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Measuring sample maximums : an application to water quality monitoring Casey, Donald B. 1982

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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,<M 71 8. Sample w i t h maximum measurement i n C2 73 v i i i 9. Sample w i t h maximum measurement not i n f i n a l composite " 77 10. Sample w i t h maximum measurement i n f i n a l composite 77 11. P r o p o r t i o n of t r i a l s the a c t u a l maximum found v e r s u s composite s i z e . (2009 t r i a l s ) 81 12. Mean a b s o l u t e range e r r o r v e r s u s composite s i z e . (2009 t r i a l s ) 84 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 ) 86 14. Record 418: Sept. 18, Hour 10... 88 15. Record 1579: Nov. 10, Hour 19 88 16. Mean squared e r r o r v e r s u s composite s i z e . (2009 t r i a l s ) 91 17. Mean squared e r r o r v e r s u s composite s i z e . (1999 t r i a l s ) 92 18. 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 . ( S t r a t i f i e d sample) 97 i x 1 9 . P r o p o r t i o n of t r i a l s the a c t u a l maximum found v e r s u s composite s i z e . ( S t r a t i f i e d s a m p l e ) . . . 9 9 20. Mean a b s o l u t e range e r r o r v e r s u s composite s i z e . ( S t r a t i f i e d sample) 9 9 21. 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 . ( S t r a t i f i e d sample) 101 22. Mean squared e r r o r v e r s u s composite s i z e f o r low a u t o c o r r e l a t i o n group. ( S t r a t i f i e d sample) 101 23. Mean squared e r r o r v e r s u s composite s i z e f o r h i g h and moderate a u t o c o r r e l a t i o n groups. ( S t r a t i f i e d sample) 102 24. P r o p o r t i o n of t r i a l s the a c t u a l maximum found v e r s u s composite s i z e 110 25. Mean a b s o l u t e range e r r o r v e r s u s composite s i z e 110 26. 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 111 x Acknowledgements I w i s h t o e x p r e s s my a p p r e c i a t i o n t o P e t e r Nemetz f o r f i r s t i n t r o d u c i n g me t o the a r e a of water q u a l i t y m o n i t o r i n g and f o r h i s p a t i e n t and thorough r e v i e w of the many d r a f t s . F u r t h e r m o r e , I would l i k e t o thank Shelby B r u m e l l e who was a c a t a l y s t f o r many of the i d e a s i n the t h e s i s . I must e x p r e s s a s p e c i a l t r i b u t e t o Dean Uyeno, whose a d v i s e , s u p p o r t , and c o n f i d e n c e w i l l remain the f o c a l p o i n t of my e x p e r i e n c e s a t the U n i v e r s i t y of B r i t i s h Columbia. I have known no f i n e r mentor. x i I . I n t r o d u c t i o n 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 t a k i n g samples and the p r o c e s s of o b t a i n i n g sample measurements a r e d i s t i n c t . In th 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 samples a re c o l l e c t e d and then must be s u b j e c t e d t o some k i n d of e v a l u a t i o n or a n a l y s i s t o produce the sample measurement. The l a t t e r p r o c e s s w i l l be r e f e r r e d t o as t e s t i n g the samples. C o n s i d e r , f o r example, a s i t u a t i o n i n q u a l i t y c o n t r o l . P r o d u c t s a r e randomly s e l e c t e d and t e s t e d u n t i l they f a i l . A sample would be the p r o d u c t i t s e l f . The sample measurement would be the l e n g t h of time or perhaps the f o r c e r e q u i r e d f o r the pr o d u c t t o f a i l . In t h i s t h e s i s , methods w i l l be d e 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. Some assumptions about the s t r u c t u r e of the s i t u a t i o n a r e needed. These assumptions w i l l form a t h e o r e t i c a l framework on which the methods proposed i n t h i s t h e s i s w i l l be based. The assumptions a r e : 1. The p r o c e s s of c o l l e c t i n g samples and the p r o c e s s of o b t a i n i n g sample measurements a re d i s t i n c t . 2. 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. 3. The sample measurements a r e 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 . The d i s t i n c t i o n between the samp l i n g p r o c e s s and the t e s t i n g p r o c e s s has a l r e a d y been d i s c u s s e d . T h i s i s a ve r y common 1 2 s i t u a t i o n . The c o s t r e f e r r e d t o i n the second assumption can be i n terms of money and/or t i m e . These f i r s t two assumptions c h a r a c t e r i z e the s i t u a t i o n s i n which the o b j e c t i v e i s m e a n i n g f u l . I f the c o s t of t e s t i n g the sample i s q u i t e s m a l l then any d e c r e a s e i n the number of t e s t s performed w i l l amount t o an i n s i g n i f i c a n t s a v i n g . A t t e n t i o n would more p r o f i t a b l y be f o c u s e d on the s a m p l i n g p r o c e s s . The c o s t of t e s t i n g i s , however, t y p i c a l l y much l a r g e r than the c o s t of s a m p l i n g . T h i s i s e s p e c i a l l y t r u e i f the t e s t i n g i n v o l v e s l a b o r a t o r y a n a l y s i s . The t h i r d and f i n a l assumption i s fundamental t o the performance of the methods t h a t w i l l be d e v e l o p e d . I t i s the presence 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 a t w i l l a l l o w us t o g a i n i n f o r m a t i o n about some of the samples i m p l i c i t l y . A l t h o u g h the above d i s c u s s i o n i s p r e s e n t e d i n a g e n e r a l c o n t e x t , the impetus comes from the area of water q u a l i t y m o n i t o r i n g . In many a p p l i c a t i o n s , water samples a r e c o l l e c t e d and then s u b j e c t e d t o l a b o r a t o r y a n a l y s i s t o d e t e r m i n e the l e v e l of a water q u a l i t y parameter. The c o s t of t h i s a n a l y s i s i s u s u a l l y much g r e a t e r than the c o s t of s a m p l i n g . The presence 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 has been noted by s e v e r a l a u t h o r s . In a d d i t i o n , the maximum sample measurement i s a p a r t i c u l a r l y v a l u a b l e s t a t i s t i c f o r the purpose of e n f o r c i n g water q u a l i t y s t a n d a r d s . I f the maximum measurement i s g r e a t e r than the a l l o w a b l e l i m i t s , then t h e r e i s no doubt t h a t a v i o l a t i o n of the s t a n d a r d s has o c c u r r e d . I f the maximum measurement i s l e s s than the 3 a l l o w a b l e l i m i t s and the s e t of samples i s r e p r e s e n t a t i v e of the system under i n v e s t i g a t i o n , then one must be s a t i s f i e d t h a t a v i o l a t i o n has not o c c u r r e d . For these reasons i t i s a l s o e s s e n t i a l 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 . Due t o the random n a t u r e of the d a t a , i t i s conceded t h a t w i t h a s i g n i f i c a n t r e d u c t i o n i n the number of t e s t s , no method e x i s t s t h a t w i l l f i n d the maximum sample measurement w i t h c e r t a i n t y . T h i s s i t u a t i o n i s analogous i n many ways t o the development of h e u r i s t i c s which have proved so u s e f u l i n i n t e g e r programming. I t w i l l be n e c e s s a r y t h a t any method f i n d the maximum sample measurement a l a r g e p r o p o r t i o n of the t i m e . When i t does n o t , i t must a t l e a s t f i n d a ' l a r g e ' v a l u e . The b e n e f i t from r e d u c i n g the number of t e s t s performed can be viewed i n two d i f f e r e n t ways. The most o b v i o u s i s t h a t a r e d u c t i o n i n the number of t e s t s performed w i l l r e s u l t i n a de c r e a s e i n c o s t s . I t may, however, be f e a s i b l e t o r e a l l o c a t e t h i s a c q u i r e d s a v i n g t o i n c r e a s e the number of samples. I n c r e a s i n g the number of samples 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 sampling from any p e r i o d i n which the system i s i n v i o l a t i o n and, c o n s e q u e n t l y , g i v e a more r e p r e s e n t a t i v e and r e l i a b l e s e t of samples. At l e a s t i m p l i c i t l y , i n f o r m a t i o n i s g a i n e d . Thus, a r e d u c t i o n i n the number of t e s t s w i l l a l l o w f o r the c o l l e c t i o n of a more p o w e r f u l s e t of samples s u b j e c t t o a f i x e d e x p e n d i t u r e . E i t h e r of the two b e n e f i t s j u s t d e s c r i b e d a re c l e a r l y 4 d e s i r a b l e . I I . Review of the L i t e r a t u r e P o l l u t i o n m o n i t o r i n g i s a broad and d i v e r s e f i e l d . A survey of the l i t e r a t u r e uncovers the seemingly u n r e l a t e d t i t l e s , " B i o a s s a y f o r a s s e s s i n g the e f f e c t of s u l f u r d i o x i d e on oat s e e d l i n g s , " " E f f e c t s of f l u c t u a t i n g s u b l e t h a l a p p l i c a t i o n s of heavy m e t a l s o l u t i o n s upon the g i l l v e n t i l a t i o n of B l u e g i l l s , " and " B e n t h i c i n v e r t e b r a t e s as water q u a l i t y i n d i c a t o r s i n the Penobscot R i v e r . " The t a s k of m o n i t o r i n g p o l l u t i o n i s complex and the t e c h n i q u e s employed a r e o f t e n problem dependent. G e n e r a l l y s p e a k i n g , however, the l i t e r a t u r e can be grouped i n t o the t h r e e major c a t e g o r i e s of a i r , l a n d , and water p o l l u t i o n . An overview of a water m o n i t o r i n g program i s p r e s e n t e d h e r e , f o l l o w e d by a more d e t a i l e d d i s c u s s i o n of the l i t e r a t u r e which i s of p a r t i c u l a r r e l e v a n c e t o the t o p i c of t h i s t h e s i s . A. Overview of a Water M o n i t o r i n g Program A water m o n i t o r i n g program i s e s s e n t i a l f o r p r o v i d i n g c u r r e n t l y n o n a v a i l a b l e i n f o r m a t i o n t o both government a g e n c i e s and i n d u s t r i a l d i s h a r g e r s . A f l o w c h a r t h i g h l i g h t i n g the main components of such a program i s p r e s e n t e d i n F i g u r e 1. The f l o w d e f i n e s a s t e p w i s e procedure f o r d e s i g n i n g a m o n i t o r i n g program. A b r i e f d i s c u s s i o n of the s t e p s i n d i c a t e d i n the f l o w c h a r t f o l l o w s . O b j e c t i v e s The o b j e c t i v e s of a m o n i t o r i n g program a r e s i t u a t i o n dependent. A l t h o u g h the i n h e r e n t d i v e r s i t y of i n t e r e s t s and 5 Objectives D e f i n i t i o n of Parameters f o r Analysis Methods of Analysis Sampling Program M/ Expression of Results A s s i m i l a t i o n of Information Modify? Development of a Water Monitoring Program Figure 1 7 g o a l s i m ply t h a t i t i s i m p o s s i b l e t o c o n s t r u c t a l i s t of a l l o b j e c t i v e s , a few of the most common a r e : 1. To conduct survey programs t o p r o v i d e i n f o r m a t i o n on c u r r e n t water q u a l i t y and q u a n t i t y . 2 . To determine q u a n t i t i e s of v a l u a b l e b y - p r o d u c t s t h a t c o u l d be r e c o v e r e d from a waste e f f l u e n t . 3. To a s s e s s p o s s i b l e d e t r i m e n t a l e f f e c t s of waste e f f l u e n t d i s c h a r g e on the q u a l i t y of the r e c e i v i n g water and on f l o r a and fauna a s s o c i a t e d w i t h t h i s waterbody. 