b l z . 1 . I n l e i d i n g 1 2 . D e f i n i t i e v a n f o u t e n 3 3 . S c h a t t e n v a n g e m i d d e l d e n e n v a r i a n t i e s 5 3 . 1 S t a t i s t i s c h e e i g e n s c h a p p e n 5 3 . 2 B e r e k e n i n g s m e t h o d e 9 4 . S c h a t t e n v a n k a n s d i c h t h e i d s f u n k t i e s 10 , 4 . 1 S t a t i s t i s c h e e i g e n s c h a p p e n 10 4 . 2 B e r e k e n i n g s m e t h o d e 1'^ 5 . S c h a t t e n v a n c o r r e l a t i e f u n k t i e s 15 5 . 1 S t a t i s t i s c h e e i g e n s c h a p p e n 17 5 . 2 B e r e k e n i n g s m e t h o d e 2 6 5 . 2 . 1 D i r e k t e s c h a t t i n g v a n c o r r e l a t i e f u n k t i e s 2 6 5 . 2 . 2 I n d i r e k t e s c h a t t i n g v a n c o r r e l a t i e f u n k t i e s 2 8 6 . H e t s c h a t t e n v a n s p e c t r a 3 2 6 . 1 S t a t i s t i s c h e e i g e n s c h a p p e n : a l g e m e e n 3 2 6 . 2 S p e c t r a l e s c h a t t e r s v i a d e i n d i r e k t e m e t h o d e 4 2 6 . 2 . 1 S t a t i s t i s c h e e i g e n s c h a p p e n 4 2 6 . 2 . 2 B e r e k e n i n g s m e t h o d e '^8 6 . 3 S p e c t r a l e s c h a t t e r s v i a d e d i r e k t e m e t h o d e 5 0 6 . 3 . 1 De r u w e p e r i o d o g r a m s c h a t t e r s 5 0 6 . 3 . 1 . 1 S t a t i s t i s c h e e i g e n s c h a p p e n 5 0 6 . 3 . 1 . 2 B e r e k e n i n g s m e t h o d e 5 6 6 . 3 . 2 D e g e m o d i f i c e e r d e p e r i o d o g r a m s c h a t t e r s 5 8 6 . 3 . 2 . 1 S t a t i s t i s c h e e i g e n s c h a p p e n 5 8 6 . 3 . 2 . 2 B e r e k e n i n g s m e t h o d e 6 2 6 . 3 . 3 D e g e m i d d e l d e p e r i o d o g r a m s c h a t t e r s 6 3 6 . 3 . 3 . 1 S t a t i s t i s c h e e i g e n s c h a p p e n 6 3 6 . 3 . 3 . 2 B e r e k e n i n g s m e t h o d e 6 7 6 . 3 . 3 . 3 O v e r l a p p e n d e r e e k s e n 6 7 6 . 3 . 3 . 4 D e k e u z e v a n d e d e e l r e e k s l e n g t e 6 8 6 . 3 . 4 De a f g e v l a k t e s p e c t r a l e s c h a t t e r s 6 9
6 . 3 . 4 . 2 B e r e k e n i n g s m e t h o d e • • • 7 2 6 . 4 B e t r o u w b a a r h e i d s i n t e r v a l l e n v o o r s p e c t r a l e s c h a t t e r s 73 6 . 5 H e t g e b r u i k v a n s p e c t r a a l - a n a l y s e p r o g r a m m a t u u r 76 6 . 6 V o o r b e e l d e n 79 7 . H e t s c h a t t e n . v a n k r u i s s p e c t r a 9 1 7 . 1 I n l e i d i n g 9 1 7 . 2 S t a t i s t i s c h e e i g e n s c h a p p e n 9 3 7 . 2 . 1 De i n d i r e k t e m e t h o d e 9 3 7 . 2 . 2 D e d i r e k t e m e t h o d e 9 4 7 . 2 . 3 A l i g n m e n t • 9 4 7 . 3 B e r e k e n i n g s m e t h o d e 9 8 7 . 3 . 1 De i n d i r e k t e m e t h o d e . . . 9 8 7 . 3 . 2 De d i r e k t e m e t h o d e 9 8 8 . S c h a t t e r s v a n c o h e r e n t i e f u n k t i e s . . 1 0 0 8 . 1 S t a t i s t i s c h e e i g e n s c h a p p e n • • • • 1 0 0 8 . 2 B e r e k e n i n g s m e t h o d e • 1 0 3 9 . H e t s c h a t t e n v a n o v e r d r a c h t s f u n k t i e s 1 0 5 9 . 1 S t a t i s t i s c h e e i g e n s c h a p p e n 1 0 5 9 . 2 B e r e k e n i n g s m e t h o d e 1 1 3 1 0 . V e r g e l i j k i n g e n i n d i r e k t e e n d i r e k t e m e t h o d e 1 1 9 1 1 . H e t s c h a t t e n v a n e x t r e m a 1 2 1 1 1 . 1 I n l e i d i n g 1 2 1 1 1 . 2 S t a t i s t i s c h e e i g e n s c h a p p e n v a n s p e c t r a l e m o m e n t e n 1 2 1 1 1 . 3 B e r e k e n i n g s m e t h o d e v a n s p e c t r a l e m o m e n t e n . . . 1 2 4 1 2 . T i j d r e e k s m o d e l l e n e n v o o r s p e l l i n g e n 4 vr
6 . 3 . 4 . 2 B e r e k e n i n g s m e t h o d e 7 2 6 . 4 B e t r o u w b a a r h e i d s i n t e r v a l l e n v o o r s p e c t r a l e s c h a t t e r s 73
6 . 5 H e t g e b r u i k v a n s p e c t r a a l - a n a l y s e p r o g r a m m a t u u r 7 6
a c o n s t a n t e ; o n d e r g r e n s d a t a - r a n g e A J . A ^ y ( r h ) , e v e n d e e l v a n k r u i s c o r r e l a t i e f u n k t i e b b o v e n g r e n s d a t a - r a n g e b . . b i a s ( o n z u i v e r h e i d ) v a n B b a n d b r e e d t e ; b a c k s h i f t - o p e r a t o r B g e q u i v a l e n t e b a n d b r e e d t e B g e q u i v a l e n t e b a n d b r e e d t e h o r e n d b i j G ^ ( f ) B g e q u i v a l e n t e b a n d b r e e d t e v a n W ( f ) B ^ e q u i v a l e n t e b a n d b r e e d t e v a n W ( f ) B ^ ' e q u i v a l e n t e b a n d b r e e d t e v a n W ( f ) B ^ B j 5 . y ( r h ) , o n e v e n d e e l v a n k r u i s c o r r e l a t i e f u n k t i e B { . . } b i a s f o u t c c o n s t a n t e ; g e s t a n d a a r d i s e e r d e b a n d b r e e d t e c c o r r e c t i e f a k t o r O c ^ 5 C j c o n s t a n t e s C ^ ( k ) , C „ ( T ) a u t o c o v a r i a n t i e f u n k t i e C j j y ( f ) c o ï n c i d e n t - s p e c t r u m ( c o - s p e c t r u m ) C ( k ) k r u i s c o v a r i a n t i e f u n k t i e d d i f f e r e n t i e - o r d e i n A R / M A - m o d e l dj^ e i n d p u n t i ^ i n t e r v a l d ( t ) , d - ^ ( t ) d a t a v e n s t e r d ^ ( t ) r e c h t h o e k i g d a t a v e n s t e r d j ( t ) c o s i n u s t a p e r D d i f f e r e n t i e - o r d e s e i z o e n - A R / M - m o d e l D ( f ) F o u r i e r g e t r a n s f o r m e e r d e v a n d ( t ) OO D ( g ) _ J f ( k ) W ( f - g ) d f D " ' - ( f ) F o u r i e r g e t r a n s f o r m e e r d e v a n d - ' - ( t ) D ^ ( f ) F o u r i e r g e t r a n s f o r m e e r d e v a n d ^ ( t ) D J ( f ) F o u r i e r g e t r a n s f o r m e e r d e v a n d j ( t ) e ^ ( £ ) ü - s t a p s - v o o r s p e l f o u t e . d . f . e q u i v a l e n t d e g r e e o f f r e e d o m E . . v e r w a c h t i n g s w a a r d e v a n f f r e q u e n t i e f ^ c u t - o f f f r e q u e n c y f | ^ d i s k r e t e f r e q u e n t i e - w a a r d e f ^ l.hf
f^y^ N y q u i s t - f r e q u e n t i e g i n t e g r a t i e v a r i a b e l e *^x(^k-^ ' •'^^^^ s p e c t r a l e s c h a t t e r G x ( f ) e e n z i j d i g s p e c t r u m G- j^ ( f ) r u w e p e r i o d o g r a m s c h a t t e r G ^ ( f ) g e m o d i f i c e e r d e p e r i o d o g r a m s c h a t t e r G ^ C p ) g e m i d d e l d e p e r i o d o g r a m s c h a t t e r G ^ ( p ) a f g e v l a k t e s p e c t r a a l s c h a t t e r G ^ ( p ) r u w e p e r i o d o g r a m s c h a t t e r v o o r i ^ d e e l r e e k s G ^ C p ) g e m o d i f i c e e r d e p e r i o d o g r a m s c h a t t e r v o o r i ^ d e e l r e e k s G ^ y ( f ) e e n z i j d i g k r u i s s p e c t r u m ^ x y ^ ^ k ^ ^ ^ ^ e s p e c t r a l e s c h a t t e r h d i s k r e e t t i j d i n t e r v a l , b e m o n s t e r i n g s i n t e r v a l h ( t ) i m p u l s r e s p o n s f u n k t i e H ( f ) o v e r d r a c h t s f u n k t i e H ^ ( p ) s u b s t i t u u t v o o r G ^ C p ) , G ^ ( p ) , G ^ C p ) o f G^^Cp) H ( f ) I g a i n f u n k t i e i i n t e g e r I a a n t a l s p e c t r a l e s c h a t t e r s j / - 1 ' k i n t e g e r K a a n t a l i n t e r v a l l e n o v e r d a t a r a n g e ; a a n t a l a u t o c o r r e l a t i e s ^ O 1 i n t e g e r L a a n t a l b e m o n s t e r i n g e n i n e e n d e e l r e e k s ; M / A m m a x i m u m a a n t a l l a g s ; s t e e k p r o e f g e m i d d e l d e ; i n t e g e r my^ k ^ s p e c t r a l e m o m e n t M m a x i m a l e l a g ; a a n t a l d e e l r e e k s e n Mj^ a a n t a l o v e r l a p p e n d e d e e l r e e k s e n n a a n t a l v r i j h e i d s g r a d e n n ( t ) r u i s n ^ , U j a a n t a l v r i j h e i d s g r a d e n N a a n t a l w a a r n e m i n g e n N £ a a n t a l w a a r d e n i n i ^ i n t e r v a l N a a n t a l w a a r d e n i n i n t e r v a l x ± 2 W 2C N.