MEASURING SAMPLE MAXIMUMS: AN APPLICATION TO WATER QUALITY MONITORING by DONALD B. CASEY B . S c , The U n i v e r s i t y of A l b e r t a , 1980 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES ( F a c u l t y Of Commerce And B u s i n e s s ' A d m i n i s t r a t i o n ) 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 the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA September 1982 © Donald B. Casey, 1982 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h C olumbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head of my Department or by h i s or her r e p r e s e n t a t i v e s . I t i s un d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . F a c u l t y of Commerce and B u s i n e s s A d m i n i s t r a t i o n The U n 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 Date: October 1, 1982 A b s t r a c t There a r e many s i t u a t i o n s i n which the p r o c e s s of o b t a i n i n g a sample and the p r o c e s s of measuring a sample a r e d i s t i n c t . In t h e s e c i r c u m s t a n c e s , i t i s u s u a l t h a t the c o s t of measuring or t e s t i n g the samples i s s i g n i f i c a n t r e l a t i v e t o the c o s t of s a m p l i n g . In t h i s t h e s i s , methods a r e de v e l o p e d f o r e s t i m a t i n g the maximum sample measurement from a f i n i t e s e t of s e q u e n t i a l samples w i t h o u t e x p l i c i t l y t e s t i n g a l l of the samples. In a d d i t i o n t o the above a s s u m p t i o n s , i t i s assumed t h a t the sample measurements are h i g h l y p o s i t i v e l y a u t o c o r r e l a t e d and t h a t i t i s d e s i r a b l e t h a t the e s t i m a t e of the maximum sample measurement be an observed v a l u e . T h i s o b j e c t i v e t o g e t h e r w i t h the s t a t e d assumptions a re of p a r t i c u l a r r e l e v a n c e t o the ar e a of water q u a l i t y m o n i t o r i n g . Two d i f f e r e n t approaches a re d e v e l o p e d . These ar e c a l l e d m a t h e m a t i c a l programming methods and composite methods. The l a t t e r approach i s i n v e s t i g a t e d e m p i r i c a l l y . S e v e r a l composite methods a r e proposed w i t h p r i m a r y f i r s t o r d e r c o m p o s i t i n g b e i n g examined i n d e t a i l . T h i s method was found t o be s u p e r i o r t o the t r a d i t i o n a l t e c h n i q u e of random sampling over a wide range of performance measures. P r i m a r y f i r s t o r d e r c o m p o s i t i n g a l s o proved v e r y e f f e c t i v e a t d e t e c t i n g extreme v a l u e s and p r o v i d e d a more e f f i c i e n t e s t i m a t e of the p o p u l a t i o n mean. When a p p l y i n g p r i m a r y f i r s t o r d e r c o m p o s i t i n g i t i s shown t h a t a l l c o m p o s i t e s s h o u l d be of e q u a l s i z e and the s m a l l e r composite s i z e s s h o u l d be p r e f e r r e d . Table of C o n t e n t s Chapter Page A b s t r a c t . . . i i L i s t of T a b l e s v L i s t of F i g u r e s v i i Acknowledgements x i I . I n t r o d u c t i o n 1 I I . Review of the L i t e r a t u r e 5 A. Overview of a Water M o n i t o r i n g Program 5 O b j e c t i v e s 5 Parameters 7 Methods of A n a l y s i s 8 Sampling Program 9 R e s u l t s 10 A s s i m i l a t i o n of I n f o r m a t i o n 10 B. R e l e v a n t L i t e r a t u r e 10 I I I . A u t o c o r r e l a t i o n , 25 A. M a t h e m a t i c a l D e f i n i t i o n 25 B. C a l c u l a t i o n 30 C. I m p l i c a t i o n s 31 Composite Methods 32 M a t h e m a t i c a l Programming Methods 33 IV. Data ...36 A. S p e c i f i c C o n d u c t i v i t y 36 B. Data C o l l e c t i o n 38 C. S t a t i s t i c a l P r o p e r t i e s 40 i i i V. Measures of E r r o r 45 A. P r o p o r t i o n 46 B. Mean Square E r r o r 47 C. Mean A b s o l u t e Range E r r o r 49 D. Maximum A b s o l u t e D e v i a t i o n 51 V I . Composite Methods 52 A. P r i m a r y F i r s t Order C o m p o s i t i n g 55 B. The E f f e c t of the A u t o c o r r e l a t i o n F u n c t i o n ....95 C. The E f f e c t of the Time Between Samples 107 D. A l t e r n a t i v e Composite Methods 113 E. S e l e c t i n g the Composite S i z e * 117 V I I . Summary 120 R e f e r e n c e s 127 Appendix A 130 i v L i s t of T a b l e s Number Page 1. Mean l e v e l of s p e c i f i c c o n d u c t i v i t y and 95% c o n f i d e n c e i n t e r v a l f o r each day of the week......41 2. Mean l e v e l of s p e c i f i c c o n d u c t i v i t y and 95% c o n f i d e n c e i n t e r v a l f o r each hour of the day 41 3. E x p e c t e d and o b s e r v e d number of t e s t s per t r i a l . (2009 t r i a l s ) 60 4. Expected minus obse r v e d average number of t e s t s per t r i a l (E(m)-A(m)) and observed minus e x p e c t e d p r o p o r t i o n of t r i a l s on which the f i n a l composite was s e l e c t e d (P'(m)-P(m)) 62 5. V a r i a n c e of composite measurements 69 6. The p r o b a b i l i t y of not s e l e c t i n g the f i n a l composite g i v e n i t has the maximum sample (Pt-'FClMF]) and the p r o b a b i l i t y of s e l e c t i n g the f i n a l composite g i v e n i t does not have the maximum sample (P[FC | -,MF]) 80 v 7. P r o p o r t i o n of t r i a l s on which the a c t u a l maximum was found f o r b a l a n c e d c o m p o s i t i n g . ( 2 0 0 9 t r i a l s ) . . 8 2 8. The mean a b s o l u t e range e r r o r (MARE) f o r b a l a n c e d c o m p o s i t i n g . (2009 t r i a l s ) 86 9. The maximum a b s o l u t e d e v i a t i o n (MAD) and the r e c o r d on which i t was i n c u r r e d . (2009 t r i a l s ) . . . . 8 9 10. The mean squared e r r o r (MSE) f o r b a l a n c e d c o m p o s i t i n g . (1999 t r i a l s ) 94 11. The p r o p o r t i o n of t r i a l s on which the a c t u a l maximum found was f o r p r i m a r y f i r s t o r d e r c o m p o s i t i n g (PFOC) and random samp l i n g 104 12. The mean a b s o l u t e range e r r o r (MARE) f o r p r i m a r y f i r s t o r d e r c o m p o s i t i n g (PFOC) and random sampl i n g 104 13. The maximum a b s o l u t e d e v i a t i o n (MAD) f o r p r i m a r y f i r s t o r d e r c o m p o s i t i n g (PFOC) and random sampl i n g 106 14. The mean squared e r r o r (MSE) f o r p r i m a r y f i r s t o r d e r c o m p o s i t i n g (PFOC) and random sampling 106 v i 15. A u t o c o r r e l a t i o n f u n c t i o n s f o r low a u t o c o r r e l a t i o n group, moderate a u t o c o r r e l a t i o n group, and 60 samples per day group 108 16. Mean squared e r r o r (MSE) f o r low a u t o c o r r e l a t i o n group, moderate a u t o c o r r e l a t i o n group, and 60 samples per day group 112 v i i L i s t of F i g u r e s Number Page 1. F l o w c h a r t of a water m o n i t o r i n g program 6 2. A u t o c o r r e l a t i o n and a u t o c o r r e l a t i o n averaged over each of the 84 days 43 3. A u t o c o r r e l a t i o n and a u t o c o r r e l a t i o n averaged over each of the 2016 hours . 43 4. Average number of t e s t s per t r i a l v e r s u s composite s i z e . (2009 t r i a l s ) 57 5. Sample w i t h maximum measurement i n C1 70 6. Two normal d e n s i t i e s w i t h d i f f e r e n t v a r i a n c e s and M,>n 70 7. Two normal d e n s i t i e s w i t h d i f f e r e n t v a r i a n c e s and M,0. In p a r t i c u l a r the s t a t i s t i c M i s p l o t t e d f o r A r a n g i n g from .01 t o 100. T h i s i s a ve r y s t r o n g a s s u mption; much more r e s t r i c t i v e than r e q u i r i n g o n l y t h a t A be l e s s than the c o n s t a n t TQ+T,. From a t h e o r e t i c a l s t a n d p o i n t , i t c o n t r a d i c t s the assumption of e x p o n e n t i a l l y d i s t r i b u t e d v i o l a t i o n p e r i o d s . In a d d i t i o n t o the above assumption, the a u t h o r s s t a t e t h a t f o r t h e i r d e r i v a t i o n " i t i s t a c i t l y assumed t h a t A0. The i m p l i c a t i o n of such an assumption i s t h a t the B ( i ) ' s a r e a l s o no l o n g e r e x p o n e n t i a l l y d i s t r i b u t e d . B e c k e r s ' r e s u l t g i v e n by (2.3) h i n g e s on the p e r i o d s of v i o l a t i o n and p e r i o d s between v i o l a t i o n s b e i n g e x p o n e n t i a l l y d i s t r i b u t e d . Thus, even under the more r e s t r i c t i v e a ssumptions t h a t were i n f a c t used, the r e s u l t g i v e n by (2.3) i s not v a l i d . A s i g n i f i c a n t p o r t i o n of the r e s e a r c h i n the ar e a of sam p l i n g f o r water p o l l u t a n t s has been d i r e c t e d t o d e t e r m i n i n g a p p r o p r i a t e sample s i z e . The use of t r a d i t i o n a l s t a t i s t i c a l t e c h n i q u e s f o r e s t i m a t i n g sample s i z e f o r a p r e - s p e c i f i e d l e v e l of p r e c i s i o n about the p o p u l a t i o n mean 1 ( c o n t ' d ) B ( i ) , i t f o l l o w s i m m e d i a t e l y t h a t A < B ( i ) + V ( i ) f o r a l l i . 14 was i n t r o d u c e d by Ward [ 4 ] . S u p p o s e i t i s d e s i r e d t o e s t i m a t e t h e a v e r a g e l e v e l , *, o f a p o l l u t a n t s o t h a t t h e r e i s a ( 1 - c ) p r o b a b i l i t y t h a t ? w i l l l i e i n an i n t e r v a l o f l e n g t h 2R. I f « 2 i s t h e v a r i a b i l i t y i n t h e l e v e l o f t h e p o l l u t a n t , t h e n e e d e d s a m p l e s i z e i s g i v e n b y , Ward a s s u m e s t h a t t h e s a m p l e i s b e i n g d rawn f r o m a n o r m a l l y d i s t r i b u t e d p o p u l a t i o n a l t h o u g h t h i s a s s u m p t i o n i s n o t n e c e s s a r y , i f t h e random s a m p l e i s s u f f i c i e n t l y l a r g e . From a s l i g h t l y d i f f e r e n t p e r s p e c t i v e , S h e r w a n i a n d M o r e a u [ 5 ] p r o p o s e t h e f o l l o w i n g m e t h o d . S u p p o s e two s a m p l e s a r e t a k e n w i t h f r e q u e n c i e s n 1 a n d n 2 p e r y e a r r e s p e c t i v e l y . L e t M, a n d M 2 be t h e o b s e r v e d means. F u r t h e r s u p p o s e t h a t n, r e p r e s e n t s some " i d e a l " v a l u e , p e r h a p s d a i l y s a m p l i n g ( n , = 3 6 5 ) . What i s t h e minimum v a l u e o f n 2 s u c h t h a t a d i f f e r e n c e o f d b e t w e e n M, a n d M 2 w i l l be d e t e c t e d w i t h p r o b a b i l i t y ( 1 - o ) ? I t i s w e l l known t h a t t h a t t h e s t a t i s t i c T d e f i n e d b y , n ( 2 . 4 ) M - M 2 T ( 2 . 5 ) 15 h a s a t d i s t r i b u t i o n w i t h ( n , + n 2 - 2 ) d e g r e e s o f f r e e d o m when S 2 i s t h e p o o l e d v a r i a n c e 2 . I f t h e d i f f e r e n c e i n t h e means i s d , |M 1-M 2|=d, t h e n t h e n u l l h y p o t h e s i s o f e q u a l means w i l l be r e j e c t e d w i t h p r o b a b i l i t y ( 1 - c ) when S e t t i n g t h e l a s t two t e r m s a t e q u a l i t y a n d s o l v i n g f o r t h e unknown n 2 w i l l y i e l d t h e minimum v a l u e a t w h i c h t h e d i f f e r e n c e w i l l be d e t e c t e d ; t h a t i s , t h e n u l l h y p o t h e s i s w i l l be r e j e c t e d . The p r o b l e m w i t h s u c h an a p p r o a c h i s t h a t i t i s u s u a l l y t h e d i f f e r e n c e f r o m t h e p o p u l a t i o n mean t h a t i s o f i n t e r e s t a n d n o t t h e d i f f e r e n c e f r o m o t h e r s a m p l e e s t i m a t e s . The u s e o f t h e t r a d i t i o n a l a p p r o a c h h a s been q u e s t i o n e d due t o t h e e m p i r i c a l e v i d e n c e o f p o s i t i v e s e r i a l l y c o r r e l a t e d d a t a . The e f f e c t o f t h i s phenomena was n o t i c e d by C u r t i s [ 6 ] a l t h o u g h he f a i l e d t o a t t r i b u t e h i s r e s u l t s t o i t . C u r t i s p r e s e n t s t h e e f f e c t o f t h e s a m p l i n g i n t e r v a l on t h e s i z e o f t h e r e s u l t i n g c o n f i d e n c e i n t e r v a l s f o r a s p e c i f i c d a t a s e t . S p e c i f i c c o n d u c t a n c e i s u s e d t o m o n i t o r t h e e f f e c t s o f s u r f a c e m i n i n g on s m a l l h e a d w a t e r s t r e a m s i n 2 S i n c e t h e o b s e r v a t i o n s a r e b e i n g drawn f r o m t h e same p o p u l a t i o n i t c a n be a s s u m e d t h a t t h e v a r i a n c e s a r e e q u a l . An e s t i m a t e o f t h e v a r i a n c e f r o m t h e s a m p l e o f s i z e n, w o u l d s u f f i c e . d > t a / 2 ( 2 . 6 ) 16 Kentucky. S p e c i f i c conductance was shown t o c o r r e l a t e v e r y h i g h l y w i t h c o n c e n t r a t i o n s of S0„, Ca, Mg, and HC0 3 i n water from both mined and unmined watersheds and can be measured r e l a t i v e l y s i m p l y and a t low c o s t . H i s r e s u l t s show o n l y s m a l l d e c r e a s e s i n the s i z e of c o n f i d e n c e i n t e r v a l s when sampli n g changed from monthly t o b i w e e k l y t o weekly. In a d d i t i o n t o the e v i d e n c e of p o s i t i v e s e r i a l c o r r e l a t i o n , Sanders and A d r i a n [7] note t h a t the m a j o r i t y of h y d r o l o g i c a l d ata e x h i b i t n o n s t a t i o n a r i t y . They a s s e r t t h a t the t r a d i t i o n a l t e c h n i q u e s a r e i n a p p r o p r i a t e due t o the v i o l a t i o n of the i m p l i c i t assumption of an independent and i d e n t i c a l l y d i s t r i b u t e d random sample. As an a l t e r n a t i v e they suggest the f o l l o w i n g p r o c e d u r e . F i r s t t r a n s f o r m the d a t a s e r i e s , u s i n g whatever methods are a p p r o p r i a t e [ 8 ] , t o a s e r i e s , W ( t ) , t h a t i s s t a t i o n a r y 3 . A B o x - J e n k i n s time s e r i e s model i s then f i t t o the d a t a and the v a r i a n c e of the w h i t e n o i s e e r r o r , S 2, i s e s t i m a t e d . T h i s p r o c e d u r e i s r e p e a t e d f o r the s e r i e s , (... W ( t - k ) , W ( t ) , W(t+k) ...) k=2, 3, 4,... (2.7) 3 Sanders and A d r i a n observed t h a t annual s e a s o n a l v a r i a t i o n was a major component of the n o n s t a t i o n a r i t y . T h i s v a r i a t i o n can be removed by f i r s t f i t t i n g a d e t e r m i n i s t i c s i n u s o i d a l f u n c t i o n of the form Y ( t ) = A ( c o s o t + C ) , where Y ( t ) i s the v a l u e of the d e t e r m i n i s t i c f u n c t i o n a t time t . These v a l u e s a r e then s u b t r a c t e d from the o r i g i n a l s e r i e s . S t a t e d more e x p l i c i t l y , i f the o r i g i n a l s e r i e s i s ( Z ( t ) } , then the nonseasonal s e r i e s i s { W ( t ) = Z ( t ) - Y ( t ) } . That i s , r e p e a t u s i n g a s e r i e s c o n s i s t i n g of every second, t h i r d , f o u r t h , . . . o b s e r v a t i o n . P l o t t i n g the e s t i m a t e d v a r i a n c e of the w h i t e n o i s e a g a i n s t k, a v a l u e can be d e t e r m i n e d , k, say, above which the v a r i a n c e i s a p p r o x i m a t e l y c o n s t a n t . T h i s i m p l i e s t h a t f o r k 2>k, the W's i n the s e r i e s , (... W ( t - k 2 ) , W ( t ) , W(t + k'2) ...) (2.8) a r e independent and i d e n t i c a l l y d i s t r i b u t e d w i t h VAR[W(t)]=S 2. The t r a d i t i o n a l t e c h n i q u e as d e s c r i b e d by Ward [4] would now be a p p l i e d t o f i n d the n e c e s s a r y sample s i z e . F o c u s i n g a t t e n t i o n on the w h i t e n o i s e v a r i a n c e i s an i n d i r e c t method of d e t e r m i n i n g the independence of the W's. A more s t r a i g h t f o r w a r d t e c h n i q u e i s merely t o t e s t i f the c o e f f i c i e n t s of the B o x - J e n k i n s model are s i m u l t a n e o u s l y z e r o . I f t r u e , the s t r u c t u r e of the model d i c t a t e s t h a t the W's are independent and i d e n t i c a l l y d i s t r i b u t e d " . One v e r y i m p o r t a n t c o n s i d e r a t i o n when u s i n g the p r o c e d u r e j u s t o u t l i n e d i s t h a t the c a l c u l a t e d sample s i z e i s v a l i d o n l y i n the range of c o n s t a n t v a r i a n c e ; t h a t i s , 4 As an example, the f i r s t o r d e r a u t o r e g r e s s i v e - moving average p r o c e s s , ARMA(1,1), i s d e f i n e d by W ( t ) = * , W ( t - 1 ) + a ( t ) - e , a ( t - l ) where the { a ( t ) } are assumed independent and i d e n t i c a l l y d i s t r i b u t e d . C l e a r l y i f the c o e f f i c i e n t s i n the model a r e z e r o , we have W(t)=a(t) and hence independent and i d e n t i c a l l y d i s t r i b u t e d by d e f i n i t i o n . Other B o x - J e n k i n s models d i f f e r o n l y i n the number of l a g s of W(t) and a ( t ) t h a t appear. 18 s a m p l e s must be a t l e a s t a p a r t . A s u p e r i o r p r o c e d u r e i s s u g g e s t e d by L o f t i s a n d Ward [ 9 ] . I t c a n be shown t h a t i n t h e p r e s e n c e o f s e r i a l c o r r e l a t i o n , t h e s a m p l e mean h a s v a r i a n c e , V a r ( X ) = ~2 n n-1 n + 2 I (n-k) p ( k ) k = l ( 2 . 