4 . To determine the s u i t a b i l i t y of water f o r an i n t e n d e d use. Parameters Once the o b j e c t i v e s have been c l e a r l y d e l i n e a t e d , the parameters of a n a l y t i c a l i n t e r e s t must be s e l e c t e d . T h i s p r o c e s s s h o u l d a l s o i n c l u d e the l e v e l s a t which the parameters a r e t o be measured; p r o c e d u r e s f o r sampling and a n a l y s i s of heavy m e t a l s , f o r example, w i l l d i f f e r between c o n c e n t r a t i o n s i n the m i l l i g r a m per l i t e r and microgram per l i t e r ranges. A g a i n , the parameters of p o s s i b l e i n t e r e s t a re d i v e r s e . They can v a r y from p h y s i c a l a t t r i b u t e s ( s a l i n i t y , e l e c t r i c a l conductance, suspended m a t t e r , . . . ) and c h e m i c a l a t t r i b u t e s ( c h e m i c a l oxygen demand, c h l o r i n e demand,...) t o p h y s i o l o g i c a l a t t r i b u t e s (odor, t a s t e , . . . ) . For each parameter of i n t e r e s t , t o l e r a n c e l i m i t s s h o u l d a l s o be s e t . To a i d i n t h i s p r o c e d u r e , " Q u a l i t y C r i t e r i a f o r Water" i s p u b l i s h e d by the E n v i r o n m e n t a l P r o t e c t i o n Agency (EPA) of 8 the U n i t e d S t a t e s . T h i s document r e p r e s e n t s a s t o c k t a k i n g by the EPA t o i d e n t i f y , on a n a t i o n a l s c a l e , the v a r i o u s water q u a l i t y c o n s t i t u e n t s . Over 50 parameters a r e examined and t o l e r a n c e l i m i t s a r e recommended. Methods of A n a l y s i s A f t e r the o b j e c t i v e s and parameters of i n t e r e s t have been i d e n t i f i e d , the methods of a n a l y s i s must be c o n s i d e r e d . Many im p o r t a n t i s s u e s must be a d d r e s s e d , the most i m p o r t a n t of which a r e h i g h l i g h t e d below. 1. I s the program t o det e r m i n e the e c o l o g i c a l impact of the water q u a l i t y l e v e l ? 2. I s the program t o d e t e c t extremes i n q u a l i t y and/or means i n q u a l i t y ? 3. S e n s i t i v i t y : how s e n s i t i v e i s the environment t o s m a l l changes i n the parameters? 4. A c c u r a c y and P r e c i s i o n : t h e s e items w i l l be dependent on the s e n s i t i v i t y of the environment and on the t o l e r a n c e l i m i t s d e c i d e d upon. 5. Measurement method: s h o u l d the a n a l y s i s be performed i n the f i e l d or i s a l a b o r a t o r y r e q u i r e d ? 6. Manual or i n s t r u m e n t t e c h n i q u e s : t h i s r e f e r s t o both the p h y s i c a l s a m p l i n g p r o c e s s and the a n a l y s i s . 7. Frequency of measurement: a t t h i s s tage o n l y a p r e l i m i n a r y e s t i m a t e i s needed, so t h a t the s i z e of the sam p l i n g program can be ap p r o x i m a t e d . 8. R a p i d i t y of a n a l y s i s : once a sample i s t a k e n , how q u i c k l y must i t undergo a n a l y s i s t o ensure 9 r e p r e s e n t a t i v e r e s u l t s ? 9 . Cost of the program. Sampling Program The parameters of i n t e r e s t and the methods of a n a l y s i s have been s e l e c t e d . These r e s u l t s , t o g e t h e r w i t h the o b j e c t i v e s , w i l l impose c o n s t r a i n t s on the s a m p l i n g program. The s a m p l i n g scheme i s then d e v e l o p e d i n an attempt t o maximize the i n f o r m a t i o n o b t a i n e d w h i l e k e e p i n g the c o s t l e s s than some budgetary upper bound. Sampling p r o c e d u r e s can be e x p e c t e d t o v a r y w i d e l y from one s i t u a t i o n t o a n o t h e r . 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 r o u t i n e f i x e d - s t a t i o n m o n i t o r i n g t o determine means i n water q u a l i t y , supplemented by s h o r t i n t e n s i v e s u r v e y s t o o b t a i n enforcement i n f o r m a t i o n . The i m p o r t a n t i s s u e s t o c o n s i d e r when d e s i g n i n g a s a m p l i n g p l a n a r e : 1. The number of s i t e s : one s i t e or network d e s i g n 2 . . L o c a t i o n of sample s i t e ( s ) 3. Sampling f r e q u e n c i e s 4 . Sample s i z e 5 . Type of s a m p l i n g : c o m p o s i t e , grab, or c o n t i n u o u s 6 . S t a b i l i t y : e l i m i n a t i o n of p o t e n t i a l v a r i a t i o n s i n the s u r r o u n d i n g c o n d i t i o n s 7 . P r e s e r v a t i o n of samples between the time a sample i s taken and the time i t i s a n a l y z e d . The t o p i c s of r e l e v a n c e t o t h i s t h e s i s i n c l u d e s a m p l i n g f r e q u e n c i e s and sample s i z e as they r e l a t e t o the d e t e c t i o n of 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 . 10 R e s u l t s R e s u l t s s h o u l d be r e p o r t e d w i t h due r e g a r d t o the p r o p o s a l s of the I n t e r n a t i o n a l O r g a n i z a t i o n of S t a n d a r d i z a t i o n . T h i s w i l l ensure u n i f o r m i t y of nomenclature and measures and thus c o m p a r a b i l i t y of f i n d i n g s . The method of c o l l e c t i o n and the s t a t i s t i c a l t r e a tment of the d a t a s h o u l d always be i n c l u d e d . A s s i m i l a t i o n of I n f o r m a t i o n An assessment i s performed a t t h i s stage t o determine i f c e r t a i n i n f o r m a t i o n i s l a c k i n g and whether the o b j e c t i v e s have been met. At t h i s j u n c t u r e , a d e c i s i o n would be made as t o whether the m o n i t o r i n g program r e q u i r e s m o d i f i c a t i o n . I f i t does, one would r e c y c l e t h rough the s t e p w i s e procedure g i v e n i n F i g u r e 1, s t a r t i n g a t the " o b j e c t i v e s " phase. The e x t e n t and scope of the m o d i f i c a t i o n s would, of c o u r s e , be dependent on the p a r t i c u l a r s i t u a t i o n . B. R e l e v a n t L i t e r a t u r e A l t h o u g h r e c e n t emphasis has been toward s a m p l i n g s t r a t e g i e s d e s i g n e d t o d e t e c t l o n g range t r e n d s and a v e r a g e s , some l i t e r a t u r e has been p u b l i s h e d w i t h the g o a l of d e t e c t i n g i n d i v i d u a l 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 . Beckers et a l . [ 1 ] , [ 2 ] , and [3] a d d r e s s the i s s u e of measuring the e f f e c t i v e n e s s of a f i x e d i n t e r v a l s a m p l i n g program w i t h t h i s g o a l i n mind. The a u t h o r s propose as a t e m p o r a l p r i o r i t y measure, the e x p e c t e d p r o p o r t i o n of v i o l a t i o n s d e t e c t e d . T h i s i s d e f i n e d a s , _ E x p e c t e d Number o f V i o l a t i o n s D e t e c t e d ^ 2 1 ) ~ E x p e c t e d Number o f V i o l a t i o n s Suppose t h a t samples a r e taken a t f i x e d i n t e r v a l s of l e n g t h A. I f a v i o l a t i o n exceeds a p e r i o d of 2 A , then a t l e a s t 2 samples w i l l be ta k e n d u r i n g t h i s p e r i o d of v i o l a t i o n . For the purposes of the measure d e f i n e d , t h i s w i l l count o n l y once i n the numerator. Thus the s t a t i s t i c M w i l l range between 0 and 1. An a p p r o x i m a t i o n t o M under a s i m p l i f y i n g a s sumption i s d e v e l o p e d i n [ 1 ] . U n f o r t u n a t e l y , the a n a l y s i s and the s t a t e d a s s u m p t i o n s a r e i n e r r o r . F o l l o w i n g the n o t a t i o n i n [ 1 ] , l e t T 0 be the e x p e c t e d i n t e r v a l between v i o l a t i o n s and T, be the e x p e c t e d d u r a t i o n of a v i o l a t i o n . The t w o - s t a t e a l t e r n a t i n g p r o c e s s of a v i o l a t e d p e r i o d f o l l o w e d by a n o n v i o l a t e d p e r i o d i s r e p r e s e n t e d a?, a Markov p r o c e s s . The tim e s spent i n each of the two s t a t e s a r e c o n s i d e r e d independent and i d e n t i c a l l y d i s t r i b u t e d e x p o n e n t i a l random v a r i a b l e s . A c o m p l e t e l y e q u i v a l e n t and more p o w e r f u l f o r m u l a t i o n i s t h a t of an a l t e r n a t i n g renewal p r o c e s s . D e f i n e the random v a r i a b l e s V ( i ) as t h e l e n g t h of t h e i ' t h v i o l a t i o n and B ( i ) as the time between the i ' t h and ( i + 1 ) ' s t v i o l a t i o n . As b e f o r e , the { V ( i ) } a r e independent e x p o n e n t i a l random v a r i a b l e s w i t h E [ V ( i ) ] = T , f o r a l l i ; the { B ( i ) } a r e independent e x p o n e n t i a l random v a r i a b l e s w i t h E [ B ( i ) ] = T 0 f o r a l l i ; the V ( i ) ' s and 12 the B ( i ) ' s a r e independent of each o t h e r . The system i s now s u c c i n c t l y e x p r e s s e d by the sequence of random v a r i a b l e s , V ( 1 ) , B ( 1 ) , V ( 2 ) , B ( 2 ) , . . . (2.2) B e c k e r s e t a l . s t a t e t h a t under the s i m p l i f y i n g assumption A<T 0 +T,, the s t a t i s t i c M i s g i v e n by, M = ? Q e x p ( - A / T Q ) - e x p ( - A / T 1 ) A" T Q / T 1 - 1 (2.3) From the d e t a i l e d d e r i v a t i o n g i v e n i n [ 1 ] , the assu m p t i o n s t a t e d i s not what i s a c t u a l l y assumed. The assu m p t i o n i n f a c t used i s t h a t i f the system i s not i n v i o l a t i o n a t time t , t hen i n the i n t e r v a l ( t , t + A ) the system w i l l e i t h e r remain not v i o l a t e d or become v i o l a t e d and remain so. Observe t h a t t h i s i s even a s t r o n g e r assumption than A < B ( i ) + V ( i ) f o r a l l i 1 . 1 A more c o n c i s e m a t h e m a t i c a l statement of the assumption can be f o r m u l a t e d as f o l l o w s . Suppose the system i s not v i o l a t e d a t time t . Then t h e r e e x i s t s a j such t h a t t l i e s i n t he i n t e r v a l c o r r e s p o n d i n g t o B ( j ) . L e t B ( j , t ) be the tim e u n t i l t he next v i o l a t i o n from t . (Due t o t h e memoryless p r o p e r t y of the e x p o n e n t i a l d i s t r i b u t i o n , B ( j , t ) i s e x p o n e n t i a l l y d i s t r i b u t e d w i t h E [ B ( j , t ) ] = T 0 ) . The assumption can now be s t a t e d a s : f o r each i and a l l t i n t h e i n t e r v a l c o r r e s p o n d i n g t o B ( i ) , A < B ( i , t ) + V ( i ) S i n c e B(i,t)£B(i) f o r a l l t i n the i n t e r v a l c o r r e s p o n d i n g t o 1 3 An i m p l i c a t i o n of such an assumption i s t h a t t h e r e can be at most one v i o l a t i o n i n the i n t e r v a l (t,t+A) f o r a l l A>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 A<T 0." Aga i n t h i s i s not the assumption t h a t i s a c t u a l l y used. ( C l e a r l y A<T 0 i m p l i e s A<T 0+T,). The assumption which i s used i s t h a t A<B(i) f o r a l l i . That i s , the time between v i o l a t i o n s must be g r e a t e r than A f o r a l l A>0. 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 <y2/N even though o n l y (N/m> 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 , <s2, need n o t be e s t i m a t e d as i t r e s u l t s o n l y i n a l i n e a r change i n s c a l e . T h a t i s , i f we d e f i n e t h e v a r i a n c e of a c o m p o s i t e measurement o f s i z e n t o be S 2 ( n ) , t h e n i t i s s u f f i c i e n t t o d e a l w i t h S 2 ( n ) / < r 2 . The v a l u e s o f S 2 ( n ) / t r 2 were c a l c u l a t e d f o r a l l t h e c o m p o s i t e s i z e s under e x a m i n a t i o n and a r e p r e s e n t e d i n T a b l e 5. C l e a r l y , as t h e s i z e of t h e c o m p o s i t e i n c r e a s e s , t h e v a r i a n c e of t h e c o m p o s i t e measurement d e c r e a s e s . The c o n s e q u e n c e of t h i s i s t h a t , f o r u n b a l a n c e d c o m p o s i t i n g , t h e f i n a l c o m p o s i t e measurement w i l l p o s s e s s a h i g h e r v a r i a n c e t h a n t h e measurements from t h e o t h e r c o m p o s i t e s . To u n d e r s t a n d t h e impact of t h i s i n c r e a s i n g v a r i a n c e on P[FC|->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 r<m. The c o m p o s i t e s w i l l be r e f e r r e d t o a s C1 and C2 r e s p e c t i v e l y . The f i r s t c a s e t o be c o n s i d e r e d i s when t h e sample w i t h t h e maximum measurement i s i n C1. T h i s c o r r e s p o n d s t o t h e e v e n t -"MF. G i v e n t h i s e v e n t , t h e e x p e c t e d c o m p o s i t e 69 Ta b l e 5 V a r i a n c e of composite measurements. Composite S 2(m)/ t f S i z e (m) 2 0.92963 3 0.49909 4 0.33037 5 0.24407 6 0.19252 7 0.15893 8 0.13541 9 0.11794 10 0.10447 1 1 0.09384 1 2 0.08525 1 3 0.07809 1 4 0.07209 1 5 0.06697 16 0.06258 1 7 0.05869 18 0.05530 1 9 . 