^ a a n t a l t o e g e v o e g d e n u l l e n z p i n t e g e r ; o r d e v a n A R - p r o c e s p ( x ) k a n s d i c h t h e i d s f u n k t i e p ( x , y ) s i m u l t a n e k a n s d i c h t h e i d s f u n k t i e
P P ( x , W ) P ( x , W x ; y , W y ) P r o b [. .] q Q Q ( f ) r r . - x y R r R C( r h ) R x y ( ^ ) s ^ x S2 S , ( f ) S , ( f ) i , ( f ) t T T ' T m a x T T ^ x ' '-K,y u u( t ) U t V v a r v d W ( T ) W o r d e v a n A R - s e i z o e n - o p e r a t o r a u t o c o r r e l a t i e m a t r i x k a n s v e r d e l i n g s f u n k t i e s i m u l t a n e k a n s v e r d e l i n g s f u n k t i e k a n s d a t o r d e v a n M A - p r o c e s o r d e v a n M A - s e i z o e n - o p e r a t o r ; t o e t s i n g s g r o o t h e i d p o s t m a n t e a u l a c k o f f i t t e s t q u a d r a t u u r - s p e c t r u m ( q u a d - s p e c t r u m ) l a g n u m m e r ; i n t e g e r s c h a t t i n g v o o r k r u i s c o r r e l a t i e c o ë f f i c i ë n t R ^ ( r h ) : s c h a t t e r v o o r a u t o c o r r e l a t i e f u n k t i e a u t o c o r r e l a t i e f u n k t i e s c h a t t i n g v o o r c y c l i s c h e a u t o c o r r e l a t i e f u n k t i e k r u i s c o r r e l a t i e f u n k t i e s e i z o e n - p e r i o d e s t a n d a a r d a f w i j k i n g s t e e k p r o e f v a r i a n t i e t w e e z i j d i g e r u w e s p e c t r a l e s c h a t t e r t w e e z i j d i g s p e c t r u m t w e e z i j d i g e c o n t i n u e v o r m v a n G^^Cf) t w e e z i j d i g e c o n t i n u e v o r m v a n G ^ ( f ) t w e e z i j d i g k r u i s s p e c t r u m t i j d v a r i a b e l e m e e t d u u r d e e l r e e k s l e n g t e e f f e k t i e v e r e g i s t r a t i e l e n g t e e f f e k t i e v e d e e l r e e k s l e n g t e p e r i o d e v a n d e l a n g z a a m s t e c o m p o n e n t E A t ^ i n t e g r a t i e - v a r i a b e l e i n g a n g s s i g n a a l w i t t e r u i s p r o c e s i n t e g r a t i e v a r i a b e l e v a r i a n t i e v a n v e r d u n n i n g s f a k t o r v e n s t e r f u n k t i e k l e i n i n t e r v a l
W ( f ) s p e c t r a a l v e n s t e r W ( f ) s p e c t r a a l v e n s t e r h o r e n d b i j G ^ C p ) W ( f ) s p e c t r a a l v e n s t e r h o r e n d b i j | x ( f ) W j ( f ) s p e c t r a a l v e n s t e r h o r e n d b i j c o s i n u s t a p e r W ( f ) s p e c t r a a l v e n s t e r Y l D ( f ) h W j ^ ( f ) r e c h t h o e k i g s p e c t r a a l v e n s t e r W ^ , W k l e i n e i n t e r v a l l e n x ( t ) t i j d s a f h a n k e l i j k e v a r i a b e l e X j ^ d a t a w a a r d e n x ^ d i s k r e t e w a a r n e m i n g e n r e e k s X j . t e b e s c h r i j v e n p r o c e s X] ^ ( t ) s t o c h a s t i s c h p r o c e s x i ( r h ) p u n t e n u i t i ^ d e e l r e e k s { x ( t ) } s t o c h a s t i s c h p r o c e s r e ë l e d e e l v a n X ^ ( f , T ) X J i m a g i n a i r e d e e l v a n X j ^ ( f , T ) X ] ; ( p ) r e ë l e d e e l v a n X ^ ( p ) X J ( p ) i m a g i n a i r e d e e l v a n X - ^ ( p ) x i ( p ) F o u r i e r c o m p o n e n t v o o r i ^ d e e l r e e k s g e d e f i n i e e r d d o o r ( 7 . 2 5 ) X^^, X ( k ) F o u r i e r c o m p o n e n t X ( f j ^ , T ) d i s k r e t e F o u r i e r g e t r a n s f o r m e e r d e v a n x ^ X ( p ) F o u r i e r c o m p o n e n t g e d e f i n i e e r d d o o r ( 6 . 5 0 ) X - ^ ( p ) F o u r i e r c o m p o n e n t v o o r i ^ d e e l r e e k s X ^ ( p ) F o u r i e r c o m p o n e n t g e d e f i n i e e r d d o o r ( 6 . 5 9 ) y ( t ) t i j d s a f h a n k e l i j k e v a r i a b e l e d i s k r e t e w a a r n e m i n g e n r e e k s { y ( t ) } s t o c h a s t i s c h p r o c e s Y-^ r e ë l e d e e l v a n Y ] ^ ( f , T ) Y j i m a g i n a i r e d e e l v a n Y - ^ ( f , T ) Y i ; ( p ) r e ë l e d e e l v a n Y ^ ( p ) Y ^ ( p ) i m a g i n a i r e d e e l v a n Y - ' - ( p ) Y - ' - ( p ) F o u r i e r c o m p o n e n t v o o r i ® d e e l r e e k s g e d e f i n i e e r d d o o r ( 7 . 2 6 ) Y-^, Y ( k ) F o u r i e r c o m p o n e n t Y ( f , T ) F o u r i e r g e t r a n s f o r m e e r d e v a n y ( t ) z s t a n d a a r d n o r m a a l v e r d e e l d e v a r i a b e l e z ( t ) t i j d s a f h a n k e l i j k e v a r i a b e l e ^ n ^ n + l Y n Zy_, Z ( k ) F o u r i e r c o m p o n e n t
Z ( f , T ) F o u r i e r g e t r a n s f o r m e e r d e v a n z ( t ) a s m o o t h i n g p a r a m e t e r ; o n b e t r o u w b a a r h e i d 3 c o r r e c t i e f a k t o r y ^ . ^ y i f ) c o h e r e n t i e f u n k t i e 6 ( t ) d e l t a f u n k t i e A b e m o n s t e r i n g s i n t e r v a l A f 1 / T A f ' 1 / ( N + ^^)h; ' / T ' e g e n o r m a l i s e e r d e r u i s f o u t c-^ g e n o r m a l i s e e r d e b i a s f o u t g e n o r m a l i s e e r d e s t a n d a a r d f o u t 6 l o o p t i j d , v e r t r a g i n g ; p a r a m e t e r M A ( 1 ) - p r o c e s 6 ^ M A - c o ë f f i c i ë n t 9 ^ y ( f ) a r g u m e n t v a n G ^ y ( f ) 8 ( B ) M A . - o p e r a t o r 9 ( B ^ ) M A - s e i z o e n - o p e r a t o r g e m i d d e l d e w a a r d e v a n x V a a n t a l v r i j h e i d s g r a d e n V p a a n t a l e . d . f . v a n G ^ ( p ) V p a a n t a l e . d . f . v a n G x ( p ) V p a a n t a l e . d . f . v a n G ^ ( p ) V p a a n t a l e . d . f . v a n G ^ ( p ) T T ( B ) g e w i c h t e n g e n e r e r e n d e f u n k t i e P j ^ , p ( k ) c o r r e l a t i e c o ë f f i c i ë n t p ^ y ( T ) k r u i s c o r r e l a t i e c o ë f f i c i ë n t O s t a n d a a r d a f w i j k i n g T t i j d v e r s c h u i v i n g (j) A R ( 1 ) - p a r a m e t e r A R - c o ë f f i c i ë n t ( j ) ( B ) A R - o p e r a t o r ( j ) ( f ) f a s e $ v a r i a b e l e $ ( B ^ ) A R - s e i z o e n - o p e r a t o r C h i - k w a d r a a t v a r i a b e l e m e a n - s q u a r e - v a l u e I | J ( B ) g e w i c h t e n g e n e r e r e n d e f u n k t i e V d i f f e r e n t i e - o p e r a t o r ? s c h a t t e r v o o r . c o m p l e x t o e g e v o e g d e v a n .