9 ) where n = number o f s a m p l e s p(k) = a u t o c o r r e l a t i o n a t l a g k a2 = w h i t e n o i s e v a r i a n c e R e s u l t s f o r c o r r e l a t e d random v a r i a b l e s , i n t h e s p i r i t o f t h e C e n t r a l L i m i t T h eorem, show t h a t t h e d i s t r i b u t i o n o f t h e s a m p l e mean w i l l be c l o s e t o n o r m a l . H e n c e , t h e t r a d i t i o n a l a p p r o a c h c a n be a p p l i e d b u t w i t h t h e s u b s t i t u t i o n o f t h e a b o v e e x p r e s s i o n f o r t h e v a r i a n c e . I n l a t e r a r t i c l e s [ 1 0 ] [ 1 1 ] , L o f t i s a n d Ward i l l u s t r a t e t h e e f f e c t o f t h e v a r i o u s l e v e l s o f s t a t i s t i c a l a s s u m p t i o n s on t h e s i z e o f t h e r e s u l t i n g c o n f i d e n c e i n t e r v a l s . The t h r e e c a s e s c o n s i d e r e d h a v e b e e n p r e v i o u s l y d i s c u s s e d , 1. o b s e r v a t i o n s a r e i n d e p e n d e n t a n d i d e n t i c a l l y d i s t r i b u t e d 2. o b s e r v a t i o n s a r e i n d e p e n d e n t a n d i d e n t i c a l l y 19 d i s t r i b u t e d when s e a s o n a l v a r i a t i o n i s removed 3. o b s e r v a t i o n s a re s e r i a l l y c o r r e l a t e d when s e a s o n a l v a r i a t i o n i s removed. In summary, the i m p l i c a t i o n s of the assumptions on the c a l c u l a t i o n s a r e as f o l l o w s : f o r case 1, c o n f i d e n c e i n t e r v a l s a r e c a l c u l a t e d by the t r a d i t i o n a l method; f o r case 2, s e a s o n a l v a r i a t i o n i s removed as d e s c r i b e d p r e v i o u s l y and then the t r a d i t i o n a l method i s a p p l i e d ; f o r case 3, the s e a s o n a l v a r i a t i o n i s removed and the a c t u a l v a r i a n c e of the sample mean of a c o r r e l a t e d s e r i e s as g i v e n by (2.9) i s s u b s t i t u t e d i n the t r a d i t i o n a l c o m p u t a t i o n s . S e v e r a l c o n c l u s i o n s a re im m e d i a t e l y a p p a r e n t . Removing s e a s o n a l v a r i a t i o n reduces the v a r i a n c e of the s e r i e s . Thus the s i z e of the c o n f i d e n c e i n t e r v a l under the assumptions of case 1 must be l a r g e r than f o r case 2. P o s i t i v e s e r i a l c o r r e l a t i o n i n c r e a s e s the e s t i m a t e of the v a r i a n c e of the sample mean and thus the c o n f i d e n c e i n t e r v a l f o r case 3 must a l s o be l a r g e r than those r e s u l t i n g from 2. S e v e r a l o t h e r o b s e r v a t i o n s can be made when the c o n f i d e n c e i n t e r v a l s i z e under the t h r e e s e t s of assumptions a r e p l o t t e d a g a i n s t the s a m p l i n g i n t e r v a l . As the sampling i n t e r v a l i n c r e a s e s , the c o r r e l a t i o n between o b s e r v a t i o n s d e c r e a s e s and the c o n f i d e n c e i n t e r v a l s f o r case 3 approach those f o r case 2. T h i s i s merely an i m p l i c a t i o n of the r e s u l t s of Sanders and A d r i a n [ 7 ] , That i s , the o b s e r v a t i o n s of the s e a s o n a l l y a d j u s t e d s e r i e s become independent and i d e n t i c a l l y d i s t r i b u t e d i f the samples are p l a c e d f a r enough 20 a p a r t . C o n v e r s e l y , s e r i a l c o r r e l a t i o n i s s t r o n g e s t when the sampli n g i n t e r v a l i s s m a l l . Thus the c o n f i d e n c e i n t e r v a l s i z e f o r case 3 i s l a r g e r than those r e s u l t i n g from cases 2 and 3 when sampl i n g f r e q u e n t l y . F i n a l l y , t h e r e e x i s t s some range over which cases 1 and 2 produce c o n f i d e n c e i n t e r v a l s of s i m i l a r s i z e . T h i s can be i n t e r p r e t e d as the e f f e c t s of s e a s o n a l v a r i a t i o n and s e a s o n a l c o r r e l a t i o n c a n c e l l i n g each o t h e r . A l t h o u g h s u g g e s t i o n s a r e p r e s e n t e d g i v i n g ranges over which the s i m p l e r assumptions of 1 and 2 may s u f f i c e , they are d a t a dependent. The s a f e and sure p o l i c y . i s t o use the procedure f o r case 3 whenever p o s s i b l e . In r e c e n t y e a r s , t h e r e has been a t r e n d towards d e s i g n i n g m o n i t o r i n g networks t o d e t e c t means i n q u a l i t y . Ward et a l . [12] p r e s e n t a method f o r a l l o c a t i n g a f i x e d number of samples, N, among the s i t e s or s t a t i o n s of a m o n i t o r i n g network. The o b j e c t i v e of the a l l o c a t i n g p l a n i s t o p r o v i d e e q u a l - s i z e d c o n f i d e n c e i n t e r v a l s about the means d e t e c t e d a t each s i t e . High v a r i a n c e s i t e s would r e c e i v e a l a r g e r p r o p o r t i o n of the samples than lower v a r i a n c e s t a t i o n s . C o n s i d e r a network c o n s i s t i n g of K s t a t i o n s d e s i g n e d t o monitor a s i n g l e parameter. L e t e 2 ( i ) be the v a r i a n c e of the parameter a t l o c a t i o n i . Then the number of samples t o be taken a t s i t e i i s g i v e n by, 21 N ( i ) = N (2.10) T h i s w i l l r e s u l t i n u n i f o r m s i z e d c o n f i d e n c e i n t e r v a l s w i t h h a l f l e n g t h , I f , however, more than one parameter i s b e i n g measured by the network, the s i t u a t i o n becomes more d i f f i c u l t . A l o c a t i o n t h a t has h i g h v a r i a n c e f o r one parameter may have low v a r i a n c e f o r a n o t h e r parameter. Complete u n i f o r m i t y of c o n f i d e n c e i n t e r v a l s a c r o s s a l l parameters i s i m p r o b a b l e . G e n e r a l l y , however, a s a m p l i n g a l l o c a t i o n can be d e t e r m i n e d t h a t w i l l produce g r e a t e r u n i f o r m i t y than e q u a l - s i z e d samples. F i r s t compute th e number of samples f o r each parameter a t each s i t e by the method o u t l i n e d above. The sample s i z e a t a s i t e i s then g i v e n by the w e i g h t e d average of t h e s e v a l u e s summed over a l l p a r a m e t e r s . T h i s can be s t a t e d m a t h e m a t i c a l l y a s , R (2.11) 22 N(i) = £ I w(j)N(i,j) j = l ( 2 . 1 2 ) where N ( i , j ) = the number of samples a l l o c a t e d t o parameter j a t s t a t i o n i N ( i ) = the "average" number of samples t o be c o l l e c t e d a t s t a t i o n i K = t h e t o t a l number of parameters t o be c o n s i d e r e d w ( j ) = a w e i g h t i n g f a c t o r f o r the r e l a t i v e importance of each parameter A more s o p h i s t i c a t e d f o r m u l a t i o n of the problem i s g i v e n i n L o f t i s and Ward [13]. A m a t h e m a t i c a l programming model i s d e v e l o p e d t h a t m i n i m i z e s the o v e r a l l d i f f e r e n c e between d e s i r e d c o n f i d e n c e i n t e r v a l w i d t h s and p r e d i c t e d c o n f i d e n c e i n t e r v a l w i d t h s s u b j e c t t o a f i x e d b udgetary c o n s t r a i n t . M a t h e m a t i c a l l y e x p r e s s e d , the o p t i m i z a t i o n problem i s : Min I I i = l j=l P K X ( i , j ) - X(j)° X(j)° ( 2 . 1 3 ) s u b j e c t t o 23 K I C ( j ) < CT i = l C ( i ) = f [ N ( i ) ] i - 1 ,2 i • • • P X ( i , j ) = g[ N ( i ) , j ] i - 1 ,2 t • • • P; j=1,2 i • • • K X(j)° = c o n s t a n t CT = c o n s t a n t a n d X ( i , j ) - X(j)° = 0 i f X ( i , j ) - x(J)° < 0 where X ( i , j ) = p r e d i c t e d c o n f i d e n c e i n t e r v a l w i d t h f o r p a r a m e t e r j a t s t a t i o n i X(j)° = d e s i r e d c o n f i d e n c e i n t e r v a l w i d t h f o r p a r a m e t e r j C ( i ) = a n n u a l c o s t o f s a m p l i n g a t s t a t i o n i CT = t o t a l a n n u a l o p e r a t i n g b u d g e t N ( i ) = number o f s a m p l e s c o l l e c t e d p e r y e a r a t s t a t i o n i P = t o t a l number o f s t a t i o n s c o n s i d e r e d K = t o t a l number o f p a r a m e t e r s c o n s i d e r e d i n t h e d e s i g n O b s e r v e t h a t t h e d i f f e r e n c e , X ( i , j ) - X(j)°, i s s e t t o z e r o i f n e g a t i v e , s o t h a t t h e o b j e c t i v e f u n c t i o n c a n n o t i m p r o v e t h r o u g h d e c r e a s i n g t h e s i z e o f c o n f i d e n c e i n t e r v a l s a l r e a d y o f a c c e p t a b l e s i z e . L o f t i s a n d Ward h a v e d e v e l o p e d a c o m p u t e r c o d e t h a t w i l l s o l v e t h i s o p t i m i z a t i o n p r o b l e m 24 u s i n g dynamic programming. I t i s apparent from a survey of the l i t e r a t u r e t h a t l i t t l e r e s e a r c h has been d i r e c t e d towards m o n i t o r i n g w i t h the g o a l of d e t e c t i n g v i o l a t i o n s of water q u a l i t y s t a n d a r d s . Moreover, some of the r e s e a r c h i n t h i s a r e a , s p e c i f i c a l l y B e c k e r s et a l . [ 1 ] , has been shown t o be i n e r r o r . A major o b j e c t i o n t o m o n i t o r i n g to d e t e c t v i o l a t i o n s i s t h a t a h i g h f r e q u e n c y and thus h i g h c o s t m o n i t o r i n g e f f o r t i s needed. T h i s concern has been e x p r e s s e d by s e v e r a l a u t h o r s , most n o t a b l y , Ward [ 4 ] , Serwani and Moreau [ 5 ] , and Ward et a l . [ 1 2 ] . The methods t h a t w i l l be proposed i n t h i s t h e s i s , however, can be a p p l i e d i n a manner t h a t w i l l reduce the c o s t of t e s t i n g the samples and, c o n s e q u e n t l y , reduce the c o s t of the m o n i t o r i n g program. I f the c o s t of the a n a l y s i s i s c o m p a r a t i v e l y h i g h then the s a v i n g may be s u b s t a n t i a l . Thus, the c o n t e n t of t h i s t h e s i s would appear t o be a s i g n i f i c a n t c o n t r i b u t i o n t o the r e s e a r c h i n t h i s a r e a . I I I . A u t o c o r r e l a t i o n The t e c h n i q u e s b e i n g e x p l o r e d i n t h i s t h e s i s h i n g e on the s e t of assumptions d e l i n e a t e d i n the i n t r o d u c t i o n . The assumption of a d i s t i n c t i o n between the s a m p l i n g p r o c e s s and the measuring p r o c e s s and the assumption c o n c e r n i n g the r e l a t i o n s h i p between the c o s t of sampling and the c o s t of t e s t i n g , a r e e x t e r n a l t o the a n a l y s i s . T o gether, they d e a l w i t h the q u e s t i o n of the a p p l i c a b i l i t y of the t e c h n i q u e s but not t h e i r performance. The assumption of h i g h p o s i t i v e a u t o c o r r e l a t i o n , t h e r e f o r e , i s the s o l e assumption on which the performance r e s t s . The c u r r e n t c h a p t e r d e a l s w i t h t h i s c o n c e p t , which i s of fundamental importance t o the r e s u l t s . The f i r s t s e c t i o n t r e a t s the mathematics of a u t o c o r r e l a t i o n and i s f o l l o w e d by the method of c o m p u t a t i o n . The f i n a l s e c t i o n e x p l a i n s the i m p l i c a t i o n of h i g h p o s i t i v e a u t o c o r r e l a t i o n i n the c o n t e x t of t h i s t h e s i s . A. M a t h e m a t i c a l D e f i n i t i o n B e f o r e the mathematics of a u t o c o r r e l a t i o n can be e x p l o r e d , s e v e r a l r e l a t e d c o n c e p t s must be d e v e l o p e d . The f i r s t , i s c o v a r i a n c e . I n t u i t i v e l y , on can t h i n k of the dependence of two random v a r i a b l e s , W, and W2, as i m p l y i n g t h a t one v a r i a b l e , say W,, e i t h e r i n c r e a s e s or d e c r e a s e s as W2 changes. The c o v a r i a n c e of W, and W2 i s merely a q u a n t i t a t i v e measure of t h i s i n t u i t i v e n o t i o n . S t a t e d m a t h e m a t i c a l l y , the c o v a r i a n c e of the random v a r i a b l e s W, and W2 i s d e f i n e d a s , 25 26 Cov[W,,W 2] = E [ ( W 1 - M , ) ( W 2 - M 2 ) ] (3.1) where j/ 1=E[W l] and n 2=E[W 2]. Suppose t h a t v a l u e s f o r W, t h a t a r e l e s s than or g r e a t e r than i t s mean, » ,, tend t o be a s s o c i a t e d w i t h v a l u e s f o r W2 t h a t a r e l e s s than or g r e a t e r than i t s mean, * 2 , r e s p e c t i v e l y . The c o v a r i a n c e would then be p o s i t i v e . I f , however, v a l u e s f o r W, t h a t a re l e s s then i t s mean, ^ , tend t o be a s s o c i a t e d w i t h v a l u e s of W2 t h a t a re g r e a t e r than i t s mean, »#2, the c o v a r i a n c e would then be negat i v e . The l a r g e r the a b s o l u t e v a l u e of the c o v a r i a n c e of W, and W2, the g r e a t e r the l i n e a r dependence between W, and W2. P o s i t i v e c o v a r i a n c e i n d i c a t e s t h a t a change i n W, w i l l t end to imply a change i n W2 i n the same d i r e c t i o n . S i m i l a r l y , n e g a t i v e c o v a r i a n c e i n d i c a t e s t h a t a change i n W, w i l l tend to be a s s o c i a t e d w i t h a change i n W2 i n the o p p o s i t e d i r e c t i o n . A c o v a r i a n c e of z e r o would i n d i c a t e no l i n e a r dependence between W, and W2. (Zero c o v a r i a n c e i s a n e c e s s a r y c o n d i t i o n f o r the independence of two random v a r i a b l e s but i s not s u f f i c i e n t ) . U n f o r t u n a t e l y , i t i s d i f f i c u l t t o employ c o v a r i a n c e as an a b s o l u t e measure of dependence because i t s v a l u e depends on the s c a l e of measurement. I t i s , t h e r e f o r e , d i f f i c u l t t o d e f i n e what one means by ' l a r g e ' or ' s m a l l ' v a l u e s , which would imply s t r o n g 27 and weak l i n e a r dependence r e s p e c t i v e l y . A u t o c o v a r i a n c e i s a concept c l o s e l y r e l a t e d t o c o v a r i a n c e . F i r s t assume t h a t we have a s t a t i o n a r y sequence of random v a r i a b l e s , Z ( t ) t=1,2,.... The p r o p e r t y of s t a t i o n a r i t y i s c h a r a c t e r i z e d by the p r o c e s s r e m a i n i n g i n e q u i l i b r i u m about a c o n s t a n t mean l e v e l . A more r i g o r o u s m a t h e m a t i c a l development can be found i n [14, pp. 3-4]. Two i m p o r t a n t i m p l i c a t i o n s of s t a t i o n a r i t y are t h a t , E [ Z ( t ) ] = „ t=1,2,. . . (3.2) and, C o v [ Z ( t ) , Z ( t + j ) ] = C o v [ Z ( t + h ) , Z ( t + h + j ) ] = r ( j ) (3.3) f o r a l l t , j , and h. That i s , the c o v a r i a n c e between any p a i r depends o n l y on the number of p e r i o d s s e p a r a t i n g them, j . The a u t o c o v a r i a n c e at l a g j i s d e f i n e d t o be r ( j ) , the p r e f i x 'auto' r e f e r r i n g t o the f a c t t h a t the c o v a r i a n c e i s between d i f f e r e n t random v a r i a b l e s i n the same sequence. Observe t h a t by d e f i n i t i o n , 28 C o v [ Z ( t ) , Z ( t ) ] = r ( 0 ) = V a r [ Z ( t ) ] (3.4) S i n c e a u t o c o v a r i a n c e s a r e merely c o v a r i a n c e s w i t h i n a s t a t i o n a r y sequence, they a r e i n t e r p r e t e d i n e x a c t l y the same manner. A p o s i t i v e v a l u e f o r y ( j ) i n d i c a t e s t h a t a h i g h e r than average o b s e r v a t i o n tends t o be f o l l o w e d by a h i g h e r than average o b s e r v a t i o n j p e r i o d s l a t e r , o r , a lower than average o b s e r v a t i o n w i l l t e n d t o be f o l l o w e d by a lower than average o b s e r v a t i o n j p e r i o d s l a t e r . On the o t h e r hand, a n e g a t i v e v a l u e f o r r ( j ) i n d i c a t e s t h a t a h i g h e r than average o b s e r v a t i o n tends t o be f o l l o w e d by a lower than average o b s e r v a t i o n j p e r i o d s l a t e r , and v i c e v e r s a . A g a i n , the l a r g e r the a b s o l u t e v a l u e , the g r e a t e r the dependence. The l a r g e s t d i s a d v a n t a g e w i t h u s i n g c o v a r i a n c e t o measure dependence, as p o i n t e d out e a r l i e r , i s t h a t i t s v a l u e depends on the s c a l e of measurement. For example, suppose t h a t measurements f o r some experiment a r e c o n v e r t e d from meters t o c e n t i m e t e r s . T h i s i s e q u i v a l e n t t o m u l t i p l y i n g a l l v a l u e s by 100. A l t h o u g h the change of s c a l e has no e f f e c t on any dependencies w i t h i n the d a t a , the e f f e c t w i l l be t o i n c r e a s e a l l c o v a r i a n c e s by a f a c t o r of 10,000! T h i s i n f l u e n c e of s c a l e can be e l i m i n a t e d by s t a n d a r d i z i n g the a u t o c o v a r i a n c e s by d i v i d i n g them a l l by the v a r i a n c e of the s e r i e s , denoted i n (3.4) as r ( 0 ) . The 29 r e s u l t i n g v a l u e s a r e r e f e r r e d t o a s a u t o c o r r e l a t i o n s . Thus, t h e a u t o c o r r e l a t i o n a t l a g j , i s d e f i n e d a s , •(j) = r ( j ) / r ( 0 ) ( 3 . 5 ) The a u t o c o r r e l a t i o n i s d i m e n s i o n l e s s and t h u s , i n d e p e n d e n t of t h e s c a l e of measurement of t h e d a t a . A l s o o b s e r v e t h a t by d e f i n i t i o n , r ( 0 ) > 0 . T h i s i m p l i e s t h a t a p o s i t i v e a u t o c o r r e l a t i o n can be i n t e r p r e t e d i n t h e same manner as a p o s i t i v e a u t o c o v a r i a n c e . L i k e w i s e , a n e g a t i v e a u t o c o r r e l a t i o n has t h e same i n t e r p r e t a t i o n a s a n e g a t i v e a u t o c o v a r i a n c e . In a d d i t i o n , r ( 0 ) has t h e p r o p e r t y t h a t r ( 0 ) > y ( j ) f o r a l l j . Hence, -1 < />(j) < 1 ( 3 . 