0.05230 20 0.04961 21 0.04718 22 0.04499 23 0.04301 24 0.04119 25 0.03953 26 0.03800 27 0.03659 28 0.03528 29 0.03406 30 0.03293 measurement of C1 w i l l be n+6 f o r some 6>0. 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 <r2 and h<r2 r e s p e c t i v e l y , then P[ Y>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,<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 a decrease 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. C o n s i d e r F i g u r e 7. F i g u r e 7A shows a v a l u e of M, t h a t i s g r e a t e r than the p o i n t a t which the d e n s i t i e s a r e e q u a l . 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 s m a l l 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 by the a r e a of the shaded r e g i o n . In F i g u r e 7B a v a l u e f o r M, l e s s than the p o i n t a t which the d e n s i t i e s a r e e q u a l i s i n d i c a t e d . A g a i n , the area of the shaded r e g i o n i s e q u a l t o the amount by which 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 s m a l l e r f o r the d e n s i t y w i t h the h i g h e r v a r i a n c e . T h i s , t o o , i s e a s i l y v e r i f i e d m a t h e m a t i c a l l y . I t has been 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 a decrease i n P[FC|-"MF]. C o n v e r s e l y , 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,<M] i s l e s s than 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,]=M<M+6 i m p l i e s t h a t + i s l e s s than p t M ^ n + 6 ] . 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 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 r e s u l t i n an i n c r e a s e i n P[ -"FC |MF] . The above a n a l y s i s has i l l u s t r a t e d 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 composite w i l l tend t o i n c r e a s e both P[FC|-«MF] and P t ^ F C l M F J . Indeed, the e f f e c t on both s h o u l d , i n the l o n g r u n , be e q u i v a l e n t . I t w i l l now be shown, however, t h a t the a d d i t i o n of one or more composites of s i z e m w i l l r e s u l t 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 f FCl^MF] w i l l i n c r e a s e w h i l e s i m u l t a n e o u s l y i n c r e a s i n g 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 . R e c o n s i d e r the f i r s t s i t u a t i o n , d e p i c t e d i n F i g u r e 5, but l e t M* be the maximum composite measurement of a l l the comp o s i t e s of s i z e m. There a re impo r t a n t d i f f e r e n c e s between the d i s t r i b u t i o n of the maximum, M*, and the d i s t r i b u t i o n of a s i n g l e o b s e r v a t i o n , M,. The most c r u c i a l h ere i s t h a t P[M*<t]<P[M,<t] f o r a l l f i n i t e t . To prove t h i s , l e t M 2* be the maximum of a l l the composites not i n c l u d i n g M t . Thus, M*=Max(M,,M2*) and we can now w r i t e , 75 P[M*<t] = P[M,<t] P[M 2*<t] (6.12) S i n c e P[M 2*<t]<1, t h i s i m p l i e s t h a t P[M*<t]<P[M t<t]. An im p o r t a n t i m p l i c a t i o n of t h i s r e s u l t i s t h a t the median of the d i s t r i b u t i o n of the maximum, Med(M*), i s t o the r i g h t of the median of which, by symmetry, i s e q u a l t o the mean, E[M-,]. To see t h i s , f i r s t o bserve t h a t Med(M*)>Med(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,<Med(M*)] (6.13) A p p l y i n g (6.12) w i t h t=Med(M*) y i e l d s , 0.5 = P[M*<Med(M*)] < P[M,<Med(M*)] (6.14) 76 w i l l produce a decrease i n P[FC|-'MF]. S i m i l a r l y , i f 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,<i/] < 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*<i/+6. C o n v e r s e l y , P[ - ,FC|MF] w i l l d e 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*<n + 6] < P[M,<»/ + 6] 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<p (1 ) <0 . 8 formed the o t h e r two. The number of hours of d a t a composing each group were 1262, 427, and 272 r e s p e c t i v e l y . 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 s t h a t were produced by these g r o u p i n g s appear i n F i g u r e 18. None of the f u n c t i o n s demonstrate a h i g h degree of v a r i a b i l i t y as i n d i c a t e d by the t i g h t n e s s of the bounding c o n f i d e n c e i n t e r v a l s . T h i s s u g g e s t s t h a t the method of s t r a t i f i c a t i o n d i d indeed form groups t h a t p o s s e s s 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 . Not s u r p r i s i n g l y , 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 of those hours w i t h /> (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 ) 0.8<p(1)<0.9 i s a b o v e t h e 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 o f t h o s e h o u r s w i t h 0.2 < p(1)<0.8. G i v e n t h i s h i e r a r c h i c a l b e h a v i o r , t h e g r o u p s w i l l be r e f e r r e d t o a s t h e h i g h a u t o c o r r e l a t i o n g r o u p , t h e m o d e r a t e a u t o c o r r e l a t i o n g r o u p , a n d t h e l o w a u t o c o r r e l a t i o n g r o u p . 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 a p p l i e d t o e a c h g r o u p s e p a r a t e l y f o r a l l c o m p o s i t e s i z e s . S t a t i s t i c s were t a b u l a t e d a n d r e c o r d e d . The c o m p l e t e s e t o f n u m e r i c v a l u e s c a n be f o u n d i n A p p e n d i x A. The r e s u l t s f o r t h e l o w a u t o c o r r e l a t i o n g r o u p a p p e a r i n T a b l e A . 1 , t h e m o d e r a t e a u t o c o r r e l a t i o n g r o u p i n T a b l e A . 3 , a n d t h e h i g h a u t o c o r r e l a t i o n g r o u p i n T a b l e A.4. The p r e s e n t a t i o n i n t h e t e x t w i l l a g a i n be p r i n c i p a l l y g r a p h i c . 98 The p r o p o r t i o n of t r i a l s on which the maximum sample measurement was found i s p r e s e n t e d i n F i g u r e 19. Not u n e x p e c t e d l y , t h i s s t a t i s t i c shows the same h i e r a r c h i c a l r a n k i n g e v i d e n t i n 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 s . For e very composite s i z e , the h i g h e r the a u t o c o r r e l a t i o n f u n c t i o n , the g r e a t e r the number of t r i a l s on which the sample w i t h the maximum measurement i s l o c a t e d . A l s o observe t h a t 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 , t h a t i s , unbalanced c o m p o s i t i n g b e i n g i n f e r i o r , i s s t i l l e v i d e n t w i t h i n each group. Now c o n s i d e r the composite s i z e s which produce b a l a n c e d c o m p o s i t i n g . I t w i l l be r e c a l l e d t h a t they occur i n p a i r s , each of which r e s u l t s i n the i d e n t i c a l number of t e s t s b e i n g performed. I t can be seen i n F i g u r e 19 t h a t , 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 s h i f t s downward, t h e r e i s a tendency f o r the s m a l l e r composite s i z e s t o p e r f o r m b e t t e r . 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 i n F i g u r e 20 a c r o s s a l l composite s i z e s . A l l of the above comments a r e s t i l l v a l i d . The lower the a u t o c o r r e l a t i o n f u n c t i o n , the poorer the performance. In f a c t , the low a u t o c o r r e l a t i o n group f a i r s much more p o o r l y . B a l a n c e d c o m p o s i t i n g i s a g a i n s u p e r i o r t o unbalanced c o m p o s i t i n g . I t i s - a l s o e v i d e n t , by r e f e r r i n g t o T a b l e A.2, T a b l e A.3, and T a b l e A.4, t h a t the lower the a u t o c o r r e l a t i o n , the g r e a t e r the v a r i a n c e of the MARE. More d r a m a t i c , however, i s the marked s u p e r i o r i t y of the s m a l l e r composite s i z e s f o r the low a u t o c o r r e l a t i o n group. In p a r t i c u l a r , a composite s i z e 99 F i g u r e 19 P r o p o r t i o n o f t r i a l s t h e a c t u a l maximum f o u n d v e r s u s c o m p o s i t e s i z e . ( S t r a t i f i e d s a m p l e ) 0? ec' 0.9 < f l U < 1-0 0.8 < p M ! < 0.9 0.2 < P u i < 0.8 , 1 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 ' 0 2 4 6 6 50 J2 14 16 18 20 22 24 26 28 30 C O M P O S I T E S J Z E F i g u r e 20 Mean a b s o l u t e r a n g e e r r o r v e r s u s c o m p o s i t e s i z e ( S t r a t i f i e d s a m p l e ) in • or CCU' 8-0.9 < p(3! < 3 .0 0.8 < o i l ! < 0.9 o 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 1 0 2 4 6 8 10 12 14 J6 18 20 22 24 26 28 30 C O M P O S I T E S ] Z E 100 of 2 accounted f o r , on average , almost 8% more of the o b s e r v a b l e e r r o r than d i d a composite s i z e of 30. The dominance of the s m a l l e r composite s i z e s w i t h r e s p e c t t o the low a u t o c o r r e l a t i o n group i s a l s o apparent f o r the maximum a b s o l u t e d e v i a t i o n (MAD), F i g u r e 21. In a d d i t i o n , the MAD f o r t h i s group i s s u b s t a n t i a l l y l a r g e r than f o r the two groups w i t h h i g 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 . These l a t t e r two groups e x h i b i t l i t t l e v a r i a b i l i t y 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 w i t h the moderate a u t o c o r r e l a t i o n group u s u a l l y h a v i n g a l a r g e r MAD. The f i n a l s t a t i s t i c t o examine i s the mean squared e r r o r (MSE). The MSE f o r the low a u t o c o r r e l a t i o n group was t y p i c a l l y two o r d e r s of magnitude l a r g e r than the MSE f o r e i t h e r of the o t h e r two groups. P r e s e n t a t i o n of the MSE f o r a l l t h r e e groups on the same graph was i m p o s s i b l e . C o n s e q u e n t l y , the MSE f o r the low a u t o c o r r e l a t i o n group i s p r e s e n t e d s e p a r a t e l y i n F i g u r e 22 w h i l e the MSE f o r the h i g h e r a u t o c o r r e l a t i o n groups appear i n F i g u r e 23. G e n e r a l l y , b a l a n c e d c o m p o s i t i n g i s s u p e r i o r t o unbalanced c o m p o s i t i n g , a l t h o u g h f o r the h i g h a u t o c o r r e l a t i o n group, composite s i z e s 17 and 18 have lower mean squared e r r o r s than a composite s i z e of 15. The h i e r a r c h i c a l r a n k i n g of the groups w i t h r e s p e c t t o MSE i s c l e a r l y e v i d e n t . The b e n e f i t of c h o o s i n g the s m a l l e r composite s i z e i s a g a i n q u i t e pronounced f o r the low a u t o c o r r e l a t i o n group, as shown i n F i g u r e 22. A l t h o u g h not i n d i c a t e d i n the f i g u r e s , t h e r e i s a l s o a h i e r a r c h i c a l r a n k i n g of the v a r i a n c e of the MSE. The 101 F i g u r e 21 Maximum a b s o l u t e d e v i a t i o n v e r s u s c o m p o s i t e s i z e . ( S t r a t i f i e d s a m p l e ) 0.9 < p t n < i.o 0.8 < p(1) < 0.9 0.2 < p i l l < 0.B I 1 1 1 1 1 I I ' \ I 1 ' 1 ' + ; \ ; \ I * ' +-+ / \ / \ / \ j I I I I I I I t - 4 - ! I I I N- I I 4-I I I I I 1 I I I I I I I I I I I I I T I I 2 4 6 8 10 12 34 36 38 20 22 C O M P O S I T E S I Z E 24 26 28 30 F i g u r e 22 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 f o r low a u t o c o r r e l a t i o n g r o u p . ( S t r a t i f i e d s a m p l e ) -0.2 < p13! < 8.0 9 2 I I I I I I I I I I I I I I I I 1 I I I I I I I I I I 1 4 6 8 JO 32 3 4 16 3 8 20 22 24 26 28 30 C O M P O S I T E S I Z E 1 0 2 F i g u r e 23 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 f o r h i g h a n d m o d e r a t e a u t o c o r r e l a t i o n g r o u p s . ( S t r a t i f i e d s a m p l e ) l o w e r 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 , t h e h i g h e r t h e v a r i a n c e . I n s u m m a t i o n , i t i s e v i d e n t t h a t t h e l o w e r 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 , t h e p o o r e r t h e p e r f o r m a n c e . T h i s a p p l i e s t o b o t h t h e a v e r a g e e r r o r a n d t h e v a r i a n c e o f t h e e r r o r a n d was c o n s i s t e n t f o r a l l o f t h e m e a s u r e s o f e r r o r . I n a d d i t i o n , b a l a n c e d c o m p o s i t i n g p r o v e d s u p e r i o r t o u n b a l a n c e d c o m p o s i t i n g i n a l l t h r e e g r o u p s . T h e s e r e s u l t s were n o t u n e x p e c t e d . More s u r p r i s i n g , h o w e v e r , was t h e s t r o n g e v i d e n c e s u p p o r t i n g t h e a d o p t i o n o f s m a l l e r c o m p o s i t e s i z e s . I t i s now p o s s i b l e t o u s e t h e s t r a t i f i e d d a t a t o g a i n some i n s i g h t i n t o how ' h i g h ' t h e a u t o c o r r e l a t i o n n e e d s t o be i n o r d e r 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 be e f f e c t i v e . 