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

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ESTIMATING THE INTENSITY FUNCTION OF THE NONSTATIONARY POISSON PROCESS by DAVID WILSON FYNN .Sc., U n i v e r s i t y o f S c i e n c e and T e c h n o l o g y Ghana, 19 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE DEPARTMENT OF MATHEMATICS (INSTITUTE OF APPLIED MATHEMATICS AND STATISTICS) We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A u g u s t , 1976 (c) David Wilson Fynn In p resent ing t h i s t he s i s in p a r t i a l f u1 f i lment 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 reference and study. I f u r t h e r agree tha t permiss ion for ex tens i ve copying of t h i s t he s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r ep re sen ta t i ve s . It i s understood that copying or p u b l i c a t i o n o f t h i s t he s i s f o r f i n a n c i a l ga in s h a l l not be al lowed without my w r i t t e n permis s ion. Department of The Un i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 i ABSTRACT L e t ' { N ( t ) , -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 w i t h i n t e n s i t y f u n c t i o n , X ( t ) > 0 , assumed i n t e g r a b l e on [-T,T] . The o p t i m a l l i n e a r e s t i m a t o r , X , o f t h e i n t e n s i t y f u n c t i o n i s con-Li s i d e r e d i n t h i s t h e s i s . C h a p t e r 1 d i s c u s s e s X as a f u n c t i o n o f h ( t ; s ) , w h i c h i s L th e unique s o l u t i o n o f t h e Fredholm i n t e g r a l e q u a t i o n o f the second k i n d , •> m(s)h. (s) + / b K ( s ; u ) h . (u)du = K ( t ; s ) , a<s<b. t a t C h a p t e r s 2 and 3 a r e r e s p e c t i v e l y devoted 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 and approximate methods 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 i l l u s t r a t e t he use o f t h e t e c h n i q u e s d e v i s e d , t h r e e n u m e r i c a l examples a r e t r e a t e d . C h apter 4 d e a l s w i t h d a t a on o i l w e l l d i s c o v e r i e s i n A l b e r t a , Canada. F i n a l l y , i n Chapter 5, the model i s a p p l i e d t o d a t a on t r a f f i c c o u n t s on the L i o n s Gate B r i d g e , Vancouver, and t o d a t a on c o a l - m i n i n g d i s a s t e r s i n G r e a t B r i t a i n . Computer programs and numerous diagrams a r e a l s o p r e s e n t e d . i i TABLE OF CONTENTS CHAPTER 1 ON THE ESTIMATION OF THE INTENSITY FUNCTION 1.0 INTRODUCTION 1 1.1 OPTIMAL LINEAR ESTIMATORS OF THE INTENSITY FUNCTION 2 1.2 THE BEST LINEAR ESTIMATOR 5 CHAPTER 2 SURVEY OF EXACT METHODS FOR SOLVING FREDHOLM INTEGRAL EQUATIONS (OF TYPE I I ) 2.0 SUMMARY 9 2.1 F I N I T E DIFFERENCE APPROXIMATIONS 12 2.2.1 Hadamard's I n e q u a l i t y 14 2.2.2 H i l b e r t S p a c e s 14 2.3 THE CLASSICAL FREDHOLM TECHNIQUES 16 2.4 HILBERT-SCHMIDT, KARHUNEN-LOEVE AND GRANDELL TYPE SOLUTIONS 2 6 2.5 GRANDELL TYPE SOLUTION 36 2.6 SOME OTHER METHODS 41 CHAPTER 3 A SURVEY OF APPROXIMATE METHODS 3.1 INTRODUCTION 4 6 3.2 QUADRATURE RULES 4 6 3.3 GENERALIZED QUADRATURE 52 3.4 COLLOCATION METHOD 54 3.5 ERROR ANALYSIS 5 8 3.6 SUMMARY AND CONCLUSION 61 i i i CHAPTER 4 COMPARISON: AN EXACT VERSUS AN APPROXIMATE SOLUTION IN A SPECIAL CASE 4.1 OUTLINE OF WHITTLE'S DERIVATION OF (1.2.6) . 63 4.2 THE EXACT SOLUTION 65 4.3 AN APPROXIMATE SOLUTION 69 4.4 COMPUTATION: OIL WELLS DISCOVERY DATA 71 CHAPTER 5 APPLICATIONS 5.1 INTRODUCTION * 7 3 5.2 ESTIMATION OF TRAFFIC DENSITIES AT THE LIONS GATE BRIDGE 75 5.3 COAL-MINING DISASTERS 79 5.4 CONCLUDING REMARKS 82 BIBLIOGRAPHY 9 3 APPENDIX COMPUTER PROGRAMS AND OUTPUT 1 THE OILWELL DISCOVERY PROCESS 95 2 THE. LIONS GATE BRIDGE PROCESS 104 3 THE COAL-MINING DISASTER PROCESS 112 i v ACKNOWLEDGEMENT I w o u l d l i k e t o e x p r e s s my g r a t i t u d e t o Dr. J i m V. Z i d e k f o r g u i d a n c e and numerous comments t h r o u g h o u t t h e w r i t i n g o f t h e m a n u s c r i p t . I a l s o a c k n o w l e d g e my i n d e b t e d n e s s t o Dr. S. W. Nash and Dr. J . M. V a r a h f o r h e l p f u l c o m m u n i c a t i o n . F o r e x p e r t t y p i n g o f d i f f i c u l t m a t e r i a l , I am most g r a t e f u l t o Hardy Bunn. I am a l s o g r a t e f u l t o W. Samaradasa and R. B r u n . F i n a l l y , I w o u l d l i k e t o t h a n k A s e e f a M e r a l i f o r p a t i e n c e , u n d e r s t a n d i n g and e n c o u r a g e m e n t i n t h e f a c e o f many s e r i e s o f e v e n t s . CHAPTER 1 'ON THE ESTIMATION OF THE INTENSITY FUNCTION OF THE NONSTATIONARY POISSON PROCESS: INTRODUCTION, PRELIMINARIES INTRODUCTION I n t h i s c h a p t e r we c o n s i d e r l i n e a r e s t i m a t o r s o f t h e i n t e n -s i t y f u n c t i o n o f 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 . U s i n g t h e f o r m u l a t i o n g i v e n by' C l e v e n s o n a nd Z i d e k ( 7 ) , we c o n s i d e r a p o i n t p r o c e s s { N ( t ) ,-T$t$T} , 0<T^°° w i t h i n d e p e n d e n t i n c r e m e n t s and w i t h p [ N ( b ) - N ( a ) = n ] = ^ [ A ( a , b ) ] R e x p [ - A (a,b) ] w h e r e A ( a , b ) = / b A ( t ) d t , -T$a<b$T, a and t h e i n t e n s i t y f u n c t i o n A ( t ) > 0 , i s assumed t o be Riemann i n t e g r a b l e on [-T,TJ. The unknown A ( t ) d o e s n o t h a v e p a r a m e t r i c f o r m . F o r d e t a i l s we r e f e r t o G r a n d e l l ( 1 0 ) . I n s e c t i o n 1.1, we c o n s i d e r l i n e a r e s t i m a t o r s o f t h e i n t e n s i t y f u n c t i o n , A ( t ) , i n 2 g e n e r a l , and two n a t u r a l e s t i m a t o r s , i n p a r t i c u l a r , t h e h i s t o g r a m and t h e m o v i n g - a v e r a g e . I n s e c t i o n 1.2, G r a n d e l l ' s c r i t e r i o n i s i m p o s e d and A ( t ) , t h e b e s t e s t i m a t o r o f A ( t ) , i s d e t e r m i n e d i n a p a r t i c u l a r f o r m . F o l l o w i n g t h e work o f C l e v e n s o n and Z i d e k ( 7 ) , a F r e d h o l m i n t e g r a l e q u a t i o n o f t h e s e c o n d k i n d i s o b t a i n e d ; t h e s o l u t i o n o f w h i c h i s r e q u i r e d i n o r d e r t o o b t a i n t h e b e s t l i n e a r e s t i m a t o r A (t) i n t h e g i v e n f o r m . 1.1 OPTIMUM LINEAR ESTIMATORS OF THE INTENSITY FUNCTION L e t {N (t) ,-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 w i t h i n t e n s i t y f u n c t i o n , A ( t ) > 0 , assumed i n t e g r a b l e on [ - T , T ] . We s eek an e s t i m a t e o f { A ( t ) ; - T $ t $ T } . F o l l o w i n g t h e a p p r o a c h o f G r a n d e l l ( 1 0 ) , as e x t e n d e d by C l e v e n s o n and Z i d e k (7 ) , A ( t ) i s assumed t o be s q u a r e i n t e g r a b l e f u n c t i o n on [-T,TJ. D e n o t i n g by E t h e e x p e c -t a t i o n w i t h r e s p e c t t o t h e j o i n t d i s t r i b u t i o n o f N and A, t h e p e r f o r m a n c e o f an e s t i m a t o r , A, i s m e a s u r e d by (1.1.1) E / ^ T [ A ( t ) - A ( t ) J 2 d t I n t h e p r e s e n t work as w e l l as i n t h e r e f e r e n c e s c i t e d a bove, t h e e s t i m a t e i s c o n s t r a i n e d t o be a l i n e a r f u n c t i o n o f t h e c o u n t i n g r e c o r d . The m o t i v a t i o n f o r t h i s c o n s t r a i n t i s t h a t i t l e a d s t o e s t i m a t o r s t h a t a r e r e l a t i v e l y e a s y t o compute i n p r a c -t i c e w i t h n u m e r i c a l a l g o r i t h m s . A l s o , two w e l l known e s t i m a t o r s , t h e h i s t o g r a m and t h e m o ving a v e r a g e , a r e l i n e a r . A d i s a d v a n -t a g e , however, i s t h a t f o r many a p p l i c a t i o n s o f i n t e r e s t , l i n e a r 3 e s t i m a t e s may n o t be s u f f i c i e n t l y a c c u r a t e . One may w i s h t o c o n s i d e r t h e e s t i m a t i o n p r o b l e m w i t h o u t t h e l i n e a r i t y c o n s t r a i n t . H e r e , t h e m a j o r d i f f i c u l t y i s a n a l y t i c i n t r a c t a b i l i t y . I n g e n e r a l , by t h e l i n e a r i t y c o n s t r a i n t , t h e e s t i m a t e A ( t ) o f A ( t ) must have t h e f o r m (1.1.2) A ( t ) = a ( t ) + h ( t , s ) d ( N ( s ) ) f o r some d e t e r m i n i s t i c f u n c t i o n a ( t ) As an example, l e t (1.1.3) a ( t ) = 0 and h ( t , s ) = r ' t r - < s < t 0, o t h e r w i s e . F o r t h i s c h o i c e , (1.1.3) x ( t ) = p f £ _ r d N ( s ) = ( N ( t ) - N ( t - r ) ) / r . T h i s i s a m o v i n g - a v e r a g e , h i s t o g r a m e s t i m a t e o f t h e i n t e n s i t y p r o c e s s . Such an e s t i m a t e i s w i d e l y u s e d b e c a u s e i t has t h e a d v a n t a g e o f r e q u i r i n g a l m o s t no knowledge a b o u t t h e i n t e n s i t y p r o c e s s b e y o n d t h a t n e e d e d t o s e l e c t t h e a v e r a g i n g t i m e r . I t i s common p r a c t i c e t o e v a l u a t e t h e m o v i n g - a v e r a g e e s t i m a t e a t d i s -c r e t e t i m e s ; f o r i n s t a n c e , a t i n t e g r a l m u l t i p l e s o f r, and t h e n t o l e a s t - s q u a r e s c u r v e f i t an assumed t i m e f u n c t i o n t o t h e s a m p l e d v a l u e s . 4 C l e v e n s o n and Z i d e k (7) assumed p r i o r k nowledge a b o u t A ( t ) i s r e p r e s e n t a b l e by t h e c o n d i t i o n t h a t {A(t);-T<t<T} i s a w i d e -s e n s e s t a t i o n a r y s t o c h a s t i c p r o c e s s w i t h c o n s t a n t mean y and c o v a r i a n c e f u n c t i o n K. DEFINITION: A s t o c h a s t i c p r o c e s s {x (t) ;-°°<t <°°} i s s a i d t o be w i d e - s e n s e s t a t i o n a r y i f i t s m e a n - v a l u e f u n c t i o n m (t) =E-{x (t) } i s a c o n s t a n t i n d e p e n d e n t o f t and i t s c o v a r i a n c e f u n c t i o n K ( t , s ) = E { x ( t ) x ( s ) }-E{x(t) }E{x (s) } i s a f u n c t i o n o n l y o f ( t - s ) and n o t t and s s e p a r a t e l y . I n (7) C l e v e n s o n and Z i d e k d e f i n e t h e h i s t o g r a m e s t i m a t o r (1.1.4) X H ( t ) = [ N ( t i ) - N ( t i _ 1 ) ] [ 2 A i ] - 1 , t ± _ 1 < t < t i , where -T=tp<t^<•..<t =T i s any p a r t i t i o n o f [-T,T] and 2 A ^ = t ^ - t ^ _ The m o v i n g - a v e r a g e e s t i m a t o r , A , i s d e f i n e d by (1.1.5) A M ( t ) = [ N ( t + A ) - N ( t - A ) ] [2 A ] " 1 , p r o v i d e d -T+ A <t <T-A. The o p t i m a l c l a s s w i d t h f o r t h e h i s t o g r a m e s t i m a t o r and t h e o p t i m a l window w i d t h f o r t h e m o v i n g - a v e r a g e e s t i m a t o r were a l s o o b t a i n e d . We do n o t , however, i n t e n d t o d i s c u s s t h e d e t a i l s h e r e . 1.2 THE BEST LINEAR ESTIMATOR I t i s a t l e a s t i n t u i t i v e l y e v i d e n t t h a t e s t i m a t o r s h a v i n g p e r f o r m a n c e s u p e r i o r t o t h e m o v i n g - a v e r a g e and h i s t o g r a m e s t i -m a t o r s c a n be d e s i g n e d i f more k n o w l e d g e a b o u t t h e s t a t i s t i c s o f t h e i n t e n s i t y p r o c e s s i s a v a i l a b l e . L e t X d e n o t e t h e l i n e a r e s t i m a t e o f X ( t ) t h a t r e s u l t s i n (1.1.2) when b o t h a ( t ) and h ( s , t ) a r e s e l e c t e d t o m i n i m i z e t h e r " 2 m e a n - s q u a r e e r r o r E [ ( A ( t ) - A ( t ) ) ] . T h i s l i n e a r minimum mean-s q u a r e e r r o r e s t i m a t e i s g i v e n by G r a n d e l l ( 1 0 ) . The m e thod we c o n s i d e r h e r e f o l l o w s t h e C l e v e n s o n - Z i d e k (7) g e n e r a l i z a t i o n o f G r a n d e l l ' s t e c h n i q u e . Thus we s e e k an e s t i m a t o r o f t h e f o r m (1.2.1) X L ( t ) = a ( t ) + / ^ T h ( t , s ) d ( N ( s ) - M ( s ) ) . H e r e , we d r o p t h e a s s u m p t i o n s i n s e c t i o n 1.1 t h a t X ( t ) i s w i d e -s e n s e s t a t i o n a r y and t h a t m ( t ) = E X ( t ) E y . We t a k e M ( s ) = ^ g m ( t ) d t . The p r o b l e m now i s t o d e t e r m i n e t h e f u n c t i o n s a ( t ) and ~ 2 • h ( t ; . ) t o m i n i m i z e t h e f u n c t i o n a l E [x^ ( t ) - X ( t ) ] . I f t h e r e s u l t i n g o p t i m a l c h o i c e s a r e , s a y , a 0 ( t ) and h ° ( t ; . ) , t h e n t h e s e w i l l i n t u r n m i n i m i z e ( 1 . 1 . 1 ) . G r a n d e l l , and s u b s e q u e n t l y C l e v e n s o n and Z i d e k , show, a f t e r some m a n i p u l a t i o n , t h a t (1.2.2) E [ X L ( t ) - X ( t ) ] 2 = ( a ( t ) - m ( t ) ) 2 + ( h ( t ; .) , h ( t ; .) ) - 2 L t ( h ( t ; . ) ) 6 w h e r e (.,.) i s d e f i n e d by (1.2.3) ( x ( . ) ,y ( . ) ) = / ^ T x ( s ) y ( s ) m ( s ) d s + / ^ T / ^ T x ( s ) y ( s ) K ( s , u ) d s d u f o r a l l f u n c t i o n s x , y f o r w h i c h (x,x)<»; and b e c a u s e K ( . , . ) i s n o n n e g a t i v e d e f i n i t e , (x,x)<°° i s an i n n e r p r o d u c t . The l i n e a r T f u n c t i o n a l i s d e f i n e d by L f c x (. ) =/_ x (s) K ( t ; s1) d s , f o r a l l x s u c h t h a t (x,x)<«>. I t i s c l e a r f r o m (1.2.2) t h a t t h e o p t i m a l c h o i c e f o r a ( t ) i s (1.2.4) a°(t)=m(t) f o r a l l t . Assume t h a t 0 < i n f m ( s ) a n d t h a t m and K a r e b o u n d e d , i t f o l l o w s s t h a t i s a c o n t i n u o u s l i n e a r f u n c t i o n a l on t h e H i l b e r t s p a c e o f f u n c t i o n s H = { x : ( x , x ) < c o } . I t i n t u r n f o l l o w s by t h e R i e s z r e p r e s e n t a t i o n t h e o r e m t h a t L t x ( . ) = ( x ( . ) , g t ( . ) ) f o r a l l xeH w h e r e g (.)eH. Thus t h e o p t i m a l c h o i c e f o r h, (1.2.5) h°(t;s)=g. ( s ) , 7 i s t h e u n i q u e s o l u t i o n i n H o f t h e i n t e g r a l e q u a t i o n , (1.2.6) x(s)m(s ) + / 2 Tx(u)K(s,u)du=K(t;s). T h i s i s t h e w e l l known F r e d h o l m i n t e g r a l e q u a t i o n o f t h e s e c o n d k i n d w h i c h o c c u r s f r e q u e n t l y i n many a r e a s o f a p p l i e d m a t h e -m a t i c s s u c h a s i n C o m m u n i c a t i o n and I n f o r m a t i o n t h e o r y . The i n t e g r a l e q u a t i o n (1.2.6) h a s b e e n s t u d i e d e x t e n s i v e l y i n c o n n e c t i o n w i t h t h e l i n e a r f i l t e r i n g p r o b l e m f o r o b s e r v a t i o n s t h a t c o n t a i n a d d i t i v e n o i s e ( s e e , f o r e x a m p l e , H. Van T r e e s ( 1 9 ) , c h a p t e r s 4 and 6 ) . The s o l u t i o n t o e q u a t i o n (1.2.6) i s o u r m a i n c o n c e r n i n t h i s w o r k a n d e x a c t m e t h o d s a s w e l l a s a p p r o x i m a t i o n t e c h n i q u e s f o r t h i s p r o b l e m a r e g i v e n i n s u b s e q u e n t c h a p t e r s . REMARK: I t i s p e r h a p s w o r t h e m p h a s i z i n g t h a t t h e o r e t i c a l l y o n l y t h e mean and c o v a r i a n c e f u n c t i o n s a r e n e e d e d t o s o l v e t h e i n t e g r a l e q u a t i o n ( 1 . 2 . 6 ) . and t h u s t o d e s i g n t h e e s t i m a t o r . H owever, f o r n u m e r i c a l and a p p r o x i m a t e s o l u t i o n s t h e d a t a i n hand i s v e r y u s e f u l . I n t h e s e q u e l ( e . g . , i n c h a p t e r 4 ) , we s h a l l c o n s i d e r t h e s p e c i a l c a s e w h e r e (1.2.7) m ( t ) = u , and K ( t ; s ) = K ( t - s ) . 8 Thus (1.2.6) becomes (1.2.8) , y x ( s ) + / j X ( u ) K ( s - u ) d u = K ( t - s ) , -T<s<T. I t f o l l o w s t h a t our l i n e a r e s t i m a t o r now t a k e s t h e s p e c i a l form i (1.2.9) A L ( t ) = y + / ^ T H ( t ; s ) d ( N ( s ) - y s ) , o b t a i n e d from (1.2.1) by s e t t i n g a 0 ( t ) = m ( t ) = y . T h i s i s the form i n w hich our model w i l l be used i n the a p p l i c a t i o n s i n c h a p t e r s 4 and 5. 