6 ) w i t h |/>(j)| = 1 i f and o n l y i f | r ( j ) | = r ( 0 ) . T h i s e x p l i c i t l y d e f i n e s t h e n o t i o n o f ' l a r g e ' and ' s m a l l ' . An a u t o c o r r e l a t i o n c l o s e t o one i n a b s o l u t e v a l u e c o r r e s p o n d s t o a ' l a r g e ' a u t o c o v a r i a n c e and t h u s , a s t r o n g l i n e a r d e p e n d e n c e . An a u t o c o r r e l a t i o n c l o s e t o z e r o i n a b s o l u t e v a l u e c o r r e s p o n d s t o a ' s m a l l ' a u t o c o v a r i a n c e and t h u s , a weak l i n e a r d e p e n d e n c e . 30 The s e t of a u t o c o r r e l a t i o n s , denoted c o l l e c t i v e l y as j=1,2,..., i s o f t e n r e f e r r e d t o as the a u t o c o r r e l a t i o n f u n c t i o n . A graph of the a u t o c o r r e l a t i o n f u n c t i o n i s sometimes c a l l e d a c o r r e l o g r a m , a l t h o u g h the term a u t o c o r r e l a t i o n f u n c t i o n can be used f o r b oth the f u n c t i o n and i t s graph w i t h o u t a m b i g u i t y . I t i s the l a t t e r nomenclature t h a t i s p r e f e r r e d i n t h i s t h e s i s . Throughout the r emainder, the a u t o c o r r e l a t i o n f u n c t i o n w i l l u s u a l l y be graphed as a smooth c u r v e , a l t h o u g h i t i s r e c o g n i z e d t h a t the f u n c t i o n i s d i s c r e t e . T h i s method of p r e s e n t a t i o n was adopted as i t was f e l t t h a t smooth c u r v e s a r e more e a s i l y compared v i s u a l l y . B. C a l c u l a t i o n E s t i m a t e s of the a u t o c o r r e l a t i o n s a r e o b t a i n e d from the measurements of the sample d a t a . Suppose t h a t N samples are t a k e n , and l e t the c o r r e s p o n d i n g measurements be Z ( 1 ) , z ( 2 ) , z ( N ) . The a u t o c o r r e l a t i o n a t l a g j , p ( j ) , i s e s t i m a t e d by r ( j ) , which i s d e f i n e d by the e x p r e s s i o n , r ( j ) = c ( j ) / c ( 0 ) j = 1, 2, . . . (3.6) c ( j ) i s an e s t i m a t e of the a u t o c o v a r i a n c e at l a g j , r ( j ) , and i s c a l c u l a t e d from the e x p r e s s i o n , 31 1 N-j c ( j ) = h I (z(t) - z) (z(t + j) - z) N t=l (3.7) where z denotes the sample mean, (3.8) Box and J e n k i n s [ 8 , pp. 33] recommend, as a r u l e of thumb, t h a t t o o b t a i n a u s e f u l e s t i m a t e of the a u t o c o r r e l a t i o n r ( j ) , t h e number of o b s e r v a t i o n s , N, s h o u l d be a t l e a s t 50 and j not l a r g e r than N/4. C. I m p l i c a t i o n s The b a s i c m a t h e m a t i c a l p r o p e r t i e s of a u t o c o r r e l a t i o n have j u s t been d i s c u s s e d . T h i s l e a d s t o a n a t u r a l q u e s t i o n , how can h i g h p o s i t i v e a u t o c o r r e l a t i o n be e x p l o i t e d when s e a r c h i n g f o r the maximum of a f i n i t e number of o b s e r v a t i o n s ? The b a s i c p r i n c i p l e can be summed up s u c c i n c t l y . A h i g h v a l u e or measurement w i l l t e n d t o f o l l o w , and be f o l l o w e d by, h i g h measurements. S i m i l a r l y , a low v a l u e or measurement w i l l t e n d t o f o l l o w , and be f o l l o w e d by, low measurements. Thus, i f a sample i s measured and shows a h i g h l e v e l , t h e samples i n a neighborhood of t h i s 32 o b s e r v a t i o n w a r r a n t f u r t h e r a t t e n t i o n . C o n v e r s e l y , a low l e v e l would i n d i c a t e t h a t t h e samples i n some n e i g h b o r h o o d of t h e sample can be i g n o r e d . The b a s i c p r i n c i p l e , a t l e a s t i n t u i t i v e l y , c o u l d h a r d l y be s i m p l e r . However, a t l e a s t two d i s t i n c t l y d i f f e r e n t a p p r o a c h e s o r c l a s s e s o f t e c h n i q u e s c a n be d e v e l o p e d . A d i s c u s s i o n of how t h i s f u n d a m e n t a l p r i n c i p l e l e a d s t o t h e d e v e l o p m e n t of e a c h c l a s s of t e c h n i q u e s f o l l o w s . Emphasis i s on t h e i m p l i c a t i o n s o f h i g h p o s i t i v e a u t o c o r r e l a t i o n and c o n s e q u e n t d e v e l o p m e n t of a p r e m i s e on w h i c h e a c h c l a s s of t e c h n i q u e s i s b a s e d and not t h e t e c h n i c a l a l g o r i t h m i c d e t a i l s . C o m p o s i t e Methods The f i r s t c l a s s of t e c h n i q u e s , and t h o s e d e a l t w i t h i n t h i s t h e s i s , w i l l be c a l l e d c o m p o s i t e methods. These combine th e i m p l i c a t i o n s o f h i g h p o s i t i v e a u t o c o r r e l a t i o n w i t h t h e method o f c o m p o s i t e s a m p l i n g . C o m p o s i t e s a m p l i n g , a common p r a c t i c e i n water p o l l u t i o n m o n i t o r i n g , i n v o l v e s t h e p h y s i c a l p o o l i n g o f a s e t of s e q u e n t i a l s a m p l e s b e f o r e any measurement i s t a k e n . T h i s w i l l y i e l d t h e a v e r a g e measurement of t h e samples t h a t were c o m p o s i t e d . Suppose t h a t we have N s a m p l e s . The b a s i s f o r t h e c l a s s of c o m p o s i t e t e c h n i q u e s i n v o l v e s f i r s t a g g r e g a t i n g t h e N s a m p l e s i n t o m s e q u e n t i a l g r o u p s . Then, f o r e a c h g r o u p , a f i x e d p o r t i o n o f 33 each sample i s p o o l e d t o form a c o m p o s i t e . The r e s u l t i n g (N/m> 5 composite samples are then measured. As a consequence of h i g h p o s i t i v e a u t o c o r r e l a t i o n , the maximum o b s e r v a t i o n among the N samples w i l l tend t o be surrounded by o b s e r v a t i o n s w i t h h i g h e r l e v e l s . T h e r e f o r e , the composite sample t h a t c o n t a i n s the maximum o b s e r v a t i o n w i l l a l s o t e n d t o have a h i g h e r measurement. T h i s s u g g e s t s t h a t the s e a r c h f o r the maximum can be c o n c e n t r a t e d on the i n d i v i d u a l o b s e r v a t i o n s t h a t formed the c o m p o s i t e ( s ) w i t h the h i g h e s t measured l e v e l ( s ) . I t i s c r u c i a l t h a t o n l y a f i x e d p o r t i o n of each sample i s p o o l e d when f o r m i n g the composites as the i n d i v i d u a l o b s e r v a t i o n s a re needed f o r l a t e r a n a l y s i s once the maximum among the co m p o s i t e s i s l o c a t e d . In summary, composite t e c h n i q u e s a re based on the f o l l o w i n g p r e m i s e : h i g h p o s i t i v e a u t o c o r r e l a t i o n i m p l i e s t h a t the o b s e r v a t i o n w i t h the h i g h e s t l e v e l w i l l t end t o appear i n the composite w i t h the h i g h e s t l e v e l . M a t h e m a t i c a l Programming Methods The second c l a s s of t e c h n i q u e s w i l l be c a l l e d m a t h e m a t i c a l programming methods and are based on one d i m e n s i o n a l o p t i m i z a t i o n methods found i n the a r e a of ma t h e m a t i c a l programming [ 1 5 ] . The N sample measurements can be thought of as N v a l u e s from a d e t e r m i n i s t i c f u n c t i o n from which the maximum i s t o be found. However, o p t i m i z a t i o n p r o c e d u r e s , i n g e n e r a l , must assume unimodal b e h a v i o r i n some neighborhood of the maximum. The l a r g e r t h i s 5 ( X ) s t a n d s f o r the s m a l l e s t i n t e g e r > x. 34 neighborhood i s , the l e s s p r e c i s e need be our i n i t i a l e s t i m a t e of the maximum's l o c a t i o n . The assumption of h i g h p o s i t i v e a u t o c o r r e l a t i o n i s i n the same s p i r i t as the assumption of u n i m o d a l i t y over a ' l a r g e ' neighborhood. A l t h o u g h i t i s d i f f i c u l t t o d e f i n e what one means by ' l a r g e ' neighborhood, i t i s p o s s i b l e t o o f f e r an i n t u i t i v e c h a r a c t e r i z a t i o n of a n e c e s s a r y c o n d i t i o n . I f a f u n c t i o n e x h i b i t s a l o t of s p i k e s 6 , i t can be unimodal f o r a t most the d i s t a n c e between c o n s e c u t i v e s p i k e s . I f the d e n s i t y of the s p i k e s i s h i g h , even t h i s upper bound on the r e g i o n of u n i m o d a l i t y w i l l o f t e n be s m a l l . Thus, f o r a f u n c t i o n t o o f t e n be unimodal over a ' l a r g e ' neighborhood, i t i s n e c e s s a r y t h a t i t not e x h i b i t a h i g h d e n s i t y of s p i k e s . T h i s p r o p e r t y i s a d i r e c t consequence of h i g h p o s i t i v e a u t o c o r r e l a t i o n ; the sample measurements i n any neighborhood w i l l t end t o be c l o s e l y r e l a t e d . T h i s i s not the case when a s p i k e o c c u r s . The assumption of h i g h p o s i t i v e a u t o c o r r e l a t i o n does not e l i m i n a t e s p i k e d b e h a v i o r but de c r e a s e s the degree t o which i t o c c u r s , both i n d e n s i t y and i n t e n s i t y . The assumption of h i g h p o s i t i v e a u t o c o r r e l a t i o n i s not e q u i v a l e n t t o the assumption of u n i m o d a l i t y over a ' l a r g e ' neighborhood but o n l y i n the same s p i r i t . The random n a t u r e of the d a t a d i c t a t e s t h a t the assumption of t r u e u n i m o d a l i t y i s u n r e a l i s t i c . However, the l a c k of s p i k e d b e h a v i o r i m p l i e s 6 I t i s s u f f i c i e n t here t o d e f i n e a s p i k e as one or two v a l u e s t h a t a re s u b s t a n t i a l l y h i g h e r than t h e i r n e i g h b o r s . 35 t h a t f l u c t u a t i o n s w i l l t end t o be s m a l l compared t o t r e n d . T h i s i s the c r u c i a l premise on which m a t h e m a t i c a l programming t e c h n i q u e s can be based. S t a n d a r d m a t h e m a t i c a l programming a l g o r i t h m s f o r f i n d i n g the maximum of a f u n c t i o n hinge on the a b i l i t y t o be a b l e t o d e f i n e a d i r e c t i o n i n which the maximum w i l l l i e . For t r u e unimodal f u n c t i o n s t h i s can be done w i t h o n l y two v a l u e s 7 . For t h i s a p p l i c a t i o n , two v a l u e s w i l l not be s u f f i c i e n t , but i f random f l u c t u a t i o n s a r e s m a l l compared t o t r e n d , i t i s p o s s i b l e t o d e f i n e a d i r e c t i o n by examining s e v e r a l v a l u e s . Thus, by m o d i f y i n g the c o n d i t i o n s f o r d e f i n i n g a d i r e c t i o n , t e c h n i q u e s can be dev e l o p e d t o s o l v e our problem based on e x i s t i n g m a t h e m a t i c a l programming a l g o r i t h m s . E x a m i n a t i o n of t h i s c l a s s of methods, a l t h o u g h p r o m i s i n g , i s beyond the scope of t h i s t h e s i s . However, i t does appear t h a t f u r t h e r study i n t h i s d i r e c t i o n i s w a r r a n t e d . 7 I f f ( - ) i s a unimodal f u n c t i o n and x, and x 2 a r e any two p o i n t s , then f ( x 1 ) < f ( x 2 ) i m p l i e s t h a t the v a l u e a t which f(«) i s maximized l i e s t o the r i g h t of x,. On the o t h e r hand, i f f ( x 1 ) > f ( x 2 ) then the v a l u e a t which f(») i s maximized w i l l l i e t o the l e f t of x 2 . IV. Data The c l a s s of methods d i s c u s s e d i n t h i s t h e s i s w i l l be a p p l i e d t o d a t a based on s p e c i f i c c o n d u c t i v i t y l e v e l s r e c o r d e d a t a p u l p m i l l i n B r i t i s h Columbia. The f i r s t p a r t of t h i s c h a p t e r d e a l s w i t h the r a t i o n a l e b e h i n d the use of s p e c i f i c c o n d u c t i v i t y i n a p o l l u t i o n m o n i t o r i n g c o n t e x t and the i m p l i c a t i o n s f o r t h i s t h e s i s . T h i s i s f o l l o w e d by a d e s c r i p t i o n of the d a t a c o l l e c t i o n and measurement p r o c e s s . The f i n a l s e c t i o n of t h i s c h a p t e r d e t a i l s the r e l e v a n t s t a t i s t i c a l p r o p e r t i e s of the d a t a . A. S p e c i f i c C o n d u c t i v i t y S p e c i f i c c o n d u c t i v i t y i s a measure.of the a b i l i t y of a g i v e n s u b s t a n c e t o conduct e l e c t r i c c u r r e n t . I t i s used by p u l p m i l l s as a measure of c h e m i c a l l o s s e s from the p u l p m i l l p r o c e s s . S e v e r a l of the advantages and d i s a d v a n t a g e s of u s i n g s p e c i f i c c o n d u c t i v i t y d a t a , i n a p o l l u t i o n m o n i t o r i n g c o n t e x t , a r e p r e s e n t e d i n Nemetz and D r e c h s l e r [16] and r e s t a t e d h e r e . C o n f i r m i n g e a r l i e r r e s u l t s by Walden et a l . [ 1 7 ] , Nemetz and D r e c h s l e r i n d i c a t e a v e r y low c o r r e l a t i o n between d a i l y m i l l c h e m i c a l l o s s e s , d a i l y suspended s o l i d s , and b i o c h e m i c a l oxygen demand (BOD 5). In a d d i t i o n , s p e c i f i c c o n d u c t i v i t y i s not c o n s i d e r e d d i r e c t l y r e l e v a n t t o p o l l u t i o n g e n e r a t i o n and c o n t r o l . As such, i t i s not measured by p r o v i n c i a l or f e d e r a l p o l l u t i o n c o n t r o l a g e n c i e s . However, Walden et a l . have shown t h a t t h e s e d a t a 36 3 7 are c l o s e l y r e l a t e d t o m i l l e f f l u e n t t o x i c i t y . T h i s i s of p a r t i c u l a r importance as Nemetz and D r e c h s l e r s t a t e t h a t " t o x i c i t y of e f f l u e n t t o a q u a t i c organisms i s one of the c e n t r a l i s s u e s i n p o l l u t i o n c o n t r o l . " The impact of t h e s e advantages and d i s a d v a n t a g e s on the c o n t e n t of t h i s t h e s i s i s p e r i p h e r a l . T h i s i s a consequence of the u n d e r l y i n g t h e o r e t i c a l s t r u c t u r e on which the r e s u l t s a r e based. Of fundamental importance i s the p r e s e n c e of h i g h p o s i t i v e a u t o c o r r e l a t i o n . As noted by s e v e r a l a u t h o r s , C u r t i s [ 6 ] , Sanders and A d r i a n [ 7 ] , and L o f t i s and Ward [ 1 0 ] , the m a j o r i t y of h y d r o l o g i c a l time s e r i e s d a t a e x h i b i t t h i s p r o p e r t y . The s p e c i f i c c o n d u c t i v i t y d a t a i s merely one such s e r i e s . I t i s the c o n t e n t i o n here t h a t the c o n c l u s i o n s can be g e n e r a l i z e d t o any time s e r i e s d a t a w i t h s i m i l a r p r o p e r t i e s . I t s h o u l d be noted t h a t the s p e c i f i c c o n d u c t i v i t y d a t a do not s a t i s f y a l l of the p r e v i o u s l y s t a t e d a s sumptions f o r t h i s t h e s i s . In p a r t i c u l a r , the c o s t of t e s t i n g or measuring a sample i s r e l a t i v e l y i n e x p e n s i v e . T h i s v i o l a t e s the assumption t h a t the c o s t of t e s t i n g a sample i s s i g n i f i c a n t r e l a t i v e t o the c o s t of o b t a i n i n g a sample. T h i s assumption i s , however, e x t e r n a l t o the a n a l y s i s . That i s , i t does not a f f e c t the performance of the methods t h a t w i l l be d i s c u s s e d but o n l y the s i t u a t i o n s i n which they w i l l be a p p l i c a b l e . The s p e c i f i c c o n d u c t i v i t y d a t a are thus a p p r o p r i a t e f o r the purposes of t h i s t h e s i s . 38 B. Data C o l l e c t i o n For an 84 day p e r i o d i n the F a l l of 1977, c o n t i n u o u s s p e c i f i c c o n d u c t i v i t y was m o n i t o r e d from a d e v i c e on a p u l p m i l l ' s p r i n c i p a l e f f l u e n t o u t f a l l t o marine water and r e c o r d e d on s t r i p c h a r t s . These c o n d u c t i v i t y l i n e s were then d i g i t i z e d 8 a t the U n i v e r s i t y of B r i t i s h Columbia t o f a c i l i t a t e n u m e r i c a l a n a l y s i s . The data were then a d j u s t e d t o account f o r the f a c t t h a t the r e c o r d i n g d e v i c e was not r e c a l i b r a t e d d a i l y . T h i s p r o c e d u r e was performed i n two s t a g e s . The f i r s t , i n v o l v e d s h i f t i n g the y - a x i s so t h a t a u s u a l l y f l a t p o r t i o n of the c o n d u c t i v i t y c u r v e , r e p r e s e n t i n g a low l e v e l of c h e m i c a l l o s s or 'normal m i l l o u t p u t ' , c o r r e s p o n d e d t o z e r o . The d a t a were t r a n s f o r m e d i n t h i s manner f o r each 24 hour p e r i o d . The t r a n s f o r m e d d a t a were then used i n the second s t a g e of the r e c a l i b r a t i o n p r o c e s s . T h i s i n v o l v e d a second s h i f t i n g of the y - a x i s but f o r the e n t i r e s e t of d a t a taken as a whole, and not j u s t f o r i n d i v i d u a l days. The f i n a l p o s i t i o n of the y - a x i s was s e l e c t e d t o maximize the c o r r e l a t i o n between d a i l y c h e m i c a l l o s s and the area under the s p e c i f i c c o n d u c t i v i t y c u r v e . Thus, the a r e a under the c u r v e i s d i r e c t l y p r o p o r t i o n a l t o c h e m i c a l l o s s . T h i s i s c r u c i a l , s i n c e s p e c i f i c c o n d u c t i v i t y 8 D i g i t i z i n g i s a p r o c e s s whereby (x,y) c o o r d i n a t e s of p a r t i c u l a r p o i n t s on a c u r v e a r e r e c o r d e d m e c h a n i c a l l y . The p o i n t s are s e l e c t e d so t h a t the c u r v e i s a p p r o x i m a t e l y l i n e a r between a d j a c e n t p o i n t s . Thus, the more c u r v a t u r e , the c l o s e r t o g e t h e r the p o i n t s need t o be. The v a l u e a t any p o i n t on the curve can t h e r e f o r e be a c c u r a t e l y e s t i m a t e d by l i n e a r i n t e r p o l a t i o n between the two a d j a c e n t d i g i t i z e d po i n t s . 