1 . I n l e i d i n g I n d i t t w e e d e d e e l v a n d e k u r s u s g a a n w e o n s b e z i g h o u d e n m e t d e t i j d r e e k s a n a l y s e . E e n t i j d r e e k s i s e e n r e a l i s a t i e v a n e e n s t o c h a s t i s c h p r o c e s e n d e t i j d r e e k s a n a l y s e b e o o g t o p g r o n d v a n é é n z o ' n r e a l i s a t i e u i t s p r a k e n t e d o e n o v e r h e t o n d e r l i g g e n d e p r o c e s . We d o e n d i t d o o r k a r a k t e r i s t i e k e g r o o t h e d e n v a n h e t p r o c e s t e b e r e k e n e n , z o a l s d e g e m i d d e l d e w a a r d e , d e v a r i a n t i e , d e c o r r e l a t i e f u n k t i e , e t c . W i l m e n d e z e k a r a k t e r i s t i e k e g r o o t h e d e n e x a k t b e r e k e n e n , d a n m o e t e n a l l e m o g e l i j k e u i t k o m s t e n v a n h e t p r o c e s b e k e n d z i j n , d i t w i l z e g g e n h e t h e l e e n s e m b l e . We h e b b e n e c h t e r s l e c h t s é é n r e a l i s a t i e , o v e r e e n b e p e r k t e t i j d s d u u r , z o d a t we s l e c h t s s c h a t t i n g e n v o o r d e g e n o e m d e k a r a k t e r i s t i e k e g r o o t h e d e n k u n n e n b e r e k e n e n . D e z e s c h a t t i n g e n h e b b e n a l l e e n d a n b e t e k e n i s , i n d i e n e n s e m b l e - g e m i d d e l d e n e n t i j d g e m i d d e l d e n a a n e l k a a r g e l i j k z i j n . D a a r o m b e p e r k e n w e o n s t o t e r g o d i s c h e p r o c e s s e n , w a a r v o o r d i t i m m e r s g e l d t . I n d e p r a k t i j k b e t e k e n t d i t d a t w e a a n n e m e n d a t h e t p r o c e s d a t w e b e s c h o u w e n e r g o d i s c h i s . I n h o o f d s t u k 2 w o r d e n e e r s t v e r s c h i l l e n d e s o o r t e n f o u t e n g e d e f i n i e e r d : d e s t a n d a a r d f o u t , d e b i a s - f o u t e n d e r o o t - m e a n - s q u a r e ( r m s ) f o u t , e n d e g e n o r m a l i s e e r d e v e r s i e s e r v a n . V e r v o l g e n s w o r d t a a n g e g e v e n h o e v e r s c h i l l e n d e k a r a k t e r i s t i e k e g r o o t h e d e n g e s c h a t k u n n e n w o r d e n . V a n d e z e s c h a t t e r s w o r d e n d e s t a t i s t i s c h e e i g e n s c h a p p e n g e g e v e n , z o a l s h u n s y s t e m a t i s c h e f o u t , v i a d e b i a s ( o n z u i v e r h e i d ) e n h u n t o e v a l l i g e f o u t , v i a d e v a r i a n t i e . V e r v o l g e n s w o r d t a a n g e g e v e n h o e d e b e r e k e n i n g s m e t h o d e i s . We b e h a n d e l e n g e m i d d e l d e n e n v a r i a n t i e s ( h o o f d s t u k 3 ) , k a n s d i c h t h e i d s f u n k t i e s ( h o o f d s t u k 4 ) , c o r r e l a t i e f u n k t i e s ( h o o f d s t u k 5 ) e n s p e c t r a ( h o o f d s t u k 6 ) . V o o r d e s p e c t r a w o r d e n v e r s c h i l l e n d e s c h a t t e r s g e g e v e n , n a m e l i j k s c h a t t e r s v i a d e i n d i r e k t e m e t h o d e e n v i a d e d i r e k t e m e t h o d e . B i j d e d i r e k t e m e t h o d e z i j n n o g t e o n d e r s c h e i d e n : r u w e p e r i o d o g r a m s c h a t t e r s , g e m o d i f i c e e r d e p e r i o d o g r a m s c h a t t e r s , g e m i d d e l d e p e r i o d o -g r a m s c h a t t e r s e n a f -g e v l a k t e s p e c t r a a l s c h a t t e r s . K r u i s s p e c t r a k o m e n i n h o o f d s t u k 7 a a n d e o r d e , t e r w i j l i n h o o f d s t u k 8 e n 9 c o h e r e n t i e f u n k t i e s r e s p e k t i e v e l i j k o v e r d r a c h t s f u n k t i e s w o r d e n b e h a n d e l d . I n h o o f d s t u k 10 w o r d t v e r v o l g e n s e e n v e r g e l i j k i n g g e m a a k t t u s s e n d e d i r e k t e e n i n d i r e k t e b e r e k e n i n g s m e t h o d e v o o r c o r r e l a t i e f u n k t i e s
e n s p e c t r a l e f u n k t i e s . E x t r e m a k o m e n i n h o o f d s t u k 11 a a n d e o r d e . I n h o o f d s t u k 12 t e n s l o t t e w o r d t i n g e g a a n o p t i j d r e e k s m o d e l l e n e n v o o r s p e l l e n . S p e c i a l e a a n d a c h t w o r d t h i e r g e g e v e n a a n d e a u t o r e g r e s s i e v e m o d e l l e n e n d e m o v i n g a v e r a g e m o d e l l e n , o o k w e l B o x - J e n k i n s - m o d e l l e n g e n o e m d , o m d a t d e z e z o ' n g r o t e r o l s p e l e n b i j d e B o x ~ J e n k i n s - m e t h o d e . D e z e m e t h o d e w o r d t o o k i n d i t h o o f d s t u k 12 b e h a n d e l d . M e t b e t r e k k i n g t o t v o o r s p e l l i n g e n w o r d e n v e r s c h i l l e n d e s m o o t h i n g - p r o c e d u r e s g e g e v e n , a l s m e d e d e w i j z e w a a r o p m e t b e h u l p v a n d e B o x - J e n k i n s - m e t h o d e k a n w o r d e n v o o r s p e l d .
2. D e f i n i t i e van f o u t e n
The accuracy o f parameter estimates based u p o n sample values can be described b y a mean square e r r o r defined as
mean square e r r o r = E[{é - <^y] P- ' f ) where Ó is a n estimator f o r O . E x p a n d i n g E q u a t i o n (2.-1) yields
E[{i> - 0)2] = E[ié - E[é] + E[é] - 0)2]
= E[{é - E m f ] + 2E[{é - E[é])(E[é] - O ) ]
+ £[(£[0] - 0
)2]
N o t e t h a t the m i d d l e t e r m i n the above expression has a f a c t o r equal to zero, n a m e l y ,
E[é - Em] = Em - E[é] = 0
Hence the mean square e r r o r reduces t o
mean square e r r o r =
£[(0 -
E[é]Y] +£[(£[0]
- O)^] (2.2) I n w o r d s , the mean square e r r o r is the sum o f t w o parts. The first p a r t is a variance t e r m w h i c h describes the r a n d o m p o r t i o n o f the e r r o r .V a r [ O ] = E[i^ - s m f ] = Em] - E ' m (2,3) a n d the second p a r t is the square o f a bias t e r m w h i c h describes the
syste-m a t i c p o r t i o n o f the e r r o r ,
b^é] = E[b^m] = E[{Em - 0)2] (2.4)
T h u s the mean square e r r o r is the s u m o f the variance o f the estimate plus the square o f the bias o f the estimate, t h a t is
£[(0 -
0)2] = V a r [ O ] + b^é] (2,5)I t is generally m o r e convenient t o describe the e r r o r o f an estimate i n terms w h i c h have the same engineering units as the parameter being estimated. T h i s can be achieved b y t a k i n g the positive square r o o t s o f the e r r o r terms i n E q u a t i o n s (z.-h) t h r o u g h (2.5). T h e square r o o t o f E q u a t i o n (2.5) yields the standard d e v i a t i o n f o r the estimate, called the standard error ( o r random error) as f o l l o w s
standard e r r o r = = ^E[^^] - £ 2 [ 0 ] (a.6) T h e square r o o t o f E q u a t i o n (2A) gives the bias error directly as
bias e r r o r = ö[Ó] = £ [ * ] - O (-2,7) T h e square r o o t o f the s u m o f the squared errors, as given by E q u a t i o n (2.5),
defines the root mean square (rms) e r r o r as
rms error = V£[(6 - 0)2] = Va2[0] + b^ê] (2.Q)
A s a f u r t h e r convenience, i t is o f t e n desirable t o define the e r r o r o f a n estimate i n terms o f a f r a c t i o n a l p o r t i o n o f the q u a n t i t y being estimated. T h i s is done by d i v i d i n g the e r r o r by the q u a n t i t y being estimated t o o b t a i n a
normalized error. F o r <1) 0, the n o r m a h z e d s t a n d a r d , bias, a n d r m s errors are given b y n o r m a l i z e d standard e r r o r = = = (S.QSI) O (D n o r m a h z e d bias e r r o r = ej, = ^ ^ ^ ^ = ^^^^ — \ (Q.Q b) 0) O n o r m a l i z e d r m s e r r o r = e = = ( 2 . q ^
N o t e t h a t the n o r m a l i z e d standard e r r o r is o f t e n called the coefficient of
variation.
I n the f o l l o w i n g chapter6,the errors associated w i t h p a r a m -eter estimates are derived i n terms o f the quantities defined b y E q u a t i o n
(2.9). G i v e n the s a m p l i n g d i s t r i b u t i o n s o f the estimators, these quantities can
be extended t o confidence i n t e r v a l statements
W i t h the exception o f spectral density a n d f r e q u e n c y response f u n c t i o n estimates, a n o r m a l i t y a s s u m p t i o n f o r the s a m p l i n g d i s t r i b u t i o n o f the p a r a m -eter estimates is usually acceptable w h e n the n o r m a l i z e d r m s e r r o r is s m a l l . Spectral density a n d f r e q u e n c y response f u n c t i o n estimates require special c o n s i d e r a t i o n because the n o r m a l i z e d r m s e r r o r f o r these estimates is o f t e n relatively large i n practice. A l o n g w i t h the d e r i v a t i o n o f e r r o r s , the consistency o f t h e various parameter estimates, is investigated.
