX,1)  CALL  SCALE(Y,N,10.0,YMIN,DY,1)  CALL A X I S ( 0 . , 0 . , CALLAXIS(0.,  ' T I M E ' , - 4 , 1 0 . , 0.,  0.,  'EXACT',  XMIN,  5, 1 0 . , 9 0 . , Y M I N ,  CALL LINE(X,Y,N,1) C A L L P L O T ( 1 2 . 0 , 0.,  -3)  RETURN END C C FUNCTION REAL  UTVAR(X,Y)  X,Y  PRO=X*Y UTVAR=SQRT(PRO) RETURN END C FUNCTION REAL  VALU(X,Y,Z)  X,Y,Z  Al=  -X*ABS(-Y-Z)  Bl=  -X*ABS(Y-Z)  DX)  IF(A1.LT.-100..AND.B1.LT.-100.) PF = IF(A1.GE.-100..AND.B1.LT.-100.)  0.  PF=EXP(Al)  DY)  108  I F ( A 1 . L T . - 1 0 0..AND.B1.GE.-100.) P F = - E X P ( B I ) IF(A1.GE.-100..AND.B1.GE.-100.) P F = E X P ( A l ) - E X P ( B I ) A2= B2  -X*(Z+Y) = - X * ( 3 * Y +Z)  IF(A2.LT.-100..AND.B2.LT.-100.)  QF=0.  IF(A2.GE.-100..AND.B2.LT.-100.)  QF=EXP(A2)  I F ( A 2 . L T . - 1 0 0 . .AND•B 2 . G E . - 1 0 0 . ) QF=  -EXP(B2)  I F ( A 2 . G E . - 1 0 0 . .AND.B 2 . G E . - 1 0 0 . ) Q F = E X P ( A 2 ) - E X P ( B 2 ) A3  = -X*(Y-Z)  B3 = - X * ( 3 * Y - Z ) IF(A3.LT.-100..AND.B3.LT.-100.)  RF = 0.  I F ( A 3 . G E . - 1 0 0 . . A N D . B 3 . L T . - 1 0 0.) RF=  EXP(A3)  I F ( A 3 . L T . - 1 0 0 . .AND.B 3 . G E . - 1 0 0 . ) RF=  -EXP(B3)  IF(A3.GE.-100..AND.B3.GE.-100.) VALU = RETURN END  ( P F + OF + R F ) / X + X  RF=EXP(A3)  - EXP(B3)  109  COMPUTER OUTPUT: THE  LIONS  GATE B R I D G E  OPTIMAL LINEAR ESTIMATOR, R E S U L T S FOR  a=2.0,  PROCESS A . T  a=0.2.  A (T)  T  A (T)  54.  167.99  -33.  118.86  53.  167.06  -32.  119.81  52.  164.14  -31.  118.14  51.  160.44  -30.  116.04  50.  154.74  -29.  117.08  49.  148.83  -28.  117.80  48.  145.01  -27.  117.38  47.  140.18  -26.  119.11  46.  136.26  "25.  120.07  45.  131.25  -24.  120.30  4-4.  127.12  -23.  121.23  43.  121.96  -22.  118.29  42.  116.97  -21.  111.63  41.  114.64  -20.  107.17  40.  111.57  "19.  106.10  39.  111.05  -18-  103.75  38.  111.39  "17.  102.03  37.  113.03  "16.  100.53  36.  114.99  "15.  99.50  35.  117.48  -14.  97.59  34.  117.46  -13.  96.70  L  L  110  A (T)  T  L  T  A (T) L  98.48  13,  95.62  -11.  100.00  14,  95.71  -10.  102.80  15,  96 . 28  -9.  104.89  16,  96.85  -8 .  108.44  17 ,  96.80  -7 .  110.80  18 ,  96 . 85  -6 .  112.50  19 ,  96.94  -5.  113.69  20,  97.86  -4.  114.57  21,  98.90  -3.  115.16  22 ,  1 0 0 .45  -2 .  113.98  23,  101.10  -1.  114.13  24 ,  105.68  0.  114.05  25.  108.08  1.  113.75  26,  109.82  2.  113.05  27 ,  111.08  3.  111.85  28,  112.17  4.  110.20  29,  113.41  . 5.  108.18  30.  114.50  6,  106.06  31,  115.23  7.  10 3.85  32.  118.75  8.  101.46  33.  121.37  9.  98.84  34.  122.75  10.  9 5 . 49  35.  124.38  11.  95.90  36.  123.76  12.  95.41  37.  119.45  -12.  Ill  T  A  Li  (T)  T  A (T) T  J-i  .38.  116.63  47.  88.13  39.  111.52  48.  89.70  40.  105.69  49.  91.04  41.  102.30  50.  89.87  42.  97.44  51.  89.08  43.  91.68  52.  85.83  44.  90.67  53.  82.93  45.  89.47  54.  80.64  46.  