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Applications of multivariate methods to inductively coupled plasma optical emmission spectra Wirsz, Douglas Franklin 1985

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APPLICATIONS OF MULTIVARIATE METHODS TO INDUCTIVELY COUPLED PLASMA OPTICAL EMISSION SPECTRA by DOUGLAS FRANKLIN WIRSZ B.Sc.(Hons.)» Simon Fr a s e r U n i v e r s i t y , 1982 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Department of Chemistry) We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA J u l y 1985 © Douglas F r a n k l i n Wirsz, 1985 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by h i s or her representatives. I t i s understood that copying or publication of t h i s thesis f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. Department Of Chemistry  The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date August 30, 1985 ABSTRACT The u t i l i t y of m u l t i v a r i a t e a n a l y s i s methods f o r t h e q u a l i t a t i v e and q u a n t i t a t i v e a n a l y s i s of complex ICP o p t i c a l e m i s s i o n s p e c t r a i s shown. A p h o t o d i o d e a r r a y i s used t o c o l l e c t d a t a w h i c h a r e s i m p l i f i e d t o a l o w r e s o l u t i o n (approx. 4 A) spectrum f o r a number of knowns and unknowns. The d i g i t i z e d s p e c t r a thus o b t a i n e d are p l o t t e d as p o i n t s i n m u l t i v a r i a t e s p a c e ( p a t t e r n s p a c e ) . The s p e c t r a o f d i f f e r e n t elements appear i n w e l l d e f i n e d r e g i o n s of p a t t e r n space. Q u a l i t a t i v e a n a l y s i s i s a c c o m p l i s h e d by methods of c l u s t e r a n a l y s i s and f a c t o r a n a l y s i s . Q u a n t i t a t i v e a n a l y s i s i s c a r r i e d out u s i n g m u l t i d i m e n s i o n a l w o r k i n g c u r v e s . The p a t t e r n space c o o r d i n a t e system i s t r a n s f o r m e d to g i v e axes c o r r e s p o n d i n g d i r e c t l y t o e l e m e n t a l c o n c e n t r a t i o n s , a l l o w i n g d i r e c t s i m u l t a n e o u s d e t e r m i n a t i o n of e l e m e n t a l c o m p o s i t i o n . The e f f e c t of l i n e i n t e n s i t i e s , d e t e c t o r dynamic range, and m a t r i x e f f e c t s a re a l s o d i s c u s s e d . i i TABLE OF CONTENTS PAGE ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES i v LIST OF FIGURES v CHAPTER I INTRODUCTION 1 1.1 GENERAL CONSIDERATIONS 1 1.2 PLASMAS AS A QUANTITATIVE SPECTRAL SOURCE .. 5 1.3 EVALUATION OF A PHOTODIODE ARRAY AS A MULTICHANNEL SIMULTANEOUS DETECTOR 15 I I THEORY 20 I I I LITERATURE REVIEW 33 3.1 DETAILED REVIEW 33 3.2 SUMMARY OF CONCEPTS FROM THE LITERATURE .... 68 IV EXPERIMENTAL 71 V RESULTS AND DISCUSSION 78 VI CONCLUSIONS 106 BIBLIOGRAPHY 109 APPENDICES 113 A GENERAL TARGET FACTOR ANALYSIS 113 B ERRORS IN THE DETERMINATION OF THE NUMBER OF FACTORS, TARGET TESTING, AND FACTOR LOADINGS 117 C QUICKFACTOR PROGRAM LISTING 120 D APL/MTS TRANSLITERATION TABLE 124 E QUICKFACTOR PROGRAM DESCRIPTION 125 i i i LIST OF TABLES PAGE I Example of a Data M a t r i x 19 I I R e s u l t s of C l u s t e r A n a l y s i s by a M i n i m a l Spanning Tree 26 I I I E i g e n v a l u e s and I n d i c a t o r s o b t a i n e d from QUICKFACTOR f o r 42 S p e c t r a 94 IV B e h a v i o r of I n d i c a t o r s near the C o r r e c t Number of F a c t o r s , F 95 V R e s u l t s of Tar g e t T e s t i n g f o r the Presence of Elements 99 VI E l e m e n t a l C o n c e n t r a t i o n s o b t a i n e d d i r e c t l y from F a c t o r L o a d i n g s 102 i v LIST OF FIGURES PAGE 1 Diagram of an I n d u c t i v e l y Coupled Plasma Torch .. 