3 . 1 S t a t i s t i s c h e e i g e n s c h a p p e n
3_. S c h a t t e n v a n gemiddelden en v a r i a n t i e s
3.1 Stati.sti.sehe eigen.schappen
Mean Values
Suppose that a single sample t i m e h i s t o r y r e c o r d x(t) f r o m a s t a t i o n a r y (ergodic) r a n d o m process {x{t)} exists over a finite t i m e T. T h e m e a n value o f {x(t)} can he estimated by
fi.=
^r^iO
dt ( 5 J ) I JoT h e t r u e mean value is
fix = E[x(t)] (1^1)
a n d is independent o f t since {x(t)} is stationary. T h e expected value o f the estimate fix is
Elflx] = E i r x ( t ) d t ^ = i rE[x(t)] d t = ^ r,,^^ d t = ( 3 . 5 )
i Jo J T Jo T Jo
since expected values c o m m u t e w i t h linear operations, as given b y E q u a t i o n ( 3 . g o ) . Hence fix is a n unbiased estimate o f fi^, independent o f T. Since fi^
is unbiased, the mean square e r r o r o f the estimate fi^ is equal t o the v a r i a n c e . V a r [fix] = E[ifix - f i x f ] = E[fix'] - ix^' ( 5 . 4 ) where f r o m E q u a t i o n ( S . l ) ,
E[fi:\ = ~ r rE[xiO x(r,)] dr, di (3.Ö") T Jo Jo
N o w the a u t o c o r r e l a t i o n f u n c t i o n Rxi^) o f a stationary r a n d o m process
{x{t)} is defined b y
RX(T) = E[x{t) x{l + r)] ( 5 . 6 )
F r o m the stationary hypothesis, R^ir) is independent o f t, a n d a n even f u n c t i o n o f T w i t h a m a x i m u m at T = 0, I t w i l l be assumed t h a t RJT) is c o n t i n u o u s a n d finite f o r a l l values o f T , a n d t h a t a l l p e r i o d i c components i n Rx{r) have been r e m o v e d at the onset. T h e autocovariance f u n c t i o n Q ( T ) is defined by
CJT) = Rxir) - fC ( 5 . ? )
I t turns o u t t h a t whenever (x^ ^ 0, i t is m o r e convenient t o w o r k w i t h Q ( T )
t h a n w i t h Rj^r). I t w i l l be assumed t h a t Q ( T ) satisfies the properties so as to m a k e {x(t)} ergodic.
I n terms o f the autocovariance f u n c t i o n ^ ( T ) , the variance (mean square e r r o r ) f r o m E q u a t i o n s ( 3 . 4 ) and ( 5 . 5 ) becomes
V a r [fix] = V f ^ C i , ? - f ) dri di = T C , ( T ) dr di
T h e last expression occurs by reversing the orders o f i n t e g r a t i o n between T a n d f a n d c a r r y i n g o u t the f i n t e g r a t i o n . T h i s changes the l i m i t s o f i n t e -g r a t i o n f o r T a n d i as s h o w n i n the sketch b e l o w so t h a t CJT) dr di Jo J -CJr) di dr 0 Jo '0 fT CJT) di dr + J-T J - T '0 f-T = (T + r) CJ,T) dr+ ( T - T ) CJr) dr J-T Jo = (T - \r\) CJr) dr J-T N o w o n l e t t i n g T tend to i n f i n i t y , E q u a t i o n ( 3 . S } becomes l i m T V a r [fix] = T-*aa CJr) dr<co
(3
.9)
s i n c e { x( t ) } i s e r g o d i c ,, w h i c h p r o v i d c s t h a t Q ( T ) a n d T CJr) are absolutely integrable over (—00, 00) to j u s t i f y passage t o the Hmit inside the i n t e g r a l sign. I n p a r t i c u l a r . E q u a t i o n ( 3 . 9 ) shows t h a t f o r large T, where|T| « T, the variance is given by
CJr) dr i&.iO)
Hence w h e n the integral is finite-valued. V a r [/<J approaches zero as T approaches i n f i n i t y , p r o v i n g t h a t fi^ is a consistent estimate o f fx^.
Example 3.1. Variance o f M e a n Value Estimates o f B a n d w i d t h L i m i t e d
W h i t e Noise. Consider the i m p o r t a n t special case where {a:(;)} is b a n d -w i d t h l i m i t e d -w h i t e noise -w i t h a mean value fx^ 9^ 0. Assume the p o -w e r spectral density f u n c t i o n is
1
G J f ) = ^ + f x J d { f ) Q < f < B B
= 0 f > B (3,11)
where B is the b a n d w i d t h . T h e associated autocovariance f u n c t i o n CJr) is given by
CJr)
Jo
Jo
G J f ) cos 2iTfr df - ixx
^ 1 , . smlrrBr = — cos 2rrfr df
Jo B 2rrBr
(3,1a)
N o t e t h a t CJO) = 1 a n d t h a t CJr) = 0 f o r T = 1/25. T h u s p o i n t s 1/25 apart are u n c o r r e l a t e d . ( T h e y w i l l be statistically independent i f {x(t)} is Gaussian.) F o r this case. E q u a t i o n (S.AO) yields the result
V a r [ ^ J ^ i
r ( ^ j l ^ ) d r
= - ^ (3.15)TLOOX 27TBr J 2BT
T h i s is the variance o f mean value estimates o f b a n d w i d t h l i m i t e d w h i t e noise w i t h b a n d w i d t h B a n d s a m p l i n g t i m e T. E q u a t i o n ('5.13) requires t h a t
T be sufficiently large so t h a t E q u a t i o n ( 5 , i p ) can replace E q u a t i o n ( S . ó ' ) .
F o r CJO) 1, E q u a t i o n (3.-15) w o u l d be
C . ( 0 ) aJ Var [/JJ
IBT 2BT
Hence, w h e n /x^ 0, the n o r m a l i z e d mean square e r r o r is given b y
, V a r [ f i x ] 1
£ = •
fXx' 2BT\fi
(-1
T h e square r o o t o f E q u a t i o n (3. •14) gives the normalized rms error e, w h i c h includes o n l y a r a n d o m e r r o r t e r m since the bias e r r o r is zero.
Mean Square Values
let x{t) be a single sample t i m e h i s t o r y record f r o m a s t a t i o n a r y (ergodic) r a n d o m process { « ( / ) } . T h e mean square value o f {«(?)} can be estimated by t i m e averaging over a finite t i m e i n t e r v a l T as f o l l o w s .
Y J = - r x \ t ) dt ( 3 . I S ) T Jo
T h e t r u e mean square value is
= E[x\t)] {5.ib)
a n d is independent o f / since {x(t)} is stationary. T h e expected value o f the estimate WJ is
E[Wx'] = - rE[xXt)] dt = - r ^ x ^ dt = (3.17)
T Jo T Jo
A
Hence WJ^ is an unbiased estimate o f T , . ^ , independent o f T. T h e m e a n square e r r o r here is given b y the variance
. V a r [YJ] = E[{WJ - W j f ] = E[YJ] - Y j
^ ''^iE[x\i) x\r])] - Y x ' ) drj di (3.1(9) ^ eT
^ r
Jo JoA s s u m e n o w t h a t {x{t)} is a Gaussian r a n d o m process w i t h mean value fXx 9^ 0. T h e n the expected value i n E q u a t i o n ( j . i ö ) takes the special f o r m
E[x%i) x%r])] = 2iRxKv - f ) - l^J) + "^J (5•^9)
F r o m the basic r e l a t i o n s h i p i n E q u a t i o n ( 5 . 7 ) ,
Hence
V a r [WJ] = l i riR.Xv - 0 - l^') dv
T Jo Jo= 2 r _ | r | \ , _^ , ^ ^ ( ^ ^ ^ ( 3 ^ ^ )
T J -T\ TJ
F o r large T, where | T | « T, the variance becomes
V a r [WJ] ( C / ( r ) + 2 ^ / C , ( T ) ) ( 3 . 1 7 )
T J—oo
T h u s YJ is a consistent estimate o f since V a r [YJ] w i l l a p p r o a c h zero as
T approaches i n f i n i t y assuming CJ(T) a n d CJr) are absolutely integrable
over (— 00, co)
Example 3 . 2 . Variance o f M e a n Square Value Estimates o f B a n d w i d t h L i m i t e d Gaussian W h i t e Noise. Consider the i m p o r t a n t special case o f b a n d w i d t h l i m i t e d Gaussian w h i t e noise where
c.« = c , o , ( ^ ' )
F o r this case, E q u a t i o n (3.ax) shows t h a tV a r [ T ; ] ^ ^ + ^ / . . ^ C . ( 0 ) (3.1^)
T h i s is the variance o f mean square value estimates o f b a n d w i d t h l i m i t e d Gaussian w h i t e noise w i t h b a n d w i d t h B a n d s a m p l i n g t i m e T. E q u a t i o n (3.24) requires t h a t T be sufficiently large so t h a t E q u a t i o n (z.Ti) can replace E q u a t i o n ( s . i l ) . Satisfactory c o n d i t i o n s i n practice are T ^ 10 | T | a n d
BT> 5.
F o r /Lix= 0, E q u a t i o n (Z.^if) becomes
BT BT
Hence w h e n /x^^O, the n o r m a l i z e d mean square e r r o r is given b y V a r [ T / ] _
V a r [ ^ ^ _ L ( 3 . ^ ^ Y J BT
T h e square r o o t o f E q u a t i o n (3.26) gives the n o r m a l i z e d rms e r r o r e. N o t e t h a t this result also represents the n o r m a l i z e d rms error o f variance estimates w h e n = 0. F u r t h e r note t h a t , as f o r m e a n value estimates, e includes o n l y a r a n d o m e r r o r t e r m since the bias e r r o r is zero.
3.2 Berekeningsmethode
C A L C U L A T I O N O F T H E M E A N V A L U E . T h e sample m e a n value is given by
where N is t h e n u m b e r o f data samples a n d >:„ are the data values. T h e q u a n t i t y rn calculated here is a n unbiased estimate o f the mean value /x.
B e r e k e n i n g v a n mean square v a l u e en v a r i a n t i e .