88.36  112  APPENDIX 3 COMPUTER PROGRAM: OF THE  THE O P T I M A L L I N E A R E S T I M A T O R I N T E N S I T Y FUNCTION  OF THE C O A L - M I N I N G D I S A S T E R P R O C E S S  C  ***  COAL M I N I N G D I S A S T E R S * EXACT SOLUTION  ***  C R E A L MU,  SIGMA, A L F A , D I V , FACT  DIMENSION P ( 1 0 0 ) ,  DS(IOO),  DIMENSION T M ( I O O ) , C E X ( 6 0 ) ,  SK(IOO), EXACT(IOO) CAT(60)  READ2, SIGMA, A L F A , D I V , FACT, AA, 2  FORMAT(6F6.2) N  = 60  M = 60 T = -30. UT = 6 1 . UT1 = 30.5 READ 4, 4  DS(I), I =  FORMAT(F6.0, F6.0) DO  5 I =  SK(I) 5  (P(I),  =  1,N P(I)/DIV  CONTINUE L  = 0  K = 0 8  IF(T.GE.30.) L  = L + 1  Y=AA*SK(L)  GOTO 20  1,N)  BB  113  Z=BB*SK(L) C  ***  C A L L F U N C T I O N UTVAR TO COMPUTE NEW MU=UTVAR(FACT,Y,Z) QMT=SQRT(MU) WK  = ALFA*SIGMA*SIGMA/MU  TTA  = SQRT(2.*WK.+  ALFA*ALFA)  VK = WK/TTA SUM= 0. J 10  = 0  J = J+l IF(J.GT.N) R =  GOTO 12  ABS(T-P(J))  S=T+P(J) PA1=-TTA*R IF(PA1.LT.-100.)  A=0.  A=EXP(PAl) PA2=-TTA*(UT+S) IF(PA2.LT.-100.)  B=0.  B=EXP(PA2) PA3=-TTA*(UT-S) IF(PA3.LT.-100.)  C=0.  C= E X P ( P A 3 ) A B C = A+ B + C IF(ABC.EQ.O.)  GOTO 10  SUM=SUM+DS(J)*ABC GOTO 1 0 C  TO COMPUTE X T ( S )  M(T)  ***  114  12  CVA TO  =  VALU(TTA,UT1,T)  COMPUTE H T ( S )  QCVA - CVA/QMT ADVA = M U * V K * ( 1 . K = K +  1  EXACT(K)  = VK*SUM + ADVA +  P R I N T 1 4 , T, 14  FORMAT(' T = T +  - QCVA)  MU  EXACT(K)  ', F 6 . 0 , F 1 0 . 2 ) 1  GOTO 8 20  NT  =  TM(1) DO  60 =  1.  21 I =  1,N  T M ( I + 1) .= T M ( I ) + 21  1.  CONTINUE DO  22 I =  1,NT  CAT(I)=TM(I) CEX(I)=EXACT(I) 22  CONTINUE *** NOW CALL  *  CALL SUBROUTINE AJOA  AJOA(CAT,CEX,NT)  TERMINATE PLOTTING CALL STOP END  TO  PLOTND  AND  STOP  **  DO  THE  PLOTTING  ***  115  S U B R O U T I N E A J O A ( X , Y, DIMENSION  N)  X ( N ) , Y(N)  CALL  SCALE(X,N,10.0,XMIN,DX,1)  CALL  SCALE(Y,N,10.0,YMIN,DY,1)  CALL A X I S ( 0 . , 0 . , CALL A X I S ( 0 . ,  ' T I M E ' , - 4 , 1 0 . , 0.,  0.,  'EXACT',  CALL LINE(X,Y,N,1) C A L L P L O T ( 1 2 . 0 , 0.,  -3)  RETURN END C C FUNCTION UTVAR(X, REAL  X,Y,Z  A = EXP(Y) B= E X P ( - Z ) C = A + B CASH =  X*C  UTVAR = C A S H RETURN END C  -  Y,  Z)  XMIN,  DX)  5, 1 0 . , 9 0 . , Y M I N ,  DY)  116  A (T) L  T  A (T) L  •30.  3.05  -5.  2 . 32  -29,  3.25  -4 .  2.31  •28,  3.40  -3.  2 . 31  •27.  3.47  -2.  2 .29  •26,  3.53  -1.  2.26  •25,  3.56  0.  2.21  •24,  3.54  1.  2 .17  •23,  3.53  2.  2 .13  •22,  3.52  3.  2 .10  •21,  3.48  4.  2.05  •20,  3.42  5.  1.97  •19 ,  3. 33  6.  1.92  •18,  3.25  7.  1.87  •17,  3.15  8.  1.85  •16,  3 . 03  9.  1.85  •15,  2.93  10.  1.88  •14,  2 . 84  11.  1.87  •13  1.71  12.  1.87  •12,  2 . 59  13.  1.89  •11,  2.50  14.  1.94  •10,  2 .44  15.  2 . 02  -9,  2.38  16.  2.10  -8,  2 . 34  17.  2.22  -7,  2 . 30  18 .  2.28  -6,  1.20  19 .  2 . 31  117  T  X  T  (T)  T  X  T  (T)  20.  2.32  26.  2.42  21.  2.34  27.  2.36  22.  2.38  28.  2.27  23.  2.40  29.  2.22  24.  2.41  30.  2.15  25.  2.43  

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