3 9 must be a measure of c h e m i c a l l o s s e s from t h e p u l p m i l l p r o c e s s . The d a i l y c h e m i c a l l o s s d a t a were r e c o r d e d i n d e p e n d e n t l y w i t h i n t h e p u l p m i l l . As a c o n s e q u e n c e of t h i s r e c a l i b r a t i o n p r o c e d u r e t h e h e i g h t of t h e c u r v e no l o n g e r had a p h y s i c a l i n t e r p r e t a t i o n . The c o n d u c t i v i t y c u r v e s t i l l , however, had t h e p r o p e r t y t h a t a p o i n t on t h e c u r v e of h e i g h t 2 i n c h e s would r e p r e s e n t a measure of t w i c e t h e s p e c i f i c c o n d u c t i v i t y o f a p o i n t w i t h h e i g h t 1 i n c h . In o t h e r words, t h e change i n s c a l e i s l i n e a r . T h i s would not have been t h e c a s e had t h e s e c o n d s t a g e of t h e r e c a l i b r a t i o n p r o c e s s n o t been p e r f o r m e d . The methods t h a t w i l l be d e s c r i b e d and t h e r e s u l t s o b t a i n e d i n t h i s t h e s i s a r e c o m p l e t e l y i n d e p e n d e n t of any l i n e a r t r a n s f o r m a t i o n o f t h e d a t a . The n u m e r i c v a l u e s w i l l l o s e p h y s i c a l i n t e r p r e t a t i o n but t h e r e s u l t s w i l l a l w a y s be c o n s i s t e n t . The r e c a l i b r a t e d d a t a a r e , t h e r e f o r e , s u f f i c i e n t . T h r o u g h o u t t h e r e m a i n d e r , a l l measurements w i l l be r e f e r r e d t o as l e v e l s of s p e c i f i c c o n d u c t i v i t y , a l t h o u g h i t i s a c k n o w l e d g e d t h a t no p h y s i c a l i n t e r p r e t a t i o n can be made. The r e c a l i b r a t e d c o n d u c t i v i t y c u r v e was t h e n u s e d t o p r o d u c e one o b s e r v a t i o n p e r m i n u t e . T h i s was a c c o m p l i s h e d by c a l c u l a t i n g t h e a r e a under t h e c u r v e o v e r one m i n u t e i n t e r v a l s . The a r e a under t h e c u r v e i s e q u i v a l e n t t o t h e a v e r a g e v a l u e o v e r t h e i n t e r v a l up t o m u l t i p l i c a t i o n by a s c a l a r . S i n c e t h e u n i t s a l r e a d y had no p h y s i c a l i n t e r p r e t a t i o n and, as j u s t s t a t e d , t h e methods t h a t w i l l be 4 0 d i s c u s s e d a r e independent of l i n e a r t r a n s f o r m a t i o n s of s c a l e , no f u r t h e r adjustment was made. C. S t a t i s t i c a l P r o p e r t i e s As p r e v i o u s l y s t a t e d , the d a t a were c o l l e c t e d from a p u l p m i l l i n B r i t i s h Columbia. W i t h such a complex o p e r a t i o n , v a r i a b i l i t y i n the d a t a can be a t t r i b u t e d t o many s o u r c e s . I t i s i m p o s s i b l e a t t h i s s tage t o t r a c e back p a r t i c u l a r v a l u e s t o s p e c i f i c s o u r c e s w i t h i n the m i l l . However, as w i t h most l a r g e s c a l e p l a n t s , s c h e d u l e s f o r p e r f o r m i n g c e r t a i n r o u t i n e t a s k s e x i s t . As a r e s u l t , the day of the week and the hour of the day a r e o b v i o u s c h o i c e s t o examine. Table 1 p r e s e n t s the mean l e v e l of s p e c i f i c c o n d u c t i v i t y and 95% c o n f i d e n c e i n t e r v a l s f o r each day of the week. The d i f f e r e n c e i n average l e v e l between ev e r y p a i r of days i s s t a t i s t i c a l l y s i g n i f i c a n t w i t h the e x c e p t i o n of the p a i r Wednesday and F r i d a y . The day showing the l a r g e s t v a r i a b i l i t y i s Saturday w i t h the v a r i a b i l i t y a c r o s s the o t h e r days b e i n g f a i r l y c o n s i s t e n t . Perhaps the most s u r p r i s i n g a s p e c t i s the d r a m a t i c drop i n the average l e v e l on Thursday. The average l e v e l shows a stea d y i n c r e a s e from Monday t o Wednesday but then drops suddenly on Thursday. T h i s i s f o l l o w e d by the h i g h e s t average l e v e l which o c c u r s on F r i d a y . I t i s q u i t e c l e a r t h a t the day of the week i s an imp o r t a n t f a c t o r i n the v a r i a b i l i t y of the d a t a . 41 T a b l e 1 Mean l e v e l of s p e c i f i c c o n d u c t i v i t y and 95% c o n f i d e n c e i n t e r v a l f o r each day of the week. Day Mean Std . E r r o r 95% Conf idence i n t e r v a l Monday 0 .01600 0. 000061 (0 .015878, 0. 016115) Tuesday 0 .01666 0. 000040 (0 .01.6585, 0. 016740) Wednesday 0 .01755 0. 000053 (0 .017444, 0. 017651) Thurday 0 .01536 0. 000063 (0 .015240, 0. 015489) Fr i d a y 0 .01803 0. 000031 (0 .017966, 0. 018088) Saturday 0 .01752 0. 000132 (0 .017266, 0. 017781) Sunday 0 .01698 0. 000042 (0 .016899, 0. 017065) 17,280 o b s e r v a t i o n s f o r each day of the week. Ta b l e 2 Mean l e v e l of s p e c i f i c c o n d u c t i v i t y and 95% c o n f i d e n c e i n t e r v a l f o r each hour of the day. Hour Mean Std . E r r o r 95% Conf idence i n t e r v 0 :00- 0 :59 0 .01448 0. 000101 (0 .014277, 0. 014674 1 :00- .1 :59 0 .01508 0. 000437 (0 .014224, 0. 01-5935 2 :00- 2 :59 0 .01463 0. 000096 (0 .01444 5, 0. 014822 3 :00- 3 :59 0 .01471 0. 000092 (0 .014531, 0. 014892 4 :00- 4 :59 0 .01578 0. 000075 (0 .015635, 0. 015930 5 :00- 5 :59 0 .01602 0. 000073 (0 .015873, 0. 016159 6 :00- 6 :59 0 .01613 0. 000070 (0 .015997, 0. 016272 7 :00- 7 :59 0 .01622 0. 000073 (0 .016080, 0. 016366 8 :00- 8 :59 0 .01720 0. 000099 (0 .017008, 0. 017398 9 :00- 9 :59 0 .01770 0. 000112 (0 .017476, 0. 017916 10 :00- 1 0 :59 0 .01753 0. 000082 (0 .017370, 0. 017690 1 1 :00- 1 1 :59 0 .01789 0. 000085 (0 .017725, 0. 018059 1 2 :00- 1 2 :59 0 .01763 0. 000077 (0 .017475, 0. 017777 1 3 :00- 1 3 :59 0 .01852 0. 000110 (0 .018301, 0. 018731 1 4 :00- 1 4 :59 0 .01863 0. 000139 (0 .018356, 0. 018900 1 5 :00- 1 5 :59 0 .01788 0. 000067 (0 .017750, 0. 018013 1 6 :00- 16 :59 0 .01793 0. 000101 (0 .017728, 0. 018125 1 7 :00- 1 7 :59 0 .01797 0. 000075 (0 .017819, 0. 018113 18 :00- 18 :59 0 .01814 0. 0001 1 0 (0 .017926, 0. 018355 19 :00- 19 :59 0 .01747 0. 000067 (0 .017338, 0. 017601 20 :00- 20 :59 0 .01730 0. 000069 (0 .017170, 0. 017439 21 :00- 21 :59 0 .01724 0. 000069 (0 .017109, 0. 017380 22 :00- 22 :59 0 .01724 0. 000081 (0 .017080, 0. 017399 23 :00- 23 :59 0 .01597 0. 000078 (0 .015816, 0. 016124 5040 o b s e r v a t i o n s f o r each hour of the day 42 The average l e v e l of s p e c i f i c c o n d u c t i v i t y and 95% c o n f i d e n c e i n t e r v a l s f o r each hour of the day a r e d i s p l a y e d i n T a ble 2. Not s u r p r i s i n g l y , the hour of the day i s a f a c t o r a f f e c t i n g the average l e v e l . A l a r g e jump i s e v i d e n t commencing at 8:00 AM. T h i s i s undoubtedly r e l a t e d t o a s h i f t s t a r t i n g t i m e . Moreover, t h e r e i s a l a r g e drop at m i d n i g h t which would i n d i c a t e the end of an e v e n i n g s h i f t . W ith the e x c e p t i o n of the hour 1:00 t o 1:59, the v a r i a b i l i t y a c r o s s c a t e g o r i e s i s a g a i n f a i r l y c o n s i s t e n t . A l t h o u g h the above d e s c r i p t i v e a n a l y s i s p r o v i d e s some i n s i g h t i n t o the o p e r a t i o n of the m i l l and c o n s e q u e n t l y the d a t a , i t i s not of d i r e c t consequence t o the t h e s i s . The most i m p o r t a n t p r o p e r t y i s the degree of p o s i t i v e a u t o c o r r e l a t i o n e x h i b i t e d by the d a t a . The f i r s t 150 l a g s of the a u t o c o r r e l a t i o n f u n c t i o n based on a l l 120,960 o b s e r v a t i o n s can be seen i n F i g u r e 2. The f i r s t o r d e r a u t o c o r r e l a t i o n i s 0.449 w i t h the v a l u e at subsequent l a g s d e c r e a s i n g v e r y s l o w l y . I n c l u d e d i n the c a l c u l a t i o n of t h i s f u n c t i o n i s the a u t o c o r r e l a t i o n between o b s e r v a t i o n s from d i f f e r e n t days of the week. I t was observed t h a t the average l e v e l s c o u l d v a r y s u b s t a n t i a l l y over t h i s c a t e g o r y . To e l i m i n a t e t h i s e f f e c t , the a u t o c o r r e l a t i o n f u n c t i o n was c a l c u l a t e d f o r each of the 84 days s e p a r a t e l y . The v a l u e s at each l a g were then averaged over the 84 days. The r e s u l t i n g f u n c t i o n w i l l be r e f e r r e d t o as the average a u t o c o r r e l a t i o n f u n c t i o n and a l s o appears i n F i g u r e 2. The s t a n d a r d e r r o r was c a l c u l a t e d f o r each averaged v a l u e and the r e s u l t i n g 95% 43 F i g u r e 2 A u t o c o r r e l a t i o n and a u t o c o r r e l a t i o n a v e r a g e d over each of t h e 84 days. ALL 320.950 OBSERVATIONS RVERRGE flUTOCORRELHnON FUNCTION 957 CONFIDENCE INTERVAL T — i — i — i — i — i — i — r 63 93 LAG F i g u r e 3 A u t o c o r r e l a t i o n and a u t o c o r r e l a t i o n averaged over each of the 2016 h o u r s . ALL 3 20 .960 OBSERVATIONS AVERAGE AUTOCORRELATION FUNCTION 957 CONFIDENCE INTERVAL I — I O _ J • Luif^ Op-' ^ 5 -CM . 1 — i — i — i — i — i — i — i — i — i — i — i — i — r I 3 5 L A G T — I — I — I — I — I — I — I — I — I — I — ! 1 9 1 ! 44 c o n f i d e n c e i n t e r v a l a l s o i n d i c a t e d . The e f f e c t i s s u b s t a n t i a l w i t h an average f i r s t o r d e r a u t o c o r r e l a t i o n of 4 0.869. The average a u t o c o r r e l a t i o n f u n c t i o n does, however, de c r e a s e more r a p i d l y . A s i m i l a r p a t t e r n appears i 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 i s c a l c u l a t e d f o r each hour of the day s e p a r a t e l y and then averaged. T h i s i s i l l u s t r a t e d i n F i g u r e 3. R e c a l l t h a t t h e r e were d i f f e r e n c e s i n the mean l e v e l of s p e c i f i c c o n d u c t i v i t y by hour of the day. The p r o p e r t y of h i g h p o s i t i v e a u t o c o r r e l a t i o n i s of fundamental importance t o the u n d e r l y i n g t h e o r e t i c a l s t r u c t u r e t h a t i s assumed. As j u s t d e s c r i b e d , the s p e c i f i c c o n d u c t i v i t y data p o s s e s s t h i s a t t r i b u t e . These d a t a a r e , t h e r e f o r e , a p p r o p r i a t e f o r the purpose of t e s t i n g and e v a l u a t i n g the methods t h a t w i l l be proposed. V. Measures of E r r o r The g o a l of t h i s t h e s i s i s the development of methods t h a t w i l l f i n d the maximum of a f i n i t e number of samples w i t h o u t measuring them a l l . I t i s conceded t h a t the random n a t u r e of the da t a w i l l guarantee t h a t no method e x i s t s t h a t w i l l always f i n d the g l o b a l maximum. Thus, t h e r e i s a need t o measure e r r o r s . T h i s s i t u a t i o n i s not d i s s i m i l a r t o the development of h e u r i s t i c s i n i n t e g e r programming. In ki n d , , no u n i v e r s a l l y a c c e p t e d s e t of s t a t i s t i c s e x i s t s on which t o judge or e v a l u a t e the r e l a t i v e performance of d i f f e r e n t methods. I t i s , i n f a c t , a r g u a b l e t h a t no such s e t e x i s t s , as any s t a t i s t i c adopted s h o u l d be problem dependent. The remainder of t h i s c h a p t e r d e t a i l s the measures of e r r o r t h a t have been adopted f o r the purposes of t h i s r e s e a r c h and the p r o p e r t i e s t h a t they p o s s e s s . The r a t i o n a l e f o r the s e l e c t i o n of measures of e r r o r can be summed up s u c c i n c t l y : d i v e r s i t y . In l i g h t of the s i t u a t i o n whereby the u n i t s i n which the d a t a a r e r e c o r d e d p r o v i d e no p h y s i c a l i n t e r p r e t a t i o n , a broad range of measures becomes e s s e n t i a l . In a d d i t i o n , i t i s im p o r t a n t t o be a b l e t o p i n p o i n t and h i g h l i g h t the p a r t i c u l a r s t r e n g t h s and weaknesses of a method. T h i s r e q u i r e s measures of e r r o r w i t h d i s t i n c t p r o p e r t i e s . Throughout the remainder, the 4 5 46 f o l l o w i n g n o t a t i o n w i l l be employed, A ( i ) = t h e a c t u a l maximum f o r the i ' t h t r i a l E ( i ) = the e s t i m a t e d maximum f o r t h e i ' t h t r i a l L ( i ) = t h e a c t u a l minimum f o r the i ' t h t r i a l N = t h e number of t r i a l s (5.1) A. P r o p o r t i o n S i n c e t h e g o a l i s t o f i n d t he sample w i t h t h e maximum measurement, perhaps the most s i g n i f i c a n t s t a t i s t i c i s the p r o p o r t i o n of t r i a l s on which t h i s i s a c h i e v e d . To d e f i n e t h i s s t a t i s t i c e x p l i c i t l y , l e t the random v a r i a b l e X ( i ) be d e f i n e d as the outcome of the i ' t h t r i a l where, X ( i ) = 1 i f E ( i ) = A ( i ) X ( i ) = 0 i f E ( i ) * A ( i ) (5.2) The p r o p o r t i o n of t r i a l s , P, on which the g l o b a l maximum i s o b t a i n e d , can now be e x p r e s s e d a s , l N (5.3) Denote the t r u e or p o p u l a t i o n p r o p o r t i o n by p. I f N i s s u f f i c i e n t l y l a r g e , then P w i l l be n o r m a l l y d i s t r i b u t e d w i t h mean p and v a r i a n c e p ( 1 - p ) / N . S i n c e p i s unknown, i t i s u s u a l t o use P as an e s t i m a t e . A l t h o u g h the s t a t i s t i c P r e v e a l s much about the t r i a l s i n w hich t h e the sample w i t h t h e maximum measurement i s 47 f o u n d , i t t e l l s l i t t l e a b o u t t h o s e t r i a l s i n w h i c h i t i s n o t . Some m e a s u r e o f d i s t a n c e f r o m t h e maximum i s c l e a r l y n e e d e d i n t h e s e s i t u a t i o n s . B. Mean S q u a r e E r r o r The mean s q u a r e e r r o r (MSE) i s c a l c u l a t e d f r o m t h e e x p r e s s i o n , 1 N 9 MSE = I ( A ( i ) - E ( i ) ) ( 5- 4> N i = l The MSE a s s i g n s more w e i g h t t o l a r g e r d i f f e r e n c e s . T h i s p r o p e r t y i s d e s i r a b l e i n t h e c o n t e x t o f p o l l u t i o n m o n i t o r i n g ; t h e damage f u n c t i o n , i n most c a s e s , b e i n g s u p e r l i n e a r . T h a t i s , t h e c o s t i n c u r r e d by an e r r o r o f 10 u n i t s w i l l be more t h a n t w i c e t h e c o s t i n c u r r e d by an e r r o r o f 5 u n i t s . The MSE h a s one m a j o r d r a w b a c k . O b s e r v e t h a t a l i n e a r c h a n g e i n s c a l e w i l l r e s u l t i n a q u a d r a t i c c h a n g e i n t h e m e a s u r e . I t w i l l be r e c a l l e d t h a t t h e s p e c i f i c c o n d u c t i v i t y d a t a h a v e been t r a n s f o r m e d i n t h i s manner. T h u s , one must be c a r e f u l n o t t o c o m p a r e t h e n u m e r i c a l v a l u e s p r o d u c e d by t h e MSE t o t h o s e p r o d u c e d by t h e o t h e r m e a s u r e s o f e r r o r . C o m p a r i n g t h e MSE a c r o s s d i f f e r e n t m e t h o d s , h o w e v e r , d o e s n o t p o s e a p r o b l e m . An a l t e r n a t i v e q u a d r a t i c m e a s u r e i s a v a i l a b l e f o r w h i c h a l i n e a r c h a n g e i n s c a l e w i l l n o t p r o d u c e a q u a d r a t i c c h a n g e 48 i n t he measure. I t i s c a l l e d t he r o o t mean squared e r r o r (RMSE) and i s d e f i n e d by the e x p r e s s i o n , RMSE = N I (A(i) - E ( i ) ) i = l ( 5 . 5 ) The RMSE has, however, one major d e f i c i e n c y . I t i s not p o s s i b l e t o r e l i a b l y e s t i m a t e the v a r i a n c e . To see t h i s , o b s e r v e t h a t the RMSE and the MSE a r e r e l a t e d by the e x p r e s s i o n , RMSE = [MSE/N] 1* ( 5 . 6 ) By d e f i n i t i o n , the MSE i s a p o s i t i v e random v a r i a b l e and t h u s , the RMSE i s d e f i n e d . The v a r i a n c e of the RMSE c o u l d be e s t i m a t e d i f the u n d e r l y i n g d i s t r i b u t i o n from which the terms ( A ( i ) - E ( i ) ) 2 a r e drawn i s known. As i n most s i t u a t i o n s , t h i s i s not the case h e r e . An a p p r o x i m a t i o n t o the d i s t r i b u t i o n of the MSE (not t h e i n d i v i d u a l terms ( A ( i ) - E ( i ) ) 2 ) can be o b t a i n e d t h r o u g h a p p l i c a t i o n of the C e n t r a l L i m i t Theorem. That i s , t h e MSE i s a p p r o x i m a t e l y n o r m a l l y d i s t r i b u t e d . T h i s r e s u l t i s of no v a l u e as the square r o o t t r a n s f o r m a t i o n i s d e f i n e d o n l y f o r p o s i t i v e random v a r i a b l e s . F o r t h e s e r e a s o n s , i t i s not p o s s i b l e t o 49 r e l i a b l y e s t i m a t e the v a r i a n c e of t h e RMSE. Due t o t h e s u p e r l i n e a r n a t u r e of the damage f u n c t i o n a s s o c i a t e d w i t h p o l l u t i o n l e v e l s , a q u a d r a t i c measure i s c o n s i d e r e d n e c e s s a r y . I t i s e s s e n t i a l f o r t h i s r e s e a r c h t h a t d i f f e r e n t methods can be compared s t a t i s t i c a l l y . Thus, i t was c o n c l u d e d t h a t the absence of an e s t i m a t e d v a r i a n c e f o r the RMSE was more of a h a n d i c a p than the q u a d r a t i c e f f e c t of the MSE on l i n e a r changes i n s c a l e . The MSE was, t h e r e f o r e , a dopted. Another v e r y p o p u l a r measure of e r r o r i s the mean a b s o l u t e e r r o r (MAE). The MAE i s c a l c u l a t e d from t h e e x p r e s s i o n , l N MAE = ± I | A ( i ) - E ( i ) | (5.7) i = l The MAE r e f l e c t s the " t y p i c a l " e r r o r . However, i t does not d i s t i n g u i s h between v a r i a n c e and b i a s . Moreover, i t i s o n l y a p p r o p r i a t e when the damage f u n c t i o n i s l i n e a r . For t h e s e r e a s o n s , t h e MAE was not i n c l u d e d . C. Mean A b s o l u t e Range E r r o r The mean a b s o l u t e range e r r o r (MARE) i s c a l c u l a t e d from the e x p r e s s i o n , 50 MARE = 1 - h A ( i ) - E ( i ) ( 5 . 