Een s c h a t t e r , van de mean square v a l u e wordt berekend v i a :
(3.28) en de v a r i a n t i e met: s N-1 N (3.29) n=1
4 . 1 S t a t i s t i s c h e e i g e n s c h a p p e n
S c h a t t e n van k a n s d i c h t h e i d s f u n k t i e s
4.1 s t a t i s t i s c h e eigenschappen
Consider a p r o b a b i l i t y density measurement o f a single sample time h i s t o r y r e c o r d x(t) f r o m a stationary (ergodic) r a n d o m process {.^•(0}• T h e p r o b a b i l i t y t h a t x{t) assumes p a r t i c u l a r a m p l i t u d e values between x — {Wl2) and X + (Wjl) d u r i n g a t i m e i n t e r v a l T m a y he estimated b y
P[x, W] = P r o b
1
TT i ' T
(AA)
where A^^ is the t i m e spent by x(t) i n this a m p h t u d e range d u r i n g the / t h entry i n t o the range, a n d 73^ = 2 ^^i- T h e r a t i o TJTis the t o t a l f r a c t i o n a l p o r t i o n o f the t i m e spent b y x{t) i n the range [x - (W/2), x + iWI2)]. I t s h o u l d be n o t e d t h a t T^ w i l l usually be a f u n c t i o n o f the a m p l i t u d e x. T h i s estimated p r o b a b i l i t y P[x, W] w i l l a p p r o a c h the true p r o b a b i l i t y P{x, W] as T ap-proaches i n f i n i t y . M o r e o v e r this estimated p r o b a b i l i t y is an unbiased esti-mate o f the true p r o b a b i l i t y . Hence
P[x, W] = E[P[x, W]] = l i m P[x, W] = l i m —* T-K» T^m T T h e p r o b a b i l i t y density f u n c t i o n p{x) is defined b y , , ,. P[x, W] ,. P[x, W] ,. , pix) = h m — - = h m - ^ - ^ — ' = h m p(x) ir->o W r - o o W T^oo fF—o ir-*o where P[x, W] W T T I F ( 4 . 3 )
is a sample estimate o f p(x). I n terms o f the p r o b a b i h t y density f u n c t i o n p(x), the p r o b a b i l i t y o f the t i m e h i s t o r y x(t) f a l l i n g between any t w o values aJi a n d x^ is given b y P r o b [xi<,x<, x^ I n p a r t i c u l a r P\x, W] = P r o b ••iZ Jxi p(x) dx ( 4 - 5 ) W W' ('a:+(JF/2) K f ) ^ ? f ( 4 . 6 ) Ja:-(jr72)
T h e n , f r o m E q u a t i o n (4. A ) , E[P[x, W]] P[x, W] 1 , , , , E[p{x)] = = = - p(i) di (4.7 ) W W W Jx~(w/2) T h u s f o r most p(x), ElPix)] ^ pix) (4.Q ) p r o v i n g t h a t p{x) is generally a biased estimate o f p{x).
T h e mean square e r r o r o f the estimate p{x) is calculated f r o m E q u a t i o n (2.6) b y
E[(p(x) - p i x ) f ] = V a r [pix)] + b'[p{x)] M ) where V a r [p{x)] is the variance o f the estimate as defined b y
V a r [pix)] = E[(p(x) - E[pix)]f] ( ^ . f o ) and b[p(x)] is the bias o f the estimate as d e f i n e d b y
b[p(x)] = E[p{x) - p(x)] (AM ) Variance of the Estimate
T o evaluate the variance o f an estimate p{x), i t is necessary t o k n o w the statistical properties o f the t i m e intervals A?, w h i c h comprise T^. U n f o r t u -nately, such t i m e statistics f o r a r a n d o m process are very d i f f i c u l t t o o b t a i n . H o w e v e r , the general f o r m o f a n a p p r o p r i a t e variance expression f o r p(x) can be established b y the f o l l o w i n g heuristic argument.
F r o m E q u a t i o n (4.4 ) , the variance o f ^ ( a ; ) is given b y
V a r [p(x)] = V a r [P(x, W)] (A-n)
where P{x, W) is the estimate o f a p r o p o r t i o n P{x, W). T h e variance f o r a p r o p o r t i o n estimate based u p o n N independent sample values is given b y ,
V . r l A » . » ' ) ] - ^ " - " ' ' " - ' ' ' ^ ' " ' "
S u b s t i t u t i n g E q u a t i o n (^.-15) i n t o E q u a t i o n (4.12), a n d assuming P(a;, W) ^ Wp{x) « 1, the variance o f p r o b a b i l i t y density estimates m a y be a p p r o x i -m a t e d b y
V a r [p{x)] ^ (4.14) NW
where is still t o be d e t e r m i n e d . N o w f r o m the t i m e d o m a i n s a m p l i n g t h e o r e m , a sample r e c o r d x(t) o f b a n d w i d t h B a n d
length T can be completely reproduced w i t h N = IBT discrete values. O f course, the N discrete values w i l l n o t necessarily be statistically independent. Nevertheless, f o r any given stationary (ergodic) r a n d o m process, each sample r e c o r d w i l l represent n = Njc^ independent sample values (degrees o f f r e e d o m ) , where c is a constant. Hence f r o m E q u a t i o n ( ^ . l ^ ) ,
N a r [ p { x ) ] ^ - ^ ^ (4.15) 2BTW
The constant c is dependent u p o n the a u t o c o r r e l a t i o n f u n c t i o n o f the data and the samphng rate. F o r continuous b a n d w i d t h l i m i t e d w h i t e noise, e x p e r i m e n t a l studies mdicate c 0.3. I f the b a n d w i d t h l i m i t e d w h i t e noise is d i g i -tized t o N = 2BT discrete values, the e x p e r i m e n t a l studies indicate c = 1.0 as w o u l d be expected f r o m the results o f E q u a t i o n (3.12 ) .
Bias of the Estimate
A n expression w i l l n o w be derived f o r the bias t e r m o f E q u a t i o n (4.11). I n terms o f the true p r o b a b i l i t y density f u n c t i o n . E q u a t i o n ( 4 . ? ) shows t h a t
Elp(x)] = ^^ p{i)di (4.1b)
By expanding p{i) i n a T a y l o r series a b o u t the p o i n t ^ = a n d r e t a i n i n g only the first three terms,
p{i) « p{x) + {i-x) p'{x) + P'\x) (4.17.)
F r o m the t w o relations fx+lWIi) and I t f o l l o w s t h a t { i - x ) d i = 0 ( 4 . i ö a ) Jx-(Wli) Ja x+lW/2) ( t _ „\1 w 3 ( ^ ^ d f = - ^ ( 4 . i ö b ) x-{wi2.) 2 24 W2 E[p{x)]^p{x) + —p"{x) (4.ic^)T h u s a first order a p p r o x i m a t i o n f o r the bias t e r m is given b y
b[p{x)]<=^'^p"{x) (4.ao)
Normalized rms Error
T h e t o t a l mean square e r r o r o f the p r o b a b i l i t y density estimate p(x) is given b y E q u a t i o n ( 4 . g ) as the s u m o f the variance defined i n E q u a t i o n
(A-i'o) a n d the square o f the bias defined i n E q u a t i o n (A-io). T h a t is,
Hence, the n o r m a l i z e d mean square e r r o r is a p p r o x i m a t e d b y
, 2 ^ , E l 2BTWp(x) 576
'PM
.p(x)^
(A.H)
T h e square r o o t o f E q u a t i o n (^.'xa) gives the n o r m a h z e d rms e r r o r e. I t is clear f r o m E q u a t i o n ( 4 . 1 1 ) t h a t there are c o n f l i c t i n g requirements o n the w i n d o w w i d t h Win p r o b a b i l i t y density measurements. O n the one h a n d , a large value o f Wis desirable t o reduce the r a n d o m error. O n the other h a n d , a s m a l l value o f W is needed t o suppress the bias error. H o w e v e r , the t o t a l e r r o r w i h a p p r o a c h zero as 0 0 i f I F is restricted so that W-^-O a n d
WT-> 0 0 . I n practice, values of W < 0.2a^ w i l l usually l i m i t the n o r m a h z e d
bias e r r o r to less t h a n one percent. T h i s is t r u e because o f the p"(x) t e r m i n the bias p o r t i o n o f the e r r o r given b y E q u a t i o n (A ll)- P r o b a b i l i t y density f u n c t i o n s o f c o m m o n ( a p p r o x i m a t e l y Gaussian) r a n d o m data d o n o t dis-p l a y a b r u dis-p t o r shardis-p dis-peaks w h i c h are i n d i c a t i v e o f a large second derivative.
Joint Probability Density Estimates
J o i n t p r o b a b i h t y density estimates f o r a p a i r o f sample t i m e h i s t o r y records,
x(t) a n d y(i), f r o m t w o s t a t i o n a r y (ergodic) r a n d o m processes {x{t)} a n d {y(t)}, m a y be defined as f o l l o w s . A n a l o g o u s t o E q u a t i o n {AA ) , let
P[x,Wx;y,W,] = '^ (4.23)
estimate the j o i n t p r o b a b i l i t y t h a t x{t) is inside the a m p h t u d e i n t e r v a l centered at x w h i l e simultaneously y{t) is inside the a m p h t u d e i n t e r v a l Wy centered at y. T h i s is measured b y the r a t i o T^JT where T^^y represents the a m o u n t o f t i m e t h a t these t w o events coincide i n t i m e T. C l e a r l y , Tx,y w i l l u s u a l l y be a f u n c t i o n o f b o t h x a n d y. T h i s estimated j o i n t p r o b a b i h t y w i l l a p p r o a c h the t r u e p r o b a b i l i t y P[x, W^; y, Wy] as T approaches i n f i n i t y , n a m e l y ,
The j o i n t p r o b a b i l i t y density f u n c t i o n p{x, y) is n o w defined by
K . , y) = h m ^ l ^ i i ^ i i l ü ^ = Hm i ^ B . ^ ^ ^ = Hm p^x, y) (4..5)
Tr»-»o irx-*o n-i-^o
where
= (4..6)
WxW, TWxWy
Assume t h a t a n d Wy are sufficiently small so t h a t the bias errors are neghgible. T h e n the mean square error associated w i t h the estimate p{x, y) w i l l be given by the variance o f the estimate. A s f o r first-order p r o b a b i l i t y density estirnates, this quanHty is d i f f i c u l t to determine precisely by theoreti-cal arguments alone. H o w e v e r , by using the same heuristic arguments w h i c h p r o d u c e d E q u a t i o n (4<4), a general f o r m f o r the variance can be a p p r o x i mated. Specifically, f o r the special case where x{t) and y{t) are b o t h b a n d -w i d t h l i m i t e d -w h i t e noise -w i t h identical b a n d -w i d t h s B,
where c is an u n k n o w n constant.