6 2 Comparison of common S p e c t r o s c o p i c C o n f i g u r a t i o n s . . 12 3 Dendrogram showing 8 E l e m e n t a l C l u s t e r s 23 4 Example of a M i n i m a l Spanning Tree 25 5 T r a n s f o r m a t i o n of the P a t t e r n Space C o o r d i n a t e System (wavelength to element space) 31 6 B l o c k Diagram of the I n s t r u m e n t a t i o n 72 7 S p e c t r a l Window (380 P h o t o d i o d e s ) f o r Cd, C r , Cu, and Fe 74 8 S p e c t r a l Window (380 P h o t o d i o d e s ) f o r Mg, N i , Sr and Zn 75 9 Reduced S p e c t r a of Fe and N i 76 10 Non L i n e a r Map showing 8 E l e m e n t a l C l u s t e r s 79 11 E i g e n v e c t o r P r o j e c t i o n of Working Curves ( e i g e n v e c t o r 1 vs e i g e n v e c t o r 2) 81 12 E i g e n v e c t o r P r o j e c t i o n of Working Curves ( e i g e n v e c t o r 1 vs e i g e n v e c t o r 3) 82 13 E i g e n v e c t o r P r o j e c t i o n of Working Curves ( e i g e n v e c t o r 2 vs e i g e n v e c t o r 3) . 83 14 E i g e n v e c t o r P r o j e c t i o n of Working Curves ( e i g e n v e c t o r 1 vs e i g e n v e c t o r 4) 84 15 E i g e n v e c t o r P r o j e c t i o n of Working Curves ( e i g e n v e c t o r 2 vs e i g e n v e c t o r 4) 85 16 E i g e n v e c t o r P r o j e c t i o n of Working Curves ( e i g e n v e c t o r 3 vs e i g e n v e c t o r 4) 86 17 Non L i n e a r Map of Working Curves 88 18 Examples of Working Curves i n 1, 2 and 3 Dimensions 89 19 R a t i o F u n c t i o n as an I n d i c a t o r 96 v CHAPTER I INTRODUCTION 1.1 GENERAL CONSIDERATIONS I n m o s t m o d e r n c h e m i c a l a n a l y s e s , i n w h i c h d a t a i s c o l l e c t e d and s t o r e d u n d e r c o m p u t e r c o n t r o l , much more r a w d a t a c a n be c o l l e c t e d i n a g i v e n p e r i o d o f t i m e t h a n c a n be i n t e r p r e t e d by e v e n t h e m o s t s k i l l e d i n d i v i d u a l i n t h a t same p e r i o d o f t i m e . A v e r i t a b l e f l o o d o f i n f o r m a t i o n i s o b t a i n e d w i t h d o z e n s o f p a r a m e t e r s w h i c h c o u l d p o s s i b l y a f f e c t a n a n a l y s i s . A t y p i c a l e x a m p l e o f t h i s i s s e e n i n t h e q u a n t i t y o f d a t a w h i c h i t i s p o s s i b l e t o m e a s u r e f r o m a n a n a l y t i c a l i n d u c t i v e l y c o u p l e d p l a s m a . L i t e r a l l y h u n d r e d s o f l i n e s c a n be o b t a i n e d f r o m a s c a n n e d e m i s s i o n s p e c t r u m o f e v e n t h e s i m p l e s t o f s a m p l e s , a p u r e a q u e o u s s o l u t i o n o f a n y o n e o f t h e many e l e m e n t s w h i c h c a n be d e t e r m i n e d by I C P - O E S . F o r many y e a r s , t h e m e t h o d u s e d t o c o n t r o l t h i s f l o o d h a s b e e n t o i g n o r e t h e v a s t m a j o r i t y o f t h e i n f o r m a t i o n c o n t a i n e d i n t h e many l i n e i n t e n s i t i e s a n d f o c u s on o n e o r p e r h a p s t w o l i n e s . T h i s i s a u n i v a r i a t e a p p r o a c h t o a n a l y s i s . I n a u n i v a r i a t e a p p r o a c h , a l l p a r a m e t e r s a r e f i x e d , e x c e p t o n e , w h i c h i s v a r i e d w h i l e a r e s p o n s e i s m e a s u r e d . I n t h i s c a s e , t h e c o n c e n t r a t i o n s i n t h e s a m p l e s m a y v a r y a n d a r e r e p r e s e n t e d by t h e e m i s s i o n i n t e n s i t y o f a s i n g l e l i n e f o r e a c h e l e m e n t o f i n t e r e s t . 