9 CHAPTER 2 SURVEY OF EXACT METHODS FOR SOLVING FREDHOLM INTEGRAL EQUATIONS (OF TYPE I I ) SUMMARY T h i s c h a p t e r i s d e v o t e d 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 m e t h o d s f o r s o l v i n g F r e d h o l m I n t e g r a l E q u a t i o n s ( o f t h e s e c o n d k i n d ) , t o g e t h e r w i t h some t h e o r e t i c a l b a c k g r o u n d , a n d a number o f a p p l i c a t i o n s . The 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 r e c e n t d e v e l o p m e n t s a r e a l s o d i s c u s s e d a n d c o m p a r e d . By means o f e x a m p l e s , some m o t i v a t i o n w i l l be g i v e n f o r t h e d i f f e r e n c e s i n t h e o r y a n d m e t h o d o l o g y u n d e r l y i n g t h e s e m e t h o d s a nd t h e i r i n v e s -t i g a t i o n . The n e c e s s a r y b a c k g r o u n d i n l i n e a r a l g e b r a w i l l be s k e t c h e d a n d some a s p e c t s o f H i l b e r t s p a c e t h e o r y w i l l be p r e -s e n t e d . S u b s e q u e n t l y , a l l i n t e g r a l o p e r a t o r s w i l l be v i e w e d as a c t i n g o n s u i t a b l e H i l b e r t s p a c e s . 1. INTRODUCTION An i n t e g r a l e q u a t i o n i s an e q u a t i o n i n w h i c h t h e unknown f u n c t i o n a p p e a r s u n d e r t h e i n t e g r a l s i g n . I n t e g r a l e q u a t i o n s h a v e b e e n e n c o u n t e r e d i n m a t h e m a t i c s f o r a number o f y e a r s , o r i g i n a l l y i n t h e t h e o r y o f F o u r i e r I n t e g r a l s . The a c t u a l d e v e l o p m e n t o f t h e t h e o r y o f i n t e g r a l e q u a t i o n s b e g a n , h o w e v e r , o n l y a t t h e end o f t h e n i n e t e e n t h c e n t u r y due t o t h e 10 works o f t h e I t a l i a n m a t h e m a t i c i a n V. V o l t e r r a , and p r i n c i p a l l y i n t h e y e a r 1900, when t h e S w e d i s h m a t h e m a t i c i a n I v a r F r e d h o l m p u b l i s h e d h i s famous work on a new method o f s o l v i n g t h e D i r i c h -l e t p r o b l e m * . From t h e n on, up t o t h e p r e s e n t , i n t e g r a l e q u a-t i o n s have been t h e s u b j e c t o f r e s e a r c h f o r numerous mathema-t i c i a n s . The t h e o r y o f i n t e g r a l e q u a t i o n s has c l o s e c o n t a c t s w i t h many d i f f e r e n t a r e a s o f m a t h e m a t i c s . I n d e e d , many p r o b l e m s o f a p p l i e d m a t h e m a t i c s c a n be s t a t e d i n t h e f o r m o f i n t e g r a l e q u a -t i o n s . To make a l i s t o f s u c h a p p l i c a t i o n s w o u l d be a l m o s t i m p o s s i b l e . S u f f i c e i t t o s a y t h a t t h e r e i s a l m o s t no a r e a o f a p p l i e d m a t h e m a t i c s and m a t h e m a t i c a l p h y s i c s where i n t e g r a l e q u a t i o n s do n o t p l a y a r o l e . I t i s w o r t h m e n t i o n i n g , a t t h i s s t a g e , t h a t i n d e a l i n g w i t h l i n e a r i n t e g r a l e q u a t i o n s t h e f u n d a m e n t a l c o n c e p t s o f l i n e a r v e c t o r s p a c e s , 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 p l a y a s i g n i f i c a n t r o l e . *"'Sur une n o u v e l l e methode p o u r l a r e s o l u t i o n du p r o b l e m e de D i r i c h l e t " . O f v e r s , a f K u n g l . V e t e n s k . Akad. F O r h . , S t o c k h o l m , "57, n r . 1 (10 J a n . 1900), 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 " . A c t a M a t h e m a t i c a , S t o c k h o l m , 27 (190 3 ) , 365-390. 11 The most f r e q u e n t l y s t u d i e d i n t e g r a l e q u a t i o n s a r e the f o l l o w i n g : (2.1.1) / b K ( t , s ) f ( s ) d s = g ( t ) 3. (2.1.2) f ( t ) + X / b K ( t , s ) f ( s ) d s = g ( t ) (2.1.3) m ( t ) f ( t ) + X / b K ( t / s ) f ( s ) d s - g ( t ) The above e q u a t i o n s a r e g e n e r a l l y known as Fredholm equa-t i o n s o f t h e f i r s t , second, and t h i r d k i n d , r e s p e c t i v e l y . The i n t e r v a l (a,b) may i n g e n e r a l be a f i n i t e i n t e r v a l o r (-°°,b], [a,°°), o r (-00 , 0 0 ) , where a and b a r e f i n i t e . We note t h a t we may d i v i d e (2.1.3) by m(t) t o reduce i t t o (2 . 1 . 2 ) . The f u n c t i o n K ( . , . ) , which i s g e n e r a l l y known as the k e r n e l , and g(.) a r e assumed known and f ( . ) i s sought. A l l the above e q u a t i o n s a r e l i n e a r ; t h a t . i s , t h e f u n c t i o n f ( . ) e n t e r s the e q u a t i o n s i n a l i n e a r manner so t h a t / b K ( t , s ) [ c 1 f 1 ( s ) + C 2 f 2 (s)] d s = C 1 / b K ( t , s ) f ( s ) d s + C 2 J f b K ( t , s ) f 2 ( s ) d s . As s t a t e d e a r l i e r , t h e e q u a t i o n s (2.1.1)-(2.1.3) a r i s e i n many s i t u a t i o n s ; i n s t a t i s t i c a l problems t h e k e r n e l s a re u s u a l l y symmetric and o f t e n a l s o n o n n e g a t i v e d e f i n i t e . 12 I n t h e n e x t s e c t i o n we s h a l l b r i e f l y s t a t e some m a t h e m a t i c a l r e s u l t s , i n c l u d i n g a d i s c u s s i o n o f t h e n e c e s s a r y b a c k g r o u n d o f H i l b e r t s p a c e t h e o r y , a s u s e f u l t o o l s i n t h e s e q u e l . The c l a s s i -c a l F r e d h o l m e x p a n s i o n t e c h n i q u e s w i l l be c o n s i d e r e d i n S e c t i o n 3. I n S e c t i o n 4 we s h a l l c o n s i d e r H i l b e r t - S c h m i d t , K a r h u n e n -L o e v e , and G r a n d e l l (7) t y p e s o l u t i o n s and t h e r e s u l t i n g e x p a n -s i o n s . We s h a l l d i s c u s s some o t h e r m e t h o d s i n t h e l i t e r a t u r e i n c l u d i n g t h e d e t e c t i o n t h e o r e t i c a p p r o a c h b y V a n t r e e s (19) , S l e p i a n ( 1 8 ) , and o t h e r s . 2. SOME USEFUL MATHEMATICAL TOOLS A l t h o u g h we a r e d e a l i n g w i t h e x a c t m e t h o d s , we f i n d i t a p p r o p r i a t e t o b r i e f l y c o n s i d e r f i n i t e d i f f e r e n c e a p p r o x i m a -t i o n s a t t h i s s t a g e b e c a u s e f i n i t e d i f f e r e n c e a p p r o x i m a t i o n s a r e n o t o n l y o f g r e a t p r a c t i c a l u t i l i t y , b u t a l s o p r o v i d e c e r t a i n i n s i g h t i n t o t h e n a t u r e o f i n t e g r a l e q u a t i o n s . We s h a l l a l s o s t a t e Hadamard's i n e q u a l i t y a s w e l l as some a s p e c t s o f H i l b e r t s p a c e t h e o r y . 2.1 F I N I T E DIFFERENCE APPROXIMATIONS I f , i n t h e e q u a t i o n (2.2.1) f ( t ) - X / j K ( t , s ) f ( s ) d s = g ( t ) 13 we r e p l a c e the i n t e g r a l by a s u i t a b l e sum: n , (2.2.2) f (t)-A L _ ^ K ( t ,£ ) f (i) = g(t) i = l f o r l a r g e n and a continuous k e r n e l K(t,s) and continuous f ( t ) , the sum i n (2.2.2) r e p r e s e n t s a c l o s e approximation to the i n t e g r a l i n (2.2.1). I f , furthermore, we e v a l u a t e (2.2.2) on l y at n d i s c r e t e p o i n t s n , . . (2.2.3) f ( i ) - A ^ i K ( i , - ) f ( i ) = g ( l ) , j = l , 2 , . . . , n , n 4-—= n n n n ^ n J i = l we have r e p l a c e d the i n t e g r a l e quation (2.2.1) by the a l g e b r a i c system (2.2.3) which may be r e w r i t t e n i n matrix form (2.2.4) (I-AH)F = G where the i - j t h element i n the matrix H i s ( — ) K ( i , — ) , and F i s J n n n a v e c t o r w i t h components f (^-) and G has components g {^ ) . To s o l v e (2.2.4) we i n v e r t the matrix and f i n d (2.2.5) F=(I-AH) - 1G. The above i n v e r s e w i l l e x i s t f o r a l l A, wi t h the e x c e p t i o n of at most n v a l u e s . These are the r o o t s of the c h a r a c t e r i s t i c d e t e r m i n a n t a l e q u a t i o n j 14 (2.2.6) I-AH = 0. I n (2.2.5) we see t h a t t h e r e may be s p e c i a l v a l u e s o f X f o r w h i c h no s o l u t i o n e x i s t s . Such v a l u e s a r e commonly known as e i g e n v a l u e s . We a l s o n o t e t h a t e v e r y f i n i t e a l g e b r a i c s y s t e m (2.2.3) n e c e s s a r i l y - has e i g e n v a l u e s , e v e n t h o u g h t h e i n t e g r a l e q u a t i o n (2.2.1) need n o t have e i g e n v a l u e s . 2.2.1 Hadairiard's I n e q u a l i t y We s t a t e t h i s as a t h e o r e m . THEOREM 2.2.1 L e t H be a m a t r i x w i t h t h e g e n e r a l e l e m e n t h . . . ID An u p p e r e s t i m a t e f o r i t s d e t e r m i n a n t i s g i v e n by So f a r we have n o t r e a l l y a d d r e s s e d o u r s e l v e s t o t h e q u e s t i o n o f what we mean by a s o l u t i o n o f an i n t e g r a l e q u a t i o n . B a s i c a l l y , o f c o u r s e , a s o l u t i o n o f any e q u a t i o n must r e d u c e t h e e q u a t i o n t o an i d e n t i t y . B u t o f t e n , however, we may impose a d d i t i o n a l r e s t r i c t i o n s on t h e s o l u t i o n , s u c h as demanding t h a t i t s h o u l d b e l o n g t o a p a r t i c u l a r c l a s s o f f u n c t i o n s . F o r t h e s e p u r p o s e s i t w i l l p r o v e t o be c o n v e n i e n t t o work i n t h e s o - c a l l e d H i l b e r t s p a c e s . (2.2.7) 2.2.2 H i l b e r t S p a c e s 15 D E F I N I T I O N 2.3.1 A l i n e a r s p a c e X i s s a i d t o be an i n n e r p r o d u c t s p a c e i f an i n n e r p r o d u c t i s d e f i n e d on i t . S u c h an i n n e r p r o d u c t a s s i g n s t o e v e r y p a i r f and g i n X a c o m p l e x number d e n o t e d b y ( f , g ) w i t h t h e f o l l o w i n g p r o p e r t i e s : 1. ( f , g ) = ( g , f ) 2. (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 i f f = 0. We n o t e t h a t ( f , f ) i s r e a l , s i n c e by p r o p e r t y (1) ( f , f ) = ( f , f ) . We l e t ( f , f ) = | | f | | a n d c a l l i t t h e Norm o f f . D E F I N I T I O N 2.3.2 L e t H be an i n n e r p r o d u c t s p a c e and i^-^ a C a u c h y s e q u e n c e i n H. Then H i s s a i d t o be a H i l b e r t s p a c e i f e v e r y C a u c h y s e q u e n c e c o n v e r g e s t o an e l e m e n t i n H. I n t h e s t u d y o f i n t e g r a l e q u a t i o n s , we f i n d t h a t t h e n o t i o n o f an i n t e g r a l o p e r a t o r i s f u n d a m e n t a l . S u c h an o p e r a t o r a s s i g n s t o an e l e m e n t f i n H a new e l e m e n t , s a y K f i n H. I f t h e o p e r a t o r s a t i s f i e s t h e c o n d i t i o n ( 2.2.1) K ( a f + 6 g ) = ctKf+BKg we s a y t h a t K i s a l i n e a r o p e r a t o r . An e x a m p l e o f a l i n e a r o p e r a t o r i s 16 K f = / j K ( t , s ) f ( s ) d s , w h e r e f e L 2 [ 0 , l ] a n d K ( t , s ) i s c o n t i n u o u s . S u c h an o p e r a t o r may o r may n o t be d e f i n e d on t h e w h o l e s p a c e H. THEOREM 2.3.1 C o n s i d e r L 2 [ a , b ] , w h e r e t h e i n t e r v a l [a,b] may be i n f i n i t e . I f / b / b | K ( t , s ) | 2 d t d s = M 2<~ a a' 1 t h e n t h e o p e r a t o r Kf = f K ( t , s ) f ( s ) d s i s b o u n d e d . a The s t a g e i s now s e t f o r o u r m a i n c o n c e r n i n t h e s e c t i o n s a h e a d . 2.3 THE C L A S S I C A L FREDHOLM TECHNIQUES To b e g i n w i t h , i n o r d e r t o f i x o u r i d e a s , we s h a l l c o n s i d e r t h e F r e d h o l m e q u a t i o n (2.3.1) f ( t ) . = g ( t ) + A / b K ( t , s ) f ( s ) d s a w i t h a Riemann i n t e g r a l i n a g i v e n i n t e r v a l ( a , b ) . F r e d h o l m was t h e f i r s t p e r s o n t o g i v e t h e s o l u t i o n o f e q u a -t i o n (2.3.1) i n t h e g e n e r a l f o r m f o r a l l v a l u e s o f t h e p a r a -m e t e r A. The r e s u l t s o f F r e d h o l m ' s i n v e s t i g a t i o n s a r e c o n t a i n e d i n t h r e e t h e o r e m s w h i c h a r e among t h e most i m p o r t a n t a n d b e a u t i f u l 17 m a t h e m a t i c a l d i s c o v e r i e s . The m e t h o d u s e d by F r e d h o l m c o n s i s t e d i n r e p l a c i n g t h e i n t e g r a l i n (2.3.1) b y a sum, t h e r e d u c t i o n o f t h i s e q u a t i o n t o a s y s t e m o f l i n e a r e q u a t i o n s a n d l e t t i n g t h e number o f t e r m s o f t h e sum t e n d t o i n f i n i t y . I n a c c o r d a n c e w i t h F r e d h o l m 1 s m e t h o d , we p a r t i t i o n t h e i n t e r v a l ( a , b ) , i n t o n e q u a l p a r t s by t h e p o i n t s c i — t i / tn r • • • / t —ID f h. 1 1 - ™ t • 1 2 n l + l l n and r e p l a c e e q u a t i o n (2.3.1) by n (2.3.2) f (t)=g(t)+Ah5 K ( t , t . ) f ( t . ) . i = l 1 1 L e t f ( t . ) = f . , K ( t . , t . ) = K . . ; t h e n we may r e w r i t e t h e s y s t e m o f I I ' J I J I 1 e q u a t i o n s a s (2.3.3) (1-AhK, , ) f,-AhK, _ f . .-AhK. f ' = g ( t , ) 11 1 12 2 i n n ^ 1 -AhK 1 4 =,-AhK „-...+ ( l - A h K ) f =g ( t .) . n l f 1 n2 nn n ^ n The s o l u t i o n s f , , f f o f (2.3.3) c a n be e x p r e s s e d i n 1 2 n t h e f o r m o f r a t i o s o f c e r t a i n d e t e r m i n a n t s by t h e common c h a r a c -t e r i s t i c d e t e r m i n a n t 18 (2.3.4) D n ( A ) = l-xhK 1 1 - A n K 1 2 -AhK I n - A h K 2 1 1-Ahk 22 •AhK , -AhK ~ n l n2 - AhK„ 2n 1- AhK nn p r o v i d e d t h a t t h i s d e t e r m i n a n t i s n o t e q u a l t o z e r o . An e x p a n s i o n o f (2.3,4) may be e x p r e s s e d i n t h e f o r m n (-Ah) n D n ( A ) = 1-Ah> K i j L+ 21 YZ1 i = l i»j=l (2.3.5) K. . K. . K . . K . . D i 33 (-Ah) +...+ n l n i l f i 2 , • • - i n 1 K. . K. . 1 1 1 1 X 1 X 2 ' K. • K. . 1 2 1 1 1 2 1 2 " K. . K. l l , l i„ n 1 n 2 K. . 1 n K i ~ i | 2 n . K, i 1 , n n F o r t h e s a k e o f s i m p l i c i t y , l e t (2.3.6) K ( t 1 , S 1 ) K ( t 1 , S 2 ) ... K ( t 1 , S n ) K ( t n , S 1 ) K ( t n , S 2 ) ... K ( t n , S n ) K : l # t 2 * ' " tn\ 1 2 n The d e t e r m i n a n t (2.3.6) i s c a l l e d F r e d h o l m ' s d e t e r m i n a n t , and t h e a b o v e s y m b o l i s t a k e n t o be d e f i n e d f o r e v e r y k e r n e l K ( t , s ) a l s o i n a m u l t i - d i m e n s i o n a l d o m a i n . 19 The f u n d a m e n t a l p r o p e r t y o f F r e d h o l m 1 s d e t e r m i n a n t i s t h a t , i f any p a i r o f arguments i n t h e u p p e r o r t h e l o w e r s e q u e n c e i s t r a n s p o s e d , t h e v a l u e o f t h e d e t e r m i n a n t c h a n g e s t h e s i g n . U s i n g t h e symbol ( 2 . 3 . 6 ) , we may w r i t e t h e e x p a n s i o n i n t h e f o r m n (-Ah) 2 n It. , t 1 (2.3.7) D n ( A ) = l - A h Y ~ K ( t i , t ± ) + 21 I K Vt. , t ./ \ l 1/ i = l " " i , j = l , ft • , t . , t \ , ( - A h ) 3 K 1 ^ r \ + ^ i , D , r = l y i ' 3 rJ Now s u p p o s e t h a t h+0 and n->-°°, t h e n e a c h o f t h e terms o f sum (2.3.7) t e n d s t o some s i n g l e , d o u b l e , t r i p l e i n t e g r a l , e t c . Thus a r i s e s t h e s e r i e s (2.3.8) D ( A ) = l - A / b K ( s , s ) d s + A 2 / b / b K -1' 2 a 2T a a \ s 1 # s 2 / d s , d s 2 3 /S-^,S2,S2i ~JTflfl / a K ( s i , S 2 , S 3 ) d s l d S 2 d s 3 + w h i c h was shown by F r e d h o l m , on t h e b a s i s o f Hadamard's t h e o r e m , t o c o n v e r g e f o r e v e r y v a l u e o f A. I n t h i s v e i n t h e F r e d h o l m 1 s f u n c t i o n D(A) may be expanded i n a c o n v e r g e n t power s e r i e s ( F r e d h o l m ' s f i r s t s e r i e s ) o f t h e f o r m 20 ( 2 . 3 . 9 ) D(A)=l+5 ( -A) i = l i ! ^ 1 ' ^ 2 ' " " * ' ^ 1 K , . S l f S 2 S . l d s 1 d s 2 . . . d s i i n an a r b i t r a r y domain Q. We now seek a s o l u t i o n o f the form ( 2 . 3 . 1 0 ) f (t)=g(t)+A j N ( t , s , X ) g ( s ) d s where the r e s o l v e n t k e r n e l N ( t , s , X ) i s the p r o d u c t ( 2 . 3 . 1 1 ) N(t,s,X)=D(t,s,X D(X) ' D(t,s,X) i s the sum o f a c e r t a i n sequence, CO j_ ( 2 . 3 . 1 2 ) D(t,s,X)=C Q(t,s)+') (~* } C i ( t , s ) i = l and ( 2 . 3 . 1 3 ) C ± ( t , s ) = * ' s f S i f S 2 S.J d S l d s 2 . . . d S i T h i s l e a d s t o the Fredholm s e r i e s : ( 2 . 3 . 