8 ) A ( i ) - L ( i ) The MARE has an advantage over the RMSE i n t h a t i t i s d i m e n s i o n l e s s . I t a l s o i n c o r p o r a t e s a p r o p e r t y i n h e r e n t i n the s t r u c t u r e of the s i t u a t i o n b e i n g a n a l y z e d ; t h i s l e a d s t o a p a r t i c u l a r l y v a l u a b l e i n t e r p r e t a t i o n . Suppose t h a t f o r the i ' t h t r i a l , a s i n g l e sample i s s e l e c t e d and measured. I f the ob s e r v e d v a l u e i s used as an e s t i m a t e f o r the maximum, the e r r o r can be no more than A ( i ) - L ( i ) . That i s , A ( i ) - L ( i ) r e p r e s e n t s the maximum o b s e r v a b l e e r r o r 9 . Thus, the MARE measures t h e p r o p o r t i o n of the maximum o b s e r v a b l e e r r o r t h a t i s a c c o u n t e d f o r . The MARE i s s i m i l a r t o a w i d e l y used measure of e r r o r c a l l e d the mean a b s o l u t e p e r c e n t a g e e r r o r (MAPE). T h i s measure i s d e f i n e d by the e x p r e s s i o n , The MAPE, l i k e t he MARE, i s d i m e n s i o n l e s s . I t can be i n t e r p r e t e d as the p r o p o r t i o n of t h e maximum v a l u e a c c o u n t e d 9 An o b s e r v a b l e e r r o r i s d e f i n e d as an e r r o r t h a t r e s u l t s from u s i n g an o b s e r v e d v a l u e as an e s t i m a t e f o r t h e maximum. N MAPE = I i = l A ( i ) - E ( i ) A ( i ) ( 5 . 9 ) 51 f o r . A l t h o u g h m e a n i n g f u l , t h i s i n t e r p r e t a t i o n i s not as v a l u a b l e i n t h i s s i t u a t i o n as t h e i n t e r p r e t a t i o n a t t a c h e d t o the MARE. In a d d i t i o n , t he MAPE has the d i s t i n c t d i s a d v a n t a g e of a t t a c h i n g h i g h e r weight t o s m a l l e r v a l u e s of A ( i ) . T h i s i s an u n d e s i r a b l e p r o p e r t y i n t h e s i t u a t i o n b e i n g s t u d i e d here as i t i s of g r e a t e r b e n e f i t t o measure the extreme v a l u e s more c l o s e l y . F o r the above r e a s o n s , the MAPE was not i n c l u d e d . D. Maximum A b s o l u t e D e v i a t i o n The maximum a b s o l u t e d e v i a t i o n (MAD) i s d e f i n e d by the e x p r e s s i o n , MAD = Max | A ( i ) - E ( i ) | (5.10) i = l , 2 , . . . N The MAD i s a measure of the worst case performance. In p r a c t i c e , t h i s i s u s e f u l i n f o r m a t i o n , even though i t does not g i v e any i n d i c a t i o n of the d i s t r i b u t i o n of the i n d i v i d u a l v a l u e s . I n many c i r c u m s t a n c e s i t i s s u f f i c i e n t t o be w i t h i n some neighborhood of the maximum. The MAD w i l l i n d i c a t e i f t h i s has happened. Another advantage of the MAD i s t h a t i t i s p a r t i c u l a r l y good a t i d e n t i f y i n g o u t l i e r s . V I . C o m p o s i t e Methods C o m p o s i t i n g i s t h e p h y s i c a l p o o l i n g o f a s e t of s a m p l e s . The measurement from t h e r e s u l t i n g c o m p o s i t e sample w i l l be t h e a v e r a g e measurement o f t h e samp l e s t h a t were p o o l e d . The p r o c e s s o f c o m p o s i t i n g i s o f t e n v a l i d f o r sampl e s c o l l e c t e d t o m o n i t o r water p o l l u t i o n , a s i n d i c a t e d by t h e g e n e r a l a c c e p t a n c e of c o m p o s i t e s a m p l i n g . In p a r t i c u l a r , c o m p o s i t i n g i s v a l i d f o r t h e s p e c i f i c c o n d u c t i v i t y d a t a [ 1 6 ] . C o m p o s i t e t e c h n i q u e s i n v o l v e t h e a g g r e g a t i n g of a l l t h e sa m p l e s i n t o g r o u p s of s e q u e n t i a l samples of some p r e d e f i n e d f i x e d s i z e . Then, f o r e a c h g r o u p , a f i x e d p o r t i o n o f e a c h sample i s p o o l e d t o form a c o m p o s i t e . As an example, suppose t h a t 100 s a m p l e s have been t a k e n and c o m p o s i t e s o f 12 o b s e r v a t i o n s a r e d e s i r e d . The f i r s t 96 sam p l e s w i l l be a g g r e g a t e d i n t o 8 g r o u p s e a c h c o n s i s t i n g o f 12 s e q u e n t i a l s a m p l e s w i t h t h e f i n a l 4 samples f o r m i n g a n i n t h g r o u p . W i t h i n e a c h g r o u p , a f i x e d p o r t i o n of e a c h sample i s p o o l e d t o f o r m n i n e c o m p o s i t e s a m p l e s . C o m p o s i t e t e c h n i q u e s a r e b a s e d on t h e f o l l o w i n g p r e m i s e : w i t h t h e p r e s e n c e of h i g h p o s i t i v e a u t o c o r r e l a t i o n , t h e sample w i t h t h e h i g h e s t measurement w i l l t e n d t o a p p e a r i n t h e c o m p o s i t e w i t h t h e h i g h e s t measurement. T h e r e f o r e , an i n d i c a t i o n of t h e l o c a t i o n of t h e sample w i t h t h e maximum measurement i s o b t a i n e d by t e s t i n g o n l y t h e c o m p o s i t e s a m p l e s . S i n c e o n l y a p o r t i o n of t h e o r i g i n a l o r base s a m p l e s was u s e d f o r t h e c o m p o s i t i n g , t h e s e samples a r e 52 5 3 s t i l l a v a i l a b l e f o r f u r t h e r t e s t i n g . F u r t h e r a n a l y s i s i s now performed on the base samples t h a t were p o o l e d t o form the c o m p o s i t e ( s ) w i t h the h i g h e s t measurement(s) t o i d e n t i f y the s p e c i f i c sample w i t h the maximum measurement. One of the most i m p o r t a n t p r o p e r t i e s of t h i s approach i s the r e t e n t i o n of an u n b i a s e d e s t i m a t e of the mean l e v e l . Suppose N samples were taken and aggregated i n t o groups of m s e q u e n t i a l samples. The r e s u l t i n g (N/mj com p o s i t e s r e p r e s e n t independent o b s e r v a t i o n s and thus the average of t h e i r measurements can be used t o e s t i m a t e the mean. Moreover, i f c2 i s the p o p u l a t i o n v a r i a n c e , the v a r i a n c e of the e s t i m a t e w i l l be samples a r e measured [18] i o • The o b s e r v a t i o n s r e s u l t i n g from the f u r t h e r t e s t i n g of the base samples f o r m i n g the c o m p o s i t e ( s ) w i t h the h i g h e s t l e v e l ( s ) cannot be used t o e s t i m a t e the mean. F i r s t l y , they cannot be used t o g e t h e r w i t h the composite measurements s i n c e the two s e t s of valu.es a r e c l e a r l y not independent. On the o t h e r hand, they cannot be used by themselves s i n c e the v a l u e s a r e b i a s e d ; the samples are s e l e c t e d because they a re more p r o b a b l e t o show h i g h e r l e v e l s . The p r i m a r y u n i t of time over which the methods w i l l be t e s t e d i s an hour. Each t r i a l w i l l , t h e r e f o r e , c o n s i s t of e s t i m a t i n g the maximum from a s e t of 60 samples. An hour was 1 0 The e x i s t e n c e of a u t o c o r r e l a t i o n i m p l i e s t h a t the o b s e r v a t i o n s a r e not s t a t i s t i c a l l y independent. However, the sample mean w i l l s t i l l be an u n b i a s e d e s t i m a t o r and the v a r i a n c e can be e s t i m a t e d w i t h a s l i g h t m o d i f i c a t i o n t o e x p r e s s i o n ( 2 . 9 ) . 54 chosen as the p r i m a r y u n i t of time f o r t h r e e r e a s o n s . The da t a must, of c o u r s e , demonstrate h i g h p o s i t i v e a u t o c o r r e l a t i o n . T h i s i s the case when the d a t a i s grouped i n one hour b l o c k s as i l l u s t r a t e d by F i g u r e 3. S e c o n d l y , s i x t y i s a r e a s o n a b l e number of samples t o d e a l w i t h from a p r a c t i c a l s t a n d p o i n t , both f o r t e s t i n g and s t o r a g e . F i n a l l y , one hour b l o c k s g i v e a d e s i r a b l y l a r g e sample s i z e , a p p r o x i m a t e l y 2000 t r i a l s . I t i s c o n j e c t u r e d t h a t the i s s u e of whether one minute s a m p l i n g over one hour b l o c k s would be common p r a c t i c e , i s i r r e l e v a n t . The i m p o r t a n t p r i n c i p l e i s not the time between o b s e r v a t i o n s but the a u t o c o r r e l a t i o n between o b s e r v a t i o n s . In o t h e r words, i f two s e t s of da t a r e p r e s e n t 60 samples per hour and 60 samples per day, r e s p e c t i v e l y , but p o s s e s s e q u i v a l e n t a u t o c o r r e l a t i o n f u n c t i o n s , the methods s h o u l d work e q u a l l y w e l l on both s e t s of d a t a . Some e m p i r i c a l e v i d e n c e w i l l be p r e s e n t e d which s u p p o r t s t h i s c l a i m . B e f o r e any a n a l y s i s was performed, seven one hour b l o c k s or r e c o r d s were d i s c a r d e d . The f i r s t t h r e e hours of November 2nd and the f i r s t two hours of September 22nd were r e j e c t e d due t o apparent equipment f a i l u r e . A l l o b s e r v a t i o n s were e s s e n t i a l l y z e r o . September 20, hour 2, and October 15, hour 2, were d e l e t e d because of o b v i o u s r e c o r d i n g e r r o r s . The f i n a l o b s e r v a t i o n f o r hour 2, October 15, was r e p o r t e d at 2.13919 which i s on the o r d e r of 10,000 ti m e s l a r g e r than the s u r r o u n d i n g v a l u e s . September 20, hour 2, i n c l u d e d an o b s e r v a t i o n r e p o r t e d a t -0.12610. In l i g h t of the proce d u r e 55 used t o r e c a l i b r a t e the d a t a , t h i s v a l u e i s s e v e r a l o r d e r s of magnitude more n e g a t i v e than c o u l d p o s s i b l y o c c u r . A. P r i m a r y F i r s t Order C o m p o s i t i n g The f i r s t method t o be c o n s i d e r e d i s a l s o the most s t r a i g h t f o r w a r d . I n i t i a l l y , the composites a r e formed, measured, and the composite w i t h the maximum l e v e l i s found. Then, a l l the base samples t h a t formed t h i s composite a re t e s t e d . The maximum sample measurement t h a t r e s u l t s w i l l be the e s t i m a t e of the maximum f o r the e n t i r e s e t of samples. The method j u s t d e s c r i b e d w i l l be c a l l e d p r i m a r y f i r s t o r d e r c o m p o s i t i n g . The word 'primary' r e f e r s t o the f a c t t h a t o n l y the base samples t h a t form the composite w i t h the maximum measurement are c o n s i d e r e d f o r f u r t h e r a n a l y s i s . R e c a l l i n g the premise on which composite t e c h n i q u e s a r e based, t h i s c l e a r l y r e p r e s e n t s the p r i m a r y c h o i c e of samples. The term ' f i r s t o r d e r ' i s used because no f u r t h e r c o m p o s i t i n g i s performed on the base samples t h a t remain. That i s , the c o m p o s i t i n g procedure i s a p p l i e d o n l y once. In o r d e r t o a s c e r t a i n how w e l l a method i s p e r f o r m i n g i t w i l l be n e c e s s a r y t o have some base a g a i n s t which i t can be compared. Assuming a b s o l u t e l y no s t r u c t u r e t o the d a t a , t h a t i s , complete randomness, and a d e s i r e t o e s t i m a t e the maximum based on an observ e d v a l u e , one c o u l d do no b e t t e r than t o sample randomly and use the maximum observ e d v a l u e . The assumption of complete randomness and the s e l e c t i o n of a random sample i s the t y p i c a l s i t u a t i o n and w i l l form the 56 base a g a i n s t which p r i m a r y f i r s t o r d e r c o m p o s i t i n g w i l l be compared. The r e s u l t s , t h e r e f o r e , w i l l demonstrate how the knowledge of the e x i s t e n c e of h i g h p o s i t i v e a u t o c o r r e l a t i o n can be used t o i n c r e a s e our a b i l i t y t o e s t i m a t e the maximum sample measurement. With the sample s i z e now f i x e d , the o n l y parameter y e t to be s p e c i f i e d i s the number of samples t h a t w i l l be used t o form the c o m p o s i t e s . For a f i x e d v a l u e of t h i s parameter, p r i m a r y f i r s t o r d e r c o m p o s i t i n g and random s a m p l i n g a r e a p p l i e d , i n t u r n , t o each of the 2009 one hour b l o c k s of samples. For each t r i a l , the number of sample measurements f o r each method was the same. For example, i f a p p l i c a t i o n of p r i m a r y f i r s t o r d e r c o m p o s i t i n g t o an hour b l o c k of d a t a t e r m i n a t e d w i t h 10 measurements, then 10 samples would be s e l e c t e d by the random sample approach. S t a t i s t i c s a re t a b u l a t e d and r e c o r d e d . T h i s p r o c e d u r e i s r e p e a t e d f o r a composite s i z e of 2 through 30 i n c l u s i v e . A complete l i s t of the r e s u l t i n g s t a t i s t i c s i s p r e s e n t e d i n Appendix A, T a b l e A.1. The volume of the numeric v a l u e s i s somewhat overwhelming. S i n c e g r a p h i c p r e s e n t a t i o n of d a t a i s p a r t i c u l a r l y u s e f u l i n c o n d e n s i n g l a r g e amounts of numeric d a t a i n t o a m e n t a l l y manageable and r e v e a l i n g form, t h i s method of p r e s e n t a t i o n was adopted. The t a b l e s of numeric r e s u l t s a r e i n c l u d e d i n the appendix f o r c o m p l e t e n e s s . The average number of t e s t s p erformed, or e q u i v a l e n t l y the average number of samples measured, i s p l o t t e d a g a i n s t the c o mposite s i z e i n F i g u r e 4. Note t h a t the average number 57 F i g u r e 4 Average number of t e s t s per t r i a l v e r s u s c o m p o s i t e s i z e . (2009 t r i a l s ) of t e s t s w i l l not always be i n t e g r a l . C o n s i d e r p r i m a r y f i r s t o r d e r c o m p o s i t i n g w i t h a c o m p o s i t e s i z e of 7. Seven i s not a p e r f e c t d i v i s o r of 60, r e s u l t i n g i n 8 c o m p o s i t e s of 7 samples and a f i n a l c o m p o s i t e of o n l y 4 samples. Thus, the number of t e s t s performed w i l l be e i t h e r 8 (the-number of c o m p o s i t e s ) p l u s 7 ( t h e number of w i t h i n c o m p o s i t e s a m p l e s ) , i f one of the f i r s t 7 c o m p o s i t e s has the h i g h e s t measurement, o r , 8 p l u s 4 i f t h e l a s t c omposite has the h i g h e s t measurement 1 1. I f the l a t t e r c o n d i t i o n o c c u r s a t l e a s t once i n a s e r i e s o f t r i a l s , the average number of 1 1 I f t h e l a s t c o m p o s i t e has t h e h i g h e s t measurement, 12 samples a r e t e s t e d . The random sample approach would then be a p p l i e d a l s o based on 12 samples. In a l l o t h e r c a s e s , the random sample approach would be a p p l i e d based on 15 samples. T h i s e n s u r e s a f a i r c o m p arison of t h e two t e c h n i q u e s . 58 t e s t s need not be i n t e g r a l . Whenever the composite s i z e i s not a p e r f e c t d i v i s o r of the t o t a l number of samples, the f i n a l composite w i l l c o n t a i n fewer samples than the r e s t . T h i s s i t u a t i o n w i l l be r e f e r r e d t o as unbalanced c o m p o s i t i n g . I f a l l the com p o s i t e s a r e formed from an e q u a l number of samples,- the s i t u a t i o n w i l l be r e f e r r e d t o as b a l a n c e d c o m p o s i t i n g . The jagged b e h a v i o r e x h i b i t e d i n F i g u r e 4 i s a d i r e c t consequence of unbalanced c o m p o s i t i n g . S t a r t i n g a t a composite s i z e of 6, the b e h a v i o r i s v i r t u a l l y t h a t of a s t e p f u n c t i o n , w i t h s h a r p r i s e s or s t e p s o c c u r r i n g a t composite s i z e s of 10, 12, 15, 20, and 30. I t i s no c o i n c i d e n c e t h a t t h e s e v a l u e s a re a l s o the o n l y v a l u e s l a r g e r than 6 t h a t a r e p e r f e c t d i v i s o r s of 60 and, t h e r e f o r e , the o n l y composite s i z e s t h a t r e p r e s e n t b a l a n c e d c o m p o s i t i n g . I f we denote the composite s i z e by m, the number of t e s t s performed per t r i a l w i l l be e x a c t l y (60/m>+m f o r b a l a n c e d c o m p o s i t i n g . For unbalanced c o m p o s i t i n g , however, s e l e c t i o n of the f i n a l composite w i l l reduce the average number of t e s t s per t r i a l below (60/m)+m. The more o f t e n the f i n a l composite i s the h i g h e s t , the g r e a t e r the r e d u c t i o n . G i v e n a composite s i z e of m, i t i s p o s s i b l e t o determine from the da t a the e x a c t p r o p o r t i o n of time t h a t the a c t u a l maximum f e l l i n the f i n a l c o m p o s i t e . Denote t h i s v a l u e by P(m). Our b a s i c premise i s t h a t the sample w i t h the maximum measurement w i l l t e n d t o f a l l i n the composite w i t h the h i g h e s t measurement. T h e r e f o r e , i t i s r e a s o n a b l e t o expect t h a t the p r o p o r t i o n of 59 t r i a l s on which the f i n a l c omposite i s chosen w i l l be c l o s e t o P(m). With the assumption t h a t the f i n a l c omposite i s chosen w i t h p r o b a b i l i t y P(m), i t i s p o s s i b l e t o c a l c u l a t e the e x p e c t e d number of t e s t s per t r i a l . I t w i l l be r e c a l l e d t h a t the number of composites i s g i v e n by (60/mj. The number of samples making up the f i n a l c o m p o s i t e , r , can now be d e f i n e d by the e x p r e s s i o n , r = 60 - [(60/m, - 1] m (6.1) The number of t e s t s per t r i a l , t h e r e f o r e , w i l l be (60/m)+m w i t h p r o b a b i l i t y 1-P(m) and (60/m>+r w i t h p r o b a b i l i t y P(iri). From t h i s p r o b a b i l i t y d i s t r i b u t i o n , the e x p e c t e d number of t e s t s per t r i a l , E(m), i s , ( (60/m,+m)(1-p(m)) + (,60/m,+r)P(m) = E(m) (6.2) The numeric v a l u e of t h i s e x p r e s s i o n f o r the composite s i z e s under e x a m i n a t i o n , 2 through 30 i n c l u s i v e , a re t a b u l a t e d i n T a b l e 3. A l s o i n c l u d e d a re the a c t u a l observed a v e r a g e s , A(m), which, f o r unbalanced c o m p o s i t i n g , are c o n s i s t e n t l y l e s s then the e x p e c t e d v a l u e s . T h i s i n d i c a t e s a tendency t o s e l e c t the f i n a l c omposite a d i s p r o p o r t i o n a t e number of Table 3 Expected and observed number of t e s t s per t r i a l . (2009 t r i a l s ) E x p e c t e d Average Composite Number Number B a l a n c e d S i z e P(m) of T e s t s of T e s t s or (m) ( r ) E(m) A(m) Unbalanced 2 2 .112 32.00 32.00 B a l a n c e d 3 3 . 