4.2 Berekeningsmethode
Consider N data values {«„}, n = \,2,. . . ,N, f r o m a t r a n s f o r m e d record x{t) w h i c h is stationary w h h zero mean x. I t f o l l o w s f r o m E q u a t i o n (4-4 ) t h a t the p r o b a b i l i t y density f u n c t i o n o f x{t) can be estimated by
p{x) = (4.aÖ) NW
where I I ' is a n a r r o w i n t e r v a l centered at x a n d N^ is the n u m b e r o f data values w h i c h f a l l w i t h i n the range x ± Wjl. Hence an estimate ^(a;) is o b -tained d i g i t a l l y b y d i v i d i n g the f u l l range o f x i n t o an a p p r o p r i a t e n u m b e r o f equal w i d t h class intervals, t a b u l a t i n g the n u m b e r o f data values i n each class i n t e r v a l , a n d d i v i d i n g b y the p r o d u c t o f the class i n t e r v a l w i d t h I K a n d the sample size N. N o t e t h a t the estimate p{x) is n o t unique since i t clearly is dependent u p o n the n u m b e r o f class intervals a n d their w i d t h selected f o r the analysis.
A f o r m a l statement o f this procedure w i l l n o w be given. L e t K denote the n u m b e r o f class intervals selected t o cover the entire range o f the data values
f r o m a to b. T h e n the w i d t h o f each i n t e r v a l is given b y
w = ^ ( 4 ^ 9 )
and the end p o i n t o f the / t h i n t e r v a l is defined b y
di = a + iW i = 0,1,2, . . . ,K (u.^ N o t e t h a t do = a and dj^ = b. N o w define a sequence o f i i ^ + 2 numbers
{JVJ; / = 0, 1, 2, . . . , ^ + 1, b y the c o n d i t i o n s No = [ n u m b e r o f x such t h a t x < C/Q] TV^i = [ n u m b e r o f x such t h a t d^ <. x <. di]
Ni = [ n u m b e r o f x such t h a t < a; < di] ( A - ^ l )
TV^^ = [ n u m b e r o f x such that (^2^-1 < * ^ ^ i f ] N^^i = [ n u m b e r o f a; such t h a t x > rfj^]
T h i s procedure w i l l sort o u t the N data values o f x so t h a t the n u m b e r sequence {Ni} satisfies
JC+l
JV = 2 Ni (A -il)
One m e t h o d o f d o i n g this s o r t i n g o n a d i g i t a l c o m p u t e r is t o examine each value a;„; n = 1, 2, . . . , A'^, i n t u r n as f o l l o w s .
!• I f *n < a, a d d the integer one t o N^.
2. I f a < a;„ < b, c o m p u t e / = («„ — a)IW. T h e n , select / as the largest integer less t h a n o r equal to / , a n d a d d the integer one t o iV,.
3. I f .r„ > b, a d d the integer one to y V ^ + i .
F o u r o u t p u t f o r m s f o r the sequence {Ni} can be used. T h e first o u t p u t is the h i s t o g r a m , w h i c h is s i m p l y the sequence {AT,} w i t h o u t changes. T h e second o u t p u t is the sample percentage o f data i n each class i n t e r v a l defined f o r i = 0 , l , 2 , . . . , K + l , h y
Pi = P r o b [di_i <x<di] = j^ is.)
{pj} defined at the m i d p o i n t s o f the K class intervals i n [a, b] b y
T h e f o u r t h o u t p u t is the sequence o f sample p r o b a b i l i t y d i s t r i b u t i o n estimates { / ( ; ) } defined at the class i n t e r v a l end p o i n t s where / = 0, 1, 2 , . . . , X + 1, b y
P(i) = P r o b [ - 00 < a; < c/J = 2 A ' = ^ i ^ - ^ ^ ) 3=0 3=0
Joint Probability Density Functions
I t f o l l o w s f r o m E q u a t i o n (•^.z6) that the j o i n t p r o b a b i h t y density f u n c t i o n o f t w o stationary records x(t) a n d y{t) can be estimated f r o m digitized data by
p(x,y) = - ^ ^ ( 4 . 5 6 )
where W^. and Wy are n a r r o w intervals centered o n x and y, respectively, a n d N^y is the n u m b e r o f pairs o f data values w h i c h simultaneously f a l l w i t h i n these intervals. Hence a n estimate p(x, y) is o b t a i n e d b y d i v i d i n g the f u l l ranges o f x a n d y i n t o a p p r o p r i a t e numbers o f equal w i d t h class intervals f o r m i n g t w o d i m e n s i o n a l rectangular cells, t a b u l a t i n g the n u m b e r o f data values i n each cell, and d i v i d i n g b y the p r o d u c t o f the cell area WJVy and the sample size A''. C o m p u t e r procedures f o r s o r t i n g the data values i n t o a p p r o -p r i a t e cells are s i m i l a r t o those o u t l i n e d f o r -p r o b a b i l i t y density estimates.
5 . 1 S t a t i s t i s c h e e i g e n s c h a p p e n
5 . 2 B e r e k e n i n g s m e t h o d e
5 . 2 . 1 D i r e k t e s c h a t t i n g v a n c o r r e l a t i e f u n k t i e s 5 . 2 . 2 I n d i r e k t e s c h a t t i n g v a n c o r r e l a t i e f u n k t i e s
5 j S c h a t t e n v a n c o r r e l a t i e f u n k t i e s
_5jJ S t a t i s t i s c h e eigenschappen
Consider n o w t w o sample time h i s t o r y records x{t) a n d y{l), f r o m t w o stationary (ergodic) r a n d o m processes, {«(/)} and
{y(t)}.
T h e next statistical quantities o f interest are the stationary a u t o c o r r e l a t i o n f u n c t i o n s , RJr) andRy(r), and the cross-correlation f u n c t i o n 5^„(T). T O s i m p l i f y the f o l l o w i n g
d e r i v a t i o n , the mean values /x^ and /j.^ w i l l be assumed t o be zero. F o r con-tinuous data, x{t) and y{t), w h i c h exist o n l y over a t i m e i n t e r v a l T, the sample cross-correlation estimate Kyir) can be defined by
1 C^"' = x{t) y(t + r)dt 0 < T < T T — T Jo (5.1 ) T - | T | J | r | — x\ T | J\rl ( t ) y ( t + T)dt - T < r < 0
T o a v o i d use o f absolute value signs, T w i l l be considered positive hence-f o r t h since a similar p r o o hence-f applies hence-f o r negative T. T h e sample a u t o c o r r e l a t i o n f u n c t i o n estimates -^^.(T) and ^ „ ( T ) are merely special cases w h e n the t w o records coincide. T h a t is,
^ ^ ( r ) =
r
« ( 0x(t
+
T)dt
0 < T < T T — T JoRJT) = — ^ 2 / ( 0
yit
+r)dt
0 < T < T T — r Jo( f f . 2 )
Thus by a n a l y z i n g the cross-correlation f u n c t i o n estimate, one derives results w h i c h are also applicable t o the a u t o c o r r e l a t i o n f u n c t i o n estimates.
I f the data exist f o r t i m e T + T instead o f o n l y f o r time T, then an alter-native d e f i n i t i o n f o r Kyir) is
RM) = i r 4 0 y(t + r) dt 0<r<T ( 5 . 3 )
T Jo
T h i s f o r m u l a has a fixed i n t e g r a t i o n t i m e T instead o f a variable i n t e g r a t i o n t i m e as i n E q u a t i o n (5.1 ) , a n d is the w a y the c o r r e l a t i o n f u n c t i o n s have been defined previously. N o t e t h a t f o r either E q u a t i o n (5.1 ) or E q u a t i o n ( S . i ) , mean square estimates o f x{t) o r y{t) are merely special cases w h e n T = 0. F o r s i m p l i c i t y i n n o t a t i o n . E q u a t i o n ( 5 . 3 ) w i l l be used i n the f o l l o w i n g development instead o f E q u a t i o n (5,1 ) . T h e same final results are o b t a i n e d f o r b o t h d e f i n i t i o n s , assuming the data exists f o r t i m e T+r.
E[Kyir)] = i rE[xit)y(t + T)]dt T Jo
1 Jo
Hence ^ ^ „ ( T ) is an unbiased estimate of R^yir), independent o f T. The mean square error is given by the variance
V a r [RM)] = E[(AM) - R M ) f ] = E [ R M ] - R j ( r )
(E[x(u) y{u + T ) X{V) y{v + T ) ] - R j { r ) ) dv du
(5.5) A t this p o i n t , i n order t o s i m p l i f y the later m a t h e m a t i c a l analysis and t o agree w i t h many physical cases o f interest, i t w i l l be assumed t h a t the r a n d o m processes {«(?)} a n d {y(t)} are j o i n t l y Gaussian f o r any set o f fixed times. T h i s restriction m a y be r e m o v e d b y substituting certain i n t e g r a b i l i t y c o n -d i t i o n s o n the non-Gaussian parts o f the r a n -d o m processes w i t h o u t altering
i n any essential way the results t o be derived. W h e n {«(/)} a n d {yit)} are j o i n t l y Gaussian, i t f o l l o w s that {x{t)} and {y{t)} are separately Gaussian.
F o r Gaussian stationary r a n d o m processes w i t h zero mean values, the f o u r t h - o r d e r statistical expression is obtained
f o l l o w s .