1 4 ) - i ( I / t , S , . . . ,S n D ( t , s , X ) = K ( t r s ) + 5 •, ... K 1 I ds ...ds i = l x ' ) J \S,S, , • • . ,S, 1 w h i c h has the same convergence p r o p e r t i e s as Fredholm's f i r s t s e r i e s . > 21 We a r e now i n a p o s i t i o n t o s t a t e t h e t h r e e F r e d h o l m t h e o r e m s . - x THEOREM 2.3.1 F r e d h o l m ' s e q u a t i o n o f t h e s e c o n d k i n d , u n d e r t h e a s s u m p t i o n t h a t t h e f u n c t i o n s g ( t ) and K ( t , s ) a r e i n t e g r a b l e , h as i n t h e c a s e D(A)=|=0 a u n i q u e s o l u t i o n , w h i c h i s o f t h e f o r m ( 2 . 3 . 1 0 ) . THEOREM 2.3.2- I f A Q i s a z e r o o f m u l t i p l i c i t y q o f D ( x ) , t h e n t h e homogeneous e q u a t i o n (2.3.15) f ( t ) = A 0 / K ( t , s ) f ( s ) d s n p o s s e s s e s a t l e a s t one, and a t most q, l i n e a r l y i n d e p e n d e n t . s o l u t i o n s j "^2.f^2' " " * ' ^  j — 1 f * * " ' ^ i '. S _. f o r j = l , 2 , . . . , i ; l ^ i $ q , n o t i d e n t i c a l l y z e r o , and any o t h e r s o l u t i o n i s a l i n e a r c o m b i -n a t i o n o f t h e s e s o l u t i o n s . D. d e n o t e s a F r e d h o l m m i n o r o f I o r d e r i r e l a t i v e t o t h e k e r n e l K ( t , s ) . THEOREM 2.3.3 F o r t h e nonhomogeneous e q u a t i o n t o p o s s e s s a s o l u -t i o n i n t h e c a s e D(AQ)=0, i t i s n e c e s s a r y and s u f f i c i e n t t h a t t h e g i v e n f u n c t i o n 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 s o l u t i o n s ^ ^ ( t ) ( j = l , 2 , . . . , i ) o f t h e a s s o c i a t e d homogeneous e q u a t i o n 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 f u n d a m e n t a l s y s t e m . The g e n e r a l s o l u t i o n t h e n has t h e form 22 (2.3.16) f ( t ) = g ( t ) + A „ I r g ( s ) d s D 1 p p \ s 1 , . . . , s p + ) C . ( j ) . . ( t ) , w h e r e $ • ( t ) = q V - S l ' L J J 3 r^. ^ f t j = i D q p l ' " - ' r i - l ' r i ' r i + l ' ,s. . t _ , \ \^*1' ' 2.3.2 F r e d h o l m ' s E q u a t i o n w i t h D e g e n e r a t e K e r n e l V7e c a l l t h e k e r n e l K ( t , s ) d e g e n e r a t e i f i t i s t h e sum o f p r o d u c t s 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 i ( S ) i + 1 S u b s t i t u t i n g (2.3.17) i n t o f ( t ) = g ( t ) + A J K ( t , s ) f ( s ) d s we n o t i c e i m m e d i a t e l y t h a t a s o l u t i o n i s t h e sum o f t h e f u n c t i o n g ( t ) a n d o f a c e r t a i n l i n e a r c o m b i n a t i o n o f t h e f u n c t i o n s IsL ( t ) : n (2.3.18) f(t)=g(t)+2 A i N i ( t ) , A i b e i n g c o n s t a n t i = l I n o r d e r t o d e t e r m i n e t h e c o n s t a n t s A^, we s u b s t i t u t e e x p r e s s i o n (2.3.18) i n t h e i n t e g r a l e q u a t i o n . We i l l u s t r a t e t h i s b y an e x a m p l e : EXAMPLE 2.3.1 S u p p o s e we a r e g i v e n t h e i n t e g r a l e q u a t i o n f ( t ) = g ( t ) + X / J ( t + s ) f (s)-ds The s o l u t i o n s h o u l d h a v e t h e f o r m •f (t)=g(t)+A 1t+A 2, whence we h a v e t h e i d e n t i t y g ( t ) +A 1t+A 2=g ( t ) + A / J (t+s) [g (s) + A ; L S + A 2 ] d s and t h e s y s t e m o f e q u a t i o n s A x (1-^-A)-A 2 A= A / J g (s) d s , —jA-^ A+A 2 ( l — ^ A ) =A/Jsg (s) ds Hence we o b t a i n A 1 and A 2 and t h e s o l u t i o n o f t h e g i v e n . i n t e g r a l e q u a t i o n i n t h e f o r m f ( t ) = g ( t ) + 2 A / l 6 A t S + 3 ( 2 - A ) ( t + l ) + 2 A g ( s ) d g  U 1 2 - 1 2 A - A Z p r o v i d e d A i s n o t one o f t h e e i g e n v a l u e s . The e i g e n v a l u e s a r e r o o t s o f t h e e q u a t i o n 12-12A-A 2=0 and h e n c e X =-6 + 4/3, A 2=-6-4/3 The c o r r e s p o n d i n g c h a r a c t e r i s t i c s o l u t i o n s a r e •f 1(t ) = c ( x 1t+i - | x 1 ), f 2(t ) = c ( x 2t+i - | x 2 ) where C i s an a r b i t r a r y c o n s t a n t . REMARK The F r e d h o l m e q u a t i o n w i t h d e g e n e r a t e k e r n e l p l a y s an i m p o r t a n t r o l e i n t h e t h e o r y and a p p l i c a t i o n s o f i n t e g r a l e q u a -t i o n s , s i n c e i t c a n e a s i l y be s o l v e d by a f i n i t e number o f i n t e g r a t i o n s . 2.3.3 E x i s t e n c e o f S o l u t i o n s . The F r e d h o l m A l t e r n a t i v e E q u a t i o n s o f t h e s e c o n d k i n d have an e x i s t e n c e t h e o r y : Suppose t h a t t h e r a n g e o f i n t e g r a t i o n i s f i n i t e , t h e n we have t h e F r e d h o l m A l t e r n a t i v e as f o l l o w s : E i t h e r : I f X i s a r e g u l a r v a l u e , t h e n t h e e q u a t i o n f (t) =g (t) +X/ bK (t', s) f (s) ds has a u n i q u e s o l u t i o n f o r any a r b i t r a r y a g ( t ) ; O r: I f X i s a c h a r a c t e r i s t i c v a l u e , t h e n t h e homogeneous e q u a t i o n (2.3.19) f ( t ) = X / b K ( t , s ) f ( s ) d s a has a f i n i t e number (p, say) o f l i n e a r l y i n d e p e n d e n t s o l u t i o n s <j>i ( t ) , . . . , <t>^  (t) . I n t h i s c a s e t h e t r a n s p o s e d homogeneous e q u a t i o n 25 (2.3.20) ¥(t)=A/ bK(t,s)¥(s)ds cl a l s o h a s p s o l u t i o n s - ^ ( t ) V ( t ) ; -and t h e nonhomogeneous P e q u a t i o n h a s a s o l u t i o n i f and o n l y i f g ( t ) i s o r t h o g o n a l t o a l l t h e V ^ , ; t h a t i s , i f and o n l y i f •f k g ( t ) V. ( t ) d t = 0 , i = l , . . . , p . ' a 1 T h i s s o l u t i o n i s c l e a r l y n o t u n i q u e , s i n c e we c a n a d d t o i t any l i n e a r c o m b i n a t i o n s o f t h e ^ ' s , and o b t a i n a n o t h e r s o l u t i o n . S t a t e d i n a n o t h e r f o r m , t h e F r e d h o l m A l t e r n a t i v e s a y s : I f t h e homogeneous e q u a t i o n (2.3.19) h a s t h e u n i q u e s o l u t i o n f = 0 , t h e n t h e nonhomogeneous e q u a t i o n i s s o l v a b l e f o r a r b i t r a r y g (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 ) . I n o t h e r w o r d s , u n i q u e n e s s i m p l i e s e x i s t e n c e . B e f o r e we t u r n t o t h e n e x t s e c t i o n we make t h e f o l l o w i n g o b s e r v a t i o n c o n c e r n i n g t h e a b o v e . REMARK The F r e d h o l m c l a s s i c a l t e c h n i q u e s y i e l d r a t h e r p r e c i s e i n f o r m a t i o n r e g a r d i n g s o l v a b i l i t y o f e q u a t i o n s a n d e x i s t e n c e a n d d i s t r i b u t i o n o f e i g e n v a l u e s , b u t o f t e n r e q u i r e r a t h e r t e d i o u s and w e a r i s o m e a n a l y s i s . C o n s e q u e n t l y F r e d h o l m ' s f o r m u l a e h a v e so f a r f o u n d o n l y 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 o r n u m e r i c a l , a p a r t f r o m p r o v i d i n g a f o u n d a t i o n f o r t h e t h e o r y o f i n t e g r a l e q u a t i o n s . 26 2.4 HILBERT-SCHMIDT, KARHUNEN-LOEVE, AND GRANDELL TYPE SOLUTIONS 2.4.1 H i l b e r t - S c h m i d t T h e o r y I n t h e f i r s t p a r t o f t h i s s e c t i o n we d e f i n e t h e s o -c a l l e d s e l f a d j o i n t compact o p e r a t o r s and o b t a i n some i n f o r m a t i o n r e g a r d i n g t h e i r e i g e n v a l u e s , e i g e n f u n c t i o n s and t h e a s s o c i a t e d e x p a n s i o n t h e o r e m s . T h e s e r e s u l t s a r e t h e n a p p l i e d t o compact i n t e g r a l o p e r a t o r s . As b e f o r e , we s h a l l be c o n c e r n e d w i t h i n t e g r a l e q u a t i o n s o f t h e t y p e (2.4.1) f ( t ) = g ( t ) + X / b K ( t , s ) f ( s ) d s i n t h e H i l b e r t s p a c e L 2 [ a , b J . The f u n c t i o n g ( t ) w i l l be assumed t o b e l o n g t o L 2 | a , b ] and t h e k e r n e l w i l l be assumed t o be s q u a r e -i n t e g r a b l e , so t h a t (2.4.2) / b / b | K ( t , s ) I 2 d t d s < « . a a DEFINITION 2.4.1 L e t K be a bounded, l i n e a r o p e r a t o r on a H i l b e r t s p a c e H. Then K w i l l be s a i d t o be a compact o p e r a t o r i f f r o m t h e s e q u e n c e {Kf } we c a n e x t r a c t a s u b s e q u e n c e {Kf } n n R t h a t i s a Cauchy s e q u e n c e , f o r any u n i f o r m l y bounded s e q u e n c e , { f „ } , i n ' H . 27 Now l e t {<j>^}'be an o r t h o n o r m a l s e t i n H, and l e t n K n f = K f / y i ( f , < t > i ) ' * i n -1,2,... w h e r e {y^} i s t h e s e q u e n c e o f e i g e n v a l u e s o f K o r d e r e d s u c h t h a t p l t > I y 2 1 > " Then i f K h a s an i n f i n i t y o f n o n z e r o e i g e n v a l u e s , t h e s e a c c u m u l a t e a t t h e o r i g i n , a n d ( K n f } i s a c o n v e r g e n t s e q u e n c e i n H. We t h u s h a v e t h e f o l l o w i n g r e s u l t s : THEOREM 2.4.1 L e t f be an e l e m e n t i n H. Then f c a n be r e p -r e s e n t e d i n t h e f o r m (2.4.3) f=> (f , *i) *±+f o w h e r e f Q i s a s u i t a b l e , e l e m e n t i n t h e n u l l s p a c e o f K ( i . e . , THEOREM 2.4.2 L e t {<j>^ } be t h e c o r r e s p o n d i n g e i g e n v e c t o r s a s s o -c i a t e d w i t h K, and s u p p o s e H i s L ~ [ a , b ] , t h e n l i m ,b ,-b n 2 ( 2 - 4 ' 4 ) P a ; a l K ( t ' s , - / _ W i * . ( t ) * . ( s : . i = l so t h a t d t d s = 0 K(t,s)=^) y ± ( | ) i ( t ) (j) i(s) i = l c o n v e r g e s i n t h e mean ( i n t h e s e n s e o f ( 2 . 4 . 4 ) ) 28 I t t h e n f o l l o w s t h a t (2.4.5) / b / b | K ( t , s ) | 2 d t d s = \ v±2 We n o t e t h a t i n t h e above r e s u l t s t h e f a c t t h a t K ( t , s ) i s s q u a r e - i n t e g r a b l e i s v i t a l , s i n c e f o r a r b i t r a r y compact o p e r a t o r s oo 2 t h e sum <• y. need n o t be f i n i t e . L_ 1 i = l . We a l s o r e c a l l t h a t o p e r a t o r s f o r w h i c h K ( t , s ) i s an k e r n e l 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 o p e r a t o r s . We now s t a t e t h e i m p o r t a n t HILBERT-SCHMIDT THEOREM 2.4.3 E v e r y f u n c t i o n f ( t ) o f t h e f o r m (2.4.6) f ( t ) = / K ( t , s ) h ( s ) d s i s a l m o s t e v e r y w h e r e t h e sum o f i t s F o u r i e r s e r i e s w i t h r e s p e c t t o t h e o r t h o n o r m a l s y s t e m <j>^(t) o f e i g e n f u n c t i o n s o f t h e sym-m e t r i c k e r n e l K. The above t h e o r e m i m p l i e s t h a t (2.4.7) f ( t ) = \ f . < j > . ( t ) c o n v e r g e s , / 1 1 i = l where t h e c o e f f i c i e n t s f ^ a r e t h e F o u r i e r c o e f f i c i e n t s o f t h e f u n c t i o n f ( t ) w i t h r e s p e c t t o t h e s y s t e m (<|>^(t)}, t h a t i s (2.4.8) f , = / f (t) <j>. (t) d t ^ 1 fi 1 i and t h e h ^ a r e t h e F o u r i e r c o e f f i c i e n t s o f t h e g i v e n f u n c t i o n h w i t h r e s p e c t t o t h e s y s t e m (<|>^(t) }: (2.4.9) h i = / h ( t ) <t>i(t) d t • C o n s e q u e n t l y , t h e f u n c t i o n f ( t ) i s a l m o s t e v e r y w h e r e e q u a l t o t h e sum o f i t s a b s o l u t e l y and u n i f o r m l y c o n v e r g e n t F o u r i e r s e r i e s : o  (2.4.10) f ( t ) = > ^ i j ) . ( t) . 2.4.2 A p p l i c a t i o n o f t h e H i l b e r t - S c h m i d t Theorem U s i n g t h e H i l b e r t - S c h m i d t Theorem, i t i s p o s s i b l e t o o b t a i n a s e r i e s e x p a n s i o n o f t h e s o l u t i o n f ( t ) o f t h e i n t e g r a l e q u a t i o n (2.4.11) f ( t ) = g ( t ) + A / K ( t , s ) f ( s ) d s w i t h r e s p e c t t o t h e s y s t e m o f e i g e n f u n c t i o n s (<j>^(t) ) o f a sym-m e t r i c k e r n e l K ( t , s ) . A s s u m i n g t h a t A i s n o t e q u a l t o any o f t h e e i g e n v a l u e s , A=|=A^ , t h e r e e x i s t s a u n i q u e s o l u t i o n f ( t ) i n L 2 ( n ) . S p e c i f i c a l l y e q u a t i o n (2.4.11) may be expanded i n t h e f o r m o  (2.4.12) f ( t ) ' g ( t l - \ ~ C <!>• (t) A / 1 1 L 30 w h i c h may be s u b s t i t u t e d i n (2.4.11) and i n t e g r a t e d t e r m by t e r m t o o b t a i n oo X (2.4.13) ) C. (1-X ) (j> . ( t ) = T K ( t , s ) g ( s ) d s x = l B u t , a c c o r d i n g t o t h e H i l b e r t - S c h m i d t t h e o r e m , we h a v e a g a i n (2.4.14) J " K ( t , s ) g ( s ) d s = \ T ^ - j t t ) , i = l w h e re t h e a r e t h e F o u r i e r c o e f f i c i e n t s o f t h e f u n c t i o n g ( t ) From (2.4.13) and (2.4.14) i t f o l l o w s t h a t (2.4.15) i = l X g i i i i ' ± ( t ) = 0 . M u l t i p l y i n g b o t h s i d e s i n t u r n by <|> 1 , <J>2 , . . . and i n t e g r a t i n g , b e c a u s e o f t h e o r t h o g o n a l i t y we o b t a i n (2.4.16) € . ( 1 - r — ) - Y ^ = 0 f o r e v e r y i . 1 1 g i C o n s e q u e n t l y C.= i X . -X l S u b s t i t u t i n g t h e s e v a l u e s i n t h e s e r i e s ( 2 . 4 . 1 2 ) , we o b t a i n t h e r e q u i r e d e x p a n s i o n o f t h e s o l u t i o n o f e q u a t i o n (2.4.11) a s an a b s o l u t e l y a nd u n i f o r m l y c o n v e r g e n t s e r i e s i n t e r m s o f e i g e n -f u n c t i o n s o f t h e k e r n e l : (2.4.17) f ( t ) = g ( t ) - x y ~ x - ^ - < ( ) i ( t ) (X=|=Xi) i = l 3 1 REMARK I f A w e r e e q u a l t o one o f t h e e i g e n v a l u e s * = * p = A p + 1 A p + r - i ' = x p + q = i w i t h r * n } §ati§fi§S i f §nd o n l y i f A . « = 4 S< = A-. . ~ n w i t h r a n k q, t h e n e q u a t i o n (2.4.16) w o u l d be =t=2 p+q=l (2.4.18) g p + i = / g C t ) <j , p + i (t) dt=0 ( i = 0 , 1 , 2 , . . . , q-1) C o n d i t i o n (2.4.18) i s i n a c c o r d a n c e w i t h t h e F r e d h o l m T h i r d T h eorem a nd i t h a s t o be e x t e n d e d t o a l l t h e e i g e n f u n c t i o n s c o r r e s p o n d i n g t o t h e v a l u e A p w h i c h i s r e p e a t e d i n t h e a b o v e s e r i e s a s many t i m e s a s t h e number o f i t s r a n k . I n t h a t c a s e t h e s o l u t i o n t a k e s t h e f o r m oo * (2.4.19) f (t)=g(t)-A^) X ^ Y 7 ( | > i ( t ) + C l , f , p ( t ) + i = l x + C 2 * p + l ( t ) + ••• + C q * P + q - l ( t ) * w h e r e > d e n o t e s t h a t 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 t h e v a l u e s o f i e q u a l t o p, p+1, p+q-1, f o r w h i c h X p - X p + l ~ X p + 2 ' • • _ X p + q - l ' w h e r e q i s t h e r a n k o f t h a t e i g e n v a l u e . By a s i m i l a r m e t hod we may f i n d t h e e x p a n s i o n o f t h e r e s v e n t k e r n e l 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 by N: (2.4.20) N ( t , s , A ) = K ( t , s ) + A / K ( t > y ) N ( y , s / A ) d y . We o b t a i n i n t h i s c a s e (2.4.21) i = l 4>i (t)=0, whence, as b e f o r e , on a c c o u n t o f t h e o r t h o g o n a l i t y o f (<j>^(t)}, i t f o l l o w s t h a t (s) (2.4.22) b . ( s ) = i v o / X . ( A.-X) 1 1 The r e q u i r e d e x p a n s i o n o f t h e r e s o l v e n t k e r n e l i s t h e r e f o r e o b t a i n e d i n t h e f o r m o  x—<i> (t)<j).(s: (2.4.23) N ( t , s , X ) = K ( t , s ) - X ) x _ ( A _ x _ }( X = A i ) , i = l t h e c o n v e r g e n c e o f w h i c h i s e v i d e n t i n v i e w o f t h e s i m i l a r <t> . (t)ty . (s) a b s o l u t e and u n i f o r m c o n v e r g e n c e o f t h e s e r i e s \ i l i^ T x . -l REMARK The e x p a n s i o n s (2.4.17) and (2.4.23) a r e c l e a r e r and seem more c o n v e n i e n t t o a p p l y t h a n F r e d h o l m ' s f o r m u l a e , b u t t h e f u n c t i o n a l a n a l y t i c t e c h n i q u e s o f t e n may n o t l e a d t o t h e quan-t i t a t i v e r e s u l t s p r o v i d e d by t h e c l a s s i c a l t e c h n i q u e s . 