1 T h i s t y p e of a n a l y s i s i s commonly a c c o m p l i s h e d w i t h e i t h e r a d i r e c t r e a d e r or a s l e w s c a n n i n g m o n o c h r o m a t o r . The a n a l y s i s l i n e i s n o r m a l l y c h o s e n on t h e b a s i s o f r e a s o n a b l e i n t e n s i t y and freedom from s p e c t r a l o v e r l a p . In many c a s e s t h i s r e s u l t s i n a c o m p r o m i s e s i n c e t h e most i n t e n s e l i n e s a r e i n t e r f e r e d w i t h , so a l e s s i n t e n s e l i n e must be c h o s e n . One means of r e d u c i n g t h e p o s s i b i l i t y o f s p e c t r a l i n t e r f e r e n c e i s t o u s e a h i g h r e s o l u t i o n m o n o c h r o m a t o r t o i s o l a t e t h e l i n e o f i n t e r e s t <1 ,2> , n e v e r t h e l e s s , s e v e r e o v e r l a p s can s t i l l o c c u r . I f t h e q u a l i t a t i v e c o m p o s i t i o n of the sample i s not a l r e a d y known, i t may be v e r y d i f f i c u l t t o c h o o s e a l i n e f o r a n a l y s i s and be s u r e o f f r e e d o m f r o m s p e c t r a l o v e r l a p due t o o t h e r e l e m e n t s i n t h e s a m p l e . I n t e r e l e m e n t c o r r e c t i o n s can be a p p l i e d but the sample must be c o m p l e t e l y c h a r a c t e r i z e d f o r s u c c e s s f u l r e s u l t s . These problems c o u l d be c i r c u m v e n t e d i f a m u l t i v a r i a t e a p p r o a c h i s u s e d . I n m u l t i v a r i a t e a n a l y s i s , a l l t h e s p e c t r a l d a t a ( w a v e l e n g t h s and i n t e n s i t i e s ) a r e u s e d , and the most d i s c r i m i n a t o r y f e a t u r e s are s e l e c t e d . The a b i l i t y t o e v a l u a t e t h e v a r i a n c e o f t h e e n t i r e d a t a s e t a l l o w s t h e d i s c o v e r y of the number of s i g n i f i c a n t parameters w i t h o u t p r i o r k n o w l e d g e or b i a s . I n t h e c a s e o f e m i s s i o n s p e c t r a , t h i s g i v e s the number of components ( e l e m e n t s ) p r e s e n t a t a d e t e c t a b l e c o n c e n t r a t i o n i n the sample. M u l t i v a r i a t e a n a l y s i s has been used i n many a r e a s , w i t h 2 e a c h u s i n g a d i f f e r e n t n o m e n c l a t u r e . The most commonly-a p p l i e d b r a n c h e s a r e p a t t e r n r e c o g n i t i o n and f a c t o r a n a l y s i s . The l i t e r a t u r e i s w e l l e s t a b l i s h e d f o r t h e s e t e c h n i q u e s . I n 1972, K o w a l s k i e t a l <3> were a b l e t o c l a s s i f y the g e o g r a p h i c source of o b s i d i a n samples u s i n g the c o n c e n t r a t i o n s o f t e n e l e m e n t s o b t a i n e d by x - r a y f l u o r e s c e n c e s p e c t r o s c o p y . I n 1978, S a x b u r g e t a l <4> s h o w e d t h a t i t was p o s s i b l e t o d i s c r i m i n a t e b e t w e e n d i f f e r e n t b r a n d s o f w h i s k e y on t h e b a s i s o f g a s c h r o m a t o g r a p h y . M o r e r e c e n t l y , i n 1 9 8 3 , McCue and M a l i n o w s k i <5> u s e d t a r g e t f a c t o r a n a l y s i s on t h e u l t r a v i o l e t s p e c t r u m of u n r e s o l v e d l i q u i d c h r o m a t o g r a p h y f r a c t i o n s to d e t e r m i n e the c o n c e n t r a t i o n of t h r e e i n d i v i d u a l components. R e l a t e d t e c h n i q u e s have a l s o be a p p l i e d to f l u o r e s c e n c e <6,7,8>, f o u r i e r t r a n s f o r m i n f r a r e d <9,10,11,12>, gas c h r o m a t o g r a p h y - m a s s s p e c t r o s c o p y <13>, n u c l e a r m a g n e t i c resonance <14>, and p y r o l y s i s mass s p e c t r o s c o p y <15> to name but a few. Two a p p r o a c h e s may be t a k e n i n t h e s t u d y of s p e c t r a l i n f o r m a t i o n u s