1 03 T h i s can be a c c o m p l i s h e d through the comparison of the r e s u l t s from the low a u t o c o r r e l a t i o n group t o the random sample method based on t h i s same group of d a t a . I f 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 b e t t e r than random s a m p l i n g , a l l e v i d e n c e i n d i c a t 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 be s u p e r i o r f o r any s e t of d a t a p o s s e s s i n g a h i g h e r a u t o c o r r e l a t i o n f u n c t i o n . A l l p r e v i o u s r e s u l t s demonstrate 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 c o m p o s i t i n g . For t h i s r e a s o n , o n l y 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 w i l l now be c o n s i d e r 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 was a p p l i e d t o the low a u t o c o r r e l a t i o n group and, i n t u r n , the random s a m p l i n g method. The p r o p o r t i o n of t r i a l s on which the sample w i t h the maximum measurement was found i s p r e s e n t e d i n T a b l e 11. T h i s i s f o l l o w e d by the MARE i n T able 12. The p r o p o r t i o n of s u c c e s s 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 i s s t a t i s t i c a l l y l a r g e r than the p r o p o r t i o n of su c c e s s e s f o r random sampling 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 l e s s than 0.0015 a t eve r y composite s i z e . The r e s u l t s f o r the MARE show s i m i l a r p r o p e r t i e s , w i t h 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 b e i n g g r e a t e r than the MARE f o r random samp l i n g a c r o s s a l l composite s i z e s . However, f o r the worst c a s e , a composite s i z e of 30, the d i f f e r e n c e i s not s i g n i f i c a n t a t an c l e v e l of 0.05. In a l l o t h e r c a s 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 i s l e s s than 0.0026 and, t h e r e f o r e , one would a c c e p t t h a t the MARE f o r pr 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 a r g e r a t any r e a s o n a b l e a 1 04 Ta b l e 1 1 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 was found 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 s a m p l i n g . Composite S i z e PFOC P r o p o r t i o n Random Sample P r o p o r t i o n 2 0.76 0.58 30 0.67 0.55 3 0.62 0.38 20 0.55 0.35 4 0.56 0.28 15 0.52 0.29 5 0.54 0.33 1 2 0.48 0.29 6 0.47 0.23 10 0.49 0.20 Tab l e 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 s a m p l i n g . PFOC Random Sample Composite S i z e MARE Sta n d a r d E r r o r ( x l O " 1 ) MARE Stand a r d E r r o r ( x l O - 1 ) 2 0.970 0.049 0.897 0.116 30 0.890 0. 1 23 0.877 0. 125 3 0.920 0.090 0.813 0. 1 47 20 0.859 0. 1 30 0.785 0. 1 55 4 0.900 0. 1 07 0.764 0. 1 56 1 5 0.839 0. 1 38 0.781 0. 1 52 5 0.872 0.119 0.781 0. 1 52 1 2 0.830 0. 1 39 0.769 0. 155 6 0.842 0. 1 25 0.752 0.161 10 0.835 0. 138 0.724 0. 166 105 l e v e l . I t s h o u l d a l s o be n o t i c e d 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 e x h i b i t s l e s s v a r i a b i l i t y 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 r e s u l t s f o r the MAD and the MSE appear i n Table 13 and T a b l e 14, r e s p e c t i v e l y . 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 has a lower MAD f o r every composite s i z e except 10, 15, and 30. However, f o r a composite s i z e of 10, the maximum d e v i a t i o n s a re i d e n t i c a l and the d i f f e r e n c e s f o r composite s i z e s 15 and 30 are n e g l i g i b l e . 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 c r o s s a l l composite s i z e s but o n l y h a l f of the d i f f e r e n c e s a r e s i g n i f i c a n t a t an a l e v e l of 0.05. For composite s i z e s 2, 3, 4, 6, and 20, one would c o n c l u d e t h a t 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 sa m p l i n g 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 s m a l l e r than 0.03. The l a r g e r number of i n s i g n i f i c a n t d i f f e r e n c e s i s , i n p a r t , due t o the l a r g e v a r i a n c e a s s o c i a t e d w i t h the MSE. The v a r i a b i l i t y of the e s t i m a t 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 i s , however, l e s s f o r a l l composite s i z e s . I t i s apparent 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 , when a p p l i e d t o the low a u t o c o r r e l a t i o n group, performs as w e l l and i n most c a s e s b e t t e r than random s a m p l i n g . In p a r t i c u l a r , f o r the s m a l l e r composite s i z e s , 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 b s t a n t i a l l y b e t t e r . I t can be c o n c l u d e d 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 s t i l l an e f f e c t i v e method even a t these low a u t o c o r r e l a t i o n l e v e l s . 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 would, t h e r e f o r e , appear t o 1 06 Tab l e 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 s a m p l i n g . PFOC Random Composite Sample S i z e MAD MAD ( X 1 Q - 1 ) ( x l O - 1 ) 2 0.087 0.456 30 0.520 0.514 3 0.264 0.501 20 0.461 0.570 4 . 0.207 0.570 15 0.466 0.459 5 0.461 0.570 12 0.453 0.465 6 0.452 0.570 10 0.570 0.570 Table 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 s a m p l i n g . PFOC Random Sample Composite S i z e MSE ( X 1 0 - « ) S t a n d a r d E r r o r ( X 1 0 - " ) MSE ( X 1 0 - « ) S t a n d a r d E r r o r (x10-«) 2 30 0.006 0.210 0. 003 0. 1 05 0.224 0.330 0.097 0.147 3 20 0.048 0.212 0.027 0.088 0.346 0.596 0.115 0. 1 79 4 1 5 0.087 0.243 0.027 0.091 0.730 0.380 0.214 0.116 5 1 2 0. 177 0.257 0.084 0.087 0.368 0.473 0. 138 0. 1 39 6 10 0.221 0.421 0.086 0. 165 0.647 0.760 0.201 0.211 1 07 be a f a i r l y r o b u s t t e c h n i q u e . C. The E f f e c t of the Time Between Samples I t has been p r e v i o u s l y mentioned t h a t the i m p o r t a n t p r i n c i p l e s h o u l d not be the time between samples but the a u t o c o r r e l a t i o n between samples. That i s , performance depends on the a u t o c o r r e l a t i o n f u n c t i o n and not the i n t e r v a l between samples. A l t h o u g h t h i s appears i n t u i t i v e l y sound, i t w i l l be d i f f i c u l t t o v a l i d a t e e m p i r i c a l l y . I d e a l l y , what i s needed i s two s e t s of d a t a , perhaps from d i f f e r e n t s o u r c e s , each c o l l e c t e d a t d i f f e r e n t s a m p l i n g i n t e r v a l s but 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 . 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 c o u l d then be a p p l i e d t o each s e t of da t a and the performances compared. ( I f the d a t a came from d i f f e r e n t s o u r c e s , o n l y d i m e n s i o n l e s s measures of e r r o r would be of u s e ) . U n f o r t u n a t e l y , t h i s type of data i s not r e a d i l y a v a i l a b l e . However, i t i s p o s s i b l e t o g a i n some ev i d e n c e from 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 . Suppose 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 i s sampled l e s s f r e q u e n t l y , say once ev e r y 24 minutes or 60 samples a day. In the l a s t s e c t i o n we e x p l o r e d the performance 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 f o r t h r e e groups of da t a p o s s e s s i n g d i s t i n c t 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 . We now have some i d e a of the way i n 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 can be expected, t o pe r f o r m f o r a s m a l l 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 based on 60 samples per hour. I f the i m p o r t a n t p r i n c i p l e i s ind e e d the a u t o c o r r e l a t i o n 108 T a b l e 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. n Low Moderate A u t o c o r r e l a t i o n A u t o c o r r e l a t i o n p(n) p(n)  60 samples per day p(n) 2 3 4 5 6 7 8 9 10 0.621 0.427 0.312 0.249 0. 183 0. 1 49 0. 129 0. 1 02 0.067 0.045 0.861 0.693 0.541 0.415 0.311 0.235 0. 180 0. 1 34 0.094 0.061 0.708 0.576 0.470 0.391 0.328 0.271 0.224 0.173 0.131 0.094 between samples, then we s h o u l d be a b l e t o i n d i c a t e the way i n 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 1 1 p e r f o r m when a p p l i e d t o 60 samples per day merely by comparing the r e s u l t i n g average a u t o c o r r e l a t i o n f u n c t i o n t o the a u t o c o r r e l a t i o n f u n c t i o n s based on 60 samples per hour. T h i s was done. S t a r t i n g a t m i d n i g h t f o r each day, samples were s e l e c t e d every 24 minut e s . 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 then a p p l i e d and s t a t i s t i c s t a b u l a t e d and r e c o r d e d . The complete s e t of v a l u e s can be found i n Appendix A, Table A.5. The f i r s t t e n l a g s of the r e s u l t i n g a u t o c o r r e l a t i o n f u n c t i o n a l o n g w i t h the f i r s t t e n 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 s f o r the low and moderate a u t o c o r r e l a t i o n groups as d e f i n e d i n the p r e v i o u s s e c t i o n a r e p r e s e n t e d i n T a b l e 15. 