2.4' THE KARHUNEN-LOEVE EXPANSION The a p p r o a c h w i l l be t o expand t h e f u n c t i o n f ( t ) i n a p a r -t i c u l a r k i n d o f s e r i e s . T h i s i s a n a l o g o u s t o a F o u r i e r s e r i e s e x p a n s i o n i n t e r m s o f s i n e and c o s i n e f u n c t i o n s w i t h a p p r o p r i a t e w e i g h t i n g c o e f f i c i e n t s ; a good r e f e r e n c e i s H e l s t r o m (126 pp.124-133) . To be more p r e c i s e , we d e s i r e an e x p a n s i o n i n t e r m s o f a s e t o f o r t h o n o r m a l f u n c t i o n s <f>^  (t) w i t h u n c o r r e l a t e d w e i g h t i n g c o e f f i c i e n t s r ^ . R e c a l l t h a t a s e t o f f u n c t i o n s {<j>i (t) : i = l , 2 , . •. . ) d e f i n e d o v e r t h e i n t e r v a l (0,T) i s o r t h o n o r m a l on t h i s i n t e r v a l i f m f l , i = j / *. (t) ** . (t)dt=. 0 1 3 lo, I f t h e o r t h o n o r m a l s e t i s c o m p l e t e , t h e n a s q u a r e - i n t e g r a b l e f u n c t i o n f ( t ) , d e f i n e d on (0,T) may be r e p r e s e n t e d as (2.4.1)' f (t)=2_r i* i(t) The c o e f f i c i e n t s r ^ may be d e t e r m i n e d by m u l t i p l y i n g e a c h s i d e o f (2.4.1)' by 4>*j (t) and i n t e g r a t i n g o v e r t h e i n t e r v a l ( 0 , T ) , / J f (t) <fr*j (t)dt=) /Q ri«|)i(t)+*j (t)dt. S i n c e t h e f u n c t i o n s a r e o r t h o n o r m a l , t h e r i g h t - h a n d s i d e i s z e r o e x c e p t f o r j = i , and so (2.4.2)' r ± = / Q f (t) * * k ( t ) d t . 3 4 F o r i l l u s t r a t i v e p u r p o s e s we c o n s i d e r t h e homogeneous i n t e g r a l e q u a t i o n where K ( t - s ) i s t h e k e r n e l and where we assume t h a t 4>^(t) i s an e i g e n f u n c t i o n , . and X^ and e i g e n v a l u e . We know t h a t f o r a p o s i t i v e d e f i n i t e k e r n e l , t h e e i g e n -v a l u e s a r e s t r i c t l y p o s i t i v e . F u r t h e r m o r e , i f K ( t , s ) i s p o s i t i v e d e f i n i t e , t h e e i g e n f u n c t i o n s f o r m a c o m p l e t e s e t . By d e f i n i t i o n , t h e s e t o f f u n c t i o n s <}>^ (t) i s s a i d t o be c o m p l e t e i f t h e o n l y f u n c t i o n g ( t ) s a t i s f y i n g f o r a l l i i s t h e f u n c t i o n g ( t ) = 0 . I n e s s e n c e t h i s means t h a t i f a f u n c t i o n g ( t ) n o t i d e n t i c a l l y z e r o , i s o r t h o g o n a l t o t h e s e t o f f u n c t i o n s <j>^(t), t h e n g ( t ) i s a l s o an e i g e n f u n c t i o n . The s i g n i f i c a n c e o f t h e c o m p l e t e n e s s p r o p e r t y o f t h e f u n c -t i o n s <f>^(t) i s t h a t t h e s e r i e s e x p a n s i o n (2.4.1) c o n v e r g e s i n t h e mean t o f ( t ) . C o n v e r g e n c e i n t h e mean means / - g ( t ) cf, i(t)dt=0 T n i = l 35 The s e r i e s e x p a n s i o n o f e q u a t i o n (2.4.1)' w i t h t h e f u n c t i o n s cj)^(t) c h o s e n a s t h e e i g e n f u n c t i o n s o f t h e i n t e g r a l e q u a t i o n (2.4.3) ' i s t h e K a r h u n e n - L o e v e e x p a n s i o n . Some o t h e r f a c t s w i l l be o f e v e n t u a l i n t e r e s t . One i s M e r c e r ' s t h e o r e m w h i c h s t a t e s i f K ( t , s ) i s p o s i t i v e s e m i d e f i n i t e i t c a n be e x p a n d e d i n t e r m s o f 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 a s o  (2.4.4) ' K ( t , s ) = > X <j) (t)<f>* . (s) . L 1 1 3 i = l . • I n some c i r c u m s t a n c e s i t i s c o n v e n i e n t t o u s e t h e i n v e r s e k e r n e l K "*"(u,v) d e f i n e d by (2.4.5/ / j ! J K ~ 1 ( t , u ) K ( u , v ) d u = 6 ( t - v ) , 0<t, v<T. The i n v e r s e k e r n e l h a s an e x p a n s i o n i n t e r m s o f <!>^(t) and X^ g i v e n by 0 0 (2.4.6)' K _ 1 ( t , u ) = ^ 4»i(t) <|)*i(u)-I=T 1 The u s e f u l n e s s o f t h e i n v e r s e k e r n e l i s , h o w e v e r , m a i n l y a n a l y t i c a l . I n p r a c t i c e , I t may be d i f f i c u l t t o d e t e r m i n e . I n summary, we may r e p r e s e n t a s q u a r e - i n t e g r a b l e f u n c t i o n o f a random p r o c e s s o v e r a f i n i t e o b s e r v a t i o n i n t e r v a l i n a s e r i e s o f o r t h o n o r m a l f u n c t i o n s . The c o e f f i c i e n t s may be made 36 t o have t h e u s e f u l p r o p e r t y o f b e i n g u n c o r r e l a t e d . REMARK Homogeneous F r e d h o l m i n t e g r a l e q u a t i o n s p l a y an i m p o r -t a n t roi© i n c o m m u n i c a t i o n t h e o r y . As a t h e o r e t i c a l t o o l , t h e y a r e u s e d i n d e t e r m i n i n g t h e K a r h u n e n - L o e v e e x p a n s i o n t h e o r y o f . a random p r o c e s s . A d i f f i c u l t a s p e c t o f t h i s t h e o r y , however, i s t h a t i t i s o f t e n d i f f i c u l t t o f i n d s o l u t i o n s t o t h e e q u a t i o n s i n v o l v e d . 2.5 GRANDELL TYPE SOLUTION In ( 1 0 ) , G r a n d e l l u s e d a method s i m i l a r t o t h a t u s e d i n c h a p t e r 1, t o o b t a i n t h e b e s t l i n e a r e s t i m a t o r . I n t h i s s e c t i o n we b r i e f l y r e v i e w G r a n d e l l 1 s method and make some comments t h e r e o n . W i t h t h e a s s u m p t i o n t h a t o n l y t h e c o v a r i a n c e i s known, G r a n d e l l a d o p t s as a c r i t e r i o n f o r t h e c h o i c e o f t h e e s t i m a t e , t h e q u a d r a t i c mean. Thus he s e e k s e s t i m a t e s o f t h e t y p e (2.5.1) X * - ( t ) = a ( t ) + / J e t ( s ) d ( N ( s ) - s ) . where t h e f u n c t i o n s a ( t ) and 3 (s) (as i n c h a p t e r 1) a r e d e t e r m i n e d so t h a t E { A * ( t ) - X ( t ) } 2 i s m i n i m i z e d . 3 7 By v i r t u e o f t h e K a r h u n e n - L o e v e e x p a n s i o n and M e r c e r ' s t h e o r e m o f t h e p r e c e d i n g s e c t i o n , t h e c o v a r i a n c e k e r n e l K ( s , t ) c a n be r e p r e s e n t e d by (t) . (s) (2.52) K ( s , t ) = x i = l t h e and <j>^  b e i n g 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 , r e s p e c t i v e l y , 2 By a s s u m p t i o n a (t) and 3 t(.s) a r e f i n i t e and E{ X * ( t ) - X (t) } e x i s t s , so 3 t ( s ) has t h e f o r m (2.5.3) 3 t ( s ) = ^ ) B i < j ) i ( s ) + b t ( s ) i = l where b t ( s ) i s o r t h o g o n a l t o ty ^ , cf> ^  t • • • • We now p r o c e e d t o m i n i m i z e t h e e x p r e s s i o n (2.5.4) E ( A * ( t ) - X ( t ) } 2 = E { ( a ( t ) - l ) + / g B f c ( s ) d N ( s ) - s ) - ( N t ) - 1 ) } 2 oo oo O g ^ = d ( a ( t ) - l ) 2 + ^ ) e i 2+/Qb 2 t(s)ds+^> i = l i=l. 1 oo O oo — * . 2 ( t ) _ 3 . + .(t) + \ — -2 y i A - " i i = l " i = l S i n c e (2.5.4) i s b e i n g m i n i m i z e d , a ( t ) = l and b t ( s ) = 0 . Thus 0 0 2 (B,-*, ( t ) ) Z E U * ( t ) - X (t) } 2=^ { 3 . 2 + - i — - } 38 so t h a t &E{A»(t)-Ut) >2 - ,« 2(@ 1-^ 1(t) ) 38. " 2 3 i + V-x x s e t t i n g t h i s e q u a l t o z e r o we o b t a i n i . 1+y. 1 w h i c h c o r r e s p o n d s t o a minimum s i n c e 3 2 E { A * ( t ) - A (t) } 2 = 2 + 2 _ > Q •30. 2 U i I t f o l l o w s t h a t ( s u b s t i t u t i o n i n ( 2 . 5 . 3 ) ) , <i>, (t)4> . (s) (2.5.5) 8^(s)=) - V r r - ^ , t w / / 1+y. and (2.5.1) i n t u r n becomes -<f>, (t)<j>, (s) (2.5.6) A*(t)=l+/g^> X 1 + y 1 d ( N ( s ) - s ) ; and 0 0 2 2 \ T ^ i ( t ) (2.5.7) E { A * (t) - A ( t ) } = > - i -i = l 1+y i I f we now m u l t i p l y e x p r e s s i o n s (2.5.2) and ( 2 . 5 . 5 ) , and i n t e g r a t e o v e r t h e i n t e r v a l ( 0 , T ) , we o b t a i n 3 9 CO 0 0 <J). (t) <|>. (u) <f) . (u) <|>. (s) r p y v v p . V W v p . V " / v p . v p -/ j 3 t ( u ) K ( u , s ) d u = / J ^ _ > d u i = i D=l 1 3 <t>i (t) (s) i = l ( l + y i ) P ± (JK (t) < p i (s) ^ - — < p i (t) ( p i (s) y. / 1+y. i = l i = l = K ( s , t ) - 6 t ( s ) Thus 6 t ( s ) i s a s o l u t i o n , o f t h e i n t e g r a l e q u a t i o n (2.5.8) B T ( S ) + / Q K ( U , S ) B T ( u ) d u = K ( t , s ) REMARK A l t h o u g h e q u a t i o n (2.5.8) i s a F r e d h o l m i n t e g r a l e q u a t i o n o f t h e s e c o n d k i n d , G r a n d e l l f a i l s t o n o t e t h i s f a c t i n (10) . G r a n d e l l shows t h a t 3 t ( t ) i s a u n i q u e s o l u t i o n . F u r t h e r , i n t h e c a s e o f a d e g e n e r a t e c o v a r i a n c e k e r n e l , t h e r e e x i s t s an u n b i a s e d e s t i m a t e o f X ( t ) . On t h e o t h e r hand, i f t h e number o f e i g e n f u n c t i o n s i s i n f i n i t e , i t i s i m p o s s i b l e t o f i n d an u n b i a s e d e s t i m a t e o f X ( t ) w h i c h i s u s e f u l f o r e v e r y t i n ( 0 , T ) . The i n t e r e s t e d r e a d e r may r e f e r t o G r a n d e l l ' s p a p e r . To i l l u s t r a t e t h i s t h e o r y , we c o n s i d e r an example. EXAMPLE ( 2 . 5 ) . Suppose t h e k e r n e l i s g i v e n as K ( s , t ) = -' p 40 As was n o t e d i n t h e p r e v i o u s s e c t i o n , a l l e i g e n v a l u e s , y (t) = y / g K ( t , s ) (f)(s)ds a r e p o s i t i v e . Thus i n t h i s example we g e t t h e e q u a t i o n (2.5.9) • (t) = yi/ J c j ) ( s ) d s , t h e o n l y s o l u t i o n b e i n g <|> (t) = C, a c o n s t a n t . The o r t h o g o n a l i t y r e q u i r e m e n t o f t h e <J>^(t) i m p l i e s T 2 2 / J c T ( t ) = Tp = 1, so t h a t * (t) = — S u b s t i t u t i o n i n e q u a t i o n (2.5.9) g i v e s /T P so t h a t Now t o g e t t h e b e s t l i n e a r e s t i m a t o r , A*, we use e q u a t i o n (2.5.6) , t h u s X* (t) = .1 + 1/T , , 0 1+(P/T) d { N ( s ) ~ s } f r o m w h i c h we o b t a i n t h e b e s t l i n e a r e s t i m a t e a s : (2.5.10) X* - P + N ( T ) P+T REMARK We n o t e t h a t i n t h e above example t h e b e s t l i n e a r e s t i -mate i s d e p e n d e n t o n l y on t h e c o v a r i a n c e f u n c t i o n and so t h e e s t i m a t e , ( 2 . 5 . 1 0 ) , i s t h e b e s t l i n e a r e s t i m a t e f o r e v e r y p r o c e s s w i t h t h e g i v e n c o v a r i a n c e f u n c t i o n . 2 . 6 SOME OTHER METHODS We now t u r n t o o t h e r s o l u t i o n t e c h n i q u e s w h i c h a p p e a r m a i n l y i n t h e e n g i n e e r i n g l i t e r a t u r e as d e s c r i b e d by Van T r e e s ( 1 9 ) , H e l s t r o m (12b) and o t h e r s . I n t e g r a l e q u a t i o n s a r e f r e q u e n t l y e n c o u n t e r e d i n t h e t h e o r y o f s i g n a l d e t e c t i o n and e s t i m a t i o n . Two s u c h e q u a t i o n s e n c o u n -t e r e d i n c o n n e c t i o n w i t h t h e d e t e c t i o n o f s i g n a l s i n n o n w h i t e n o i s e a r e : (2.6.1) / j F ( t - s ) f ( s ) d s = A f ( t ) 0<t<T where f ( t ) and A a r e t o be d e t e r m i n e d , and (2.6,2) / j ! J F ( t - s ) f ( s ) d s = g ( t ) where f ( t ) i s t o be d e t e r m i n e d . By a s s u m p t i o n t h e c o v a r i a n c e f u n c t i o n i s t h e sum o f p a r t s c o r r e s p o n d i n g t o w h i t e and non-w h i t e n o i s e , so N 0 R ( t - s ) = ( t - s ) + K ( t - s ) and s u b s t i t u t i n g t h i s i n t o e q u a t i o n (2,6.2) p r o d u c e s t h e i n t e g r a l e q u a t i o n N 0 T (2.6.3) ( t ) + / j K ( t - s ) f ( s ) d s = g ( t ) 0<t<T w h i c h we i m m e d i a t e l y r e c o g n i z e as a F r e d h o l m i n t e g r a l e q u a t i o n o f t h e s e c o n d k i n d . As m e n t i o n e d i n s e c t i o n s (2.3) and ( 2 . 4 ) , a s o l u t i o n t o (2.6.3) w i l l g e n e r a l l y e x i s t u n l e s s (-N n/2) i s an e i g e n v a l u e 4.3 o f t h e homogeneous i n t e g r a l e q u a t i o n ( 2 . 6 . 1 ) . S i n c e K ( t - s ) i s a p o s i t i v e - d e f i n i t e k e r n e l , t h e i n t e g r a l e q u a t i o n c a n n o t have a n e g a t i v e e i g e n v a l u e , and t h e r e i s no t r o u b l e a b o u t t h e e x i s t e n c e o f a s o l u t i o n . 2.6.1 A p p l i c a t i o n s o f F o u r i e r T r a n s f o r m s I n numerous a p p l i c a t i o n s , i n t e g r a l e q u a t i o n s o f t h e t y p e o  (2.6.4) / _ O O K ( t - s ) f ( s ) d s = g ( t ) a r e e n c o u n t e r e d . The i n t e g r a l on t h e l e f t i s a c o n v o l u t i o n . I f K ( T ) , g ( x ) e I>2 I"- 0 0 / 0 0] ' w e c a n u s e the Fourier transform o f b o t h s i d e s t o o b t a i n (2.6.5) / 2 n G ( K ) G ( f ) = G ( g ) , and so (2.6.6) G ( f ) = , ./2JIG(K) p r o v i d e d G(K) does n o t v a n i s h . I f t h e r i g h t s i d e , as a f u n c t i o n o f u, i s i n L 2 [ - O o , c o ] we f i n a l l y o b t a i n ( 2 . 6 . 7 ) f . - i - G M l j l f ) • EXAMPLE 2.6.2. C o n s i d e r (2.6.0) f ( t ) - A / M £ I 1 S l f ( s ) d s = g ( t ) By a d i r e c t i n t e g r a t i o n G ( e - j t | = {VA l + u 2 s o t h a t G ( f ) - A / 2 n ' - ^ < 3 ( f ) = G(g) l + u and so , 2 ( 2 . 6 . 9 ) : G ( f ) = G(g) l + u -2A w h e r e we r e q u i r e ^K<\% Then _ _ + ,v/2/n . 1,°° p - i u t , l + u -2A - 2 0 1 l + u -2A and f o r ; > i < ' j ' w e o b t a i n t h e s o l u t i o n (2.6.10) ' f = ^ w h e r e y = /1-2A REMARK T e c h n i q u e s and p r o p e r t i e s o f L a p l a c e , H a n k e l , and M e l l i n t r a n s f o r m s ( H e l s t r o m ( 1 2 b ) ) c a n be d e r i v e d by r e l a t i n g them t o F o u r i e r t r a n s f o r m s ; we do n o t , h o w e v e r , i n t e n d t o d i s -c u s s t h e s e t r a n s f o r m s i n t h e p r e s e n t w o r k . 2.6.3 E q u a t i o n s w i t h S e p a r a b l e K e r n e l s I n t h i s c a s e we may expand t h e k e r n e l i n t h e f o r m n (2.6.11) K ( t , s ) = y Aj ^ ( t ) ^ ( s ) , j = l w here A and <f>j (t) a r e t h e 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 o f K ( t , s ) . Thus we c a n use t h e t e c h n i q u e s o f s e c t i o n s (2.4) and (2.5) t o s o l v e t h e g i v e n i n t e g r a l e q u a t i o n . REMARK L a c k o f s p a c e p r e v e n t s us f r o m d i s c u s s i n g a l l t h e numerous methods w h i c h a r e f o u n d i n t h e l i t e r a t u r e . A n o t e -w o r t h y o m i s s i o n f o r f i n i t e o b s e r v a t i o n and n o n s t a t i o n a r y p r o -c e s s e s , i s t h e s t a t e - v a r i a b l e f o r m u l a t i o n o f Kalman and Bucy (6b) w h i c h l e a d s t o a c o m p l e t e s o l u t i o n . 46 CHAPTER 3 A SURVEY OF APPROXIMATE METHODS FOR THE SOLUTION OF FREDHOLM INTEGRAL EQUATIONS OF THE SECOND KIND 3.1 INTRODUCTION S i n c e e x a c t s o l u t i o n s o f t h e F r e d h o l m i n t e g r a l e q u a t i o n a r e u s u a l l y d i f f i c u l t t o o b t a i n , i t i s n a t u r a l t o c o n s i d e r , as we do h e r e , t h e p o s s i b i l i t y o f f i n d i n g a p p r o x i m a t e s o l u t i o n s . I n a p r i v a t e c o m m u n i c a t i o n , P r o f e s s o r James V a r a h s u g g e s t e d t h e use o f q u a d r a t u r e methods o f t h e f o r m i = l i . e . , t h e i n t e g r a l i s r e p r e s e n t e d by a w e i g h t e d sum o f v a l u e s o f t h e i n t e g r a n d a t a f i n i t e number o f p o i n t s t ^ . I n t h i s c h a p t e r we e x p l o r e t h i s and o t h e r a p p r o x i m a t e methods; i n g e n e r a l , t h e n u m e r i c a l methods a r e e x p l a i n e d w i t h o u t p r o o f o f t h e i r v a l i d i t y , f o r w h i c h we c a n r e f e r t o t h e c i t e d l i t e r a t u r e ( 6 ) , (12) , (14) . I n s e c t i o n (3.2) we s h a l l d e a l w i t h s i m p l e q u a d r a t u r e r u l e s ; i n p a r t i c u l a r t h e T r a p e z i u m and Simpson's r u l e s . We s h a l l b r i e f l y c o n s i d e r g e n e r a l i z a t i o n s t o t h e s e r u l e s i n s e c t i o n ( 3 . 3 ) , and t h e n c o n s i d e r t h e method o f c o l l o c a t i o n i n s e c t i o n ( 3 . 4 ) . I n s e c t i o n (3.5) we t a c k l e t h e q u e s t i o n o f e r r o r s and t h e s u b s e q u e n t s u g g e s t e d improvements on t h e methods d i s c u s s e d t h u s f a r . F i n a l l y , n (3.11) 4 7 i n s e c t i o n (3.6) we s h a l l m e n t i o n o t h e r methods w h i c h a r e n o t d i s c u s s e d i n d e t a i l h e r e . Then we s h a l l end t h e c h a p t e r w i t h a summary and some c o n c l u d i n g r e m a r k s . 