i n g m u l t i v a r i a t e t e c h n i q u e s . I n t h e f i r s t , many e l e m e n t a l c o n c e n t r a t i o n s a r e f i r s t d e t e r m i n e d by c o n v e n t i o n a l methods (one l i n e p e r e l e m e n t , e a ch measured s e p a r a t e l y ) . These e l e m e n t a l c o n c e n t r a t i o n s are then used as t h e d a t a f o r f u r t h e r m u l t i v a r i a t e a n a l y s i s . P a t t e r n 3 r e c o g n i t i o n i s t h e n used t o i d e n t i f y c l a s s e s of s a m p l e s (e.g. d i f f e r e n t s o u r c e s of o b s i d i a n ) or f a c t o r a n a l y s i s can be used to d e t e r m i n e the number of c l a s s e s (e.g. the number of d i f f e r e n t g e o g r a p h i c a l s o u r c e s of o b s i d i a n ) . Much of the work p r e v i o u s l y m e n t i o n e d has been o f t h i s t y p e . T h i s a p p r o a c h has a l s o been used w i t h e l e m e n t a l c o n c e n t r a t i o n s o b t a i n e d from ICP s p e c t r a to c l a s s i f y samples of b r a s s <16>. I n t h e s e c o n d a p p r o a c h , t h e w a v e l e n g t h and i n t e n s i t y i n f o r m a t i o n f r o m a l a r g e s p e c t r a l segment i s used f o r a number of d i f f e r e n t samples, w i t h each wavelength g i v i n g a d i s c r e t e i n t e n s i t y measurement. F a c t o r a n a l y s i s a p p l i e d d i r e c t l y t o t h i s s p e c t r a l i n f o r m a t i o n then g i v e s the number of components and t h e i r c o n c e n t r a t i o n s w i t h o u t the need f o r c o n c e r n a b o u t s p e c t r a l o v e r l a p s and i n t e r f e r e n c e s . Such d i r e c t f a c t o r a n a l y s i s of ICP s p e c t r a to d e t e r m i n e e l e m e n t a l c o n c e n t r a t i o n s has not been p r e v i o u s l y a t t e m p t e d . T h i s p a per d e m o n s t r a t e s t h e f e a s i b i l i t y o f t h i s method f o r q u a l i t a t i v e and q u a n t i t a t i v e a n a l y s i s u s i n g a c o n v e n t i o n a l ICP source and an i n e x p e n s i v e low r e s o l u t i o n p o l y c h r o m a t o r based on a photodiode a r r a y d e t e c t o r . 4 1.2 PLASMAS AS A QUANTITATIVE SPECTRAL SOURCE P l a s m a s have commonly been r e f e r r e d t o as t h e f o u r t h s t a t e of matter a f t e r s o l i d , l i q u i d , and gas. Chen <17> has d e f i n e d a p l a s m a as "a q u a s i n e u t r a l gas o f c h a r g e d and n e u t r a l p a r t i c l e s which e x h i b i t s c o l l e c t i v e b e h a v i o r " . The most common p l a s m a used i n a n a l y t i c a l c h e m i s t r y i s t h e i n d u c t i v e l y c o u p l e d argon plasma, i n which a f l o w of argon gas a t a t m o s p h e r i c p r e s s u r e i s i n d u c t i v e l y h e a t e d by an RF f i e l d t o a t e m p e r a t u r e i n t h e v i c i n i t y o f 8000 K. The ICP c o n v e n t i o n a l l y used i n a n a l y t i c a l c h e m i c a l work has t h e f o l l o w i n g d e s i g n . The p l a s m a t o r c h c o n s i s t s of t h r e e c o n c e n t r i c t u b e s ( F i g u r e 1). The outermost tube c o n t a i n s a f l o w of argon gas w h i c h s p i r a l s up t h e t u b e and l e n d s s t a b i l i t y t o t h e a r g o n p l a s m a by p r o d u c i n g a v o r t e x a t t h e t o p of t h e t u b e . The n e x t t u b e c a r r i e s a d d i t i o n a l a r g o n gas, and t h e i n n e r m o s t tube c a r r i e s the sample a e r o s o l , t o be d i s c u s s e d l a t e r . To i n i t i a t e t h e g e n e r a t i o n of a p l a s m a f r o m a r g o n , a n e u t r a l g a s , t h e main gas f l o w i s seed e d w i t h i o n s p r