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 based on 60 samples per day i s s t r i c t l y g r e a t e r 109 t h a t 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 low a u t o c o r r e l a t i o n group. Thus, 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 the former group s h o u l d r e s u l t i n s u p e r i o r performance on a l l s t a t i s t i c s . In a d d i t i o n , one would expect the v a r i a n c e of the MARE and the MSE t o be l e s s f o r the 60 samples per day d a t a . The 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 moderate a u t o c o r r e l a t i o n group does n o t , u n f o r t u n a t e l y , l i e s t r i c t l y above the 60 sample per day a u t o c o r r e l a t i o n f u n c t i o n . I t i s s u b s t a n t i a l l y l a r g e r f o r the f i r s t 3 l a g s then s l i p s below at l a g 5 and remains s l i g h t l y lower f o r the remainder. T h i s t ype of r e l a t i o n s h i p was not e x p l o r e d i n the p r e v i o u s s e c t i o n and, t h e r e f o r e , we cannot say w i t h c e r t a i n t y what t o e x p e c t . However, w i t h the f i r s t 3 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 f o r the moderate a u t o c o r r e l a t i o n group b e i n g so much l a r g e r , i t would seem r e a s o n a b l e t o expect a u s u a l l y s u p e r i o r performance f o r t h i s group. The p r o p o r t i o n of t r i a l s on 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 found the maximum sample i s p r e s e n t e d i n F i g u r e 24. With the e x c e p t i o n of a composite s i z e of 2, the p r o p o r t i o n of s u c c e s s e s based on the 60 samples per day group i s above 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 low a u t o c o r r e l a t i o n group. As c o n j e c t u r e d , the p r o p o r t i o n of su c c e s s e s f o r the moderate a u t o c o r r e l a t i o n group i s u s u a l l y , but not s t r i c t l y , l a r g e r . Thus, w i t h r e s p e c t t o the p r o p o r t i o n of s u c c e s s e s , the obser v e d r e s u l t s agree w i t h the h y p o t h e s i z e d r e s u l t s . The same can be s a i d f o r the MARE 110 F i g u r e 24 P r o p o r t i o n o f t r i a l s t h e a c t u a l maximum f o u n d v e r s u s c o m p o s i t e s i z e . o e r -o ec O O • — I O 1 — . _ O o — O . Q i o Q _ _ n? o IN) o — o C3 TIME UNJT ONE DAT 0.8 < p tJ) < 0.9 0.2 < PIJ! •= 0.8 " 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 0 2 4 6 8 30 12 14 36 18 20 22 24 26 28 COMPOSITE S I Z E 3 0 F i g u r e 25 Mean a b s o l u t e r a n g e e r r o r v e r s u s c o m p o s i t e s i z e , — OD -er o -in -c r I \ 1 \ o ~ \ I C\l-c r I I i . a ~ Ul -D i e r _ n e e T \ o ~ CO -ec o ~ ec " C3 T i l l " TIME UNJT ONE DRY 0 . 8 < p l l ! < 0 . 9 0 . 2 < P 11! < 0.R I I I I I I I I I I I I I I ' ' ' ' ' 1 ' 111 F i g u r e 26 Maximum a b s o l u t e d e v i a t i o n v e r s u s c o m p o s i t e s i z e . TIME U N I T ONE: DAT 0.8 < pi)) < 0.9 0.2 < p U) < 0.8 I I I I I I I I I I I I I I I I I I I T I I I I I I I I 2 4 6 6 10 12 14 1 6 16 20 22 24 26 28 30 COMPOSITE SJZE w h i c h i s p r e s e n t e d i n F i g u r e 25 and t h e MAD w h i c h a p p e a r s i n F i g u r e 26. As e x p e c t e d , t h e MARE f o r t h e 60 s a m p l e s p e r day g r o u p l i e s b e t w e e n t h e MARE f o r t h e o t h e r two g r o u p s . By r e f e r r i n g t o T a b l e A.2, T a b l e A . 3 , a n d T a b l e A.5 i n t h e A p p e n d i x , i t i s e v i d e n t t h a t t h e h y p o t h e s i s c o n c e r n i n g t h e v a r i a n c e s o f t h e MARE i s a l s o t r u e . T h a t i s , t h e v a r i a n c e o f t h e MARE f o r t h e low a u t o c o r r e l a t i o n g r o u p i s s t r i c t l y l a r g e r t h a n t h e v a r i a n c e o f t h e MARE f o r t h e 60 s a m p l e s p e r d a y g r o u p w h i c h i s u s u a l l y l a r g e r t h a n t h e v a r i a n c e o f t h e MARE f o r t h e m o d e r a t e a u t o c o r r e l a t i o n g r o u p . The MAD f o r t h e 60 s a m p l e s p e r day g r o u p i s s t r i c t l y l e s s t h a n t h e MAD f o r t h e l o w a u t o c o r r e l a t i o n g r o u p a n d i n most c a s e s i s much l e s s . When c o m p a r e d t o t h e m o d e r a t e a u t o c o r r e l a t i o n g r o u p , 1 12 Table 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. Composite Low Moderate 60 samples S i z e A u t o c o r r e l a t i o n A u t o c o r r e l a t i o n per day (m) MSE St.Dev. MSE St.Dev. MSE St.Dev. ( X 1 0 - " ) ( X 1 0 , - • ) (xlO ,-•) 2 0.006 0.049 0.002 0.018 0.005 0.020 3 0.048 0.447 0.003 0.026 0.007 0.021 4 0.087 0.439 0.003 0.019 0.077 0.447 5 0. 177 1 .392 0.003 0.022 0.028 0. 156 6 0.221 1.414 0.003- 0.025 0.033 0.203 7 0.314 1 .933 0.003 0.020 0.030 0. 1 58 8 0.298 1 .864 0.004 0.025 0.096 0.490 9 0. 129 0.660 0.004 0.028 0.045 0. 176 10 0.421 2.729 0.003 0.024 0.013 0.035 1 1 0.247 1 .532 0.003 0.021 0.045 0.208 12 0.257 1 .461 0.002 0.016 0. 100 0.489 13 0.480 2.912 0.003 0.020 0.063 0.286 1 4 0.474 2.955 0.003 0.019 0.057 0.265 1 5 0.243 1 .499 0.003 0.022 0.050 0.262 1 6 0.318 1 .951 0.004 0.025 0.054 0.265 1 7 0.452 2.781 0.004 0.030 0.080 0.318 18 0.378 2.489 0.004 0.029 0.057 0.201 1 9 0.376 2.525 0.004 0.032 0.034 0.101 20 0.212 1 .450 0.001 0.004 0.024 0.080 21 0.223 1 .469 0.001 0.009 0.043 0.255 22 0.249 1 .532 0.002 0.013 0.297 2.248 23 0.291 1 .680 0.002 0.013 0.299 2.248 24 0.332 1 .974 0.002 0.013 0.050 0.258 25 0.565 3.173 0.002 0.013 0.069 0.272 26 0.574 3.208 0.002 0.017 0.080 0.295 27 0.565 3.220 0.004 0.032 0.087 0.328 28 0.576 3.241 0.005 0.034 0.082 0.335 29 0.584 3.262 0.005 0.034 0.478 3.499 30 0.210 1 .737 0.002 0.012 0.045 0.258 the MAD f o r the l a t t e r group i s u s u a l l y l a r g e r . The r e s u l t s f o r the MSE, T a b l e 16, are not q u i t e as c l e a n . The MSE f o r the 60 samples per day group i s l a r g e r than the MSE f o r the moderate a u t o c o r r e l a t i o n group f o r a l l composite s i z e s . The same can be s a i d f o r the v a r i a b i l i t y of the MSE, w i t h the e x c e p t i o n of composite s i z e s 2 and 3, i n 1 1 3 which the d i f f e r e n c e s a re n e g l i g i b l e . However, the MSE, and the v a r i a n c e of t h i s e s t i m a t e , f o r the 60 samples per day group i s not s t r i c t l y s m a l l e r than the MSE and i t s v a r i a n c e f o r the low a u t o c o r r e l a t i o n group. For the v a s t m a j o r i t y of the composite s i z e s , the MSE and i t s v a r i a n c e a r e much l e s s f o r the 60 samples per day group, but f o r composite s i z e s 22, 23, and 29, the low a u t o c o r r e l a t i o n group performs somewhat b e t t e r . We have now seen t h a t a l l of the s t a t i s t i c s tend t o agree w i t h the h y p o t h e s i z e d behavior.. A l t h o u g h t h i s i s not an e x p l i c i t v a l i d a t i o n , i t i s e v i d e n c e t h a t s u p p o r t s the c l a i m t h a t the time between samples i s not a f a c t o r a f f e c t i n g performance. D. A l t e r n a t i v e Composite Methods Only one s p e c i f i c composite method has been examined i n t h i s t h e s i s . However, t h e r e a r e many a l t e r n a t i v e c o m p o s i t i n g schemes. S e v e r a l w i l l be d i s c u s s e d i n t h i s s e c t i o n . With the knowledge g a i n e d from the study 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 , i t w i l l be p o s s i b l e , i n most c a s e s , t o comment on the e x p e c t e d performance. One method t h a t w i l l improve upon the performance e x h i b i t e d by 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 t o a p p l y 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 then t e s t a d d i t i o n a l samples. The l o g i c a l c h o i c e f o r the s e a d d i t i o n a l samples would be the samples t h a t formed the composite w i t h the second h i g h e s t measurement. Thus, one method t h a t w i l l 1 1 4 g u a r a n t e e improved p e r f o r m a n c e i s t o f i n d t h e c o m p o s i t e s w i t h t h e two h i g h e s t measurements and t h e n t e s t a l l t h e s a m p l e s t h a t made up t h e s e two c o m p o s i t e s . In o u r c u r r e n t n o m e n c l a t u r e t h i s t e c h n i q u e would be c a l l e d s e c o n d 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 t e r m ' s e c o n d a r y ' r e f e r s t o t h e s e l e c t i o n of t h e f i r s t and s e c o n d c o m p o s i t e s most l i k e l y t o c o n t a i n t h e maximum sample f o r f u r t h e r e x a m i n a t i o n . T h i s method o b v i o u s l y i n v o l v e s a t r a d e o f f between p e r f o r m a n c e and t h e number of t e s t s . I t w i l l g u a r a n t e e s u p e r i o r p e r f o r m a n c e t o 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 but w i l l a l s o g u a r a n t e e t h a t more t e s t s w i l l be p e r f o r m e d . The l a r g e r t h e c o m p o s i t e s i z e , t h e g r e a t e r t h e a d d i t i o n a l number of t e s t s . T h i s t e c h n i q u e , t h e r e f o r e , would seem most a p p r o p r i a t e when a p p l i e d w i t h s m a l l e r c o m p o s i t e s i z e s . I f t h e c o m p o s i t e s i z e i s s m a l l enough, i t might even be r e a s o n a b l e t o e x t e n t t h i s t e c h n i q u e beyond t h e s e l e c t i o n of o n l y two c o m p o s i t e s . One would e x p e c t , however, t h a t t h e m a r g i n a l improvement i n p e r f o r m a n c e w i l l d i m i n i s h as t h e number of c o m p o s i t e s s e l e c t e d f o r f u r t h e r e x a m i n a t i o n i n c r e a s e s . A c o n c u r r e n t b e n e f i t of t h i s a p p r o a c h i s t o a l l e v i a t e some of t h e d i f f i c u l t i e s a s s o c i a t e d w i t h u n b a l a n c e d c o m p o s i t i n g . I t has been d e m o n s t a t e d t h a t t h e f i n a l c o m p o s i t e f o r u n b a l a n c e d c o m p o s i t i n g schemes 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 . By s e l e c t i n g more t h a n one c o m p o s i t e f o r f u r t h e r e x a m i n a t i o n , however, 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 among t h e c o m p o s i t e s o f e q u a l s i z e w i l l a l w a y s be i n c l u d e d . 115 A t r a d e o f f between performance and the number of t e s t s was j u s t proposed i n which i n c r e a s i n g the number of t e s t s r e s u l t e d i n an i n c r e a s e i n performance. I f , however, performance from 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 more than a c c e p t a b l e , then i t may be d e s i r a b l e t o reduce the number of t e s t s , and c o n s e q u e n t l y the c o s t , i f the s a c r i f i c e i n performance i s not too g r e a t . One p o s s i b l e approach i s t o per f o r m c o m p o s i t i n g a second time on the samples t h a t formed the c omposite w i t h the h i g h e s t measurement. For example, i f we have 60 samples and use a composite s i z e of 30, two comp o s i t e s a r e formed and the maximum composite measurement found. We now have 30 samples from which we want t o l o c a t e the sample w i t h the maximum measurement. T h i s i s i d e n t i c a l t o the o r i g i n a l s i t u a t i o n except t h a t the number of samples has been d e c r e a s e d t o 30. Thus, c o m p o s i t i n g t e c h n i q u e s can be a p p l i e d t o these r e m a i n i n g samples. I f 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 a p p l i e d t o the 30 samples w i t h a composite s i z e of 3, the number of t e s t s performed o v e r a l l w i l l o n l y be 15. T h i s r e p r e s e n t s a decrease of more than 50% over the number of t e s t s r e s u l t i n g from 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 a p p l i e d i n i t i a l l y t o the 60 samples w i t h a composite s i z e of 30. We have p r e v i o u s l y seen t h a t the composite w i t h the maximum measurement c o n t a i n e d the sample w i t h the maximum measurement on 82% of the t r i a l s f o r a composite s i z e of 30 and 75% of the t r i a l s f o r a composite s i z e of 3. One would expect the t e c h n i q u e j u s t d e s c r i b e d t o f i n d the sample w i t h the maximum measurement on 1 1 6 a p p r o x i m a t e l y 75% of 82% or 61.5% of the t r i a l s . T h i s r e p r e s e n t s a decrease i n performance of 25% w i t h a