1 26 23.00 23.00 B a l a n c e d 4 4 . 1 45 19.00 19.00 B a l a n c e d 5 5 . 1 58 1 7.00 1 7.00 B a l a n c e d 6 6 . 1 67 1 6.00 1 6.00 B a l a n c e d 7 4 . 1 45 15.57 15.42 Unbalanced 8 4 . 1 45 15.42 15.17 Unbalanced 9 6 . 1 67 1 5.50 15.36 Unbalanced 1 0 10 .221 1 6.00 1 6.00 B a l a n c e d 1 1 5 . 1 58 1 6.05 1 5.58 Unbalanced 1 2 1 2 .246 17.00 1 7.00 B a l a n c e d 1 3 8 .2 02 1 6.99 16.69 Unbalanced 1 4 4 . 1 45 17.55 16.47 Unbalanced 1 5 1 5 .276 19.00 1 9.00 B a l a n c e d 16 1 2 .246 19.02 18.84 Unbalanced 1 7 9 .209 • 19.33 18.72 Unbalanced 18 6 . 1 67 20.00 18.56 Unbalanced 19 3 . 1 26 20.98 18.43 Unbalanced 20 20 .334 23.00 23.00 B a l a n c e d 21 18 .314 23.06 22.93 Unbalanced 22 1 6 .291 23.25 22.87 Unbalanced 23 1 4 .266 23.61 22.80 Unbalanced 24 1 2 .246 24.05 22.86 Unbalanced 25 1 0 . 221 24.69 22.86 Unbalanced 26 8 .202 2.5.36 22.95 Unbalanced 27 6 . 1 67 26.49 22.97 Unbalanced 28 4 . 1 45 27.52 22.92 Unbalanced 29 2 .112 28.98 23.02 Unbalanced 30 30 .444 32.00 32.00 B a l a n c e d t i m e s i n these s i t u a t i o n s . For b a l a n c e d c o m p o s i t i n g the average number of t e s t s per t r i a l must always be (60/m)+m The r e l a t i o n s h i p between b a l a n c e d and unbalanced c o m p o s i t i n g i s the most r e v e a l i n g b e h a v i o r e x h i b i t e d i n F i g u r e 4 and Table 3 and w i l l be examined more c l o s e l y . C o n s i d e r the s e t s of composite s i z e s 12 through 14, 15 th r o u g h 19, and 20 through 29. The composite s i z e s of 12, 61 15, and 20 r e p r e s e n t b a l a n c e d c o m p o s i t i n g w h i l e the r e s t r e p r e s e n t unbalanced c o m p o s i t i n g . Note t h a t each s e t of composite s i z e s r e s u l t s i n the f o r m a t i o n of the same number of i n i t i a l c o m p o s i t e s , 5, 4, and 3 r e s p e c t i v e l y . The f i r s t p o i n t t o observe i s t h a t w i h i n each s e t , the average number of t e s t s performed per t r i a l f o r unbalanced c o m p o s i t i n g i s c o n s i s t e n t l y l e s s than the e x p e c t e d number of t e s t s per t r i a l . In o t h e r words, unbalanced c o m p o s i t i n g tends t o r e s u l t i n the f i n a l c o m p o s i t e b e i n g chosen a d i s p r o p o r t i o n a t e number of t i m e s . The o t h e r i n t e r e s t i n g f e a t u r e t o observe i s t h a t the d i f f e r e n c e between the exp e c t e d number of t e s t s per t r i a l , E(m), and the average number of t e s t s per t r i a l , A(m), i n c r e a s e s m o n o t o n i c a l l y . These d i f f e r e n c e s a re h i g h l i g h t e d i n Table 4. I t i s e v i d e n t t h a t w i t h i n each set of composite s i z e s j u s t d e f i n e d , the s i z e of the f i n a l c omposite d e c r e a s e s as the composite s i z e i n c r e a s e s . T h i s i n c r e a s i n g d i f f e r e n c e between the composite s i z e and the s i z e of the f i n a l composite can be thought of as an i n c r e a s e i n the imbalance of the c o m p o s i t i n g scheme. The e x p r e s s i o n s 'as the composite s i z e i n c r e a s e s ' and 'as the s i z e of the f i n a l c o mposite d e c r e a s e s ' w i l l be used i n t e r c h a n g e a b l y t o d e s c r i b e t h i s w i t h i n - s e t phenomena. I t i s a l s o t r u e t h a t the d i f f e r e n c e between the p r o p o r t i o n of t r i a l s on which the f i n a l c omposite i s a c t u a l l y chosen and the p r o p o r t i o n of t r i a l s . o n which i t s h o u l d be chosen, P(m), i n c r e a s e s m o n o t o n i c a l l y . To see t h i s , f i r s t l e t P'(m) be d e f i n e d as the o b s e r v e d p r o p o r t i o n 62 Tab l e 4 Exp e c t e d minus observed average number of t e s t s per t r i a l (E(m)-A(m)) and obser v e d minus e x p e c t e d p r o p o r t i o n of t r i a l s on which the f i n a l composite was s e l e c t e d (P'(m)-P(m)). Composite I n i t i a l S i z e Composites E(m)-A(m) P'(m)-P(m) (m) ( r ) ( 60/m) 7 4 9 0.15 0.050 8 4 8 0.25 0.063 9 6 7 0.14 0.047 1 1 5 6 0.47 0.078 1 3 8 5 0.30 0.060 1 4 4 5 1 .08 0. 1 08 1 6 1 2 4 0.18 0.045 1 7 9 4 0.61 0.076 18 6 4 1 .44 0. 1 20 1 9 3 4 2.55 0. 1 59 21 18 3 0.13 0.043 22 1 6 3 0.35 0.058 23 1 4 3 0 . 8 1 0.090 24 1 2 3 ' 1.19 0.099 ,25 10 3 1 .82 0.121 26 8 3 2.41 0. 1 34 27 6 3 3.52 0. 1 68 28 4 3 4.60 0. 192 29 2 3 5.96 0.221 of t r i a l s on which the f i n a l composite was a c t u a l l y s e l e c t e d . A(m) and P'(m) are r e l a t e d by the f o l l o w i n g i d e n t i t y , 63 ( (60/m)+m)(i-P'(m)) + (,60/m,+r)P'(m) = A(m) (6.3) T h i s i d e n t i t y can be s o l v e d f o r P'(m) which y i e l d s the e x p r e s s i o n , P'(m) = ( (60/m, + m - A(m)) / (m - r ) (6.4) By d e f i n i t i o n , E(m) i s e q u a l t o e q u a t i o n ( 6 . 2 ) . In a s i m i l a r f a s h i o n , t h i s i d e n t i t y can be s o l v e d f o r P(m), Agai n by d e f i n i t i o n , P'(m)-P(m) i s the d i f f e r e n c e between the p r o p o r t i o n of t r i a l s on which the f i n a l composite had the h i g h e s t l e v e l and the p r o p o r t i o n of t r i a l s i t s h o u l d have had the h i g h e s t l e v e l . By s u b t r a c t i n g (6.5) from ( 6 . 4 ) , t h i s d i f f e r e n c e can be d e f i n e d i n terms of E(m) and A(m), the v a l u e s a t hand. T h i s s u b t r a c t i o n r e s u l t s i n the e x p r e s s i o n , P(m) = ((60/m, + m - E(m)) / (m - r ) (6.5) P'(m) - P(m) = (E(m) - A(m)) / (m-r) ( 6 . 6 ) The r e s u l t i s not d e f i n e d f o r b a l a n c e d c o m p o s i t i n g , t h a t i s , r=m. T h i s d i f f e r e n c e , f o r a l l unbalanced composite s i z e s , appears i n Table 4. Thus we can now be much more p r e c i s e than s a y i n g merely t h a t the f i n a l composite i s s e l e c t e d a d i s p r o p o r t i o n a t e number of t i m e s . In f a c t , t h i s ' d i s p r o p o r t i o n ' i n c r e a s e s as the composite s i z e , m, i n c r e a s e s , or e q u i v a l e n t l y , as the s i z e of the f i n a l c o m p o s i t e , r , d e c r e a s e s . B e f o r e o f f e r i n g an e x p l a n a t i o n f o r t h i s b e h a v i o r , some n o t a t i o n i s r e q u i r e d . The event t h a t the sample w i t h the maximum measurement i s i n the f i n a l composite w i l l be denoted as MF. I f the maximum sample measurement i s e l s e w h e r e , t h i s event w i l l be denoted as -"MF. FC w i l l s t a n d f o r the f i n a l composite h a v i n g the maximum composite measurement, w h i l e ""FC w i l l denote the event t h a t the f i n a l c o mposite does not have the h i g h e s t composite measurement. Two t y p e s of e r r o r can be a s s o c i a t e d w i t h the f i n a l c o m p o s i t e . The f i r s t , i s t h a t the sample w i t h the maximum measurement i s i n the f i n a l c o mposite but i t does not have the h i g h e s t composite measurement. T h i s can be s u c c i n c t l y e x p r e s s e d as the event [MF and _ 1 F C ] , The second type of e r r o r i s t h a t the f i n a l c o mposite has the h i g h e s t composite 6 5 measurement but does not c o n t a i n the sample w i t h the maximum sample measurement. In our event n o t a t i o n t h i s i s [-•MF and F C ] . P [ • ] w i l l r e f e r t o the p r o b a b i l i t y of the event e n c l o s e d i n the b r a c k e t s o c c u r r i n g . Assume now t h a t the composite s i z e i s f i x e d a t m. Observe t h a t i n our new n o t a t i o n , P(m) i s e q u i v a l e n t t o P[MF] and P'(m) i s e q u i v a l e n t t o P [ F C ] . Thus, P'(m) - P(m) = P[FC] - P[MF] ( 6 . 7 ) Our g o a l i s t o e x p l a i n the o b s e r v e d i n c r e a s e i n t h i s d i s p r o p o r t i o n ' as the composite s i z e i n c r e a s e s . A p p l y i n g the Law of T o t a l P r o b a b i l i t y t o both events g i v e s the i d e n t i t i e s , P[FC] P[FC and MF] + P[FC and -MF] P[MF] P[MF and FC] + P[MF and -FC] ( 6 . 8 ) In t a k i n g the d i f f e r e n c e , P[FC and MF] i s e l i m i n a t e d y i e l d i n g the e x p r e s s i o n , 66 P[FC] - P[MF] = P[FC and -MF ] - P[MF and -FC] (.6.9) In o t h e r words, the d i f f e r e n c e P'(m)-P(m) i s a d i r e c t consequence of the r e l a t i v e o c c u r r e n c e of both t y p e s of e r r o r s . The p o s i t i v e d i f f e r e n c e observed i n d i c a t e s t h a t the f i r s t type of e r r o r , [FC and -"MF ], i s more p r e v a l e n t . T h i s i s m erely a more p r e c i s e statement of comments made p r e v i o u s l y . Now, however, we a r e i n a p o s i t i o n t o e x p l i c i t l y e x p l o r e t h i s r e l a t i o n s h i p much more c l o s e l y . E x p r e s s i o n (6.9) can be r e w r i t t e n u s i n g the d e f i n i t i o n of c o n d i t i o n a l p r o b a b i l i t y , P[FC]-P[MF] = PfFCl-MF] P[-MF] - P[MF|-FC] P[MF] (6.10) E s t i m a t e s of P[MF] have a l r e a d y been p r e s e n t e d i n Table 3. W i t h i n each s e t of composite s i z e s which c o r r e s p o n d t o the same number of i n i t i a l c o m p o s i t e s b e i n g formed, P[MF] d e c r e a s e s as the composite s i z e i n c r e a s e s . C o n s e q u e n t l y , P[-,MF] = 1-P[MF] i n c r e a s e s as the composite s i z e i n c r e a s e s . To un d e r s t a n d the impact of t h i s , suppose t h a t PfFCl^MF] = P[-FC|MF] = P" f o r a l l m. S u b s t i t u t i n g P" i n t o ( 6 . 1 0 ) , the e x p r e s s i o n reduces t o , 67 P[FC]-P[MF] = P" (1-2P[MF]) (6.11) I t i s apparent t h a t any d e c r e a s e i n P[MF] w i l l r e s u l t i n an i n c r e a s e i n the d i f f e r e n c e . Thus, under t h e s e c i r c u m s t a n c e s , an i n c r e a s e i n the composite s i z e , or e q u i v a l e n t l y , a d e c r e a s e i n the s i z e of the f i n a l c o m p o s i t e , w i l l produce an i n c r e a s e in. the d i f f e r e n c e d e f i n e d by (6.1.1). However, i t w i l l be argued t h a t b oth P[-MF] and PtFCl-MF] i n c r e a s e w i t h the c omposite s i z e and both P[MF] and P[-FC|MF] may d e c r e a s e as the composite s i z e i n c r e a s e s . T h i s w i l l , of c o u r s e , a c c e n t u a t e the change i n the d i f f e r e n c e g i v e n by (6.10) as the c omposite s i z e changes. The b e h a v i o r of P[FC|-MF] and P[-FC|MF] w i t h r e s p e c t t o composite s i z e depends on the i n t e r a c t i o n of two f a c t o r s , the v a r i a n c e and the number of i n i t i a l c o m p o s i t e s . The f i r s t f a c t o r t o be examined i s the v a r i a n c e , and the immediate q u e s t i o n i s how the v a r i a n c e of a composite measurement changes as the s i z e of the composite changes. The measurement from any composite i s merely the sample mean and, as such, the v a r i a n c e i s g i v e n by ( 2 . 9 ) . The e s t i m a t e of the v a r i a n c e depends o n l y on the sample s i z e , the t h e o r e t i c a l a u t o c o r r e l a t i o n f u n c t i o n , and the p o p u l a t i o n v a r i a n c e c2 . The sample s i z e i s merely the number of samples i n the c o m p o s i t e . The t h e o r e t i c a l a u t o c o r r e l a t i o n f u n c t i o n 68 can be e s t i m a t e d by t h e a v e r a g e o b s e r v e d a u t o c o r r e l a t i o n f u n c t i o n . F o r t h e p u r p o s e s h e r e , t h e p o p u l a t i o n v a r i a n c e , MF] and P[>FC|MF], two c a s e s must be c o n s i d e r e d . F o r t h e f o r t h c o m i n g a rguments t h e mean w i l l be assumed t o be a p p r o x i m a t e l y n o r m a l l y d i s t r i b u t e d . In a d d i t i o n , i t w i l l be assumed t h a t t h e maximum sample measurement w i l l be g r e a t e r t h a n t h e p o p u l a t i o n mean, ». S i m i l a r a r guments h o l d f o r t h e c a s e when t h e maximum sample measurement i s l e s s t h a n » . Be c a u s e of t h i s r e d u n d a n c y and t h e f a c t t h a t t h e p r o b a b i l i t y of t h i s l a t t e r s i t u a t i o n a r i s i n g i s s m a l l , i t was not i n c l u d e d . Suppose t h a t we have two c o m p o s i t e s , one of s i z e m and t h e o t h e r o f s i z e r0. The ex p e c t e d c o m p o s i t e measurement of C2 w i l l remain a t M . T h i s s i t u a t i o n i s d e p i c t e d i n F i g u r e 5. I f , however, C2 has a h i g h e r measurement, event FC o c c u r s . L e t the random v a r i a b l e M, be the measurement of C1. I f M,^, then an i n c r e a s e i n the v a r i a n c e of the composite measurement of C2 w i l l produce an i n c r e a s e i n the p r o b a b i l i t y t h a t C2 w i l l have a h i g h e r measurement, t h a t i s , an i n c r e a s e i n PfFCl-'MF]. To F i g u r e 5 Sample w i t h maximum measurement i n C1. u F i g u r e 6 Two n o r m a l d e n s i t i e s w i t h d i f f e r e n t v a r i a n c e s a n d M, [l M i 71 F i g u r e 7 Two normal d e n s i t i e s w i t h d i f f e r e n t v a r i a n c e s and M t<«. Mi LI u n d e r s t a n d why t h i s i s the case c o n s i d e r F i g u r e 6. Both diagrams show two normal d e n s i t i e s , w i t h the same mean but d i f f e r e n t v a r i a n c e s , superimposed. At p o i n t A the two d e n s i t y f u n c t i o n s a r e e q u a l . F i g u r e 6A shows a v a l u e f o r M, t h a t i s g r e a t e r than A. C l e a r l y the p r o b a b i l i t y of o b s e r v i n g a v a l u e l a r g e r than M, i s g r e a t e r f o r the d e n s i t y t h a t has the h i g h e r v a r i a n c e . The amount by which i t i s g r e a t e r i s e q u a l t o the a r e a of the shaded r e g i o n . In F i g u r e 6B a v a l u e of M, l e s s than A i s i n d i c a t e d . A l t h o u g h l e s s o b v i o u s , the p r o b a b i l i t y of o b s e r v i n g a v a l u e l a r g e r than Mt i s g r e a t e r f o r the d e n s i t y t h a t has the h i g h e r v a r i a n c e . A g a i n , the amount by which i t i s g r e a t e r i s e q u a l t o the a r e a of the shaded r e g i o n . T h i s r e s u l t can a l s o be s u b s t a n t i a t e d 72 m a t h e m a t i c a l l y . I t can e a s i l y be shown t h a t f o r any h>1, i f X and Y a r e n o r m a l l y d i s t r i b u t e d w i t h mean » and v a r i a n c e s t ]>P[X>t ] f o r a l l t>»i. In a s i m i l a r manner, i t can be demonstrated t h a t i f M,(. then an i n c r e a s e i n the v a r i a n c e of the composite measurement of C2 w i l l produce an i n c r e a s e i n P[FC|-MF]. However, E[M, ] = v + 6>i> which i m p l i e s t h a t P[M,|/]. T h e r e f o r e , i t i s more p r o b a b l e t h a t we are i n a s i t u a t i o n i n which an i n c r e a s e i n the v a r i a n c e of the f i n a l composite measurement w i l l produce an i n c r e a s e i n P t F C l ^ M F ] . I t w i l l be r e c a l l e d t h a t the v a r i a n c e of the f i n a l composite has been seen t o 73 F i g u r e 8 Sample w i t h maximum measurement i n C2. C l C2 Ll + b i n c r e a s e as the s i z e , r , d e c r e a s e s . In summary, as the s i z e of the f i n a l c o mposite d e c r e a s e s , t h e r e w i l l t e n d t o be an i n c r e a s e i n P[FC| - ,MF], Assume now t h a t we a r e i n t h e s i t u a t i o n i n which the sample w i t h the maximum measurement i s i n C2. That i s , the event MF has o c c u r r e d . The e x p e c t e d c o m p o s i t e measurement of C1 w i l l now be t> w h i l e the e x p e c t e d composite measurement of C2 w i l l be n+6. T h i s s i t u a t i o n i s d e p i c t e d i n F i g u r e 8. Event -«FC w i l l o c c u r when the compos i t e measurement of C2 i s l e s s t han M,. Observe i n F i g u r e 8, t h a t i f M,<»i+6 then an i n c r e a s e i n t he v a r i a n c e of t h e c o m p o s i t e measurement of C2 w i l l i n c r e a s e the p r o b a b i l i t y of •'FC o c c u r r i n g . That i s , an i n c r e a s e i n P[-'FC|MF], I f , however, M,>»«+6, then an i n c r e a s