3.2 QUADRATURE RULES We summarize t h e p r i n c i p a l f o r m u l a e o f t y p e (3.1.1) w i t h t h e i r e r r o r t e r m s . L e t us w r i t e I ( f ) f o r t h e i n t e g r a l / f ( t ) d t . c l F o r t h e r e p e a t e d f o r m s , t h e i n t e r v a l [ a , b j i s d i v i d e d i n t o n e q u a l s t e p s o f l e n g t h h, so t h a t h = ( b - a ) / n , and t h e f o r m u l a i s a p p l i e d o v e r s u b - I n t e r v a l s . The e r r o r depends on some h i g h o r d e r d e r i v a t i v e o f f ( t ) a t a p o i n t £ i n [a,bj , where we assume t h a t t h i s d e r i v a t i v e i s c o n t i n u o u s . The f o l l o w i n g t h r e e f o r -mulae use e q u a l l y s p a c e d p o i n t s , so we l e t t ^ = a + i h . i ~ 1 ^  2 / • • • / n • (3.2.1) The M i d - P o i n t R u l e n where I (f) = h ^ f ( t i _ j s ) - + R 1 1 2 II R-^  = 2-^(b-a)h f " ( £ ) , i s t h e r e m a i n d e r t e r m (3.2.2) The T r a p e z i u m R u l e n-1 1 ( f ) = |h{f(a)+2^> f ( t i ) + f (b) }+R 2 i = l 48 where R 2 - ™ ( b - a ) h 2 f " (5) (3.2.3) Simpson's R u l e 1 1 - i 2 n 2 ^ } 1 ( f ) = | h { f ( a ) + 4 ^ ~ f ( t 2 i _ 1 ) + 2 ^ > f ( t 2 i ) + f (b) }+R 3 i = l i = l where R 3 = - y i o ( b " a ) h 4 f l V ( 5 ) REMARK I n f o r m u l a ( 3 . 2 . 3 ) , n must be e v e n . The above f o r m u l a e a l l u s e e q u a l l y s p a c e d p o i n t s , and h i g h e r - o r d e r f o r m u l a e o f t h e same t y p e may be d e r i v e d . However, t h e above t h r e e a r e t h e most i m p o r t a n t i n p r a c t i c e . (3.2.4) F o r m u l a t i o n o f D i s c r e t e E q u a t i o n s F r e d h o l m e q u a t i o n s o f t h e s e c o n d k i n d c a n be a p p r o x i m a t e d i n a s t r a i g h t f o r w a r d way by means o f q u a d r a t u r e f o r m u l a e . The g e n e r a l c a s e i s now c l e a r . We c h o o s e any q u a d r a t i v e f o r m u l a n / b f ( t ) d t = >^ w i f ( t i ; I=T i n v o l v i n g t h e n p o i n t s t ^ and t h e c o r r e s p o n d i n g w e i g h t s w^. The g e n e r a l F r e d h o l m ' i n t e g r a l e q u a t i o n o f t h e s e c o n d k i n d 4.9 (3.3.1) / b K ( t , s ) f ( s ) d s + g ( t ) = f ( t ) cL i s t h e n r e p l a c e d by a s y s t e m o f n l i n e a r a l g e b r a i c e q u a t i o n s f o r t h e unknowns f ( t ^ ) ( t o i n d i c a t e t h a t t h i s i s o n l y an a p p r o x i m a -t i o n t o t h e i n t e g r a l e q u a t i o n f ( t ) has b e en r e p l a c e d by f ( t ) ) . W r i t t e n i n m a t r i x f o r m , t h i s s y s t e m o f e q u a t i o n s becomes (3.3.2) ( I - K D ) f = g, where t h e m a t r i x K has t h e e l e m e n t s K . . = K ( t . , t . ) and D has t h e 13 1 3 d i a g o n a l e l e m e n t s w^. The s o l u t i o n o f t h e s e e q u a t i o n s , f ^ , rep-r e s e n t s t h e a p p r o x i m a t e v a l u e s o f f ( t ) a t t h e p o i n t s t = t ^ . We i l l u s t r a t e t h e above methods by two s i m p l e examples EXAMPLE 3.2.1 C o n s i d e r f ( t ) = t + / j K ( t , s ) f ( s ) d s where t h e k e r n e l i s o f t h e f o r m t ( l - s ) i f t<s K ( t , s ) = { s ( 1 _ t ) i f t > s ( H i l d e r b r a n d (13) n o t e s t h a t t h i s k e r n e l i s w e a k l y s i n g u l a r . ) Take t ^ = 0, ^2 = \' ^5 ~ ^ a n d n ~ \ ( t r a p e z i u m r u l e ) Then K±± = t ± ( 1 - t ^ , K±2 = t 1 ( l - t 2 ) , . . . , K 3 5 = t j ( l - t 5 ) . . . i . e . , K ± j = t i ( l - t j ) , i < j , i , j = 1,...,5. 50 so t h a t K 0 0 0 0 0 0 3 / i d 1/8 i / 1 6 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 = d i a g h ( l / 2 , 1, 1, 1, 1/2) Hence f r o m ( I - K D ) f = g w i t h g1= [o 1/4 1/2 3/4 l] , we o b t a i n , t h e s y s t e m o f e q u a t i o n s 61/6 4 f 2 - l / 3 2 f 3 - l / 6 4 f 4 - l / 3 2 f 2 + 1 5 / 1 6 f 3 - l / 3 2 f 4 - l / 6 4 f 2 - l / 3 2 f 3 + 6 1 / 6 4 f 4 = 0 = 1/4 = V 2 = 3/4 = 1 The s o l u t i o n t o t h i s s e t o f e q u a t i o n i s f = -K 0.0000 f 2 0.2943 *3 0.5702 *4 0.8104 f 5 1.0000 -EXAMPLE 3.2.2 Suppose we a r e r e q u i r e d t o f i n d an a p p r o x i m a t e s o l u t i o n o f t h e i n t e g r a l e q u a t i o n 51 /JK(+.,S) f ( s ) d s + g ( t ) = f ( t ) . We u s e Simpson's r u l e t o a p p r o x i m a t e t h e i n t e g r a l i n t h e f o r m / J u ( t ) d t = l / 6 { u ( o ) + 4 u ( l / 2 ) + u ( l ) } + E 2 w h e r e E 2 r e p r e s e n t s t h e e r r o r o r r e m a i n d e r t e r m . N e g l e c t i n g t h i s r e m a i n d e r f o r t h e moment, we o b t a i n t h e r e l a t i o n l / 6 ( K ( t , o ) f ( o ) + 4 K ( t , | ) f ( | ) + K ( t , l ) f (1) } + g ( t ) = f ( t ) . I n t h i s e q u a t i o n we w r i t e t = 0, j and 1 s u c c e s s f u l l y , a nd o b t a i n 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 ) + 4 K ( | , | ) f ( | ) + K ( | ) , l ) f (1) } + g ( l / 2 ) = f ( l / 2 ) l / 6 { K ( l , o ) f ( o ) + 4 K ( l , | ) f ( l / 2 ) + K ( l , l ) f (1) }+g(D = f ( l ) w h i c h may be w r i t t e n as ( I - K D ) f = g w i t h D = d i a g ( l / 6 , 4/6, 1 / 6 ) . We c a n , t h e r e f o r e , s o l v e f o r t h e unknown v a l u e s f , , f _ and f _ 52 3.3 GENERALIZED QUADRATURE Assume f e c [ a , b ] and <j> (t) i s L e b e s q u e i n t e g r a b l e on f a , b j . C o n s i d e r t h e p r o b l e m o f n u m e r i c a l l y i n t e g r a t i n g f (t) <\> (t) o v e r [ a , b j . The a p p r o a c h h e r e i s t o u s e p i e c e w i s e p o l y n o m i a l i n t e r -p o l a t i o n t o d e v e l o p t h e g e n e r a l i z a t i o n s o f t h e t r a p e z o i d a l r u l e and Simpson's r u l e . 3.3.1 The G e n e r a l i z e d T r a p e z o i d a l R u l e As i n t h e p r e c e d i n g s e c t i o n , l e t h = ( b - a ) / n and d e f i n e t ^ = a + i h , i = 0, 1, n. L e t f n ( t ) be t h e p i e c e w i s e l i n e a r i n t e r p o l a t i o n f u n c t i o n o f f ( t ) a t t h e n o d a l p o i n t s t Q , t-^, t ; i . e . , (3.3.1) f n ( t ) = i { ( t ± - t ) f ( t ± _ 1 ) + . ( t - t i _ 1 ) f ( t ± ) }, t i _ 1 < t $ t ± i = l , . . . , n . S u b s t i t u t i n g (3.3.1) i n t o / b f n (t) <j> (t) d t , we o b t a i n n (3.3.2) / b f n ( t ) <j)(t)dt = y i a ± f ( t i _ 1 ) + 6 i f ( t ± ) } i = l where a . and 3. a r e g i v e n by: x 1 3 2 (3.3.3) - i ^ i _ i ( v t ) t ( t , a t i 6 . . ( t _ t • ) m ) d t 53 REMARKS F o r t h i s f o r m o f q u a d r a t u r e , i t i s n e c e s s a r y t o e v a l u -a t e t h e i n t e g r a l s o f <j> (t) and tty (t) o v e r a r b i t r a r y i n t e r v a l s . S i n c e u s u a l l y t h e s i n g u l a r i t y o f an i n t e g r a n d c a n be i s o l a t e d as a s i m p l e f u n c t i o n , t h e s e i n t e g r a t i o n s may, i n g e n e r a l , n o t be d i f f i c u l t . We a l s o n o t e t h a t when o> (t) =1, we o b t a i n t h e o r d i n a r y t r a p e z o i d a l r u l e s i n c e i n t h i s c a s e a . = B. = £ 1 l 2 3.3.2 The G e n e r a l i z e d Simpson's R u l e Here we l e t h = ( b - a ) / 2 n , n > l , and d e f i n e f as t h e p i e c e w i s e q u a d r a t i c i n t e r p o l a t i o n f u n c t i o n t o f on t ^ , t ^ , t 2 n ; ^n b e i n g q u a d r a t i c on e a c h s u b i n t e r v a l ^2i-2' t 2 i ^ ' """=^' n * The q u a d r a t u r e f o r m u l a beomces: n (3.3.4) / b f n ( t ) . < f ) ( t ) d t = ^ t a i f ( t i _ 2 ) + B i f ( t 2 i _ 1 ) + Y i f ( t 2 i ) } i = l where a i = z V t 2 , 1 ( t - ^ . X t - t ^ J M t J d t , 2h 2 i - 2 (3.3.5) &i = Z^fl2i ( t - t 2 i _ 2 ) ( t - t 2 i ) * ( t ) d t h 2 i - 2 Y. = — / ^ 2 i ( t - t . ) ( t - t . ) <j> ( t ) d t 2h 2 i - 2 We s h a l l s e e t h e u s e f u l n e s s o f t h e s e g e n e r a l i z a t i o n s when we a r e d e a l i n g w i t h e r r o r a n a l y s i s i n s e c t i o n ( 3 . 5 ) . 3.4 COLLOCATION METHOD The above d i r e c t methods o f t e n have t h e d i s a d v a n t a g e o f c o n c e a l i n g anomalous e r r o r s . I n a s e n s e , t h e y a r e t o o g e n e r a l , and need t o be m o d i f i e d i n o r d e r t o t a k e a d v a n t a g e o f any s p e c i a l f e a t u r e s ' w h i c h t h e p a r t i c u l a r e q u a t i o n may p o s s e s . F o l l o w i n g G r e e n ( 1 2 ) , l e t <j>^ , t)>2, $n f o r m a s e t o f l i n e a r l y i n d e p e n d e n t f u n c t i o n s on [ a , b ] . We assume t h a t t h e <JK 1 s f o r m p a r t o f an o r t h o n o r m a l b a s i s f o r L 2 [a.,b] . Now c o n -s i d e r t h e i n t e g r a l e q u a t i o n (3.4.1) f ( t ) = g ( t ) + A / b K ( t , s ) f ( s ) d s , and assume an a p p r o x i m a t e s o l u t i o n o f t h e f o r m n (3.4.2) f (n) i = l where t h e C.'s a r e u n d e t e r m i n e d c o n s t a n t c o e f f i c i e n t s . x S u b s t i t u t i n g (3.4.2) i n t o (3.4.1) y i e l d s n n i = l i = l where e d e n o t e s t h e e r r o r i n v o l v e d as a r e s u l t o f a s s u m i n g t h e 55 s o l u t i o n f ( n ) - W e a i m t o c h o o s e t h e c o e f f i c i e n t s i n s u c h a way as t o m i n i m i z e t h e e r r o r e n I n t h i s v e i n we c h o o s e a s e t o f p o i n t s t ^ , t 2 , t f i , and d e t e r m i n e t h e c o e f f i c i e n t s by t h e r e q u i r e m e n t t h a t e ^  (t) s h o u l d v a n i s h a t e a c h o f t h e s e p o i n t s . L e t and b . = g ( t . ) 3 3 a.. = 4>. ( t . ) - A / b K ( t . ,s) <!>. ( s ) d s , x j ± j a j x s u b s t i t u t i n g t h e s e i n t o t h e e x p r e s s i o n ( 3 . 4 . 3 ) , we o b t a i n a s y s t e m o f n l i n e a r e q u a t i o n s : n (3.4.4) 2 a i j c i = b j j = 1, 2, n i = l f r o m w h i c h t h e C^'s a r e t o be d e t e r m i n e d , Now t o s o l v e ( 3 . 4 . 4 ) , we c h o o s e t h e p o i n t s t j f r o m t h e n u m e r i c a l d a t a a t hand, and t h e numbers a.. a r e o b t a i n e d by u s e o f a q u a d r a t u r e f o r m u l a . C o n d i t i o n s f o r s o l v a b i l i t y : ( i ) The d e t e r m i n a n t o f t h e s y s t e m (3.4.4) must be n o n - z e r o . ( i i ) A t l e a s t one o f t h e b . ' s must be n o n - z e r o . 3 I f t h e s e c o n d i t i o n s a r e s a t i s f i e d , t h e s o l u t i o n o f (3.4.4) g i v e s f i n t h e f o r m o f a p o l y n o m i a l a p p r o x i m a t i o n . ' We now g i v e an example. \ 56 EXAMPLE 3.3.1 C o n s i d e r t h e e q u a t i o n (3.4.5) f ( t ) = t + / j ! j K ( t , s ) f ( s ) d s w i t h s s< t K ( t ' s ) = { t s>t Suppose we have t h e p o i n t s . • ' _ n 1 1 3 , t ± / s i u, 4 , 2 , 4 , i . L e t us assume a s o l u t i o n o f t h e f o r m f ( 3 ) ( t ) = c 1 + c 2 t + c 3 t 2 w i t h t h e c o n d i t i o n t h a t t h e e r r o r s h o u l d v a n i s h a t t h e p o i n t s t-^=0, t 2= 2 :- t 3 = l . We u s e t h e a n a l y t i c f o r m o f t h e k e r n e l t o s i m p l i f y t h e c o m p u t a t i o n . The c o e f f i c i e n t s a . . a r e g i v e n b y a, . - l - / o J s d s - / ^ t . d s ID 0 j t_. j a n . - t . - / ^ j s 2 d s - / ? ~ t . 2] D 0 j t . j and s d s i D 2 t 3 1 ~> a 3 j = t J~-/"QD S d s - / t t j S ' d s b . = t . D D 57 Thus l l l W^sds-O/Jds = 1 l12 , .1/2 , 1,1 , W 0 s d S _ 2 ; i / 2 d s = 5/8 = u ^ , e t c . Hence t h e s y s t e m (3.4.4) assumes t h e f o r m 120 C.j + 52 C 2+17 C 3 6 C±+ C 2 + 9 C 3 = 0 = 96 = 12 w h i c h l e a d s t o t h e a p p r o x i m a t i o n : f ( 3 ) (t) = 1.988t-0.434t' w h i c h i s a u t o m a t i c a l l y b e s t f o r t h e c h o s e n p o i n t s t=0, where f^=0, f 3 = 0 . 8 3 6 , f^=1.554, and g i v e s t h e s o l u t i o n s 1, f 2 = 0.470; f 4 = 1.247 a t t h e p o i n t s t 2 = ^- and t^=3/4, r e s p e c t i v e l y . Thus t h e a p p r o x i -mate s o l u t i o n t o e q u a t i o n (3.4.5) i s : (3.4.6) 0.000 0.470 0 . 836 1.247 1.554 REMARKS The c o l l o c a t i o n method s u f f e r s f r o m t h e d i s a d v a n t a g e t h a t an e x a c t m a t c h i n g o f t h e s o l u t i o n a t c e r t a i n p o i n t s does n o t c o n t r o l t h e s i z e o f t h e d e v i a t i o n between t h e e x a c t and t h e a p p r o x i m a t i n g s o l u t i o n s a t o t h e r p o i n t s ( u n l e s s we want t o c h o o s e a l l t h e g i v e n p o i n t s w h i c h may n o t be c o m p u t a t i o n a l l y f e a s i b l e ) . 58 A t any r a t e , t h e c o l l o c a t i o n method i s w o r t h c o n s i d e r i n g b e c a u s e a t l e a s t i t i s an improvement on t h e d i r e c t (and w i d e l y used) methods o f s u c c e s s i v e a p p r o x i m a t i o n s and t h e q u a d r a t u r e methods. 3.5 ERROR ANALYSIS 3.5.1 We now c o n s i d e r t h e p r o b l e m o f e s t i m a t i n g t h e a c c u r a c y o f t h e r e s u l t o f t h e c a l c u l a t i o n f o r any o f t h e q u a d r a t u r e methods d i s c u s s e d . I n t h e s e q u e l we s h a l l d e n o t e t h e norm o f any f u n c t i o n 4- (t) by | U| | = /b|<j, (t) | d t d R e t u r n i n g t o t h e p r o b l e m o f e s t i m a t i o n we know t h a t t h e f u n c t i o n f ( t ) i s r e q u i r e d t o s a t i s f y t h e i n t e g r a l e q u a t i o n (3.5.1) J b K ( t , s ) f ( s ) ds+g(t) = f (t) a. The computed f u n c t i o n f ( t ) a c t u a l l y s a t i s f i e s (3.5.2) / b K ( t , s ) f ( s ) d s + g ( t ) = f ( t ) - E ( t ) cl where E (t) d e n o t e s t h e e r r o r t e r m . 5? I f we w r i t e e ( t ) = f ( t ) - f ( t ) f o r t h e e r r o r i n o u r r e s u l t , we f i n d by s u b t r a c t i o n t h a t (3.5.3) / b K ( t , s ) e ( s ) d s = e ( t ) + E ( t ) . ci I n t h e u s u a l way, w r i t i n g K f o r t h e i n t e g r a l o p e r a t o r : K ( f ) = / b K ( t , s ) f ( s ) d s , t h e e r r o r i s g i v e n by (3.5.4) e ( t ) = - ( 1 - K ) - 1 E ( t ) , and s o (3.5.5) || e|| <|| ( 1 - K ) _ 1 | | ||E||. Now ||E|| c a n be e s t i m a t e d . E ( t ) w i l l u s u a l l y be e x p r e s s e d i n t e r m s o f a d e r i v a t i v e , e i t h e r o f t h e c o m p l e t e i n t e g r a n d i n t h e n o n s i n g u l a r c a s e , o r p e r h a p s o f f ( t ) o n l y i n a s i n g u l a r c a s e . U n d e r s u i t a b l e a s s u m p t i o n s a b o u t t h e s m o o t h n e s s o f t h e v a r i o u s t e r m s we c a n e s t i m a t e t h e m a g n i t u d e s o f t h e d e r i v a t i v e s by e x a m i n i n g f i n i t e d i f f e r e n c e s . I t r e m a i n s t o e s t i m a t e || (1-K) | | . A w e l l known r e s u l t [6] , [9] , s t a t e s t h a t (3.5.6) 60 where K i s any bounded l i n e a r o p e r a t o r i n a Banach s p a c e , p r o -v i d e d t h a t | |K | | <1. I f we c h o o s e t h e norm | | f | | =max | f | , o v e r t h e i n t e r v a l [ a , b ] , t h e n t h e norm o f K i s (3.5.7) | | K | | = / b | K ( t , s ) | d s , and (3.5.6) h o l d s p r o v i d e d I I K I We t h e r e f o r e have a r i g o r o u s bound f o r t h e e r r o r i n t h e r e s u l t . 3.5.2 G e n e r a l i z e d E r r o r T u r n i n g t o t h e c o r r e s p o n d i n g e r r o r t e r m o f t h e g e n e r a l i z e d q u a d r a t u r e r u l e s , and u s i n g t h e n o t a t i o n o f s e c t i o n ( 3 . 3 ) , we have (3.5.8) E ( f ) = / b{ f ( t ) - f (t)}4> ( t ) d t , and n a. n (3.5.9) | E n ( f ) | < | U| | || f - f j | • A s s u m i n g t h a t f " ^ i s c o n t i n u o u s , and u s i n g t h e e r r o r f o r -mula f o r L a g r a n g e i n t e r p o l a t i o n on e a c h s u b i n t e r v a l [ t _ . _ ^ , t j ] , A t k i n s o n (3) has shown t h a t t h e g e n e r a l i z e d t r a p e z o i d a l r u l e has t h e e r r o r bound (3.5.10) | E n ( f ) | <^h 2| | f 1 ! L | | |U||. 