r e d u c t i o n i n the number of t e s t s a t more than 50%. U s i n g our c u r r e n t nomenclature t h i s t e c h n i q u e w i l l be c a l l e d p r i m a r y second o r d e r c o m p o s i t i n g , the term 'second o r d e r ' r e f e r r i n g t o the f a c t t h a t c o m p o s i t i n g was performed t w i c e . The g r e a t e r the s i z e of the com p o s i t e , the g r e a t e r the p o s s i b l e s a v i n g s i n the number of t e s t s . T h i s approach, t h e r e f o r e , would seem most a p p r o p r i a t e when a p p l i e d w i t h l a r g e r composite s i z e s . I f the composite s i z e at the second c o m p o s i t i n g i s l a r g e enough, i t might even be r e a s o n a b l e t o a p p l y c o m p o s i t i n g more than t w i c e . Each a d d i t i o n a l a p p l i c a t i o n would r e s u l t i n a d e c r e a s e i n the number of t e s t s accompanied by a de c r e a s e i n performance. I t i s i m p o r t a n t t o observe t h a t the two approaches j u s t d i s c u s s e d can be blended t o g e t h e r . The f i r s t approach i s concerned w i t h the s e l e c t i o n of the com p o s i t e s f o r f u r t h e r e x a m i n a t i o n w h i l e the second approach d e a l s w i t h t h i s subsequent e x a m i n a t i o n . I t i s c o n c e i v a b l e t h a t through the s e l e c t i o n of more than one c o m p o s i t e , f o l l o w e d by f u r t h e r c o m p o s i t i n g , t h a t an i n c r e a s e i n performance and a decrease i n the number of t e s t s c o u l d be o b t a i n e d . The number of p o s s i b l e c o m b i n a t i o n s of the two approaches i s , u n f o r t u n a t e l y , l a r g e . F u r t h e r study of these blended or h y b r i d t e c h n i q u e s , 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 . 1 1 7 E. S e l e c t i n g t h e C o m p o s i t e S i z e A l l e v i d e n c e p o i n t s t o t h e a d o p t i o n of b a l a n c e d c o m p o s i t i n g schemes o v e r u n b a l a n c e d c o m p o s i t i n g . T h i s i s recommended h e r e . I t has been d e m o n s t r a t e d t h a t b a l a n c e d c o m p o s i t e s i z e s o c c u r i n p a i r s , e a c h r e s u l t i n g i n t h e i d e n t i c a l number of t e s t s b e i n g p e r f o r m e d . The s m a l l e r of t h e two s i z e s has been s e e n t o be s u p e r i o r and s h o u l d be p r e f e r r e d . A s a m p l i n g scheme f o r a water q u a l i t y m o n i t o r i n g p r o g r a m . s h o u l d a t t e m p t t o m a x i m i z e th e i n f o r m a t i o n o b t a i n e d w h i l e k e e p i n g t h e c o s t l e s s t h a n some b u d g e t a r y upper bound. I f e v e r y sample i s t e s t e d , t h e n t h e c o s t of t h e program w i l l be a l i n e a r f u n c t i o n of t h e number of samples t a k e n . S p e c i f y i n g t h e number of s a m p l e s c o m p l e t e l y d e t e r m i n e s t h e number of t e s t s . T h i s i s not t h e c a s e f o r c o m p o s i t e methods. T h e r e i s , i n f a c t , a t r a d e o f f between t h e s e two p a r a m e t e r s . T e s t i n g e v e r y sample i s e q u i v a l e n t t o a 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 one. A d j u s t i n g t h e c o m p o s i t e s i z e can r e d u c e t h e number of t e s t s and c o n s e q u e n t l y t h e c o s t of t h e p r ogram. T h i s s a v i n g may, however, be r e a l l o c a t e d t o i n c r e a s e t h e number o f s a m p l e s t a k e n . I n c r e a s i n g t h e number of s amples w i l l i n c r e a s e t h e r e l i a b i l i t y of t h e p r o g r a m . T h a t i s , i n c r e a s e t h e p r o b a b i l i t y t h a t a sample w i l l be t a k e n from any p e r i o d e x h i b i t i n g a h i g h l e v e l o f p o l l u t i o n . T h i s can e q u i v a l e n t l y be i n t e r p r e t e d as a more r e p r e s e n t a t i v e sample. The g r e a t e r t h e r e d u c t i o n i n t h e number of t e s t s t h r o u g h a d j u s t i n g t h e c o m p o s i t e s i z e , t h e 1 18 g r e a t e r t h i s r e l i a b i l i t y . U n f o r t u n a t e l y , we have seen t h a t t h e p e r f o r m a n c e 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 d e c r e a s e s a s t h e number o f t e s t s d e c r e a s e s . T h u s , t h e r e i s a t r a d e o f f between t h e r e l i a b i l i t y of t h e sample and t h e p e r f o r m a n c e o f t h e method employed. B a l a n c i n g t h i s t r a d e o f f i s s o m e t h i n g t h a t w i l l be l e f t t o t h e e x p e r i e n c e of t h e d e s i g n e r of t h e water q u a l i t y m o n i t o r i n g p r o g r a m . T h e r e a r e , however, s e v e r a l f a c t o r s t h a t s h o u l d be kept i n mind. I t s h o u l d be n o t e d t h a t a l l r e s u l t s have shown 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 was s u p e r i o r t o random s a m p l i n g f o r e v e r y c o m p o s i t e s i z e . The s e l e c t i o n of t h e c o m p o s i t e s i z e , t h e r e f o r e , may not be as c r u c i a l as i t a p p e a r s . I f t h e s i t u a t i o n b e i n g m o n i t o r e d i s p r o n e t o l a r g e e x t r e m e s of s h o r t d u r a t i o n , t h e n t h e number of samples s h o u l d be kept t o a maximum. We have seen t h a t c o m p o s i t e methods a r e v e r y e f f e c t i v e a t f i n d i n g extreme v a l u e s w i t h any c o m p o s i t e s i z e . T h e r e f o r e , i t i s f a r more i m p o r t a n t t o i n c r e a s e t h e p r o b a b i l i t y of s a m p l i n g from t h e s e p e r i o d of h i g h p o l l u t i o n l e v e l s . I f t h e c o s t of t e s t i n g i s much g r e a t e r t h a n t h e c o s t o f s a m p l i n g , t h e n s m a l l r e d u c t i o n s i n t h e number o f t e s t s w i l l p r o d u c e d a marked i n c r e a s e i n t h e r e l i a b i l i t y o f t h e sample. In t h i s s i t u a t i o n i t i s p r o b a b l y s u f f i c i e n t t o c o n s i d e r o n l y a s m a l l r a n g e o f c o m p o s i t e s i z e s . A n o t h e r f a c t o r t o be aware of when s e l e c t i n g t h e c o m p o s i t e s i z e i s t h e e s t i m a t e o f t h e p o p u l a t i o n mean and t h e e s t i m a t e o f t h e p o p u l a t i o n v a r i a n c e . Suppose t h a t N 119 s a m p l e s a r e t a k e n a nd t h a t t h e c o m p o s i t e s i z e m 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 . B r u m e l l e e t a l . [18] h a v e shown t h a t t h e a v e r a g e o f t h e N/m c o m p o s i t e m e a s u r e m e n t s i s an u n b i a s e d e s t i m a t e o f t h e p u p u l a t i o n mean a n d h a s v a r i a n c e « 2/N where ez i s t h e p o p u l a t i o n v a r i a n c e . The a u t h o r s h a v e a l s o shown t h a t an u n b i a s e d e s t i m a t e o f t h e p o p u l a t i o n v a r i a n c e i s g i v e n b y , m N/m _ 2 (6.17) where M, i s t h e measurement o f t h e f i r s t c o m p o s i t e , M 2 i s t h e m easurement o f t h e s e c o n d c o m p o s i t e , a n d s o f o r t h . M i s t h e a v e r a g e o f a l l t h e c o m p o s i t e m e a s u r e m e n t s . B r u m e l l e e t a l . go on t o show t h a t t h e v a r i a n c e o f t h i s e s t i m a t e i s , V a r ( S ' ) = 3 (m-1) a N/m m m (N/m-3)a (N/m-1) (6.18) w h e r e i s t h e f o u r t h c e n t r a l moment. I t c a n be d e m o n s t r a t e d t h a t f o r f i x e d N, (6.18) i s an i n c r e a s i n g f u n c t i o n o f m i n t h e i n t e r v a l [ 0 , N ] . I n f a c t , t h e r a t e o f i n c r e a s e i s e x p o n e n t i a l . T h u s , f o r a f i x e d number o f s a m p l e s , t h e s m a l l e r t h e c o m p o s i t e s i z e , t h e more e f f i c i e n t t h e e s t i m a t e o f t h e v a r i a n c e . V I I . Summary The r e s u l t s i n t h i s t h e s i s have a d i r e c t i mpact on t h e a r e a of water q u a l i t y m o n i t o r i n g . The o b j e c t i v e o f t h e t h e s i s was t h e d e v e l o p m e n t of methods t o e s t i m a t e t h e maximum sample measurement from a s e t o f 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 o f t h e s a m p l e s . The methods were b a s e d on a t h e o r e t i c a l framework c o m s i s t i n g of t h e f o l l o w i n g a s s u m p t i o n s : 1. The p r o c e s s of c o l l e c t i n g s amples and t h e p r o c e s s of o b t a i n i n g sample measurements a r e d i s t i n c t . 2. 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 t h e c o s t of o b t a i n i n g a sample. 3. The sample measurements a r e 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 . I t was seen t h a t t h e o b j e c t i v e and t h e a s s u m p t i o n s were of p a r t i c u l a r r e l e v a n c e t o t h e a r e a of water q u a l i t y m o n i t o r i n g . The a s s u m p t i o n of t h e 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 was f u n d a m e n t a l t o t h e p e r f o r m a n c e of any t e c h n i q u e . The e x i s t e n c e o f t h i s 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 water q u a l i t y m o n i t o r i n g d a t a had been n o t e d by s e v e r a l a u t h o r s . T h i s a s s u m p t i o n l e d t o t h e d e v e l o p m e n t of two e n t i r e l y d i f f e r e n t a p p r o a c h e s . The f i r s t , w h i c h was i n v e s t i g a t e d i n t h i s t h e s i s , were c o m p o s i t e methods. T h i s c l a s s o f t e c h n i q u e s i n v o l v e d a g g r e g a t i n g samples i n t o s e q u e n t i a l g r o u p s , f o r m i n g a c o m p o s i t e sample from e a c h g r o u p , and t h e n m e a s u r i n g t h e c o m p o s i t e s a m p l e s . 1 20 121 T h i s a p p r o a c h was b a s e d on t h e p r e m i s e t h a t i n 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 would 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 u s , t h e c o m p o s i t e measurements gave an i n d i c a t i o n a s t o 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. The s e c o n d c l a s s of t e c h n i q u e s were c a l l e d m a t h e m a t i c a l programming methods. I t was s u g g e s t e d t h a t i f f l u c t u a t i o n s were s m a l l compared t o t r e n d , t h e n i t would be p o s s i b l e t o m o d i f y 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 t o s e a r c h f o r t h e maximum sample measurement. These methods were n o t i n v e s t i g a t e d f u r t h e r i n t h i s t h e s i s . I t was c o n c e d e d t h a t , due t o t h e random n a t u r e of t h e d a t a , no method w h i c h r e s u l t e d i n a s i g n i f i c a n t r e d u c t i o n i n t h e number of t e s t s would f i n d t h e maximum sample measurement w i t h c e r t a i n t y . Thus, t h e r e was a need t o measure e r r o r s . Many measures of e r r o r were c o n s i d e r e d and f i n a l l y f o u r were s e l e c t e d . T h i s g r o u p i n c l u d e d t h e p r o p o r t i o n of 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 was f o u n d , 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 s q u a r e d e r r o r , and t h e mean range e r r o r . The l a t t e r measure o f e r r o r was d e v e l o p e d s p e c i f i c a l l y f o r t h i s s t u d y . The methods d e v e l o p e d i n t h i s t h e s i s were a p p l i e d t o d a t a b a s e d 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 C o l u m b i a . P r e l i m i n a r y a n a l y s i s d e m o n s t r a t e d t h a t t h e s e d a t a met t h e n e c e s s a r y c o n d i 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 . The d a t a were, t h e r e f o r e , 1 22 a p p r o p r i a t e f o r t h e p u r p o s e of t e s t i n g and e v a l u a t i n g t h e methods p r o p o s e d . As p r e v i o u s l y m e n t i o n e d , t h i s t h e s i s i n v e s t i g a t e d c o m p o s i t e methods. A t t e n t i o n was c o n c e n t r a t e d on 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 i s method i n v o l v e d f i r s t a g g r e g a t i n g t h e samples i n t o s e q u e n t i a l g r o u p s o f e q u a l s i z e . 