e 74 i n the v a r i a n c e a s s o c i a t e d w i t h the composite measurement of C2 w i l l produce a de c r e a s e i n Pf-^FCJMF]. But E[M,]=MMed(M! )=E[M, ] i f and o n l y i f , and we have our r e s u l t . T h i s s i t u a t i o n i s d e p i c t e d i n F i g u r e 9. The event FC w i l l now o c c u r i f the f i n a l composite measurement i s g r e a t e r than M*. F o l l o w i n g t h e same l i n e of argument used i n the f i r s t s i t u a t i o n , M*<^ i m p l i e s t h a t an i n c r e a s e i n the v a r i a n c e of the f i n a l composite measurement 0.5 < P[M,p then an i n c r e a s e i n t h i s v a r i a n c e r e s u l t s i n an i n c r e a s e i n P[FC|- ,MF]. But, we now have the r e l a t i o n s h i p , P[M*<„] < P[M,n] < P[M*>n] (6.15) Thus a d d i t i o n a l c o m p o s i t e ( s ) of s i z e m r e s u l t i n an i n c r e a s e i n the a l r e a d y f a v o u r a b l e p r o b a b i l i t y t h a t an i n c r e a s e i n the v a r i a n c e of the f i n a l c omposite w i l l produce an i n c r e a s e i n P[FC|-'MF]. C o n s i d e r now the e f f e c t of the a d d i t i o n a l c o m p o s i t e ( s ) on the second case which i s d e p i c t e d i n F i g u r e 10. For the event ~>FC t o o c c u r , the f i n a l c o m p o s i t e measurement w i l l have t o be l e s s than M*. Thus, Pf-'FClMF] w i l l i n c r e a s e as the v a r i a n c e of the f i n a l composite measurement i n c r e a s e s o n l y when M*n+6. Now, however, the f o l l o w i n g i n e q u a l i t i e s h o l d , P[M*M+6] > P[M,>n+6] (6.16) In o t h e r words, the a d d i t i o n a l c o m p o s i t e ( s ) has r e s u l t e d i n F i g u r e 9 Sample w i t h maximum measurement not i n f i n a l c omposit MRXIMUM ) J L + b FINAL / COMPOSITE / F i g u r e 10 Sample w i t h maximum measurement i n f i n a l c o m p o s i t e . MAXIMUM M L FINRL COMPOSITE Li + b 78 a r e d u c t i o n i n the p r o b a b i l i t y t h a t Pt-FClMF] w i l l i n c r e a s e as the v a r i a n c e of the f i n a l c o mposite i n c r e a s e s . E q u i v a l e n t l y , the a d d i t i o n a l c o m p o s i t e ( s ) has r e s u l t e d i n an i n c r e a s e i n the p r o b a b i l i t y t h a t P[-"FC|MF] w i l l d e c r e a s e as the f i n a l composite v a r i a n c e i n c r e a s e s . A l t h o u g h i t was known t h a t P[M1<»i+6] was g r e a t e r than P[M,>»/+6], the same cannot be s a i d f o r M*. I t i s q u i t e c o n c e i v a b l e , i n f a c t , t h a t the r e v e r s e i n e q u a l i t y may h o l d f o r M*. That i s , P[M*<»+6] may be l e s s than P[M*>n+6]. It t h i s were the c a s e , P[>FC|MF] would ten d t o decrease as the v a r i a n c e of the f i n a l c omposite i n c r e a s e d . L e t us s t o p f o r a moment and r e i t e r a t e . The impetus of the above d i s c u s s i o n was t o p r o v i d e a r a t i o n a l e f o r the b e h a v i o r e x h i b i t e d i n F i g u r e 4, T a b l e 3, and T a b l e 4. The f i r s t s t e p was the development of (6 . 1 0 ) . The components of t h i s e x p r e s s i o n were then examined s e p a r a t e l y . I t was then shown t h a t as the s i z e of the f i n a l composite d e c r e a s e d , P[-MF] i n c r e a s e d and hence P[MF ] = 1-P[-MF ] d e c r e a s e d . U s i n g the d e f i n i t i o n of the v a r i a n c e g i v e n by ( 2 . 9 ) , i t was then demonstrated t h a t a r e d u c t i o n i n the s i z e of the f i n a l c o m p o s i t e produced an i n c r e a s e i n the v a r i a n c e of the f i n a l c o mposite measurement. I t was then argued t h a t t h r o u g h the i n t e r a c t i o n of t h i s change i n v a r i a n c e ' w i t h the number of c o m p o s i t e s , PfFCl-MF] must i n c r e a s e f a s t e r than P[-"FC|MF] as the f i n a l composite s i z e s d e c r e a s e s . P u t t i n g a l l these r e s u l t s t o g e t h e r , i t i s e v i d e n t t h a t the d i f f e r e n c e d e f i n e d by (6.10) must i n c r e a s e as the f i n a l c omposite s i z e 79 d e c r e a s e s . That i s , the b e h a v i o r e x h i b i t e d i n F i g u r e 5, Table 3, and Table 4 i s c o n s i s t e n t w i t h the above t h e o r y . I t i s p o s s i b l e t o e s t i m a t e Pt-'FClMF] and P[FC| _ ,MF] from the d a t a by merely c o u n t i n g t h e i r o c c u r r e n c e s . The v a l u e s P[MF] a l r e a d y appear i n Table 3 under P(m). T h i s was done f o r composite s i z e s r e p r e s e n t i n g unbalanced c o m p o s i t i n g . The observed p r o p o r t i o n s appear i n T a b l e 6. As e x p e c t e d , w i t h i n each s e t of composites c o r r e s p o n d i n g t o the same number of i n i t i a l c o mposites b e i n g formed, P|>FC|MF] i n c r e a s e s as the s i z e of the f i n a l c omposite d e c r e a s e s . On the o t h e r hand, P[FC|- ,MF] d e c r e a s e s as the s i z e of the f i n a l composite i n c r e a s e s . T h i s , t o o , i s c o n s i s t e n t w i t h the above t h e o r y . The p r o p o r t i o n of t r i a l s on which the a c t u a l or g l o b a l maximum was found i s p l o t t e d a g a i n s t the composite s i z e i n F i g u r e 11. With r e s p e c t t o t h i s s t a t i s t i c , p r i m a r y f i r s t o r d e r c o m p o s i t i n g performs s u b s t a n t i a l l y b e t t e r than random sampl i n g f o r every composite s i z e . At the v e r y w o r s t , p r i m a r y f i r s t o r d e r c o m p o s i t i n g f i n d s the a c t u a l maximum i n 32% more t r i a l s than does random s a m p l i n g . With a composite s i z e of 6, the d i f f e r e n c e i s as h i g h as 45%. From a s t a t i s t i c a l s t a n d p o i n t , the d i f f e r e n c e s are s i g n i f i c a n t t o a c o n f i d e n c e l e v e l f a r e x c e e d i n g 0.9999. C o n s i d e r a g a i n the s e t s of composite s i z e s 12 through 14, 15 t h r o u g h 19, and 20 through 29. For b r e v i t y , l o c a t i n g the sample w i t h the a c t u a l or g l o b a l maximum w i l l be termed a s u c c e s s on t h a t t r i a l . The f i r s t p o i n t t o observe i s t h a t , w i t h i n each s e t , the p r o p o r t i o n of s u c c e s s e s f o r unbalanced 80 T a b l e 6 The p r o b a b i l i t y of not s e l e c t i n g the f i n a l composite g i v e n i t has the maximum sample ( P t - F C l M F ] ) and the p r o b a b i l i t y of s e l e c t i n g the f i n a l c omposite g i v e n i t does not have the maximum sample (P[FC|-MF]). Composite S i z e (60/m, P[-FC|MF] P[FC|-MF] (m) ( r ) 7 4 9 0.065 0.068 8 4 8 0.055 0.082 9 5 7 0.110 0.077 1 1 5 6 0.050 0. 1 02 1 3 8 . 5 0.113 0.103 1 4 4 5 0.038 0. 133 1 6 1 2 4 0. 1 72 0.116 1 7 9 4 0.121 0. 1 28 18 6 4 0.065 0. 1 57 19 3 4 0.016 0. 185 21 18 3 0. 1 68 0. 1 37 22 16 3 0. 1 56 0. 154 23 14 3 0. 138 0. 172 24 12 3 0.119 0. 170 25 1 0 3 0. 106 0. 186 26 8 3 0.081 0. 188 27 6 3 0.045 0.210 28 4 3 0.024 0.228 29 2 3 0.004 0.248 c o m p o s i t i n g i s c o n s i s t e n t l y l e s s than the p r o p o r t i o n of s u c c e s s e s f o r the composite s i z e which r e p r e s e n t s b a l a n c e d c o m p o s i t i n g . In o t h e r words, unbalanced c o m p o s i t i n g does not pe r f o r m as w e l l w i t h r e s p e c t t o t h i s s t a t i s t i c . The o t h e r i n t e r e s t i n g f e a t u r e t o observe i s t h a t the d e c l i n e i n performance i s v i r t u a l l y monotonic. That i s , as the d i f f e r e n c e between the composite s i z e , m, and the s i z e of the f i n a l composite i n c r e a s e s , the p r o p o r t i o n of s u c c e s s e s 81 F i g u r e 11 P r o p o r t i o n of t r i a l s t h e a c t u a l maximum found v e r s u s c o m p o s i t e s i z e . (2009 t r i a l s ) PR1MRRY FIRST ORDER COMPOSITING 95; CONFIDENCE INTERVAL .1 o o C3 d e c r e a s e s . I n l i g h t o f t h e p r e v i o u s a n a l y s i s o f F i g u r e 4, t h i s r e s u l t i s n o t u n e x p e c t e d . T a b l e 3 i l l u s t r a t e d t h a t t h e p r o p o r t i o n o f t r i a l s on w h i c h t h e f i n a l c o m p o s i t e was c h o s e n when i t d i d n o t c o n t a i n t h e maximum s a m p l e i n c r e a s e s a s t h e s i z e o f t h e f i n a l c o m p o s i t e d e c r e a s e d . The c o n s e q u e n c e o f t h i s i s , o f c o u r s e , a r e d u c t i o n i n t h e p r o p o r t i o n o f s u c c e s s e s , a n d w o u l d c o n t r i b u t e t o a m o n o t o n i c d e c r e a s e i n t h e p r o p o r t i o n o f s u c c e s s e s . T h e s e r e s u l t s s u g g e s t t h a t b a l a n c e d c o m p o s i t i n g s h o u l d a l w a y s be p r e f e r r e d t o u n b a l a n c e d c o m p o s i t i n g , a t l e a s t w i t h r e s p e c t t o t h i s s t a t i s t i c . F o c u s w i l l now be c o n c e n t r a t e d on o n l y t h e c o m p o s i t e s i z e s t h a t r e p r e s e n t b a l a n c e d c o m p o s i t i n g . The p r o p o r t i o n o f s u c c e s s e s a s s o c i a t e d w i t h 82 T a b l e 7 P r o p o r t i o n o f t r i a l s on w h i c h t h e a c t u a l maximum was f o u n d f o r b a l a n c e d c o m p o s i t i n g . (2009 t r i a l s ) Number o f C o m p o s i t e P r o p o r t i o n T e s t s S i z e 32 2 0.83 30 0.82 23 3 0.75 20 0.77 19 4 0.75 15 0.73 17 5 0.72 12 0.72 1 6 6 0.71 10 0.71 t h e s e c o m p o s i t e s i z e s a p p e a r s t o be a f u n c t i o n of t h e number of t e s t s p e r f o r m e d . The g r e a t e r t h e number o f samples t e s t e d , t h e l a r g e r t h e p r o p o r t i o n o f t r i a l s on w h i c h t h e sample w i t h t h e maximum measurement i s f o u n d . T a b l e 3 r e v e a l s t h a t t h e r e a r e a l w a y s two d i s t i n c t b a l a n c e d c o m p o s i t e s i z e s t h a t r e s u l t i n t h e same number of t e s t s b e i n g p e r f o r m e d . F o r example, a c o m p o s i t e s i z e of 2 and a c o m p o s i t e s i z e of 30 b o t h w i l l r e s u l t i n a t o t a l of 32 t e s t s b e i n g p e r f o r m e d p e r t r i a l . S i m i l a r l y , t h e p a i r s 3 and 20, 4 and 15, 5 and 12, and 6 and 10 c o r r e s p o n d t o 23, 19, 17, and 16 t e s t s p e r t r i a l r e s p e c t i v e l y . The p r o p o r t i o n o f s u c c e s s e s w i t h i n e a c h p a i r a r e h i g h l i g h t e d i n T a b l e 7. The w i t h i n - p a i r p r o p o r t i o n s a r e v e r y s i m i l a r w i t h none of t h e d i f f e r e n c e s b e i n g s i g n i f i c a n t a t an o l e v e l of 0.05. T h i s s u g g e s t s t h a t t h e p r o p o r t i o n of s u c c e s s e s may be a f u n c t i o n s o l e l y of t h e 83 number of t e s t s performed. The mean a b s o l u t e range e r r o r (MARE) i s p l o t t e d a g a i n s t the composite s i z e i n F i g u r e 12. Many of the p r o p e r t i e s j u s t d e s c r i b e d f o r the p r o p o r t i o n of s u c c e s s e s a l s o a p p l y t o the MARE. P r i m a r y f i r s t o r d e r c o m p o s i t i n g performs s u b s t a n t i a l l y b e t t e r than random s a m p l i n g f o r every composite s i z e . At w o r s t , a composite s i z e of 30, one would have t o a c c e p t t h a t the mean v a l u e of the MARE f o r p r i m a r y f i r s t o r d e r c o m p o s i t i n g was l a r g e r than the mean v a l u e f o r random s a m p l i n g . a t an a l e v e l of 0.01. For a l l o t h e r composite s i z e s , the s i g n i f i c a n c e p r o b a b i l i t y would be l e s s than 0.0001. Random sampling a l s o e x h i b i t s a somewhat l a r g e r v a r i a n c e . The e x a c t numeric v a l u e s can be found i n Appendix A, Table A.1. These r e s u l t s i n d i c a t e t h a t even when p r i m a r y f i r s t o r d e r c o m p o s i t i n g does not f i n d the maximum v a l u e , i t does seem t o f i n d a ' l a r g e ' v a l u e . The r e l a t i o n s h i p between b a l a n c e d and unbalanced c o m p o s i t i n g , e v i d e n t i n the p r o p o r t i o n of s u c c e s s e s , i s a l s o apparent i n F i g u r e 12. B a l a n c e d c o m p o s i t i n g performs c o n s i s t e n t l y b e t t e r than unbalanced c o m p o s i t i n g . W i t h i n each se t of composite s i z e s which form the same number of i n i t i a l c o m p o s i t e s , performance d e c r e a s e s as the s i z e of the f i n a l c o mposite d e c r e a s e s . Thus, t h i s s t a t i s t i c a l s o i n d i c a t e s t h a t b a l a n c e d c o m p o s i t i n g s h o u l d be p r e f e r r e d t o unbalanced composit i n g . The performance of the random sampling method t r a c k s the r e l a t i o n s h i p between b a l a n c e d and unbalanced composite 84 F i g u r e 12 Mean a b s o l u t e range e r r o r v e r s u s c o m p o s i t e s i z e . (2009 t r i a l s ) CD CM-CO Q B ) " UJo> QL PRJMRRY FIRST ORDER COMPOSITING 95J CONFIDENCE: INTERVAL cr I I I I I I I I I I I I I I I I I I I I I I I I l 0 2 A 6 6 10 12 14 16 18 20 22 24 26 28 30 C O M P O S I T E S I Z E s i z e s j u s t d e s c r i b e d f o r p r i m a r y f i r s t o r d e r c o m p o s i t i n g . That i s , the performance f o r unbalanced c o m p o s i t i n g d e c r e a s e s as the s i z e of t h e f i n a l c o m p o s i t e , r , d e c r e a s e s r e l a t i v e t o m. I t w i l l be r e c a l l e d t h a t i f the f i n a l c o m p o s i t e was chosen d u r i n g a p p l i c a t i o n of p r i m a r y f i r s t o r d e r c o m p o s i t i n g , t h e n (60/m.)+r t e s t s would have been made. The random sample method would then be a p p l i e d based on (60/m>+r samples. F o r example, c o n s i d e r the co m p o s i t e s i z e s of 28 and 29. For a co m p o s i t e s i z e of 28, the random sample method would be a p p l i e d based on 3+28=31 or 3+4=7 samples. A co m p o s i t e s i z e of 29 would r e s u l t i n the a p p l i c a t i o n of random s a m p l i n g w i t h 3+29=32 or 3+2=5 samples. The d i f f e r e n c e i n the e x p e c t e d v a l u e of the maximum sampled 85 measurement between 32 and 31 samples would be much s m a l l e r than the d i f f e r e n c e i n t h e e x p e c t e d v a l u e of the maximum sampled measurement between 7 and 5 samples. S i n c e the f i n a l c o mposite i s chosen on a s i m i l a r number of t r i a l s f o r both a composite s i z e of 28 and a composite s i z e of 2 9 1 2 , one would expect m=29 t o p e r f o r m more p o o r l y than m=28 w i t h r e s p e c t t o the MARE. A t t e n t i o n w i l l now be g i v e n o n l y t o the composite s i z e s t h a t r e p r e s e n t b a l a n c e d c o m p o s i t i n g . R e s u l t s f o r the p r o p o r t i o n of s u c c e s s e s i n d i c a t e d t h a t performance might be r e l a t e d s o l e l y t o the number of t e s t s and, t h e r e f o r e , not the i n i t i a l c o mposite s i z e . T h i s i s not t r u e f o r the MARE. The numeric v a l u e s a r e h i g h l i g h t e d i n Table 8. The v a l u e of the MARE f o r the s m a l l e r composite s i z e w i t h i n each p a i r i s s t a t i s t i c a l l y l a r g e r w i t h a s i g n i f i c a n c e p r o b a b i l i t y e x c e e d i n g 0.01. Thus, s m a l l e r composite s i z e s p e r f o r m b e t t e r w i t h r e s p e c t t o t h i s s t a t i s t i c . However, i t s h o u l d be noted t h a t no l e s s than about 95% of the o b s e r v a b l e e r r o r i s a c c o u n t e d f o r u s i n g any of the b a l a n c e d composite s i z e s . In p a r t i c u l a r , by p e r f o r m i n g as few as 16 t e s t s , 96% of the o b s e r v a b l e e r r o r , on the average, can be accounted f o r . I t i s a l s o e v i d e n t i n T a b l e 8 t h a t the v a r i a b i l i t y of the MARE i s l e s s f o r the s m a l l e r composite s i z e s . T h i s , t o o , i s 1 2 The p r o p o r t i o n of t r i a l s on which the f i n a l composite was a c t u a l l y chosen can be d e t e r m i n e d by adding the column under P(m), T a b l e 3, t o the column P'(m)-P(m), T a b l e 4. T h i s w i l l show t h a t the p r o p o r t i o n of t r i a l s on which the f i n a l c o mposite was s e l e c t e d i s almost c o n s t a n t w i t h i n each s e t of unbalanced composite s i z e s . T a b l e 8 The mean a b s o l u t e range e r r o r (MARE) f o r b a l a n c e d c o m p o s i t i n g . (2009 t r i a l s ) Number of T e s t s Composite S i z e MARE St a n d a r d E r r o r ( X 1 0 - 2 ) 32 2 0.990 0.097 30 0.961 0.276 23 3 0.978 0.168 20 0.953 0.285 19 4 0.974 0.191 15 0.948 0.299 17 5 0.967 0.220 12 0.950 0.285 1 6 6 0.961 0.242 -10 0.952 0.278 F i g u r e 13 Maximum a b s o l u t e d e v i a t i o n v e r s u s composite s i z e . (2009 t r i a l s ) CM o' a Ccco-8 H RRMDOM SAMPLE PRJ.HRRY F J R S T ORDER C O M P O S I T I N G I I "I I I I I I I I I I I I I I I I I I I I I I I I I I I | 2 4 6 8 10 12 14 16 IB 20 22 24 26 28 30 C O M P O S I T E S J Z E 87 d e s i r a b l e as the p r o b a b i l i t y of o b s e r v i n g l a r g e e r r o r s w i l l be l e s s . The t h i r d measure of e r r o r t o be examined i s the maximum a b s o l u t e d e v i a t i o n (MAD). The v a l u e of the MAD f o r each composite s i z e i s p l o t t e d i n F i g u r e 13 f o r both p r i m a r y f i r s t o r d e r c o m p o s i t i n g and random s a m p l i n g . C l e a r l y , the MAD f o r p r i m a r y f i r s t o r d e r c o m p o s i t i n g i s much l e s s than the MAD f o r random s a m p l i n g . T h i s i m p l i e s t h a t p r i m a r y f i r s t o r d e r c o m p o s i t i n g w i l l always f i n d a ' l a r g e ' v a l u e and thus p r o v i d e a more r e l i a b l e e s t i m a t e . A more i n t e r e s t i n g p i c t u r e emerges by l o o k i n g a t the r e c o r d number or hour i n which the MAD o c c u r r e d . These a r e summarized i n T a b l e 9. Two r e c o r d s , 418 and 1579, account f o r a l l of the l a r g e a b s o l u t e d e v i a t i o n s i n c u r r e d by the random sample method. These r e c o r d s do n o t , however, cause the same d i f f i c u l t i e s f o r p r i m a r y f i r s t o r d e r c o m p o s i t i n g . To u n d e r s t a n d why i t i s t h a t p r i m a r y f i r s t o r d e r c o m p o s i t i n g i s much s u p e r i o r i n these c a s e s , f i r s t c o n s i d e r the graphs of the d a t a from t h e s e hours t h a t appear i n F i g u r e 14 and F i g u r e 15. Both d a t a s e r i e s are c h a r a c t e r i z e d by v e r y low i n i t i a l l e v e l s f o l l o w e d by a s i n g l e extreme v a l u e which i s f o l l o w e d by m o d e r a t e l y low v a l u e s t h a t s t e a d i l y d e c r e a s e . By s a m p l i n g randomly, t h e r e i s a l a r g e p r o b a b i l i t y t h a t the sample w i t h the extreme v a l u e w i l l not be s e l e c t e d . For i n s t a n c e , i f 20 samples a r e s e l e c t e d from each hour, the p r o b a b i l i t y of not f i n d i n g both of the extremes and, hence, i n c u r r i n g a l a r g e e r r o r , i s 1 - ( 1 / 3 ) 2 = 8/9. T h i s p r o b a b i l i t y w i l l i n c r e a s e as F i g u r e 14 Record 418: S e p t . 