o d u c e d by t h e momentary d i s c h a r g e of a T e s l a c o i l . The gas passes up the t o r c h and r e a c h e s t h e r e g i o n o f t h e l o a d c o i l . R a d i o f r e q u e n c y e n e r g y a t 27 Mhz and a p p r o x i m a t e l y 1.5 KW i s f e d t o t h e l o a d c o i l f r o m an RF g e n e r a t o r t h r o u g h an i m p e d e n c e matching network. The seed e l e c t r o n s from the i n t i a l T e s l a d i s c h a r g e a r e a c c e l e r a t e d by t h e i n t e n s e t i m e v a r y i n g magnetic f i e l d o r i e n t e d a x i a l l y i n s i d e the l o a d c o i l s . The 5 PLASMA 4 > A Neb .< Main f < Aux Figure 1. Diagram of an Inductively Coupled Plasma Torch, 6 a c c e l e r a t i o n of t h e e l e c t r o n s l e a d s t o o hmic h e a t i n g and f u r t h e r i o n i z a t i o n of the argon. The plasma thus generated c o n s i s t s of a r g o n i o n s ( A r + ) and f r e e e l e c t r o n s . Once i n i t i a t e d , i t s u s t a i n s i t s e l f as a w e a k l y i o n i z e d gas a t a t e m p e r a t u r e of 6000 t o 10000 K f o r as l o n g as t h e RF e n e r g y i s s u p p l i e d . An aqueous s a m p l e i s i n t r o d u c e d i n t o t h e p l a s m a by means of a n e b u l i z e r , which c r e a t e s a sample a e r o s o l . T h i s s a m p l e a e r o s o l f l o w s up t h e c e n t r a l t u b e , and t h e c o o l gas punches a c h a n n e l i n t o the hot plasma, g i v i n g i t a t o r o i d a l or doughnut shape n e a r t h e b o t t o m and a c o n i c plume n e a r t h e t o p . As t h e s a m p l e t r a v e l s t h r o u g h t h e p l a s m a i t goes through the s t a g e s of d e s o l v a t i o n , m o l e c u l a r d i s s o c i a t i o n , i o n i z a t i o n and e x c i t a t i o n t o g i v e c h a r a c t e r i s t i c a t o m i c e m i s s i o n s p e c t r a o f a l l t h e e l e m e n t s (M) p r e s e n t i n t h e s a m p l e . Due t o t h e h i g h t e m p e r a t u r e s i n t h e p l a s m a , t h e e m i s s i o n s p e c t r a of the f i r s t i o n i z e d s t a t e (M +) i s commonly seen a l o n g w i t h t h a t of t h e atom. The r a t i o o f i o n i c t o a t o m i c s p e c t r a l i n t e n s i t i e s a r e d i r e c t l y r e l a t e d t o t h e i o n i z a t i o n energy of the element, w i t h those w i t h the l o w e s t i o n i z a t i o n energy g i v i n g the h i g h e s t i o n i c p o p u l a t i o n s and h e n c e t h e most i n t e n s e i o n i c s p e c t r a . I n t h e r e g i o n a p p r o x i m a t e l y 16 mm above the l o a d c o i l (normal a n a l y t i c a l z o ne) t h e e m i s s i o n f r o m t h e s a m p l e i s optimum f o r most elements i n c o n v e n t i o n a l m a t r i c e s , and t h i s i s where s p e c t r a 7 are c o l l e c t e d . The s p a t i a l d i s t r i b u t i o n of i o n i c and a t o m i c s p e c i e s t h r o u g h o u t t h e p l a s m a can be v e r y c o m p l e x , and has been s t u d i e d i n d e t a i l by C a u g h l i n and B l a d e s <18>. D a h l q u i s t and K n o l l <19> e v a l u a t e d many of the common a n a l y t i c a l a p p l i c a t i o n s o f a c o n v e n t i o n a l i n d u c t i v e l y c o u p l e d plasma. The d e t e r m i n a t i o n of the major components (Na, K, P, Ca, Mg) and t r a c e e l e m e n t s ( F e , Cu, Zn, Mn, Pb, Cd, Co, C r , N i , V, T i , A l , S r , and Ba) i n o r c h a r d l e a v e s , b o v i n e l i v e r a nd s o i l e x t r a c t s was f o u n d t o c o m p a r e f a v o r a b l y i n a c c u r a c y , p r e c i s i o n and s e n s i t i v i t y w i t h o t h e r m u l t i e l e m e n t t e