61 REMARK We n o t e t h a t t h e o r d e r t h a t o f t h e o r d i n a r y t r a p e z o i d a l b elow, t h i s w i l l n o t be t r u e f o r q u a d r a t u r e r u l e s . o f c o n v e r g e n c e i s t h e same as r u l e , b u t , as we s h a l l f i n d t h e g e n e r a l i z a t i o n o f a l l C o n s i d e r i n g Simpson's r u l e , t h e c o r r e s p o n d i n g bound f o r t h e e r r o r i s : (3.5.11) | E (f) I <^|h 3| | f 1 1 1 ! I ' I I * I I . n 27 As h i n t e d e a r l i e r , we f i n d t h a t , u n l i k e t h e c a s e o f t h e t r a p e -3 z o i d a l r u l e , t h e g e n e r a l i z e d Simpson's r u l e o n l y has an h o r d e r o f c o n v e r g e n c e whereas t h e r e g u l a r Simpson's r u l e i s o f a h i g h e r o r d e r h 4 . Thu s , a t l e a s t , t h e g e n e r a l i z e d Simpson's r u l e p r o v i d e s an improvement on t h e o r d i n a r y method. 3.6 SUMMARY AND CONCLUSION We have c o n s i d e r e d i n some d e t a i l t h e t r a p e z i u m and Simp-s o n ' s r u l e s p l u s t h e i r g e n e r a l i z a t i o n s as c o n v e n i e n t q u a d r a t u r e methods f o r t h e n u m e r i c a l a p p r o x i m a t i o n t o t h e s o l u t i o n o f t h e F r e d h o l m i n t e g r a l e q u a t i o n o f t h e s e c o n d k i n d . We f o u n d t h a t t h e method o f c o l l o c a t i o n may y i e l d a s i g n i f i c a n t improvement on t h e r e s u l t s o f t h e f o r m e r methods. 62 We do n o t c l a i m t o have m e n t i o n e d a l l , o r e v e n most, o f t h e a v a i l a b l e methods. I n d e e d , f o r q u a d r a t u r e r u l e s , some o t h e r methods go by v a r i o u s names a s : S i m p l e Gauss r u l e , " T h r e e -E i g h t h s " r u l e , Newton-Cotes r u l e , Radau q u a d r a t u r e , L o b a t t o r u l e , e t c . A l t h o u g h we do n o t have an o p p o r t u n i t y t o d i s c u s s a l l methods h e r e , y e t i t i s w o r t h e x p l o r i n g some f u r t h e r p o s s i b i l i -t i e s . F o r example, t h e r e a r e e x p a n s i o n methods s u c h as t h e G a l e r k i n and t h e R a y l e i g h - R i t z t e c h n i q u e s , a l l o f w h i c h may be f o u n d i n t h e c i t e d l i t e r a t u r e . I n c o n c l u s i o n , we w i s h t o p o i n t o u t t h a t f i n d i n g e x p l i c i t s o l u t i o n s o f i n t e g r a l e q u a t i o n s i s i n g e n e r a l d i f f i c u l t . O n l y i n e x c e p t i o n a l c a s e s c a n s u c h s o l u t i o n s be f o u n d . G e n e r a l l y , v a r i o u s a p p r o x i m a t e and n u m e r i c a l methods have t o be u s e d . We do n o t f e e l , however, t h a t an a p p r o x i m a t e method i s i n any way t o o i n f e r i o r t o one g i v i n g an e x a c t s o l u t i o n . A s o l u -t i o n i n c l o s e d f o r m may c e r t a i n l y be c o n v e n i e n t , b u t i s r a r e l y a b s o l u t e l y n e c e s s a r y , b e a r i n g i n mind t h a t t h e i n t e g r a l e q u a t i o n , c o n s i d e r e d as a model f o r some r e a l s y s t e m , i s a l m o s t c e r t a i n l y o n l y an a p p r o x i m a t e r e p r e s e n t a t i o n o f t h e s y s t e m i n t h e f i r s t p l a c e . 63 CHAPTER 4 COMPARISON: AN EXACT VERSUS AN APPROXIMATE SOLUTION IN A SPECIAL CASE INTRODUCTION W h i t t l e (22) does o b t a i n B a y e s l i n e a r e s t i m a t o r s and equa-t i o n ' (1.2.6) f o r c e r t a i n s p e c i a l c a s e s . W h i t t l e ' s work s u g g e s t s a method o f o b t a i n i n g t h e o p t i m a l l i n e a r e s t i m a t o r u s i n g l i n e a r s m o o t h i n g t e c h n i q u e s . We s h a l l u s e a s l i g h t l y m o d i f i e d f o r m o f W h i t t l e ' s method t o o b t a i n an e x a c t s o l u t i o n t o t h e F r e d h o l m i n t e g r a l e q u a t i o n (1.2.6) and c o n s e q u e n t l y o b t a i n t h e l i n e a r e s t i m a t o r ( 1 . 2 . 1 ) . To i l l u s t r a t e t h e u s e o f t h e t e c h n i q u e s d e v i s e d , a n u m e r i c a l example i s t r e a t e d . The d a t a c o n s i s t s o f s u c c e s s i v e 30-day t o t a l s o f o i l w e l l s d i s c o v e r e d by w i l d c a t e x p l o r a t i o n i n A l b e r t a f o r t h e p e r i o d 1953 t o 1971. C l e v e n s o n and Z i d e k (7) c o n s i d e r t h i s p r a c t i c a l p r o b l e m and compute t h e a p p r o x i m a t e o p t i m a l l i n e a r e s t i m a t o r A . We compare o u r e x a c t and C l e v e n s o n - Z i d e k ' s a p p r o x i m a t e r e s u l t s . 4.1 A BRIEF OUTLINE OF WHITTLE'S DERIVATION OF (1.2.6) E s t i m a t e A ( t ) by a l i n e a r s m o o t h i n g f o r m u l a e so t h a t A (t) L w i l l be e s t i m a t e d by s t a t i s t i c s o f t h e t y p e 64 (4.1.1) X L ( t ) = / b w t ( s ) d N ( s ) , f o r w h i c h , c o n d i t i o n a l l y , g i v e n X (4.1.2) EX (t) = /bw. ( s ) X ( s ) d s , j_i a L. and (4.1.3) v a r X T (t) = / b w 2 ( s ) X ( s ) d s . J_i a t Assume t h a t t h e p a r t i c u l a r f u n c t i o n , X ( t ) , i s a member o f a p o p u l a t i o n o f s u c h f u n c t i o n s , so t h a t t h e X ( t ) ' s have an a p r i o r i d i s t r i b u t i o n o f v a l u e s . The optimum l i n e a r e s t i m a t o r w i l l t h e n be o b t a i n e d by m i n i m i z i n g (4.1.4) C*[t;w] = E [ / w 2 X d s + ( / w X d s ) 2 - 2 X ( t ) / w X d s + X 2 ( t ) ] , where E d e n o t e s e x p e c t a t i o n w i t h r e s p e c t t o p r i o r d i s t r i b u t i o n o f t h e X ( t ) 1 s . I f E X ( t ) = p ( t ) and E [ x ( t ) X ( s ) ] = y ( t , s ) t h e n m i n i m i z i n g (4.1. w i t h r e s p e c t t o w^_(s) y i e l d s t h e i n t e g r a l e q u a t i o n (4.1.5) y ( s ) w f c ( s ) + / p ( s , u ) w ( u ) d u = u ( t , s ) . 65 By a n o r m a l i z a t i o n o f t h e f o r m C t ( s ) s w t(s) /{y (§)/y (t) r ( t , s ) = y ( t , s ) / / y (t) y (s) we g e t (4.1.6) y ( t , s ) = r ( t , s ) / y ( t ) y ( s ) , (4.1.7) w t ( s ) = C t ( s ) /y ( t ) / y (s) ' S u b s t i t u t i n g t h e s e i n t o (4.1.5) we o b t a i n , a f t e r some m a n i p u l a -t i o n , (4.1.8) C t ( s ) + / b r ( s ; u ) ? t ( u ) d u = r ( t ; s ) , w h i c h i s o b v i o u s l y o f t h e r e q u i r e d f o r m . 4.2 THE EXACT SOLUTION We s e e k t h e o p t i m a l l i n e a r e s t i m a t o r o f t h e f o r m (4.2.1) X L ( t ) = y + / ^ T h ( t ; s ) d ( N ( s ) - y s ) where h ( t ; s ) i s t h e s o l u t i o n o f t h e i n t e g r a l e q u a t i o n (4.2.2) y h t ( s ) + / ^ T K ( s ; u ) h t ( u ) d u = K ( t ; s ) , -T<s<T. 66 C o n s i d e r t h e s p e c i a l c a s e where t h e k e r n e l K ( t ; s ) depends o n l y upon ( t - s ) so t h a t (4.2.2) may be w r i t t e n as (4.2.3) y h ( s ) + / b K ( s - u ) h ( u ) d u = K ( t - s ) , a where, f o r c o n v e n i e n c e , we have d r o p p e d t h e s u b s c r i p t , t , o f h and [ a , b ] = [ - T , T ] . We f u r t h e r s p e c i a l i z e o u r r e s u l t s by a s s u m i n g C o v ( A ( t ) , A ( s ) ) = E A ( t ) A ( s ) - E A ( t ) E A ( s ) = y ( t , s ) - y 2 2 / . s 2 -a t - s = a p ( t - s ) = a e 1 \ so t h a t /+. \ 2^,-alt-sL 2 y ( t , s ) = a 6 ' n+y -. S i n c e K ( t , s ) = U ( t ' S ) , / y ( t ) y ( s ) we o b t a i n (4.2.4) K ( t , s ) = K ( t - s ) = —^e H+y W h i t t l e t r e a t s t h e s p e c i a l c a s e where t h e k e r n e l i s o f t h e f o r m K ( t ) = a( Y+ee~ a' t' ) 67 w h i c h i n o u r example means t h a t ay = y and ag = a 2 / y F o l l o w i n g W h i t t l e we f i n d t h a t t h e i n t e g r a l e q u a t i o n (4.2.3) c a n be c o n v e r t e d by r e p e a t e d d i f f e r e n t i a t i o n t o (4.2.5) h" ( s ) - 0 2 h ( s ) = -2^— 6 ( t - s ) + y a 2 f / b h ( u ) d u - l l , y L a. 2 2 w i t h 0 2 = 2 ^ - a + a * y The 6 - f u n c t i o n a r i s e s h e r e b e c a u s e o f t h e d i s c o n t i n u i t y o f t h e d e r i v a t i v e o f e ~ a l t - s l a t s - t . W h i t t l e a s s e r t s t h a t (4.2.5) must h o l d w i t h o u t t h e 6 - f u n c t i o n f o r s ^ t , however f o r s=t t h e d e r i v a -2 - , t i v e s o f h ( s ) and —6 a ' ~ s l must have s t e p d i s c o n t i n u i t i e s o f y ^ t h e same m a g n i t u d e . E q u a t i o n (4.2.5) has a g e n e r a l s o l u t i o n o f t h e f o r m (4.2.6) h ( s ) = ^ - ^-C^^+^^KQ&^^-^+R. We d e t e r m i n e t h e q u a n t i t i e s P, Q and R by s u b s t i t u t i n g (4.2.6) b a c k i n t o (.4.2.3) t o g e t ( 4 . 2 . 7 ) A e - ® 1 s _ t 1 + p e " ® ( s ~ a > + Q e " ( b " s ) + R + / b r ( £ l e - a | s-u| + u } u e - 0 | s-u| + p e - 0 (u-a) + Q e - G ( b - u ) + R ) ] d u a 2 / 3 - a I s - t I . 68 2 w h e r e A R e c a l l i n g t h a t ,[a,b] = [-T,T] , and p e r f o r m i n g t h e i n t e g r a t i o n i n v o l v e d , (4.2.7) y i e l d s t h e s o l u t i o n , (4.2.8) h t ( s ) = x [ e-0|s-t| + e-0(t-a)-8(s-a) + e-®(b-t)-0(b-s)-| > I f (4.2.8) i s s u b s t i t u t e d i n t o (4.2.1) we o b t a i n (4.2.9) A L ( t ) = y + i 1 + i 2 + i 3 , w h e r e w i t h . f b p - © | s - t | d ( N ( s ) _ y S ) , r b p - ® ( t + s - 2 a ) d ( N ( s ) _ u S ) x 2 a x / b e - © ( - t - s + 2 b ) d ( N ( s ) _ y S ) I 3 a 2 2 aa „2 2aa . 2 A = -; Q = +a y G y The e x p r e s s i o n (4.2.9) i s t h e e x a c t o p t i m a l l i n e a r e s t i m a t o r o f t h e i n t e n s i t y f u n c t i o n , A ( t ) , and i t i s t h e f o r m w h i c h i s u s e d t o o b t a i n t h e e x a c t s o l u t i o n t o t h e o i l w e l l d i s c o v e r y p r o b l e m . F i g u r e (b) d i s p l a y s t h e e x a c t a n d t h e a p p r o x i m a t e r e s u l t s g r a p h i c a l l y f o r c o m p a r i s o n . A C o m p u t e r P r o g r a m t o compute (4.2.9) i s g i v e n i n A p p e n d i x ( 1 ) . 69 4.3 AN APPROXIMATE SOLUTION OF 1.2.6 T h i s p r e s e n t a t i o n f o l l o w s C l e v e n s o n and Z i d e k ( 7 ) . Denote by h T ( t ; s ) t h e s o l u t i o n o f (4.3.1) yx(s)+/'^ Tx ( u ) K ( s - u ) d u = K ( t - s ) , -T<s<T, and l e t h O T ( t ; s ) be o u r a p p r o x i m a t i o n t o h ( t ; s ) ; t h a t i s , h ( t ; s ) i s t h e s o l u t i o n o f o  CO (4.3.2) y x ( s ) + / x ' ( u ) K ( s - u ) d u = K ( t - s ) , -co< s<oo, — CO co 2 w i t h / • x (s) ds<co. — CO U s i n g F o u r i e r t r a n s f o r m t e c h n i q u e s , C l e v e n s o n and Z i d e k (7) show t h a t e T ( t ; s ) = h T ( t ; s ) - h o o ( t ; s ) ^ 0 as Trvco f o r e a c h f i x e d ( t , s ) u n d e r s u i t a b l e r e g u l a r i t y c o n d i t i o n s . T h e i r argument a l s o g i v e s a bound f o r e T i n t h e f o r m , 1 -1 (4.3.3) | e T ( t ; s ) | < A y " 1 { / | u | > T h 2 D ( t ; u ) d u } 2 { T 2 + B y _ 1 } , w i t h A = {/°° Is Ik 2 ( s ) d s } 2 — CO 1 I 70 and B = {/°° K 2 ( s ) d s } 2 C o r r e s p o n d i n g t o e q u a t i o n s (4.3.1) and ( 4 . 3 . 2 ) , l e t (4.3.4) X T ( t ) = u + / ^ T h T ( t ; s ) d ( N ( s ) - M s ) and (4.3.5) X (t) = y+/ h ( t ; s ) d ( N ( s ) - y s ] — OO 0 0 r e s p e c t i v e l y be t h e o p t i m a l l i n e a r e s t i m a t o r and t h e a p p r o x i m a -t i o n t o A . Then by a p p l y i n g t h e bound i n i n e q u a l i t y ( 4 . 3 . 3 ) , a bound f o r E I A I=E I X ( t ) - x (t) I i s o b t a i n e d i n t h e f o r m : 1 1 1 o  1 1 _1 (4.3.6) E | A | < 4 T A y ~ 1 { / | i , J i 2 ( t ; u ) d u } 2 {T 2 + B y _ 1 } . 1 1 U > i o  C l e v e n s o n and Z i d e k (7) o b s e r v e t h a t t h i s bound i s s m a l l o n l y when t i s n o t t o o n e a r t h e b o u n d a r i e s o f t h e o b s e r v a t i o n p e r i o d , [-T,T]. Thus X^ w i l l n o t be a good a p p r o x i m a t i o n t o X T n e a r t h e b o u n d a r i e s o f t h a t p e r i o d . Note t h a t t h i s i s e q u i v a -l e n t t o o u r remark a f t e r ( 4 . 2 . 8 ) . F u r t h e r , C l e v e n s o n and Z i d e k g i v e t h e l a r g e t i m e , T, a p p r o x i m a t i o n o f t h e o p t i m a l l i n e a r e s t i m a t o r XT (t) as Li 71 (4.3.7) X j t ) - U + e / ^ e - 7 ^ S ' d ( N ( s ) - y s ) , -T<t<T, where ^ Q 2 - l n . , 2 -1 -1,"2 3 = a y (l+2a y j ) and Y = c t(l+2a 2y "^ a To d e t e r m i n e t h e a c c u r a c y o f t h e a p p r o x i m a t i o n (4.3.7) t h e bound g i v e n i n i n e q u a l i t y (4.3.6) was e v a l u a t e d , and f o u n d t o be (4.3.8) P c o s h ( s y t ) , where .^ ~2 -1 2 -1 -2vT P = '4AT{T +By } 3 Y 6 • 4.4 COMPUTATION: OIL WELLS DISCOVERY DATA From t h e d a t a t h e mean, y, was e s t i m a t e d as y-0.70. We 2 i i u s e d C o v ( X ( t ) , A ( s ) ) = a p ( | t - s | ) , where p(u) = e a l u l , w i t h a=0.05, was c h o s e n w i t h o u t e x p l i -c i t l y u s i n g t h e d a t a . The f o r m o f p i s t h a t o f a o n e - s t e p a u t o -r e g r e s s i v e p r o c e s s ; we b e l i e v e t h a t p i s s y m m e t r i c and d e c r e a s i n g on [0,°°). A l s o t o e x p r e s s o u r u n c e r t a i n t y a b o u t t h e c h o i c e * 2 y=0.70, t h e v a l u e a =0.25 was c h o s e n , w i t h s u p p o r t f r o m t h e 2 d a t a , as an e s t i m a t e o f o . Time was m e a s u r e d i n 30-day i n t e r v a l s and t h e r e were 108 o b s e r v a t i o n s i n t h e p e r i o d [ - T , T J . F o r t h i s p a r t i c u l a r 72 c a s e T=110.5, and w i t h r e f e r e n c e t o t h e c o n s t a n t s i n s e c t i o n ( 4 . 3 ) , t h e f o l l o w i n g v a l u e s w e r e o b t a i n e d : A - 5//2, B = / 5 / 2 , 3 = 0.0913, y = 0.195. The b ound g i v e n i n i n e q u a l i t y (4.3.6) was f o u n d t o be l e s s t h a n 0.01 p r o v i d e d j t | < 8 7 . As m e n t i o n e d e a r l i e r , we d i s p l a y i n F i g u r e (B) f o r c o m p a r i -s o n t h e e x a c t l i n e a r e s t i m a t o r and t h e l a r g e t i m e a p p r o x i m a t i o n . REMARK The g r a p h s i n F i g u r e (b) s u p p o r t t h e a b o v e c a l c u l a -t i o n s ; t h a t i s , t h e b o u n d i s s m a l l o n l y when t i s n o t t o o n e a r t h e b o u n d a r i e s o f t h e o b s e r v a t i o n p e r i o d , [-T,T]. T h i s i n d i -c a t e s t h a t X m w i l l n o t be a g ood a p p r o x i m a t i o n t o t h e o p t i m a l l i n e a r e s t i m a t o r , X T , n e a r t h e b o u n d a r i e s o f t h a t p e r i o d . 73 CHAPTER 5 APPLICATIONS 5.1 INTRODUCTION The main purpose o f t h i s c h a p t e r i s t o a p p l y t h e t e c h n i q u e s d e v e l o p e d t h u s f a r t o p r a c t i c a l s i t u a t i o n s w h i c h may be s l i g h t l y d i f f e r e n t 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 4. R e c a l l t h a t we have been concerned w i t h the o p t i m a l l i n e a r e s t i m a t o r , A , o f t h e i n t e n s i t y f u n c t i o n , A ( t ) , o f 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 . I t has been 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 t h e s o l u t i o n o f t h e i n t e g r a l e q u a t i o n (5.1.1) m(s)h. ( s ) + / b K ( s ; u ) h . (u)du = K ( t ; s ) a<s<b, L. a t where m(s) = y i s a c o n s t a n t ( i n c h a p t e r 4 ) . I n many cases t h e assumption t h a t m(t) i s a c o n s t a n t o v e r 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 . I t i s t h e r e f o r e o f i n t e r e s t t o st u d y o t h e r s p e c i a l s i t u a t i o n s where we drop 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 t he a p p l i c a t i o n o f our g e n e r a l model t o such p r a c t i c a l s i t u a t i o n s , c o n s i d e r the i n t e g r a l e q u a t i o n (5.1.1) where m(t) i s not a c o n s t a n t b u t i s a p r e -s c r i b e d f u n c t i o n o f 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 p r o p e r t y i s 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 t y p e . However, by 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 t h e k e r n e l , K ( t ; s ) , i t i s always p o s s i b l e t o r e w r i t e such an e q u a t i o n i n the form o f the second t y p e . I n p a r t i c u l a r , when m(t) i s p o s i t i v e t h r o u g h o u t t h e i n t e r v a l [ a , b ] , H i l d e r b r a n d ( 1 3 ) , shows t h a t the e q u a t i o n (5.1.1) can be r e w r i t t e n i n the form (5.1.2) / m T ^ ) h t ( s ) + / b K ( S ; U ) ^ M h t i u ) d U = K j t ^ f / m (s) m (u) /m (s) o r (5.1.3) x ( s ) + / b r ( s ; u ) x (u)du - g ( s ) . "c a. • "c u Thus i n t h i s form one must recompute x (s) h (s) = r /m(s) a f t e r x t ( s ) has been found. Having shown t h a t by a p p r o p r i a t e l y r e d e f i n i n g the f u n c t i o n s i n v o l v e d , we can r e w r i t e (5.1.1) i n a s u i t a b l e form as a second t y p e e q u a t i o n , we g i v e two p r a c t i c a l examples o f such s i t u a t i o n s . 