18, Hour 10 CNJ o i — i • C J ° J U J i I I M I I I I M I H I I I I I I I I I I I I I I I H I I II I I I I I I I I I I I i 4 0 4 4 4 8 52 56 60 I I I I I I I I I I I I I I I I ' ' F i g u r e 15 Record 1579: Nov. 10, Hour 19 i 'I'i'i'i'i1 • • H + H - H - H - H 11 11 11 11 11 11 11 11 H I 1 I I I I I I I I I I I I I 1 1 1 1 1 1 I I I" ! 12 16 20 24 28 32 36 10 44 48 52 56 60 M I N U T E S 89 T a b l e 9 The maximum a b s o l u t e d e v i a t i o n (MAD) and the r e c o r d on which i t was i n c u r r e d . (2009 t r i a l s ) Composite ze (m) MAD Record Day Hour MAD Record Day Hour 2 0. 036 449 Sept. 19 1 7 0. 236 418 Sept. 18 10 3 0. 037 1018 Oct. 15 10 0. 237 418 Sept. 18 10 4 0. 039 1018 Oct. 1 5 10 0. 237 418 Sept. 18 1 0 5 0. 055 446 Sept. 19 1 4 0. 237 418 Sept. 18 10 6 0. 058 1906 Nov. 24 10 0. 240 418 Sept. 18 10 7 0. 059 1 906 Nov. 24 1 0 0. 1 98 1 579 Nov. 1 0 19 8 0. 097 1 1 43 Oct. 20 15 0. 195 1 579 Nov. 10 19 9 0. 081 404 Sept. 17 20 0. 205 1 579 Nov. 10 19 10 0. 085 1 1 42 Oct. 20 1 4 0. 195 1579 Nov. 10 19 1 1 0. 058 1906 Nov. 24 10 0. 237 418 Sept. 18 10 1 2 0. 085 1 1 42 Oct. 20 1 4 0. 109 1 331 Oct. 28 1 1 1 3 0. 195 1579 Nov. 10 1 9 0. 1 98 1 579 Nov. 10 19 1 4 0. 1 95 1579 Nov. 10 19 0. 240 418 Sept. 18 10 1 5 0. 1 10 1 331 Oct. 28 1 1 0. 238 418 Sept. 18 10 1 6 0. 097 1 1 43 Oct. 20 1 5 0. 236 418 Sept. 18 1 0 1 7 0. 097 1 1 43 Oct. 20 1 5 0. 241 418 Sept. 18 10 18 0. 097 1 1 43 Oct. 20 1 5 0. 236 418 Sept. 18 10 1 9 0. 1 09 1 331 Oct. 28 1 1 0. 236 418 Sept. 18 10 20 0. 1 1 0 1 331 Oct. 28 1 1 0. 1 95 1 579 Nov. 10 19 21 0. 1 1 0 1 331 Oct. 28 1 1 0. 195 1 579 Nov. 10 19 22 0. 1 10 1 331 Oct. 28 1 1 0. 237 418 Sept. 18 1 0 23 0. 097 1 1 43 Oct. 20 1 5 0. 237 418 Sept. 18 10 24 0. 097 1 1 43 Oct. 20 1 5 0. 195 1 579 Nov. 1 0 19 25 0. 097 1 1 43 Oct. 20 1 5 0. 237 418 Sept. 18 1 0 26 0. 097 1 1 43 Oct. 20 1 5 0. 1 95 1 579 Nov. 10 19 27 0. 097 1 1 43 Oct. 20 1 5 0. 238 418 Sept. 18 10 28 0. 109 1 331 Oct. 28 1 1 0. 1 97 1 579 Nov. 10 19 29 0. 1 09 1 331 Oct. 28 1 1 0. 1 09 1 331 Oct. 28 1 1 30 0. 1 1 0 1 331 Oct. 28 1 1 0. 195 1 579 Nov. 10 1 9 the number of t r i a l s w i t h an extreme v a l u e i n c r e a s e s . For p r i m a r y f i r s t o r d e r c o m p o s i t i n g , the consequence of extreme v a l u e s i s e n t i r e l y o p p o s i t e . The s i n g l e extreme v a l u e w i l l t end t o dominate the measurement of any composite t h a t i n c l u d e s i t , r e s u l t i n g i n the s e l e c t i o n of the composite and the subsequent l o c a t i o n of the sample w i t h the extreme measurement. In f a c t , the l a r g e r the extreme v a l u e , the more l i k e l y i t i s t o be d e t e c t e d . T h i s i s a v e r y 90 d e s i r a b l e p r o p e r t y and r e p r e s e n t s a s i g n i f i c a n t advantage over random s a m p l i n g . In F i g u r e 13 i t can be seen t h a t the s m a l l e r composite s i z e s have the lowest MAD. T h i s i s d i r e c t l y r e l a t e d t o the d e t e c t i o n of extreme v a l u e s . The s m a l l e r the composite s i z e , the more overwhelming the extreme v a l u e w i l l be on the composite measurement. Thus, the s m a l l e r composite s i z e s w i l l be more l i k e l y t o d e t e c t even m o d e r a t e l y h i g h s p i k e s . The f i n a l measure of e r r o r t o examine i s a q u a d r a t i c measure, the mean squared e r r o r (MSE). The average l e v e l s and 95% c o n f i d e n c e i n t e r v a l s f o r p r i m a r y f i r s t o r d e r c o m p o s i t i n g and random samp l i n g a r e p r e s e n t e d i n F i g u r e 16. For every composite l e v e l , the MSE f o r p r i m a r y f i r s t o r d e r c o m p o s i t i n g i s l e s s than the MSE f o r random s a m p l i n g . T h i s measure a l s o e x h i b i t s a l a r g e degree of v a r i a b i l i t y f o r random s a m p l i n g . S i n c e the MSE i s s e n s i t i v e t o l a r g e v a l u e s , these r e s u l t s would i n d i c a t e t h a t random sampling produces f a r more ' l a r g e ' e r r o r s . However, the p a t t e r n of the f l u c t u a t i o n s f o r both random samp l i n g and p r i m a r y f i r s t o r d e r c o m p o s i t i n g bear a s t r o n g s i m i l a r i t y t o the b e h a v i o r of t h e i r r e s p e c t i v e maximum a b s o l u t e d e v i a t i o n s . T h i s would suggest the the MSE i s b e i n g overwhelmed by these extreme v a l u e s . To a s s e s s the impact, a l l the r e c o r d s on which a MAD was i n c u r r e d (Table 9) were e l i m i n a t e d and the MSE was r e c a l c u l a t e d . These r e s u l t s appear i n F i g u r e 17. The former c o n c l u s i o n s s t i l l a p p l y . The MSE f o r p r i m a r y f i r s t o r d e r c o m p o s i t i n g i s l e s s than the MSE f o r random sampling a t CM —I CM 00 —I i o<=> X o — o i RANDOM SAMPLE 95/ CONFIDENCE INTERVAL PRIMARY FIRST ORDER COMPOSITING 95/ CONFIDENCE INTERVAL f t A * \ i I I I i i i I I I I I i I I I 1 i i I I I l l l 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 C O M P O S I T E S I Z E V O Figure 16: Mean squared error versus composite size.(2009 t r i a l s ) -RRNDOM SAMPLE 95/ CONFIDENCE INTERVAL PRIMARY FIRST ORDER COMPOSITING CONFIDENCE INTERVAL F i g u r e 17: Mean s q u a r e d e r r o r v e r s u s c o m p o s i t e s i z e . (1999 t r i a l s ) vo to 93 eve r y composite s i z e and random samp l i n g s t i l l e x h i b i t s much l a r g e r v a r i a b i l i t y . T h e r e f o r e , i t can be c o n c l u d e d t h a t random samp l i n g w i l l r e s u l t i n more ' l a r g e ' e r r o r s . F o c u s i n g a t t e n t i o n i n F i g u r e 17 s o l e l y on the MSE f o r p r i m a r y f i r s t o r d e r c o m p o s i t i n g , b a l a n c e d c o m p o s i t i n g a g a i n appears s u p e r i o r t o unbalanced c o m p o s i t i n g . With the e x c e p t i o n of a composite s i z e of 21 which has a MSE s l i g h t l y b e t t e r than t h a t f o r a composite s i z e of 20, b a l a n c e d c o m p o s i t i n g performs b e t t e r then unbalanced c o m p o s i t i n g . However, a s t r i c t monotonic i n c r e a s e i n the MSE i s not e v i d e n t . Among the composite s i z e s r e p r e s e n t i n g b a l a n c e d c o m p o s i t i n g , the s m a l l e r composite s i z e s have lower mean squared e r r o r s . The numeric v a l u e s a re p r e s e n t e d i n Ta b l e 10. For the p a i r s 2 and 30, 3 and 20, and 4 and 15, the MSE f o r the s m a l l e r composite s i z e i s s t a t i s t i c a l l y l e s s than the MSE f o r the l a r g e r composite s i z e a t an a l e v e l of 0.05. The same can not be s a i d f o r the p a i r s 5 and 12, and 6 and 10. The s t a n d a r d e r r o r s of the mean squared e r r o r a l s o appear i n Ta b l e 10. In a d d i t i o n t o the s m a l l e r composite s i z e s h a v i n g lower average l e v e l s of MSE, the v a r i a b i l i t y of the e r r o r i s a l s o l e s s . T h i s c o m b i n a t i o n of lower mean l e v e l s of the MSE c o u p l e d w i t h lower v a r i a b l i l i t y i n d i c a t e t h a t , a t l e a s t w i t h r e s p e c t t o t h i s s t a t i s t i c , s m a l l e r b a l a n c e d composite s i z e s s h o u l d be p r e f e r r e d . In summary, p r i m a r y f i r s t o r d e r c o m p o s i t i n g was s u p e r i o r t o random samp l i n g on a l l c o u n t s . For eve r y 94 Tab l e 10 The mean squared e r r o r (MSE) f o r b a l a n c e d c o m p o s i t i n g . (1999 t r i a l s ) Number of T e s t s Composite S i z e MSE ( X 1 0 - 5 ) S t a n d a r d E r r o r ( X 1 0 - 5 ) 32 2 0.039 0.021 30 0.838 0.268 23 3 0. 187 0.082 20 0.819 0.247 19 4 0.258 0.088 1 5 0.756 0.222 1 7 5 0.389 0. 1 42 12 0.598 0.157 1 6 6 0.474 0. 1 46 10 0.901 0.268 composite s i z e , the MSE and the MAD f o r p r i m a r y f i r s t o r d e r c o m p o s i t i n g were l e s s and the p r o p o r t i o n of s u c c e s s e s and the MARE were g r e a t e r than the c o r r e s p o n d i n g v a l u e s f o r random s a m p l i n g . P r i m a r y f i r s t o r d e r c o m p o s i t i n g a l s o r e s u l t e d i n lower v a r i a n c e s f o r the MSE and MARE. In a d d i t i o n , a more e f f i c i e n t e s t i m a t e of the p o p u l a t i o n mean was o b t a i n e d . Another s i g n i f i c a n t advantage of p r i m a r y f i r s t o r d e r c o m p o s i t i n g was the a b i l i t y t o d e t e c t extreme v a l u e s . In f a c t , the l a r g e r the extreme, the more l i k e l y i t was t o be d e t e c t e d . When a p p l y i n g p r i m a r y f i r s t o r d e r c o m p o s i t i n g , unbalanced c o m p o s i t i n g s h o u l d be a v o i d e d . Unbalanced c o m p o s i t i n g e x h i b i t e d p o o r e r performance on a l l measures of e r r o r except the maximum a b s o l u t e d e v i a t i o n . I t was a l s o 9 5 shown t h a t t h e p r o b a b i l i t y of s e l e c t i n g t h e f i n a l c o m p o s i t e when, i n f a c t , i t d i d n o t c o n t a i n t h e maximum sample i n c r e a s e d a s t h e d e g r e e of i m b a l a n c e i n c r e a s e d . Among c o m p o s i t e s i z e s r e p r e s e n t i n g b a l a n c e d c o m p o s i t i n g , t h e r e was an i n d i c a t i o n t h a t s m a l l e r c o m p o s i t e s i z e s s h o u l d be p r e f e r r e d . The s m a l l e r c o m p o s i t e s i z e s r e c o r d e d s u p e r i o r v a l u e s f o r t h e maximum a b s o l u t e d e v i a t i o n , t h e mean a b s o l u t e r a n g e e r r o r , and t h e mean s q u a r e d e r r o r . The l a t t e r two s t a t i s t i c s a l s o e x h i b i t e d l o w e r v a r i a b i l i t y . Among t h e s e s m a l l e r c o m p o s i t e s i z e s , t h e g r e a t e r t h e number of t e s t s , t h e b e t t e r t h e p e r f o r m a n c e . B. The E f f e c t of t h e A u t o c o r r e l a t i o n F u n c t i o n The p r e s e n c e o f ' h i g h ' p o s i t i v e a u t o c o r r e l a t i o n i s t h e o n l y a s s u m p t i o n a f f e c t i n g t h e p e r f o r m a n c e o f c o m p o s i t e t e c h n i q u e s . No a t t e m p t has been made t o q u a n t i f y t h i s s t a t e m e n t . T h i s l e a d s t o many v a l i d q u e s t i o n s . How w i l l d i f f e r e n t a u t o c o r r e l a t i o n f u n c t i o n s a f f e c t p e r f o r m a n c e ? I s i t p o s s i b l e t o c h a r a c t e r i z e o r c l a s s i f y them? What e f f e c t does t h e d u r a t i o n of t h e ' h i g h ' p o s i t i v e a u t o c o r r e l a t i o n have on p e r f o r m a n c e ? The answers t o t h e s e q u e s t i o n s a r e beyond t h e scope of t h i s t h e s i s . They would be b e s t a p p r o a c h e d t h r o u g h a M o n t e - C a r l o s i m u l a t i o n s t u d y i n w h i c h d a t a would be g e n e r a t e d a c c o r d i n g t o a b r o a d range of a u t o c o r r e l a t i o n f u n c t i o n s . However, i t i s p o s s i b l e t o g a i n some i n s i g h t s from t h e d a t a a t hand. 96 I d e a l l y , d a t a would be g e n e r a t e d a c c o r d i n g t o a known f i x e d a u t o c o r r e l a t i o n f u n c t i o n . However, a crude a p p r o x i m a t i o n i s a v a i l a b l e . I t w i l l be n e c e s s a r y t o somehow group the d a t a a c c o r d i n g t o a u t o c o r r e l a t i o n f u n c t i o n s ; a l l r e c o r d s p o s s e s s i n g s i m i l a r a u t o c o r r e l a t i o n f u n c t i o n s b e i n g grouped t o g e t h e r . Comparisons c o u l d then be made between t h e s e groups. The average a u t o c o r r e l a t i o n f u n c t i o n has a l r e a d y been p l o t t e d i n F i g u r e 3. The f u n c t i o n shows a c o n s i s t e n t v a r i a n c e through the t e n l a g s p r e s e n t e d . T h i s would seem t o i n d i c a t e t h a t most a u t o c o r r e l a t i o n f u n c t i o n s e x h i b i t a s i m i l a r r a t e of d e c r e a s e . T h e r e f o r e , i t was d e c i d e d t h a t the d a t a c o u l d be s t r a t i f i e d a c c o r d i n g t o the f i r s t o r d e r a u t o c o r r e l a t i o n s , i . e . , />(1). Three groups were formed; a l l hours w i t h 0 . 9 (1 ) < 1 . 0 formed the f i r s t group w h i l e those w i t h 0 . 8 (1 ) <0 . 9 and those w i t h 0 . 2

(1 ) >0.9 i s above the o t h e r average a u t o c o r r e l a t i o n f u n c t i o n s a t every l a g . L i k e w i s e , The average a u t o c o r r e l a t i o n f u n c t i o n f o r the group w i t h 97 F i g u r e 18 A v e r a g e a u t o c o r r e l a t i o n f u n c t i o n s . ( S t r a t i f i e d s a m p l e ) 0.8cncncocr>cncncocncncncncncncn(£I 88888888888888888888888888888 i i i i i i i i i i i i i i i i • i i i i i i i i i i i 88888888888888888888888888888 u u u u u u u u u i o u a g i i o a i i s m n ^ i i i n u i a i u i m - J i f i u i o O O t o i o i o o D O B c o c o i o O A O i - a o B O A c n o o i b w - ' ^ O O O O i O u N < i u i n t i O ~ i u o u t i u t o < i i o o a o o i < i u o o o O i O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 0^u i-^cjcncjoooMOcn^co^cn-*>aDON3uicn(o&ui(o O - * A ( D A U I I 0 - ' - J D 1 I I I I 0 9 1 O 0 D U O U ( 0 ( 0 l 0 J k U I ) g i ) l f t a D M O O O O O O O O O O O O O O O O O O O O O O O O O O O O O o o o g oooo O O O O 88 88 o o o o 5 5 5 5 o o 5 < 5 o o 5 o g — « - - » * o o o o o o o o o o o o o o o o o o o S o o o o , . . U U U M U I O ' U U M M U U U M U ' > U - > 0 - ' " ' ' 0 0 0 ^ j ( O O c j ^ ^ c o O c J i u ^ c n o ^ i o c n o ^ - k O o u i K ) u i - . t o c n c j u i c j c n ^ j f o O u ^ ^ i - ^ ^ ^ Q ' c n o o ^ - ^ A C D M c n t o Q o - . c j i c o O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 88888888888888888888888888888 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O & ^ t U U U M U t U t ^ y U U « l 0 C D . - > M ' U ' - " O O O U O ' 0 ' - ' I S ( I I O < 0 O 0 0 ^ U U ' * , > D O M I 0 k & M O ( X l l l M ~ j r o ^ < D ^ t o . u . « k - u a D - » - u . t k c j o o < o - * ^ ^ C D < j > ^ < » ) c f t G d < o c o - * i - j Cf t(DM ( 0 - tCOK)CDCJ-»Ul ( 0 - » r O C 0 c n O G O C J (O - » C 0 ( £ ICDC J^l~JCd ( J 1 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O °,88888?QQ' 88888: '98?' &-ucftco^co<7)A(0cnuiaDcnco(0(ji(ocr>&O'Ji~JaD(jicj-.^icjio^Jcocdui-*a3(jcacj(0(o O O O O O O O O O O O O O O O O O O O O O O O O O O O O O §§§i§§§§§i l §"i §8888§8§88888888 u)uiouiOcnOMcno(Ouicjcncnaicnoxkcn(o->oocnaicnuicn O - » < J i - u m O N ) ' * i - » < T > ~ « i ( 0 c j O - * - * c o r o < 0 m c n * j O 4 c f > - » ( 0 c j ' > j U ) &a)&uiO(oiMOO-4(0 o o u i > i ( a u i f t ' 9 i u i O f t i o& ' b O ( O u i ) ) u u i o o u ( ) i ^ o i f i O O O O O O O O O O O O O O O O O O O O O O O O O O O O O — O O O O O — - » - * - * 0 0 0 — -• — O O O O O O O O O O O '00(aifiio(aiD^->-'Oia(Oio->4 -40oou>- - i^j>ioc i i&&->j^^cnu>-4»co^im u O O > < o o o o t u M O O O O M O i o « i i i i « « < i ' i i g i n « g i M - J ^ 0 J U l U l U I U t U M » O - - I U l U 1 6 i N ) M a j t 0 C J ( 0 U l C D U l ( J l - » t J ) O 0 J O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O V O U U I O U I O U U I D I D U I J k U U U I O O U I D O V I O U U I i l U U &(Ocnaaui^i4^^i-j&ivcncnocnaB(Da>(o~4«.«&cB (o~i~)~jcn i f i - > a i & o u i l i u u i a i o-'-*>4 ' j k i i i M O u i o n i o u i ) i u u u ^ O O O O O O O O O O O O O O O O O O O O O O O O O O O O O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o C 0 < O < 0 < 0 < 0 < 0 ( O < O < D < O < D < 0 { 0 < 0 < D C D < 0 < O ( O < 0 < 0 < O < O ( 0 < 0 ( O < 0 ( 0 ( 0 • ' U O l ' U ' 4 l > O l i l U 0 I U < l « 0 I U I > I O O U » N > l - 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