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 o f e a c h sample was p o o l e d t o form a c o m p o s i t e s a mple. The c o m p o s i t e sample w i t h t h e h i g h e s t measurement was t h e n f o u n d and a l l o f t h e base samples t h a t formed t h i s c o m p o s i t e were t e s t e d . The maximum measurement o b s e r v e d became t h e e s t i m a t e of t h e o v e r a l l maximum sample measurement. Assum i n g no s t r u c t u r e t o t h e d a t a , one c o u l d do no b e t t e r t h a n t o sample r a n d o m l y and use t h e maximum o b s e r v e d v a l u e . T h i s i s t h e t y p i c a l s i t u a t i o n . 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 compared t o t h i s t e c h n i q u e . I t was o b s e r v e d 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 was much s u p e r i o r t o random s a m p l i n g w i t h r e s p e c t t o a l l f o u r m e asures of e r r o r i n t erms of b o t h t h e e r r o r l e v e l and t h e v a r i a b i l i t y of t h e e r r o r . In a d d i t i o n , 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 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 t h e p o p u l a t i o n mean. A n o t h e r i m p o r t a n t a d v a n t a g e d e m o n s t r a t e d by 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 t h e a b i l i t y t o d e t e c t e x t r e m e v a l u e s . In f a c t , t h e l a r g e r t h e e x t r e m e , t h e more l i k e l y i t was t o be d e t e c t e d . , , . ! , : : l | When t h e c o m p o s i t e s i z e was n o t a p e r f e c t d i v i s o r of t h e number of s a m p l e s , t h e f i n a l c o m p o s i t e was formed from 123 fewer s a m p l e s . T h i s s i t u a t i o n was r e f e r r e d t o as u n b a l a n c e d c o m p o s i t i n g . I f t h e c o m p o s i t e s i z e was a p e r f e c t d i v i s o r of t h e number o f s a m p l e s , t h e s i t u a t i o n was 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 . A l l e v i d e n c e i n d i c a t e d t h a t u n 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 a v o i d 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 , i t was d e m o n s t r a t e d t h a t t h e y o c c u r i n p a i r s , e a c h o f w h i c h r e s u l t e d i n t h e i d e n t i c a l number of t e s t s b e i n g p e r f o r m e d . The s m a l l e r c o m p o s i t e s i z e i n e a c h p a i r u s u a l l y r e s u l t e d i n s u p e r i o r p e r f o r m a n c e . 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 p e r f o r m e d , t h e b e t t e r t h e p e r f o r m a n c e . F u r t h e r r e s e a r c h i s n e eded t o d e t e r m i n e t h e 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 on t h e p e r f o r m a n c e of c o m p o s i t e methods. T h i s would, i d e a l l y , t a k e t h e form o f 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 r a n g e 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, some i n s i g h t s were g l e a n e d from 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 . The d a t a were formed i n t o g r o u p s a c c o r d i n g t o s i m i l a r i t i e s i n t h e l e v e l of t h e i 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 . T h i s p r o d u c e d t h r e e g r o u p s w i t h 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 a t d i s t i n c t l y d i f f e r e n t l e v e l s . 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 a p p l i e d t o e a c h g r o u p . Not u n e x p e c t e d l y , t h e r e s u l t s d e m o n s t r a t e d t h a t t h e l o w e r 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 , t h e p o o r e r t h e p e r f o r m a n c e . In a d d i t i o n , i t was s e e n 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 p e r f o r m e d as w e l l and i n most c a s e s b e t t e r t h a n random s a m p l i n g even when a p p l i e d t o t h e g r o u p w i t h t h e l o w e s t a v e r a g e a u t o c o r r e l a t i o n 1 24 f u n c t i o n . In p a r t i c u l a r , f o r t h e s m a l l e r c o m p o s i t e s i z e s , 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 b s t a n t i a l l y b e t t e r . T h i s would s u g g e s t 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 a r o b u s t t e c h n i q u e and 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 . I t was a r g u e d t h a t t h e t i m e between samples was an i r r e l e v a n t f a c t o r , t h e i m p o r t a n t p r i n c i p l e b e i n g t h e a u t o c o r r e l a t i o n between s a m p l e s . E m p i r i c a l e v i d e n c e s u p p o r t i n g t h i s p o s i t i o n was p r e s e n t e d . 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 was sampled a t 24 m i n u t e i n t e r v a l s f o r p e r i o d s of one day. Based on t h e r e s u l t i n g 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 and t h e e x p e r i e n c e a c q u i r e d from one m i n u t e s a m p l i n g f o r p e r i o d s o f one h o u r , p e r f o r m a n c e c h a r a c t e r i s t i c 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 were p r e d i c t e d . A l l of t h e s t a t i s t i c s , w i t h i n r e a s o n , a g r e e d w i t h th e h y p o t h e s i z e d b e h a v i o r . S e v e r a l a l t e r n a t i v e c o m p o s i t e methods were p r o p o s e d . Improved p e r f o r m a n c e would be g u a r a n t e e d i f more samples were examined t h a n t h o s e t h a t formed 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 c o m p o s i t e measurement. The l o g i c a l c h o i c e were t h e samples t h a t formed t h e c o m p o s i t e w i t h t h e s e c o n d h i g h e s t measurement. T h i s method was c a l l e d s e c o n d a r y f i r s t o r d e r c o m p o s i t i n g . Improved p e r f o r m a n c e was o b t a i n e d a t t h e e x p e n s e of a d d i t i o n a l t e s t s . T h i s a p p r o a c h c o u l d be e x t e n d e d t o t h e samples t h a t formed t h e c o m p o s i t e w i t h t h e t h i r d h i g h e s t c o m p o s i t e measurement, t h e f o u r t h h i g h e s t c o m p o s i t e measurement, and so on. A c o m p l e t e l y d i f f e r e n t a p p r o a c h was 125 c o n s i d e r e d t h a t would r e d u c e t h e number of t e s t s a t t h e c o s t of a d e c r e a s e i n p e r f o r m a n c e . T h i s was a c c o m p l i s h e d by f i r s t i s o l a t i n g t h e samples t h a t f o r m e d 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 and t h e n 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 t o t h e s e s a m p l e s . Thus, c o m p o s i t i n g was p e r f o r m e d t w i c e . T h i s method was c a l l e d p r i m a r y s e c o n d o r d e r c o m p o s i t i n g . The method c o u l d be e x t e n d e d by i n c r e a s i n g t h e number o f t i m e s c o m p o s i t i n g was p e r f o r m e d . The most p r o m i s i n g a l t e r n a t i v e , however, was a b l e n d i n g o f t h e above two a p p r o a c h e s . I t i s c o n c e i v a b l e t h a t t h r o u g h t h e s e l e c t i o n of more t h a n one c o m p o s i t e f o l l o w e d by f u r t h e r c o m p o s i t i n g t h a t an i n c r e a s e i n p e r f o r m a n c e and a d e c r e a s e i n t h e number of t e s t s c o u l d be o b t a i n e d . F u r t h e r r e s e a r c h 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 . C o m p o s i t e methods o f f e r many a d v a n t a g e s o v e r random s a m p l i n g . I n t r o d u c t i o n o f t h i s a p p r o a c h w i l l r e s u l t i n a r e d u c t i o n i n c o s t s a n d / o r a more r e p r e s e n t a t i v e sample. Extreme v a l u e s a r e more e a s i l y d e t e c t e d . F o r a f i x e d number of t e s t s , t h e e s t i m a t e of t h e sample mean i s more e f f i c i e n 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 n p a r t i c u l a r , a p p e a r s t o be a r o b u s t t e c h n i q u e a s i t p e r f o r m e d b e t t e r t h a n random s a m p l i n g even a t l o w e r l e v e l s of a u t o c o r r e l a t i o n . I n h e r e n t i n c o m p o s i t e methods a r e o t h e r v a l u a b l e p r o p e r t i e s . The number o f t e s t s p e r f o r m e d i s a c o n s t a n t and i s known b e f o r e any of t h e a n a l y s e s . T h i s w o u l d n o t be t h e c a s e f o r m a t h e m a t i c a l programming methods. W i t h knowledge of t h i s i n f o r m a t i o n , t h e c o s t o f t h e m o n i t o r i n g p r o g ram i s 1 26 known a t t h e d e s i g n s t a g e . A n o t h e r a d v a n t a g e i s t h a t i f t h e o b s e r v e d maximum i s i n v i o l a t i o n of a water q u a l i t y s t a n d a r d , t h e n i t i s p o s s i b l e t o expand t h e t e s t i n g t o a l a r g e r n e i g h b o r h o o d o f t h i s sample so t h a t an a s s e s s m e n t of t h e d u r a t i o n o f t h e v i o l a t i o n c a n be made. A l t h o u g h t h e methods p r o p o s e d i n t h i s t h e s i s were a p p l i e d t o t h e s p e c i f i c a r e a of water q u a l i t y m o n i t o r i n g , t h e y were b a s e d on a g e n e r a l t h e o r e t i c a l framework. I t i s c o n j e c t u r e d t h a t t h e s e methods have a p p l i c a b i l i t y beyond w a t e r q u a l i t y m o n i t o r i n g t o any s i t u a t i o n i n w h i c h t h e t h e o r e t i c a l a s s u m p t i o n s a r e met. M o r e o v e r , t h e methods w i l l be e q u a l l y e f f e c t i v e i n s e a r c h i n g f o r t h e minimum sample measurement i f a p p r o p r i a t e . 127 R e f e r e n c e s [1] B e c k e r s C. V., Chamberlain S. G., and G r i m s r u d G. P., Q u a n t i t a t i v e Methods f o r P r e l i m i n a r y D e s i g n of Water  Q u a l i t y S u r v e i l l a n c e Systems, U. S. E n v i r o n m e n t a l P r o t e c t i o n Agency, Report No. EPA-R5-72-001, 1972. [2] B e c k e r s C. V. and C h a m b e r l a i n , S. G., Design of C o s t - E f f e c t i v e Water Q u a l i t y S u r v e i l l a n c e Systems, U. S. E n v i r o n m e n t a l P r o t e c t i o n Agency, Report No. EPA-600/5-74-004, 1974. [3] Chamberlain S. G., Beckers C. V., G r i m s r u d G. P., and S h u l l R. D., " Q u a n t i t a t i v e Methods of P r e l i m i n a r y D esign of Water Q u a l i t y S u r v e i l l a n c e Systems," Water  Resources B u l l e t i n , V o l . 10, No. 2, pp. 199-217, A p r i l 1 974. [4] Ward, R. C , Data A c q u i s i t i o n Systems i n Water Q u a l i t y Management, E n v i r o n m e n t a l P r o t e c t i o n Agency, Report No. EPA-R5-73-014, 1973. [5] Sherwani, J . K. and Moreau, D. H., S t r a t e g i e s f o r Water  Q u a l i t y M o n i t o r i n g , Report No. 107, Water Resources Research I n s t i t u t e , U n i v e r s i t y of N o r t h C a r o l i n a , R a l e i g h , 1975. [6] C u r t i s , W. R., "Sampling f o r Water Q u a l i t y , " P r o c e e d i n g s of the 8th IMR Symposium, September 20-24, pp. 237-244, 1976. 128 [7] Sanders, T. G. and A d r i a n , D. D., "Sampling Frequency f o r R i v e r Q u a l i t y M o n i t o r i n g , " , Water Resources  R e s e a r c h , V o l . 