75 5.2 ESTIMATION OF TRAFFIC DENSITIES AT THE LIONS GATE BRIDGE Volumes o f d a t a have been c o l l e c t e d f o r t h e d i s t r i b u t i o n o f c a r s on t h e L i o n s Gate B r i d g e i n V a n c o u v e r . F i g u r e (£) g i v e s f i v e - m i n u t e c o u n t s o f t r a f f i c f o r t h e t o t a l t r a f f i c (southbound) c r o s s i n g a d e t e c t o r on a " t y p i c a l " day i n 1974. An e s t i m a t o r o f t h e i n t e n s i t y f u n c t i o n X (t) was s o u g h t . To s a y t h e l e a s t , t h i s i n t e n s i t y f u n c t i o n r e f l e c t s t h e e f f e c t s o f w e a t h e r , t i m e o f day, and o t h e r exogenous v a r i a b l e s and knowledge a b o u t t h e s e e f f e c t s a r e u s e f u l i n d e c i s i o n - m a k i n g . F o r example, s h o u l d a n o t h e r l a n e be added t o t h e e x i s t i n g o n e s ? A l o o k a t t h e c o u n t s a t an i n d i v i d u a l d e t e c t o r c o n v i n c e s one t h a t , on t h e L i o n s G a t e B r i d g e , c o u n t s a r e h i g h l y r e p r o d u c i b l e f r o m day t o day. "In p a r t i c u l a r , on w o r k i n g d a y s one w o u l d e x p e c t t r a f f i c t o be m a i n l y composed o f c a r s w h i c h p a s s t h e p a r t i c u l a r l o c a t i o n o f t h e d e t e c t o r a t n e a r l y t h e same t i m e e v e r y day; t h u s t h e r e a p p e a r s t o be an u n d e r l y i n g s c h e d u l e . I f one were t o make t r a f f i c c o u n t s o v e r a t i m e w h i c h i s l a r g e compared w i t h t h e u n c e r t a i n t y i n an i n d i v i d u a l a r r i v a l t i m e , t h e v a r i a n c e i n c o u n t s f r o m one day t o t h e n e x t s h o u l d be q u i t e s m a l l . I n o r d e r t o t r e a t t h e a p p a r e n t u n d e r l y i n g s c h e d u l e m a t h e m a t i c a l l y , a r e a l i z a t i o n o f 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 i s assumed t o be o b s e r v e d . T h i s e n a b l e s us t o a p p l y o u r model and t h e t e c h -n i q u e s d e v e l o p e d i n t h e p r e v i o u s c h a p t e r s t o s t u d y t h e u n d e r -l y i n g i n t e n s i t y p r o c e s s . S p e c i f i c a l l y , we seek a l i n e a r 76 e s t i m a t o r , X , o f t h e i n t e n s i t y f u n c t i o n , X ( t ) , i n t h e f o r m (5.2.1) X L ( t ) = y t + / ^ T h ( t ; s ) d ( N ( s ) - H ( s ) ) ; w h e r e h ( t ; s ) i s a s o l u t i o n o f ( 5 . 1 . 1 ) . N o t e t h a t h e r e we w r i t e s u b s c r i p t e d y^ t o i n d i c a t e t h a t i n t h i s e x a m p l e we d r o p t h e a s s u m p t i o n t h a t m ( t ) = y i s c o n s t a n t . W i t h t h e a s s u m p t i o n s i n c h a p t e r 4 a b o u t t h e k e r n e l , we r e q u i r e t h a t (5.2.2) C o v ( X ( t ) ,X ( S) ) = a2(ta^ S and .2 (5.2.3) K ( t ; s ) = K ( t - s ) - °y(\ t - s | ) +y f cy, U t We a r e now f a c e d w i t h t h e p r o b l e m o f f i n d i n g e s t i m a t e s 2 f o r t h e c o n s t a n t s o and a , and a l s o a method o f o b t a i n x n g t h e y ^ ' s . T h i s i s w h e r e , f o l l o w i n g t h e B a y e s i a n r e c i p e , i n t r o -s p e c t i o n comes i n . The m o t i v a t i o n f o r B a y e s i a n m e t h o d s i s e s s e n t i a l l y t h e d e s i r e t o b a s e c a l c u l a t i o n s a n d d e c i s i o n s on any a v a i l a b l e i n f o r m a t i o n , w h e t h e r i t i s s a m p l e i n f o r m a t i o n o r i n f o r m a t i o n o f some o t h e r n a t u r e , s u c h as t h a t b a s e d on p a s t e x p e r i e n c e . We s h a l l u s e e m p i r i c a l B a y e s 1 m e t h o d s a s s u g g e s t e d by B a r n e t t ( 4 , pp. 1 8 9 - 2 0 0 ) , t h e 1974 d a t a a t h a n d , a n d a l s o some v a l u e s p u b l i s h e d by L e a , N.D. and A s s o c i a t e s (16b) w h e r e d a t a c o l l e c t e d f r o m t h e same l o c a t i o n was u s e d . 77 CALCULATIONS: Consider t h e data i n T a b l e 5.1. TABLE 5.1 HOURLY VARIATION: LIONS GATE BRIDGE, 1966 Time o f Day Volume 9 - 10 1,550 10 11 1,240 11 - 12 1,140 12 - 13 950 13 - 14 1,400 14 - 15 1,150 15 - 16 • 1,175 16 - •17 1,300 17 - 18 1,300 S o u r c e : T r a f f i c U n i t , C i t y o f V a n c o u v e r I n o r d e r t o u s e t h i s i n f o r m a t i o n t o o b t a i n t h e p r i o r mean u t f o r t h e 1974 d a t a , i t i s c o n v e n i e n t t o l e t y^ change h o u r l y , t h u s i n o u r c a l c u l a t i o n s y^. v a r i e s f o r e v e r y 1 2 t h t s i n c e we a r e d e a l i n g w i t h f i v e - m i n u t e v e h i c l e c o u n t s . One i n t u i t i v e l y a p p e a l i n g p o s s i b i l i t y w o u l d be t o d i v i d e t h e 1966 f i g u r e s by 12 and u s e t h e r e s u l t i n g v a l u e s as t h e ^ ' s . B u t t h i s c a n n o t be e n t i r e l y s a t i s f a c t o r y ; one w o u l d e x p e c t t h a t t h e volume o f t r a f f i c m i g h t have i n c r e a s e d s i n c e 1966. C a l c u l a t i o n s show t h e 78 h o u r l y volume f o r 1974 was, on the a v e r a g e , 1,650, w h i l e t h a t o f 1966 was 1,050; thus i n t h e r a t i o o f 11:7. Hence, i n o r d e r t o s t e p up the 1966 v a l u e s , the c o r r e c t i o n f a c t o r 11/7 was used i n t h e c a l c u l a t i o n o f t h e y^ _. However, t o make room f o r s a m p l i n g 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 f a c t o r 3/2 was used i n the computer program p r e s e n t e d i n Appendix ( B ) . R e c a l l t h a t our c h o i c e o f p(u)=<S 01 I u I i s i n the form o f a o n e - s t e p a u t o r e g r e s s i v e p r o c e s s . T h i s knowledge may be e x p l o i t e d i n a r r i v i n g a t a v a l u e f o r a, and a l s o i n c h o o s i n g an e s t i m a t e f o r 2 a . Computer programs were run f o r the f o l l o w i n g v a l u e s o f a and a : 0 . 01 0.01 0.05 0.2 The r e s u l t i n g e s t i m a t e s o f t h e i n t e n s i t y f u n c t i o n , X ( t ) , a r e shown by the graph of F i g u r e s (D, E ) . REMARK The graphs i n d i c a t e t h a t , as i n the o i l w e l l example f o r f i x e d a, as a i n c r e a s e s , the e s t i m a t o r X becomes i n c r e a s i n g l y d a t a - s e n s i t i v e , and hence i r r e g u l a r . 1.0 2.0 2.0 2.0 79 We now c o n s i d e r a n o t h e r example where t h e mean E A ( t ) = m ( t ) i s a p r e s c r i b e d n o n c o n s t a n t f u n c t i o n o f t . 5.3, COAL-MINING DISASTERS F i g u r e (F) g i v e s t h e numbers, i n s u c c e s s i v e 400-day p e r i o d s , o f c o a l - m i n i n g d i s a s t e r s i n G r e a t B r i t a i n f o r t h e p e r i o d 1875 t o 1951. The d a t a a r e t a k e n f r o m Cox and L e w i s ( 8, pp. 2 - 6 ) . A d i s a s t e r i s d e f i n e d as a m i n i n g a c c i d e n t i n v o l v i n g t h e d e a t h o f 10 o r more men. Cox and L e w i s d i s c u s s more f o r m a l s t a t i s t i c a l methods f o r a n a l y z i n g t h i s s e t o f d a t a ; h e r e as an a l t e r n a t i v e t o Cox and L e w i s a n a l y s i s , we a p p l y o u r m o d e l . I t i s hoped t h a t t h i s example w i l l i l l u s t r a t e i n a l i t t l e more d e t a i l t h e s o r t o f p r o b l e m d i s c u s s e d i n t h i s work. I n p a r -t i c u l a r , t h i s s e c t i o n may be r e g a r d e d as an e x t e n s i o n t o t h e p r e c e d i n g s e c t i o n where e m p i r i c a l B a y e s i d e a s were c i t e d as 2 j u s t i f i c a t i o n f o r o u r c h o i c e o f a and a. Suppose t h e assump-t i o n s o f s e c t i o n (5.2) s t i l l h o l d w i t h t h e e x c e p t i o n t h a t we' now r e q u i r e m(t) t o be a c o n t i n u o u s f u n c t i o n o f t . F i g . (F) s u g g e s t s c o n s i d e r a t i o n o f t h e f u n c t i o n (5.3.1) C o s h t = ^(e^e t) , <t<°°. CO 80 S i n c e , as F i g . (F) i n d i c a t e s , t h e a v e r a g e r a t e o f o c c u r r e n c e o f d i s a s t e r s i s d e c r e a s i n g w i t h t i m e , t h e p o s i t i v e s i d e o f th e f u n c t i o n (5 .3 .1) i s u n s a t i s f a c t o r y . L e t us c h o o s e , i n s t e a d m(t) t o have t h e f o r m (5 .3 .2) m(t) = | ( e a t + e " b t ) , - l $ t < l where 0<a,b$l and b>a. The r e s t r i c t i o n on t i s t o p r e v e n t m(t) f r o m b e c o m i n g t o o l a r g e The r e s t r i c t i o n s on a and b a r e imposed f o r s i m i l a r r e a s o n s . Note t h a t w i t h t h e s e r e s t r i c t i o n s m(t) t a k e s t h e f o l l o w i n g f o r m k l \ 1 \ l ~l ~ i 1 1 1 ~l ~ 1 - 1 0 1 ^ t 81 The s a m p l e mean o f t h e g i v e n d a t a i s a b o u t 2; w i t h t h i s k n o w l e d g e , and a l s o t o s i m p l i f y t h e c a l c u l a t i o n s , we c h o o s e 3=0.5 and b - 1 . 0 . U s i n g t h e t e c h n i q u e s o f s e c t i o n ( 5 . 2 ) , "2 2 o =0.25 i s c h o s e n a s an e s t i m a t e o f a . A l s o , t h e v a l u e a=0.05 i s f o u n d t o be a g ood c h o i c e . COMPUTATION: As h a s a l r e a d y b e e n s a i d , t h e f o r m o f t h e k e r n e l and t h e m e t h o d o f c a l c u l a t i o n a r e e s s e n t i a l l y t h o s e o f t h e p r e -c e d i n g s e c t i o n . I t i s i n t e r e s t i n g t o n o t e t h a t i n c o n f o r m i t y w i t h t h e r e s t r i c t i o n on t i n ( 5 . 3 . 2 ) , i n t h e s u b p r o g r a m t o c ompute t h e f u n c t i o n m ( t ) , e a c h t i m e v a l u e t i s d i v i d e d by 30. N o t e t h a t i n t h i s e x a m p l e t e [ - 3 0 , 3 0 j , w h i l e f o r m ( t ) we r e q u i r e t e [ - i , l ] . A n o t h e r i n t e r e s t i n g f e a t u r e o f t h e p r e s e n t e x a m p l e i s t h a t , c u r i o u s l y , t h e v a l u e s a=1.75 and b=1.0 w e r e u s e d w i t h t h e same ~ 2 v a l u e s o f a and a as a b o v e . I t t u r n e d o u t t h a t b o t h p r o g r a m s p r o d u c e d v a l u e s o f t h e e s t i m a t o r o f t h e i n t e n s i t y f u n c t i o n , A ( t ) , w h i c h a r e p r e t t y much t h e same. N e v e r t h e l e s s , t h e c o s t and t i m e o f c o m p u t a t i o n a s g i v e n by t h e c o m p u t e r a r e a s f o l l o w s : V a l u e s Time ( s e c . ) C o s t ($) a=0.5; b=1.0 2.784 1. 34 a=0.75; b=1.0 2.791 1. 39 Thus one i s p e r s u a d e d t o s t i c k t o t h e v a l u e s a=0.5 and b=1.0, and t h i s i s w h a t was done h e r e . 82 F i g u r e s (G) a n d (H) d i s p l a y A_ f o r v a r i o u s v a l u e s o f a and a so t h a t one c a n see t h e e f f e c t s o f v a r y i n g o and a. Here a g a i n i t i s s e e n t h a t f o r f i x e d a, an i n c r e a s e i n a c a u s e s t h e e s t i m a t o r t o become more d a t a - s e n s i t i v e and v i c e v e r s a . CONCLUDING REMARKS C h a p t e r 2 p r e s e n t s v a r i o u s e x a c t methods o f s o l v i n g t h e F r e d h o l m i n t e g r a l e q u a t i o n o f t y p e I I e n c o u n t e r e d i n o u r q u e s t f o r t h e o p t i m a l l i n e a r e s t i m a t o r o f t h e i n t e n s i t y f u n c t i o n , A , o f 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 . S e v e r a l a p p r o x i m a t i o n t e c h n i q u e s a r e a l s o p r e s e n t e d i n c h a p t e r 3. T h e s e c h a p t e r s a r e i n t e n d e d as a s u r v e y , n e c e s s a r i l y i n c o m p l e t e , o f some o f t h e a v a i l a b l e methods o f s o l v i n g t h e F r e d h o l m t y p e e q u a t i o n s w h i c h f r e q u e n t l y o c c u r i n many a r e a s o f a p p l i e d m a t h e m a t i c s . We a g r e e w i t h G r a n d e l l ( 1 0 ) , W h i t t l e ( 2 2 ) , and C l e v e n s o n and Z i d e k (7) t h a t o n l y t h e o p t i m a l l i n e a r e s t i m a t o r , A w h i c h i j i s Bayes w i t h r e s p e c t t o a r e s t r i c t e d c l a s s o f l i n e a r e s t i m a t i o n , i s g e n e r a l l y a p p l i c a b l e when A i s any s e c o n d o r d e r p r o c e s s . Thus, i n c h a p t e r s 1, 4 and 5 t h e main e m p h a s i s was p l a c e d on o b t a i n i n g t h e o p t i m a l l i n e a r e s t i m a t o r , A , w h i c h i n v o l v e s t h e n u m e r i c a l s o l u t i o n o f e q u a t i o n ( 1 . 2 . 6 ) . I n t h e n u m e r i c a l e x a m p l e s , a s p e c i a l f o r m o f t h e k e r n e l was c h o s e n t o r e f l e c t o u r b e l i e f a b o u t t h e c o v a r i a n c e s t r u c t u r e . The c a l c u l a t i o n s r e q u i r e d t o o b t a i n A l i n t h e n u m e r i c a l example where 83 t h e a s s u m p t i o n o f c o n s t a n t mean, m ( t ) = y , i s d r o p p e d a r e much more e x t e n s i v e t h a n t h o s e r e q u i r e d f o r t h e s i m p l e c a s e w h e r e t h e a s s u m p t i o n i s u p h e l d . To t h e a u t h o r ' s k n o w l e d g e , t h e e x a c t t e c h n i q u e s c o n s i d e r e d h e r e had n o t b e e n a p p l i e d t o r e a l - l i f e s i t u a t i o n s b e f o r e t h i s w o r k was s t a r t e d , e v e n t h o u g h t h e m e t h o d s h a d f r e q u e n t l y b e e n s u g g e s t e d i n t h e l i t e r a t u r e . I t i s g r a t i f y i n g t o n o t e t h a t i n c h a p t e r 4, a c o m p a r i s o n o f t h e a p p r o x i m a t e e s t i m a t o r , and t h e e x a c t o p t i m a l l i n e a r e s t i m a t o r , A shows t h a t t h e b o u n d , i g i v e n by C l e v e n s o n and Z i d e k ( 7 , p. 2 1 ) , i s s a t i s f i e d i n t h e c a s e o f t h e o i l w e l l d i s c o v e r y e x a m p l e . A n o t h e r i n t e r e s t i n g o b s e r v a t i o n i s t h a t , i n c h o o s i n g '2 y , a , a and t h e o t h e r c o n s t a n t s w h i c h o c c u r i n o u r m o d e l , t h e B a y e s i a n a p p r o a c h i s a v e r y u s e f u l t o o l . A t l e a s t i t i s i f one may u s e e m p i r i c a l B a y e s m e t h o d s as was done h e r e . I t i s w o r t h n o t i n g t h a t i n some s i t u a t i o n s i t i s much more s a t i s f a c t o r y n o t t o assume t h a t t h e mean i s c o n s t a n t s i n c e , f r o m o u r r e s u l t s , i t a p p e a r s t h a t s u c h an a s s u m p t i o n may c a u s e t h e e s t i m a t o r s o f A t o become more d a t a - s e n s i t i v e . A w o r d o f c a u t i o n i s i n o r d e r h e r e a b o u t t h e c h o i c e o f t h e mean f u n c t i o n , m ( t ) ; an " o f f - t h e - m a r k " c h o i c e i s v e r y l i k e l y t o y i e l d i n t a n g i b l e r e s u l t s . A c a r e f u l e x a m i n a t i o n o f t h e d a t a , a t l e a s t i n g r a p h i c a l f o r m , i s u s u a l l y h e l p f u l . I n many s p e c i a l c a s e s , h o w e v e r , one may s a f e l y m a i n t a i n t h e a s s u m p t i o n t h a t m ( t ) = y , a c o n s t a n t . A l s o one 84 may use t h e l a r g e t i m e a p p r o x i m a t e e s t i m a t o r , A , i n s t e a d o f t h e o p t i m a l l i n e a r e s t i m a t o r , A T , s i n c e t h e s e methods o f t e n y i e l d f a i r l y good r e s u l t s f o r t h e e s t i m a t i o n o f t h e i n t e n s i t y f u n c t i o n , A ( t ) . C O 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. 