E[x{u) y{u + T) X(,V) y{v + T ) ]
= RJir) + RJV - u) Ry(v - u ) + RM - u + r) RM - « - r )
( 5 . 6 ) Hence the variance expression m a y be w r i t t e n as
V a r [RM)\ = i f ^ ( ^ i * ^ - " ) Ry(^ - ") T Jo Jo
-b RM - " + 7-) RyJyV - " - r)) dv du
'l-^^yRJi)Ryii)
+ RM + r) Ryji - r)) di (5.7 )
T h e second expression occurs f r o m the first b y l e t t i n g i = v — u, di = dv, a n d then reversing the order o f i n t e g r a t i o n between | a n d u. N o w ,
^ Jo Jo ^
TJ
l i m T V a r [RM)] = T->oo
( R j i ) Ryii) + RM + r) RyJi - r)) di< CO
(5.9 ) assuming RJi)Ry{i) a n d Kyi^) Ryxi^) are absolutely integrable over
(—00, oo). T h i s proves t h a t RM) is a consistent estimate of RM)> a n d f o r
V a r [RM)] ' ^ i r ( U i ) U i ) + u a + r) R y j i - r)) di (5.g )
Several special cases o f E q u a t i o n (5.9 ) are w o r t h y o f note. F o r autocorre-l a t i o n estimates, E q u a t i o n ( 5 . ^ ) becomes
V a r [RJr)] ^ ^ r ( R M ) + K(i + r) RJi - r)) di (5.10)
I J—oo
A t the zero displacement p o i n t T = 0,
V a T [ R x m > ^ ^ r Rj'(i)di (6.11) 1 J—oo
T h e assumption t h a t U ^ ) approaches zero f o r large T shows t h a t
R M ) » U i + r) RJi - r) f o r r » 0 (5.12.)
Hence f o r large T,
Yar [ÊJr)] ^ ^ r R M ) di (5.13) 1 J — oo
w h i c h is o n e - h a l f the value o f E q u a t i o n ( S . H ) .
Example 5,1. Variance o f C o r r e l a t i o n Estimates o f B a n d w i d t h L i m i t e d W h i t e
Noise. F o r b a n d w i d t h l i m i t e d w h i t e noise w i t h a mean value /^^ = 0, a b a n d w i d t h B, a n d c o v e r i n g a t i m e i n t e r v a l T, the variance f o r a l l T is c o n -servatively given b y
V a r [RJr)] ^ { R M ) + (5.14)
T h i s reduces to E q u a t i o n (5.2.5) at the p o i n t T = 0. S i m i l a r l y , w h e n b o t h x{t) a n d y{t) are samples o f l e n g t h T f r o m b a n d w i d t h l i m i t e d w h i t e noise w i t h m e a n values = = 0 and i d e n t i c a l b a n d w i d t h s B, i t f o l l o w s t h a t
V a r [RM)] (RM Rv(0) +
U\r)) (SAS)
lol
E q u a t i o n (5.15) requires t h a t The sufficiently large such t h a t E q u a t i o n (5.9 )
can replace E q u a t i o n ( 5 . ? ) . Satisfactory c o n d i t i o n s i n practice are T> 10 \T\
and BT > 5.
F o r /x^ = fly = 0 a n d R^y 5^ 0, the n o r m a l i z e d m e a n square e r r o r is given b y
^ V a r [ÊM)] ^ M ( , + RMRJO)] ( 5 , , 6 )
R j ( r ) 2BT\ RJ(T) J
T h e square r o o t o f E q u a t i o n ( 5 . i b ) gives the n o r m a l i z e d rms e r r o r e, w h i c h includes o n l y a r a n d o m e r r o r t e r m since the bias e r r o r is zero i f the r e c o r d l e n g t h is longer t h a n T + r .
V o o r b e e l d 5 . 2 S t a t i s t i s c h e e i g e n s c h a p p e n , v a n g e m e t e n t u r b u l e n t e s c h u i f s p a n n i n g e n M . b . v . b i j v o o r b e e l d L a s e r - D o p p l e r m e t i n g e n k u n n e n i n e e n b e p a a l d p u n t i n e e n v l o e i s t o f t w e e s n e l h e i d s k o m p o n e n t e n u ( t ) ( i n s t r o o m r i c h t i n g ) e n v ( t ) ( l o o d r e c h t o p s t r o o m r i c h t i n g ) g e m e t e n w o r d e n . D e z e k o m p o n e n t e n z i j n t e s p l i t s e n i n e e n g e m i d d e l d e s n e l h e i d e n e e n f l u k t u a t i e t . g . v . t u r b u l e n t i e ( u ' r e s p . v ' ) . De t u r b u l e n t e s c h u i f s p a n n i n g i s d a n g e d e -f i n i e e r d a l s T = p u ' v ' ( 5 . 1 7 ) w a a r b i j d e h o r i z o n t a l e s t r e e p t i j d s m i d d e l i n g o v e r e e n i n p r i n c i p e o n -e i n d i g l a n g -e m -e -e t t i j d v o o r s t -e l t . O n d -e r d -e a a n n a m -e d a t u ( t ) -e n v ( t ) t -e b e s c h o u w e n z i j n a l s d e r e a l i s a t i e s v a n t w e e s t a t i o n a i r e e r g o d i s c h e p r o -c e s s e n { u ( t ) } e n { v ( t ) } m e t g e m i d d e l d e w a a r d e n e n y ^ i s x / p o p t e v a t t e n a l s d e k o v a r i a n t i e t u s s e n { u ( t ) } e n { v ( t ) } : J = E { ( u ( t ) - u ) ( w ( t ) - y ) } = C ( 0 ) ( 5 . 1 8 ) p — u — w u v m e t C ^ ^ ( 9 ) d e k r u i s k o v a r i a n t i e f u n k t i e t u s s e n { u ( t ) } e n { v ( t ) } , g e d e f i -n i e e r d a l s C , , ( 9 ) = E { ( u ( t + 0 ) - y „ ) ( v ( t ) - y ) } ( 5 . 1 9 ) u v u — w O m d a t T v o l g e n s ( 5 . 1 7 ) g e d e f i n i e e r d i s v o o r o n e i n d i g e m e e t t i j d T , t e r -w i j l i n d e p r a k t i j k m e t i n g e n v a n u ( t ) e n v ( t ) s l e c h t s e e n e i n d i g e l e n g t e h e b b e n , z a l o p b a s i s v a n d e z e m e t i n g e n h o o g u i t e e n s c h a t t i n g v a n T g e -m a a k t k u n n e n w o r d e n . D e z e s c h a t t i n g x k a n w o r d e n v e r k r e g e n m . b . v . d e a l s v o l g t g e d e f i n i e e r d e s c h a t t e r ƒ ( u ( t ) - m ^ ) ( v ( t ) - m ^ ) d t ( 5 . 2 0 ) o m e t 1 ^ m ^ = Y •/• u ( t ) ( 5 . 2 1 ) o e n
1 ^ = ^ ƒ v ( t ) d t ( 5 . 2 2 ) O M e r k o p d a t f e e n s t o c h a s t i s c h e v a r i a b e l e i s : d o o r d e e i n d i g e m e e t t i j d z a l d e u i t k o m s t v a n ( 5 . 2 0 ) v o o r e l k p a a r r e a l i s a t i e s u ( t ) e n v ( t ) v e r -s c h i l l e n d z i j n l I n v e r b a n d m e t d e n a u w k e u r i g h e i d s e i s e n w e l k e i n d e p r a k t i j k a a n g e m e t e n s c h u i f s p a n n i n g e n w o r d e n g e s t e l d , i s h e t v a n b e l a n g d e i n v l o e d v a n d e m e e t d u u r T o p z o w e l d e v a r i a n t i e a l s d e b i a s v a n T / p t e k e n n e n . D i t z a l h i e r o n d e r u i t g e w e r k t w o r d e n . A l l e r e e r s t w o r d t ( 5 . 2 0 ) g e s c h r e v e n a l s ƒ ( u ( t ) - y ^ ) ( v ( t ) - y ^ ) d t - ( y ^ - m ^ ) ( y ^ - m ^ ) ( 5 . 2 3 ) O D a n g e l d t E i l / p } = C ^ ^ ( O ) - E { ( y ^ - m ^ ) ( y ^ - m ^ ) } ( 5 . 2 4 ) z o d a t d e b i a s i n x g e l i j k i s a a n B { x } = - p E { ( y ^ - m ^ ) ( y ^ - m ^ ) } T T = - p E ƒ ( u ( t ) - y ^ ) d t . ^ ƒ ( v ( t ) - y ^ ) d t } o O T T = - p { — ƒ ƒ E { ( u ( t ) - y ) ( v ( s ) - y ^ ) d t d s } O O T T = - p { — ƒ ƒ C ( t - s ) d t d s } „ 7 U V T o o T = - ^ ( 1 - ^ ) C ^ ^ ( e ) d 0 ( 5 . 2 5 ) w a a r b i j i n d e l a a t s t e s t a p g e b r u i k i s g e m a a k t v a n d e z e l f d e t r a n s f o r m a -t i e a l s i n h o o f d s -t u k 3 , f o r m u l e ( 3 . 8 ) . F o r m u l e ( 5 . 2 5 ) k a n w o r d e n g e s c h r e v e n a l s T „ C ( 9 ) B { x } = - 1 ƒ ( 1 - i f L ) ^ d 0 ( 5 . 2 6 ) - T u v w a a r u i t b l i j k t d a t t e n g e v o l g e v a n d e e i n d i g e m e e t d u u r d e t u r b u l e n t e s c h u i f s p a n n i n g s y s t e m a t i s c h w o r d t o n d e r s c h a t . V o o r v o l d o e n d g r o t e T k a n ( 5 . 2 6 ) w o r d e n b e n a d e r d d o o r