14, No. 4, pp. 569-576, August 1978. [8] Box, G. P. and J e n k i n s , G. M., Time S e r i e s A n a l y s i s :  F o r e c a s t i n g and C o n t r o l , Holden-Day, I n c . , San F r a n c i s c o , C a l i f o r n i a , 1976. [9] L o f t i s , J . C. and Ward, R. C , " S t a t i s t i c a l T r a d e o f f s i n M o n i t o r i n g Network D e s i g n , " P r o c e e d i n g s of the AWRA  Symposium on E s t a b l i s h m e n t of Water Q u a l i t y M o n i t o r i n g  Programs, San F r a n s i s c o , C a l i f o r n i a , June 12-14, 1978. [10] L o f t i s , J . C. and Ward R. C., "Sampling Frequency S e l e c t i o n f o r R e g u l a t o r y Water Q u a l i t y M o n i t o r i n g , " Water Resources B u l l e t i n , V o l . 16, No. 3, pp 501-507, June 1980. [11] L o f t i s , J . C. and Ward R. C , "Water Q u a l i t y M o n i t o r i n g - Some P r a c t i c a l Sampling Frequency C o n s i d e r a t i o n s , " E n v i r o n m e n t a l Management, V o l . 4, No. 6, pp. 521-526, 1980. [12] Ward R. C , L o f t i s , J . C , N i e l s e n , K. S., and Anderson, R. D., " S t a t i s t i c a l E v a l u a t i o n of Sampling F r e q u e n c i e s i n M o n i t o r i n g Networks," J o u r n a l of Water  P o l l u t i o n C o n t r o l F e d e r a t i o n , V o l . 51, No. 9, pp. 2292-2300, Sept. 1979. [13] L o f t i s , J . C. and Ward R. C , " C o s t - E f f e c t i v e S e l e c t i o n of Sampling F r e q u e n c i e s f o r R e g u l a t o r y Water Q u a l i t y M o n i t o r i n g , " Environment I n t e r n a t i o n a l , V o l 3., pp. 297-302, 1980. 129 [14] F u l l e r , W. A., I n t r o d u c t i o n t o S t a t i s t i c a l Time S e r i e s , John W i l e y & Sons, I n c . , T o r o n t o , 1976. [15] A v r i e l , M., N o n l i n e a r Programming: A n a l y s i s and M e t h o d s , P r e n t i c e - H a l l , I n c . , Englewood C l i f f s , New J e r s e y , 1976. [16] Nemetz, P. N. and D r e c h s l e r , H. D., "The Role of E f f l u e n t M o n i t o r i n g i n E n v i r o n m e n t a l C o n t r o l , " Water,  A i r , and S o i l P o l l u t i o n , V o l . 10, pp. 477-497, 1978. [17] Walden, C. C , Howard, T. E., and S h e r i f f , W. J . , "The R e l a t i o n of K r a f t M i l l O p e r a t i n g and P r o c e s s Parameters t o P o l l u t i o n C h a r a c t e r i s t i c s of the M i l l E f f l u e n t s , " P u l p and Paper Maqazine of Canada, 1971. [18] B r u m e l l e . S., Casey, D. B., and Nemetz, P. N., "On E s t i m a t i n g Means and V a r i a n c e s from Composite Samples," Working Paper, 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 , U n i v e r s i t y of B r i t i s h C olumbia, Vancouver, Canada, August, 1982. 1 30 A p p e n d i x A T h i s a p p e n d i x i n c l u d e s T a b l e A.1, T a b l e A.2, T a b l e A.3, T a b l e A.4, and T a b l e A.5. A l l t h e s e t a b l e s f o l l o w t h e i d e n t i c a l f o r m a t . E a c h t a b l e c o n s i s t s of t h r e e p a g e s . Over e a c h column i s a f i e l d i d e n t i f i e r o r v a r i a b l e l a b e l . A c o m p l e t e d e s c r i p t i o n of t h e v a r i a b l e l a b e l s f o r e a c h page f o l l o w s . 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 d e n o t e d by i t s acronym, PFOC. 131 Appendix A Table D e s c r i p t i o n Page 1_ F i e l d Format D e s c r i p t i o n N 14 Number of o b s e r v a t i o n s i n sample INC 14 O b s e r v a t i o n number increment ( the sampled o b s e r v a t i o n s a r e 1, 1+INC, 1 + 2 * I N C . ) M 14 Number of o b s e r v a t i o n s t o group i n t o c o m p osites I SEED 14 Random number seed NVAL 16 Number of s e t s of N o b s e r v a t i o n s used AUTOUB F6.2 Ex c l u d e a l l o b s e r v a t i o n s f o r which the AUTOLB F6.2 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 not between the s e two v a l u e s NTESTM F7.2 Average number of e v a l u a t i o n s PROPCM F9.6 P r o p o r t i o n of times a c t u a l maximum found by PFOC PROPRM F9.6 P r o p o r t i o n of tim e s a c t u a l maximum found by random sample MSECM F12.9 Mean squared e r r o r by PFOC MSECS F12.9 C o r r e s p o n d i n g s t a n d a r d d e v i a t i o n MSERM F12.9 Mean squared e r r o r by random sample MSERS F12.9 C o r r e s p o n d i n g s t a n d a r d d e v i a t i o n MADC F9.6 Maximum a b s o l u t e d e v i a t i o n by PFOC MADR F9.6 Maximum a b s o l u t e d e v i a t i o n by random sample Appendix A ~ Table D e s c r i p t i o n — Page 2 F i e l d Format D e s c r i p t i o n MARECM F9. 6 Mean a b s o l u t e range e r r o r by PFOC MARECS F9. 6 C o r r e s p o n d i n g s t a n d a r d d e v i a t i o n MARERM F9. 6 Mean a b s o l u t e range e r r o r by random sample MARERS F9. 6 C o r r e s p o n d i n g s t a n d a r d d e v i a t i o n AUTO1M F9. 6 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 AUTO1S F9. 6 C o r r e s p o n d i n g s t a n d a r d d e v i a t i o n AUT02M F9. 6 Average second ; o r d e r a u t o c o r r e l a t i o n AUTO2S F9. 6 C o r r e s p o n d i n g s t a n d a r d d e v i a t i o n AUTO3M F9. 6 Average t h i r d o r d e r a u t o c o r r e l a t i o n AUTO3S F9. 6 C o r r e s p o n d i n g s t a n d a r d d e v i a t i o n AUT04M F9. 6 Average f o u r t h o r d e r a u t o c o r r e l a t i o n AUT04S F9. 6 C o r r e s p o n d i n g s t a n d a r d d e v i a t i o n AUT05M AUTO5S F9.6 F9.6 Average f i f t h o r d e r a u t o c o r r e l a t i o n C o r r e s p o n d i n g s t a n d a r d d e v i a t i o n 1 33 A p p e n d i x A T a b l e D e s c r i p t i o n — Page 3 F i e l d Format D e s c r i p t i o n AUT06M F9. 6 Average s i x t h o r d e r a u t o c o r r e l a t i o n AUTO6S F9. 6 C o r r e s p o n d i n g s t a n d a r d d e v i a t i o n AUTO7M F9. 6 Average seventh o r d e r a u t o c o r r e l a t i o n AUTO7S F9. 6 C o r r e s p o n d i n g s t a n d a r d d e v i a t i o n AUTO8M F9. 6 Average e i g h t h i o r d e r a u t o c o r r e l a t i o n AUTO8S F9. 6 C o r r e s p o n d i n g s t a n d a r d d e v i a t i o n AUT09M F9. 6 Average n i n t h o r d e r a u t o c o r r e l a t i o n AUTO9S F9. 6 C o r r e s p o n d i n g s t a n d a r d d e v i a t i o n AUTOOM F9. 6 Average t e n t h o r d e r a u t o c o r r e l a t i o n AUTOOS F9. 6 C o r r e s p o n d i n g s t a n d a r d d e v i a t i o n FILLER 2X DATE 6A4 Date and time of run IRECC 15 Record number from which the MADC was r e t u r n e d IRECR 15 Record number from which the MADR was r e t u r n e d a»cncncncn<na>cncncocr>cncncocncn<j><T>cncncncncn<ncncn2!5?2? 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 U M M M M U M M M U U - ' - ' - ' - ' - ' - " - ' - " 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 U U U N N M U N U U M N M U U U M N U U M U U N M M U U U 88888888888888888888888888888 (0(OU)(0(0(0(0(0<0<OCO(0(0(0(0(04£I<0(0(0(0(0(0(0(0(0(OU>(£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<jiNj^ 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 s D i u i i n u i $ i b u i u i m o a ( 0 ( a t o o < i ) C D < n i i & i i b u u & ~ J O a o 0 > 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~J<ncnGD(O(b~'JGa a i W O W U I 3 1 - A O ( I ! f c ( 0 U 0 ) O I D ^ 0 ) O U I U I - ' f 0 I B I I l ( ! l - O l J l Ocx)&NJNj(jicocna)cn^>aD(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 - 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• > | n l f l l n l C l c u l C ^ ^ ^ ^ ^ s a a o a l a " a l 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 inioininininuoinuoininuoinuouiinininininininininOT C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C N C N C M C N C M C M C M G O 0 0 C D 0 0 O D O D 0 0 0 0 C O 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 — \ \ — S S S N \ \ V \ \ \ w \ \ \ —\ \ \ \ \ \ •v. \ -» •»•»•»•» •» "« sT 1 • » 1 •» -» cs C 9 C S C D 1 3 1 3 C 5 C 3 cs o a a a cs a cs is a C S O cs C 5 cs cs cs « 3 C S C S cs 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 < < < <t < < < < < < < < < < < < < < < < < < < < < < < < < 1 - 1 - 1 - f- 1 - i- (- t-t- t- t- 1- h- 1 - 1 - 1 -< < < < < < < < < < < < < < < < < < < < < < «a < < < < < < to 1 0 to C O C O l / > C O C O to to to to to to to to to to to to to to to to to to to to to to r~ r~ i~- t» f- !•» t- r» r» f- r- r- r- r- r~ fs r- h- t- t- r~ r~ r~ r~ t» f-O in in in in m in in in in m m in in L O L O in in in in m in L O L O in in in L O L O in O C O C O C O C O co C O C O C O C O C O C O C O C O C O C O C O C O C O D C O C O C O C O C O C O C O C O C O C O f- I P ip ip ip ip I P ip ip I P tp ip ip ip I P I P ip I P L P L P L P L P C P I P L P L P L P L P L P L P AU 0 0 0 0 co 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 C O ao C O C O 0 0 0 0 0 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 • f v t ' » T t ' i * J > J - f t | v f t t » f ' » - I ' . t , " - l - / - J v t w - H f ' - t t - f i t atQ^aiatototaiaiaiataiatCiatatatatctotatataiaiaiataiaiatot 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 66666666666666666606600666606 C M C M C N C M C M C M C M C M C N C M C M C M C M C M C M C M C M C M C M C M C M C M C N C M C N C M C M C M C N i n i n i n i n i n i n i n i n L n i n i n i n i n L n i n L O L O i n i n i n i n i n i n L O L n L n i o L ^ I P L P L P L P I P I P C P L P L P L P L P I P L P L P I P I P L P L P L P L P L P L P I P I P C P L P L P L P C P 0 ) C 7 ) C T ) C n C T ) C T } C T ) C * ) C T ) C A C T ) C T ) 0 ) C T > C T ) C T ) C n C * ) C T > C A L ^ 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 n L n L n L n i n i n i n L O i n L n L n i n i n i n i n i n L n L O i n L n u o i n L n i n i n L n L O i n i n C M C N C N C M C M C M C M C N C N C M C M C N C N C N C M C M C M C M C M C M C M C M C M C M C M C N C M C M C M C O C O C O D C O C O D C O C O C O C O C O C O C O C O C O C O C O C O C O C O C O C O C O C O C O C O C O 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 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 M C M C M C N C M C M C N C M C M C M C M C M C M C M C M C N C M C M C M C M C M C M C N C N C M C N C M C M C M 6 6 0 0 6 0 0 6 6 0 6 6 0 6 6 0 0 6 6 0 0 6 6 0 0 6 6 0 6 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 M C N C M C N C M C M C M C M C M C N C M C N C M C N C M C M C N C M C N C M C M C M C N C M C M C N C N C N C N ( O C O C O C O C O C O C O C O C O C O C O C O C O C O C O C O C O C O C O C O C O C O C O C O C O C O C O C O 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 IPIPIPIPtPiPLPLPlPlPIPIPIPIPIPCPIPIPIPLPIPID 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 M C M C M C M C N C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C N C M C M C M 66666666666666666666666666666 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 M C N C M C M C M C M C N C M C M C M C M C M C M C M C M C M C N C M C M C N C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C N C M C M C N C M C M C M C M C M 66 6 6 6666666666666666666666666 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 LPIPIPIPIPIPlPtPIPIPIPIPIPIPiPtPiPlPiPIPIPIPlPtt 0)0>C7>o)cno>cno)C^ L O L n i n L n L O L O L O L n L O L O i n i n L O L O L n L O i n i n L n i o i n i n i n L O i n i n L n L n i n C M C M C M C N C M C M C N C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M O . C M C . C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C N C N C M C M C N C M C M C M 66666666666666666666666666666 0 0 0 0 0 0 0 0 0 0 0 0 C D O D 0 0 0 0 0 0 0 0 C O 0 0 0 0 C O C O 0 0 0 0 0 0 0 0 O D 0 0 C O C O C O 0 0 0 0 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 C M C M C M C N C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C M C N C M C M C M C M C M C M C M C 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 

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