5 m i n u t e v e h i c l e c o u n t s — — ro ro •£» co ro o~> O o o o o o o o CO CO 7 0 V 6 0 -o o °- I 50 c £ CL in 140 ° 130 E 120 I 0 00& \ - 5 4 - 5 © 9 A M a = 0.01 , cr = 2 .0 \ X X a = 0 . 0 1 , a- = l . o \ \ 10AM \ 1 I HAM 12N00N 1 P M 1.30PM P P ^ 3Pk 4 PM i 5 , P M i 1 1 ! I 1 1 I 1 1 l - 4 0 - 3 0 - 2 0 - 1 0 0 10 T i m e of d a y 2 0 3 0 40 50 54 6 P M D. The optimal linear estimator of the intensity function of the Lions Gate bridge process using various prior parameters. 89 -a o 190 180 I 70 160 150 h \ " \ \ \ S. 140 a> | 130 in v_ <D CL a a 120 0 o 1 0 0 a; % 90 2 80 70 60 \ a = 0.05, cr = 2.0 \ /—• \ / va = 0.02, cr = 2.0 .•1 1 / \ /: \ \ V \ 10AM HAM I2N00N IPM 1.30PM 2PM 3,PM 4PM 5PM 1 54-50 -40 -30 -20 -10 1 6 10 9AM T i m e of day 20 30 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 8 >» O TD 7 o o 6 cu CL 5 (/> l . co 4 to o t o 3 TO ^_ o 2 co I E 2 0 0 10 20 30 Time (in 400day units) 40 50 60 F. Cool- mining disasters in Great Britain for the.period 1875 -1951 Numbers in successive 400-day periods. 4.5 4.0 po 3.5 CU D. 3.0 > » o X3 O o 2.5 I— cu CL in 2.0 cu CO i sa 1 .5 o 1.0 cu E. 3 2 0.5 0 - 30 1875 - 2 0 - 10 0 Y e a r 10 20 30 1951 G. 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K o p a l , Z., N u m e r i c a l A n a l y s i s ; 2nd E d n . , W i l e y , N.Y. 16b, L e a , N.D., and A s s o c i a t e s , " M e a s u r e s t o i m p r o v e bus t r a n s i t and t r a f f i c f l o w a c r o s s t h e f i r s t n a r r o w s b r i d g e " , (May, 1 9 6 7 ) . 1 6 c . L o e v e , M., P r o b a b i l i t y T h e o r y ; D. Van N o s t r a n d ( 1 9 5 5 ) , N.Y., 4 6 4 - 9 3 . 17. N e w e l l , G.F. and G.A. S p a r k s , " S t a t i s t i c a l p r o p e r t i e s o f t r a f f i c , c o u n t s " ; S t o c h a s t i c P o i n t P r o c e s s e s ( L e w i s ) ; W i l e y I n t e r s c i e n c e ( 1 9 7 2 ) , N.Y. 18. S l e p i a n , D. and T.T. K a d o t a , " F o u r I n t e g r a l e q u a t i o n s o f d e t e c t i o n t h e o r y " ; S.I.A.M. J . A p p l . M a t h . , V o l . 17, No. 6 ( 1 9 6 9 ) , 1 102-17. 19. Van T r e e s , H.D., D e t e c t i o n , E s t i m a t i o n and M o d u l a t i o n T h e o r y , P a r t I ( C h a p s . 4 and 6 ) , W i l e y ( 1 9 6 8 ) , N.Y. 20. V a r a h , J . , P r i v a t e c o m m u n i c a t i o n . 21. W i n k l e r , R.L. and W.C. H a y s , S t a t i s t i c s , P r o b a b i l i t y , I n f e r e n c e and D e c i s i o n ( 1 9 7 0 ) ; N.Y., 444-504. 22. W h i t t l e , P., "On s m o o t h i n g o f p r o b a b i l i t y d e n s i t y f u n c t i o n s " ; J . R . S . S . B, 20, No. 2 ( 1 9 5 8 ) , 334-43. 95 APPENDIX 1 COMPUTER PROGRAM: THE OPTIMAL LINEAR ESTIMATOR OF THE INTENSITY FUNCTION OF THE OILWELL DISCOVERY PROCESS C *** OILWELL DISCOVERY *** C *** EXACT SOLUTION USING WHITTLE'S SUGGESTION *** REAL MU,SIGMA,ALFA,EXACT DIMENSION P ( 1 2 0 ) , D S ( 1 2 0 ) , EXACT(225) DIMENSION T M ( 2 2 5 ) , C E X ( 2 2 5 ) , CAT(225) READ2, MU, SIGMA, ALFA 2 FORMAT(3F4.2) N=108 READ4, ( P ( I ) , D S ( I ) , 1=1,N) 4 FORMAT(F5.0, F3.0) WK = ALFA * SIGMA* SIGMA/MU TTA = SQRT(2.*WK + ALFA*ALFA) VK = WK/TTA PRINT6, WK,TTA,VK 6 FORMAT(' ',3F10.4) UT1 = 110.5 UT = 221. T= - 1 1 0 . K=0 8 I F ( T . G T . 1 1 0 . ) GOTO 20 SUM= 0. J = 0 10 J = J + l 96 I F ( J . G T . N ) GOTO 12 R = A B S ( 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) I F ( P A 3 . L T . - 1 0 0 . ) C=0 C= 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 = MU*VK*(1. - CVA) K = K + 1 EXACT(K) = VK*SUM + ADVA PRINT14, T, EXACT(K) 14 FORMAT( 1 ', F6 . 0 , F 1 0 . 2 ) T .= T + 1 GOTO 8 20 NT=221 M=NT - 1 N=NT 97 TM(1)=1. DO 22 1=1,M •TM(I+1)=TM(I)+1. CONTINUE DO 2 4 1=1,NT C A T ( I ) = T M ( I ) C E X ( I ) = E X A C T ( I ) CONTINUE *** NOW CALL SUBROUTINE AJOA TO DO THE PLOTTING CALL AJOA(CAT,CEX,NT) TERMINATE PLOTTING AND" STOP ** CALL PLOTND STOP END . SUBROUTINE A J O A ( X , Y, N) DIMENSION 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 . , 'TIME*, - 4 , 1 0 . , 0., XMIN, DX) CALL A X I S ( 0 . , 0., 'EXACT', 5, 1 0 . , 9 0 . , YMIN, DY) CALL L I N E ( X , Y , N , 1 ) CALL P L O T ( 1 2 . 0 , 0., -3) RETURN END 98 FUNCTION VALU(X,Y,Z) REAL X,Y,Z A l = -X*ABS(-Y-Z) B l = -X*ABS(Y-Z) I F ( A 1 . L T . - 1 0 0..AND.B1.LT.-100.) IF(A1.GE.-100..AND.B1.LT.-10 0.) I F ( A l . L T . - l O O . . A N D . B l . G E . - l O O . ) I F ( A l . G E . - l O O . . A N D . B 1 . G E . - 1 0 0 . ) A2= -X*(Z+Y) B2 = - X * ( 3 * Y +Z) I F ( A 2 . L T . - 1 0 0..AND.B2.LT.-100.) IF(A2.GE.-100..AND.B2.LT.-100.) I F ( A 2 . L T . - 1 0 0. .AND.B 2.GE.-100.) IF(A2.GE.-100..AND.B2.GE.-100.) 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 . ) IF(A3.GE.-100..AND.B3.LT.-100.) IF(A3.LT.-100..AND.B3.GE.-100.) I F ( A 3 . G E . - 1 0 0 . .AND.B 3.GE.-10 0.) VALU = ,(PF +.QF + R F ) / X + X RETURN END PF = 0. P F = E X P ( A l ) P F = - E X P ( B l ) P F = E X P ( A l ) - E X P ( B l ) QF=0. QF=EXP(A2) QF= -EXP(B2) QF=EXP(A2) - E X P ( B 2 ) RF = 0 . RF= EXP(A3) RF= -EXP(B3) RF=EXP(A3) - EXP(B3) 99 COMPUTER OUTPUT: OILWELL DISCOVERIES; THE APPROXIMATE AND THE EXACT LINEAR ESTIMATORS, X AND X T . o  L RESULTS FOR o=0.5, a=0.05. T A L ( T ) T A L ( T - 1 1 0 . •1.21 0. 50 - 8 9 . 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 -8 6 . 0 .93 0.93 -106 . 0.96 0. 63 - 8 5 . 0 .92 0.92 - 1 0 5 . 0. 88 0. 61 -84 . 0.91 0.91 -104 . 0.83 0.62 - 8 3 . 0. 86 0.86 - 1 0 3 . 0. 81 0. 64 -82 . 0 .84 0.84 -102. 0.83 0. 68 - 8 1 . 0 . 81 0 . 82 - 1 0 1 . . 0. 83 0. 72 - 8 0 . 0.82 0. 82 -100. 0.87 0. 78 -79 . 0.81 0 .82 -99 . 0.83 0.76 - 7 8 . 0.77 0.77 -98. 0.79 0.73 -77. 0. 75 0.76 -97. 0. 77 0.72 -7 6 . 0.73 0.73 -96. 0.78 0. 74 - 7 5 . 0.69 0.69 -9 5 . 0. 82 0 . 79 -74 . 0.68 0.68 -94. 0. 79 0. 76 - 7 3 . 0. 69 0.70 - 9 3 . 0. 78 0.76- - 7 2 . 0 .73 0.73 -92. 0.80 0. 78 - 7 1 . 0.79 0. 80 - 9 1 . 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 - 6 5 . 0.93 0.94 -64. 0.81 0.81 - 6 3 . 0.71 0.71 -6 2 . 0.64 0.64 - 6 1 . 0.59 0.59 - 6 0 . 0.56 0.56 -59 . 0.55 0.55 -5-8. 0.56 0.56 - 5 7 . 0.59 0.59 -56. 0.60 0.60 - 5 5 . 0.53 0.53 -54. 0.47 0.48 - 5 3 . 0.43 0.44 - 5 2 . 0.41 0.42 - 5 1 . . 0.37 0.37 -50. 0.33 0.34 -49 . 0.31 0.32 -4 8 . 0.30 0.31 -47. 0.30 0.30 -46. 0.31 0.31 -45 . 0.32 0.33 -44. 0.35 0.36 - 4 3 . 0.39 0.40 - 4 2 . 0.44 0.45 - 4 1 . 0.51 0.52 T A (T) A T (T) - 4 0 . 0.60 0.61 - 3 9 . ' 0.67 0.68 -3 8 . 0.77 0.77 -37. 0.86 0.86 -36. 0.94 0.94 - 3 5 . 0.98 0.99 -34. 0.96 0.96 - 3 3 . 0.93 0.93 - 3 2 . 0.90 0.90 - 3 1 . 0.86 0.87 - 3 0 . 0.86 0.87 - 2 9 . 0.86 0.86 -28. 0.85 0.85 - 2 7 . 0.86 0.87 -26 . 0.88 0.88 - 2 5 . 0.93 0.93 -24. 0.90 "0.90 - 2 3 . 0.87 0.87 - 2 2 . 0.87 0.87 - 2 1 . 0.87 0.87 -20. 0.89 0.90 - 1 9 . 0.88 0.89 -1 8 . 0.90 0.91 - 1 7 . 0.96 0.96 -16. 1.01 1.01 101 T A (T) A (T) °o L - 1 5 . 0.99 0.99 -14. 0.97 0.98 - 1 3 . 0.99 0.99 -1 2 . 0.93 0.94 - 1 1 . 0.88 0.88 -10. 0.85 0.86 - 9 . 0.86 0.86 - 8 . 0.86 0.86 - 7 . 0.85 0.86 - 6 . 0.85 0.85 - 5 . 0.87 0.87 -4. 0.92 0.93 - 3 . 0.94 0.94 - 2 . 0.95 0.96 - 1 . 0.93 0.93 0. 0.86 0.87 1. 0.83 0.84 2. 0.80 0.80 3. 0.76 0.76 4. 0.70 0.71 5. 0.68 0.68 6. 0.67 0.68 7. 0.7 0 0.70 8. 0.74 0.75 9. 0.74 0.75 T A (T) A (T) a> L 10. 0.77 0.78 1 1 . 0.83 0.83 12. 0.88 0.88 13. 0.92 0.93 14. 0.89 0.90 15. 0.90 0.90 16. 0.9 3 0.94 17. 0.93 0.94 18. 0.89 0.90 19. 0.89 0.89 20. 0.87 0.88 21. 0.90 0.90 22. 0.91 0.92 23. 0.93 0.93 24. 0.91 0.91 25. 0.84 0.85 26. 0.81 0.82 27. 0.81 0.82 28. 0.84 0.84 29. 0.89 0.90 30. 0.98 0.99 31. 1.04 1.04 32. 1.06 1.07 33. 1.08 1.09 34. 1.15 1.15 102 T A (T) A (T) CO J_, 35. 1.18 1.19 36. 1.18 1.19 37. 1.23 1.24 38. 1.26 1.26 39. 1.15 1.15 40. 1.04 1.05 4 1 . 0.94 0.94 42. 0.87 0.88 43. 0.83 0.84 44. 0.79 0.80 45. 0.78 0.78 46. 0.79 0.80 47. 0.80 0.80 48. 0.76 0.77 49. 0.71 0.72 50. 0.70 0.70 51. .0.67 0.67 52. 0.62 0, 63 53. 0.57 0.57 54. 0.53 0.53 55. 0.51 0.52 56. 0.48 0.48 57. 0.46 0.46 58. 0.45 0.46 59. 0.43 0.43 T A (T) A (T) CO J_, 60. 0.42 0.42 61. 0.42 0.43 62. 0.40 0.41 63. 0.37 0.37 64. 0.34 0.34 65. 0.32 0.33 66. 0.32 0.32 67. 0.32 0.33 68. 0.34 0.34 69. 0.37 0.37 70. 0.41 0.41 71. 0.42 0.43 72. 0.42 0.42 73. 0.43 0.44 74. 0.45 0.46 75. 0.46 0.46 76. 0.48 0.49 77. 0.52 0.52 78. 0.57 0.58 79. 0.65 0.65 80. 0.67 0.68 81. 0.69 0.69 82. 0.66 0.66 83. 0.61 0.62 84. 0.59 0.59 103 T X (T) X T (T) 85. 0.55 0.55 86. 0.53 0.54 87. 0.53 0.53 88. 0.55 0.55 89. 0.59 0.59 90. 0.61 0.61 91. 0.62 0.62 92. 0.58 0.57 93. 0.52 0.52 94. 0.4 9 0.4 7 95. 0.43 0.41 96. 0.39 0.37 97. 0.36 0.33 T X (T) X T (T) 98. 0.34 0.31 99. 0.33 0.29 100. 0.34 0.29 1 0 1 . 0.35 0.29 102. 0.38 0.30 103. 0.42 0.32 104. 0.47 0.36 105. 0.54 0.40 106. 0.63 0.46 107. 0.71 0.50 108. 0.78 0.52 109. 0.83 0.52 110. 0.89 0.50 1 0 4 APPENDIX 2 COMPUTER PROGRAM: THE OPTIMAL LINEAR ESTIMATOR OF THE INTENSITY FUNCTION OF THE LIONS GATE BRIDGE PROCESS ** LIONS GATE BRIDGE EXACT SOLUTION ** DATA FROM 9A.M. TO 6 P.M. REAL MU,SIGMA,ALFA,FACT FACT I S A MULTIPLYING FACTOR TO SCALE UP M(T) DIMENSION P ( 1 2 0 ) , D S ( 1 2 0 ) , S K ( 1 0 ) , E X A C T ( 1 2 0 ) DIMENSION T M ( 1 2 0 ) , C E X ( 1 0 9 ) , CAT(109) READ 2, SIGMA,ALFA,FACT FORMAT(3F6.3) N=109 M=9 T=-54 . UT=109. UT1=54.5 READ 4 , ( P ( I ) , D S ( I ) , 1 = 1,N) FORMAT(F6.0,F6.0) READ5, ( S K ( I ) , 1=1,M) FORMAT(F6.01) L=0 K=0 KT=0 L=L+1 Y=SK(L) 105 CALL FUNCTION UTVAR TO COMPUTE NEW M(T) MU=UTVAR(FACT,Y) RMT=SQRT(MU) WK=ALFA*SIGMA*SIGMA/MU TTA=SQRT(2.*WK + ALFA*ALFA) VK=WK/TTA 8 I F ( T . G T . 5 4 ) GOTO 20 ** CHANGE M(T) AFTER EVERY 12TH INTERVAL I F ( K T . G T . 1 2 ) GOTO 7 SUM = 0 . J=0 10 J=J+1 I F ( J . G T . N ) GOTO 12 -R=ABH;(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) 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-) GOTO 10 SUM = SUM + D S ( J ) * A B C 106 GOTO 10 ** TO COMPUTE XT(S) 12 CVA = VALU(TTA, UT1, T) ** TO COMPUTE HT(S) AFTER CALCULATING XT(S). ** HAVE = CVA/RMT ADVA = MU*VK*(1. - HAVE) K=K+1 EXACT(K)=VK*SUM + ADVA + MU PRINT14, T, EXACT(K) 14 FORMAT(' ' , F 6 . 0,F10, 2) KT=KT+1 T=T+1 GOTO 8 20 NT=109 TM(1)=1. DO 21 1=1,N TM ( I + 1 ) = T M ( I ) + 1 . 21 CONTINUE DO 22 1=1,NT C A T ( I ) = T M ( I ) C E X ( I ) = E X A C T ( I ) 22 CONTINUE ** NOW CALL SUBROUTINE AJOA TO DO THE PLOTTING ** CALL AJOA(CAT,CEX,NT) * TERMINATE PLOTTING AND STOP ** CALL PLOTND STOP 107 END C SUBROUTINE A J O A ( X , Y, N) DIMENSION 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 . , 'TIME', - 4 , 1 0 . , 0., XMIN, DX) C A L L A X I S ( 0 . , 0., 'EXACT', 5, 1 0 . , 9 0 . , YMIN, DY) CALL L I N E ( X , Y , N , 1 ) CALL P L O T ( 1 2 . 0 , 0., -3) RETURN END C C FUNCTION UTVAR(X,Y) REAL X,Y PRO=X*Y UTVAR=SQRT(PRO) RETURN END C FUNCTION VALU(X,Y,Z) REAL X,Y,Z A l = -X*ABS(-Y-Z) B l = -X*ABS(Y-Z) I F ( A 1 . L T . - 1 0 0 . . A N D . B 1 . L T . - 1 0 0 . ) PF = 0. IF(A1.GE.-100..AND.B1.LT.-100.) P F = E X P ( A l ) 108 I F ( A 1 . L T . - 1 0 0..AND.B1.GE.-100.) IF(A1.GE.-100..AND.B1.GE.-100.) A2= -X*(Z+Y) B2 = - X * ( 3 * Y +Z) IF ( A 2 . L T . - 1 0 0 . . A N D . B 2 . L T . - 1 0 0 . ) IF(A2.GE.-100..AND.B2.LT.-100.) I F ( A 2 . L T . - 1 0 0 . .AND•B 2.GE.-100.) I F ( A 2 . G E . - 1 0 0 . .AND.B 2.GE.-100.) 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 . ) IF(A3.GE.-100..AND.B3.LT.-10 0.) I F ( A 3 . L T . - 1 0 0 . .AND.B 3.GE.-100.) IF(A3.GE.-100..AND.B3.GE.-100.) VALU = (PF + OF + R F ) / X + X RETURN END PF = - E X P ( B I ) P F = E X P ( A l ) - E X P ( B I ) QF=0. QF=EXP(A2) QF= -EXP(B2) QF=EXP(A2) - EXP(B2) RF = 0. RF= EXP(A3) RF= -EXP(B3) RF=EXP(A3) - EXP(B3) 109 COMPUTER OUTPUT: LIONS GATE BRIDGE PROCESS THE OPTIMAL LINEAR ESTIMATOR, A T . RESULTS FOR a=2.0, a=0.2. A L ( T ) T A L ( T ) 54. 167.99 -33. 118.86 53. 167.06 - 3 2 . 119.81 52. 164.14 - 3 1 . 118.14 51. 160.44 - 3 0 . 116.04 50. 154.74 - 2 9 . 117.08 49. 148.83 - 2 8 . 117.80 48. 145.01 - 2 7 . 117.38 47. 140.18 -26. 119.11 46. 136.26 " 2 5 . 120.07 45. 131.25 -24. 120.30 4-4. 127.12 - 2 3 . 121.23 43. 121.96 - 2 2 . 118.29 42. 116.97 - 2 1 . 111.63 41. 114.64 - 2 0 . 107.17 40. 111.57 " 1 9 . 106.10 39. 111.05 -18- 103.75 38. 111.39 "17. 102.03 37. 113.03 "16. 100.53 36. 114.99 " 1 5 . 99.50 35. 117.48 -14. 97.59 34. 117.46 - 1 3 . 96.70 110 T A L ( T ) T A L ( T ) -12. -11. -10. -9. -8 . -7 . -6 . -5. -4. -3. -2 . - 1 . 0. 1. 2. 3. 4. . 5. 6, 7 . 8 . 9. 10. 11. 12. 98.48 100.00 102.80 104.89 108.44 110.80 112.50 113.69 114.57 115.16 113.98 114.13 114.05 113.75 113.05 111.85 110.20 108.18 106.06 10 3.85 101.46 98.84 95. 49 95.90 95.41 13, 14, 15, 16, 17 , 18 , 19 , 20, 21, 22 , 23, 24 , 25. 26, 27 , 28, 29, 30. 31, 32. 33. 34. 35. 36. 37. 95.62 95.71 96 . 28 96.85 96.80 96 . 85 96.94 97.86 98.90 100 .45 101.10 105.68 108.08 109.82 111.08 112.17 113.41 114.50 115.23 118.75 121.37 122.75 124.38 123.76 119.45 I l l T A (T) T A T (T) L i 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 5 1 . 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: THE OPTIMAL LINEAR ESTIMATOR OF THE INTENSITY FUNCTION OF THE COAL-MINING DISASTER PROCESS C *** COAL MINING DISASTERS * EXACT SOLUTION *** C REAL MU, SIGMA, ALFA, DIV, FACT DIMENSION P ( 1 0 0 ) , D S ( I O O ) , S K ( I O O ) , EXACT(IOO) DIMENSION TM(IOO), C E X ( 6 0 ) , CAT(60) READ2, SIGMA, ALFA, DIV, FACT, AA, BB 2 FORMAT(6F6.2) N = 60 M = 60 T = - 3 0 . UT = 61. UT1 = 30.5 READ 4, ( P ( I ) , D S ( I ) , I = 1,N) 4 FORMAT(F6.0, F6.0) DO 5 I = 1,N S K ( I ) = P ( I ) / D I V 5 CONTINUE L = 0 K = 0 8 I F ( T . G E . 3 0 . ) GOTO 20 L = L + 1 Y=AA*SK(L) 113 Z=BB*SK(L) C *** CALL FUNCTION UTVAR TO COMPUTE NEW M(T) *** 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 = 0 10 J = J + l I F ( J . G T . N ) GOTO 12 R = A B S ( T - P ( J ) ) S=T+P(J) PA1=-TTA*R I F ( P A 1 . L T . - 1 0 0 . ) A=0. A = E X P ( P A l ) PA2=-TTA*(UT+S) I F ( P A 2 . L T . - 1 0 0 . ) B=0. B=EXP(PA2) PA3=-TTA*(UT-S) I F ( P A 3 . L T . - 1 0 0 . ) C=0. C= EX P ( P A 3 ) ABC = A+ B + C IF(ABC.EQ.O.) GOTO 10 SUM=SUM+DS(J)*ABC GOTO 10 C TO COMPUTE XT(S) 114 12 CVA = VALU(TTA,UT1,T) TO COMPUTE HT(S) QCVA - CVA/QMT ADVA = MU*VK*(1. - QCVA) K = K + 1 EXACT(K) = VK*SUM + ADVA + MU PRINT14, T, EXACT(K) 14 FORMAT(' ', F 6 . 0 , F 1 0 . 2 ) T = T + 1 GOTO 8 20 NT = 60 TM(1) = 1. DO 21 I = 1,N TM(I + 1) .= TM(I) + 1. 21 CONTINUE DO 22 I = 1,NT C A T ( I ) = T M ( I ) C E X ( I ) = E X A C T ( I ) 22 CONTINUE *** NOW CALL SUBROUTINE AJOA TO DO THE PLOTTING *** CALL AJOA(CAT,CEX,NT) * TERMINATE PLOTTING AND STOP ** CALL PLOTND STOP END 115 SUBROUTINE A J O A ( X , Y, N) DIMENSION 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 . , 'TIME', - 4 , 1 0 . , 0., XMIN, DX) CALL A X I S ( 0 . , 0., 'EXACT', 5, 1 0 . , 9 0 . , YMIN, DY) CALL L I N E ( X , Y , N , 1 ) CALL P L O T ( 1 2 . 0 , 0., -3) RETURN END C C FUNCTION UTVAR(X, Y, Z) REAL X,Y,Z A = E X P ( Y ) B= E X P ( - Z ) C = A + B CASH = X*C UTVAR = CASH -RETURN END C 116 A L ( T ) T A L ( T ) •30. -29, •28, •27. •26, •25, •24, •23, •22, •21, •20, •19 , •18, •17, •16, •15, •14, •13 •12, •11, •10, -9, -8, -7, -6, 3.05 3.25 3.40 3.47 3.53 3.56 3.54 3.53 3.52 3.48 3.42 3. 33 3.25 3.15 3 . 03 2.93 2 . 84 1.71 2 . 59 2.50 2 .44 2.38 2 . 34 2 . 30 1.20 - 5 . -4 . - 3 . - 2 . - 1 . 0. 1. 2. 3. 4 . 5. 6. 7. 8 . 9. 10. 1 1 . 12. 13. 14. 15. 16. 17. 18 . 19 . 2 . 32 2.31 2 . 31 2 .29 2.26 2.21 2 .17 2 .13 2 .10 2.05 1.97 1.92 1.87 1.85 1.85 1.88 1.87 1.87 1.89 1.94 2 . 02 2.10 2.22 2.28 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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