^ «> C ( 9 ) -oo u v z o d a t d e r e l a t i e v e s y s t e m a t i s c h e f o u t w o r d t g e g e v e n d o o r ™ C ( 9 ) -oo u v M e r k o p d a t t e n e i n d e t e b e r e k e n e n d e g e h e l e k r u i s k o v a r i a n t i e f u n k t i e C ^ ^ ( 9 ) b e k e n d m o e t z i j n . D e v a r i a n t i e v a n T^/p k a n a l s v o l g t w o r d e n b e r e k e n d . U i t ( 5 . 2 3 ) v o l g t v a r { r / p } = v a r { A } + v a r { B } - 2 c o v { A , B } ( 5 . 2 8 ) m e t 1 A = - ƒ ( u ( t ) - y ^ ) ( v ( t ) - M ^ ) d t O ( 5 . 2 9 ) B = ( y ^ - m ^ ) ( y ^ - m ^ ) A a n g e t o o n d k a n w o r d e n d a t v o o r v o l d o e n d g r o t e T d e b i j d r a g e v a n v a r { A } a a n ( 5 . 2 8 ) o v e r h e e r s e n d i s . D a a r u ( t ) y ^ = v i ' ( t ) e n v ( t ) y ^ = v ' ( t ) z o -d a t E { u ' ( t ) } = E { v ' ( t ) } = O v o l g t u i t ( 5 . 5 ) T T v a r { A } = -L f f T O O ^ E { u ' ( t ) v ' ( t ) u ' ( s ) v ' ( s ) } - R ^ , ^ , ( 0 ) d t d s ( 5 . 3 0 ) D e f i n i e e r n u h e t s i g n a a l z ( t ) a l s h e t p r o d u k t v a n u ' ( t ) e n v ' ( t ) . D a n g e l d t R ^ ( t - s ) = E { u ' ( t ) v ' ( t ) u ' ( s ) v ' ( s ) } E { z ( t ) } = E { u ' ( t ) v ' ( t ) } = \ , ^ , ( 0 ) = C ^ ^ ( O ) ( 5 . 3 1 ) C ^ ( t - s ) = R ^ ( t - s ) - R ^ , ^ , ( 0 ) m e t R ( 9 ) e n C ( 9 ) d e a u t o k o r r e l a t i e r e s p , a u t o k o v a r i a n t i e - f u n k t i e v a n z z z ( t ) . S u b s t i t u t i e v a n ( 5 . 3 1 ) i n ( 5 . 3 0 ) g e e f t
T T v a r { A } = - ! - ƒ ƒ c ( t - s ) d t d s T o o T = Y / ( 1 - ^ c ^ ( 0 ) d e V o o r v o l d o e n d g r o t e T w o r d t d i t v a r { A } ^ - f C ( 0 ) d e X z —oo z o d a t 2 ° ° v a r { T } - ^ ƒ C ^ ( 0 ) d 0 ( 5 . 3 2 ) —oo M e r k o p d a t t e n e i n d e d e v a r i a n t i e i n t e b e r e k e n e n d e g e h e l e k o v a r i a n -t i e - f u n k -t i e v a n h e -t p r o d u k -t v a n u ' ( -t ) e n v ' ( -t ) b e k e n d m o e -t z i j n . N . B . D e z e k o v a r i a n t i e f u n k t i e d i e n t n i e t v e r w a r d t e w o r d e n m e t d e k r u i s -k o v a r i a n t i e - f u n -k t i e t u s s e n u ( t ) e n v ( t ) i A a n n e m e n d d a t i j( t ) e n v ( t ) G a u s s i s c h z i j u , v o l g t u i t ( 5 . 6 ) R ^ ( 0 ) = 2 R2, ^ , ( 0 ) + R ^ , ( 0 ) R ^ ( 0 ) z o d a t C J O ) = C ^ ^ ( O ) + ( 5 . 3 3 ) M e t ( 5 . 1 8 ) v o l g t d a n 2 " C ( 0 )
v a r l x } . f ( a X ^ ) / ^ ^^'^'^
P -oo z D e v a r i a n t i e i n x i s d u s k e n n e l i j k a f h a n k e l i j k v a n x z e l f . B e s c h o u w n u h e t v o l g e n d e k o n k r e t e g e v a l . I n e e n t u r b u l e n t e g r e n s l a a g b o v e n e e n g l a d d e b o d e m w o r d e n u ' ( t ) e n v ' ( t ) g e m e t e n o p e e n a f s t a n d y v a n d e b o d e m . D e g r e n s l a a g d i k t e i s 6 e n d e g e m i d d e l d e s n e l h e i d b u i t e n d e g r e n s l a a g i s . D o o r A n t o n i a e n v a n A t t a T l z i j n v o o r y = 0 , 0 8 6 ( d u s v l a k b o v e n d e b o d e m ) o n d e r a n -d e r e -d e v o l g e n -d e f u n k t i e s b e p a a l -d c . „ ( 0 ) u v ^ u v ^ O ) C J 0 ) ( z i e f i g u u r 5 . 1 , o n d e r s t e h e l f t ) ( z i e f i g u u r 5 . 1 , b o v e n s t e h e l f t ) C J O ) w a a r b i j 0 i s u i t g e d r u k t i n e e n h e d e n v a n ^j- s e c .F i g . 5.1 G e n o r m a l i s e e r d e k r u i s k o v a r i a n t i e f u n k t i e t u s s e n u en v ( o n d e r s t e h e l f t ) en g e n o r m a l i s e e r d e a u t o k o v a r i a n t i e -f u n k t i e v a n h e t produkt u*v* (bovenste h e l -f t ) a l s f u n k t i e van Uj^6/6 ( o n t l e e n d aan ) •
U i t d i t f i g u u r v o l g e n d e b e n a d e r i n g e n C ( 0 ) 2 U , ^ = e . p { - ^ | e | ) ( 5 . 3 5 ) U V e n » c ( 0 ) . O z 1 S u b s t i t u t i e v a n ( 5 . 3 5 ) i n ( 5 . 2 7 ) g e e f t = - ( S / T U , ( 5 . 3 7 ) b 1 S u b s t i t u t i e v a n ( 5 . 3 6 ) i n ( 5 . 3 4 ) r e s u l t e e r t i n v a r { f } ^ 1 ( « - ^ ) D a a r t e v e n s g e s t e l d k a n w o r d e n ^ u = = ^ 1 p 2 2 T < = ^ 2 ^ . = - 2 p
m e t d e s c h u i f s p a n n i n g s s n e l h e i d e n c , e n c k o n s t a n t e n v o l g t v a r m . I ( , . c j c j 3 I z o d a t d e r e l a t i e v e t o e v a l l i g e f o u t i n d e s c h a t t i n g v a n x g e l i j k i s a a n De b e p a l e n d e g r o o t h e i d i n z o w e l d e r e l a t i e v e s y s t e m a t i s c h e a l s d e r e l a -t i e v e -t o e v a l l i g e f o u -t i s d u s 6 / U j T . D a a r 6 / U j d i m e n s i e s e c o n d e h e e f t , w o r d e n d e f o u t e n d u s b e p a a l d d o o r d e v e r h o u d i n g v a n e e n t i j d s c h a a l , d i e b e p a a l d w o r d t d o o r d e g r o o t s t e a a n -w e z i g e -w e r v e l s , e n d e m e e t t i j d . ( E i n d e v o o r b e e l d ) . T o t s l o t d i e n t t . a . v . h e t s c h a t t e n v a n k o v a r i a n t i e f u n k t i e s n o g h e t v o l -g e n d e t e w o r d e n o p -g e m e r k t . B i j d e i n t e r p r e t a t i e v a n g e s c h a t t e k o v a r i a n t i e - f u n k t i e s z a l v o o r i e d e r e X n i e t a l l e e n m e t d e v a r i a n t i e v a n d e s c h a t t e r C_ ( x ) , m a a r t e v e n s m e t d e k o v a r i a n t i e t u s s e n C ^ ( T ) e n d e d a a r n a a s t g e l e g e n s c h a t t e r s C ^ ( x + 0 ) r e k e n i n g m o e t e n w o r d e n g e h o u d e n . I n s o m m i g e g e v a l l e n k a n d e z e k o v a r i a n -t i e z o s -t e r k z i j n , d a -t e e n g e s c h a -t -t e k o v a r i a n -t i e - f u n k -t i e e e n o s c i l l e r e n d v e r l o o p b l i j f t v e r t o n e n , t e r w i j l d e w e r k e l i j k e k o v a r i a n t i e - f u n k t i e C (x) a l l a n g i s u i t g e d e m p t . D i t f e n o m e e n k a n s n e l a a n l e i d i n g g e v e n t o t h e t t r e k k e n v a n v e r k e e r d e k o n k l u s i e s . ( 5 . 3 8 )
5•2 Berekeningsmethode
T w o methods w i l l be discussed f o r the d i g i t a l c o m p u t a t i o n o f corre-l a t i o n f u n c t i o n estimates. T h e first m e t h o d is the standard a p p r o a c h o f e s t i m a t i n g the c o r r e l a t i o n f u n c t i o n b y direct c o m p u t a t i o n o f average p r o d -ucts a m o n g the sample data values. T h e second m e t h o d is the r o u n d a b o u t a p p r o a c h o f first c o m p u t i n g a p o w e r spectrum by direct F o u r i e r t r a n s f o r m procedures, and then c o m p u t i n g the inverse F o u r i e r t r a n s f o r m o f the p o w e r spectrum.
5.2.1 D i r e k t e s c h a t t i n g van c o r r e l a t i e f u n k t i e s
A u t o c o r r e l a t i e f u n k t i e
F o r A f d a t a values {x„}, n = 1,2, . . . , N, f r o m a t r a n s f o r m e d r e c o r d x(t) w h i c h is s t a t i o n a r y w i t h x = 0, the estimated a u t o c o r r e l a t i o n f u n c t i o n at the displacement rh is defined by the f o r m u l a
= RJrh) = '^Ix,-x„.,, r = 0 , 1 , 2, . . . , m ( 6 . 3 9 ) N — r n=i
where r is the lag n u m b e r , m is the m a x i m u m lag n u m b e r , a n d R,. is the esti-mate o f the true value R^ at l a g r, c o r r e s p o n d i n g t o the displacement rh. N o t e t h a t the m a x i m u m l a g n u m b e r m is related t o the m a x i m u m t i m e displacement o f the estimate b y
T m a x = T,„ = mh ( 5 . 4 0 )
T h e sample a u t o c o r r e l a t i o n f u n c t i o n m a y also be defined at l a g r b y
2 i V - r
R, = RJrh) = - I x„x„^, r = 0 , 1 , 2, . . . , m ( 5 . 4 1 ) N n=l
where the d i v i s i o n b y remains constant instead o f c h a n g i n g t o A'' — r as i n E q u a t i o n ( 5 . 5 9 ) . Use o f this e q u a t i o n gives a biased estimate o f the a u t o -c o r r e l a t i o n f u n -c t i o n . H o w e v e r , f o r N large a n d m s m a l l w i t h respe-ct t o N, the values o b t a i n e d b y using E q u a t i o n (5.4)) d i f f e r very l i t t l e f r o m those o b -t a i n e d b y use o f E q u a -t i o n ( 5 . 3 9 ) .