c h n i q u e s , s u c h a s n e u t r o n a c t i v a t i o n a n a l y s i s , e n e r g y d i s p e r s i v e X r a y f l u o r e s c e n c e , and f l a m e a t o m i c a b s o r p t i o n s p e c t r o m e t r y . For purposes of m u l t i e l e m e n t a n a l y t i c a l measurements, a s e t o f s t a n d a r d e x p e r i m e n t a l and o p e r a t i o n a l p a r a m e t e r s , i n c l u d i n g o b s e r v a t i o n o f t h e e m i s s i o n f r o m t h e c e n t r a l a n a l y t i c a l c h a n n e l a t 16 mm above t h e l o a d c o i l has been found t o r e p r e s e n t an e x c e l l e n t compromise y i e l d i n g adequate d e t e c t i o n l i m i t s f o r most e l e m e n t s . Thus one s e t of p a r a m e t e r s may be u s e d f o r a l l e l e m e n t s . T h i s i s i n c o n t r a s t t o o t h e r e m i s s i o n and a b s o r p t i o n methods, where optimum c o n d i t i o n s may vary w i d e l y from element to element. The ICP has many o t h e r a d v a n t a g e s o v e r f l a m e s and f u r n a c e s . The c o n t i n u u m and a r g o n l i n e i n t e n s i t y i s g r e a t e s t n e a r t h e l o a d c o i l . I t i s v e r y l o w i n t h e n o r m a l a n a l y t i c a l zone, and t h i s l e a d s t o good s i g n a l t o background 8 r a t i o s . The d e t e c t i o n l i m i t s a c h i e v e d are g e n e r a l l y b e t t e r t h a n f l a m e s o r s p a r k s , and e q u i v a l e n t t o f u r n a c e a t o m i c a b s o r p t i o n i n many cases. L i m i t s range from 0.1 to 10 p a r t s per b i l l i o n , w i t h e x c e l l e n t a c c u r a c y and p r e c i s i o n (approx. 1%). The a n a l y t e r e g i o n i s surrounded by hot o p t i c a l l y t h i n g a s e s . T h i s g i v e s v e r y l i t t l e s e l f a b s o r p t i o n w i t h i n t h e p l a s m a , and as a r e s u l t , w o r k i n g c u r v e s can be l i n e a r o v e r up to 5 o r d e r s of magnitude. The wide dynamic range a l l o w s f o r the s i m u l t a n e o u s d e t e r m i n a t i o n of major, minor and t r a c e components, and samples can thus be i n t r o d u c e d w i t h l i t t l e or no d i l u t i o n . The ICP s o u r c e i s v e r y s t a b l e compared t o f l a m e s and s p a r k s , t h u s r e d u c i n g t h e n e e d f o r f r e q u e n t r e s t a n d a r d i z a t i o n . S i n c e t h e p l a s m a i s g e n e r a t e d w i t h o u t e l e c t r o d e s , t h e r e i s no p o s s i b i l i t y o f e l e c t r o d e c o n t a m i n a t i o n which may be found i n a r c and spark methods. I n t e r e l e m e n t e f f e c t s , a l s o known as m a t r i x e f f e c t s , i n t e r f e r e w i t h t h e l i n e a r i t y o f t h e i n t e n s i t y v s c o n c e n t r a t i o n r e l a t i o n s h i p , and a r e due t o t h e p r e s e n c e o f com p o n e n t s o t h e r t h a n t h e a n a l y t e i n t h e s a m p l e . The f o r m a t i o n o f r e f r a c t o r y c o m p o u n d s o r o x i d e s , and c o m p l e x a t i o n w i t h m a t r i x components can l e a d to a decrease i n e m i s s i o n i n t e n s i t y . T h i s i s a m a j o r p r o b l e m i n f l a m e s . I n t h e I C P , t h e h i g h t e m p e r a t u r e l e a d s t o c o m p l e t e 9 d i s s o c i a t i o n and a t o m i z a t i o n . The i n e r t ( A r ) a t m o s p h e r e a l s o a s s i s t s i n t h e p r e v e n t i o n of o x i d e f o r m a t i o n . As an e x a m p l e , c o m p l e x a t i o n w i t h p h o s p h o r u s or a l u m i n u m i s not seen s i n c e t h e s e c o m p l e x e s a r e d i s s o c i a t e d i n t h e p l a s m a b e f o r e r e a c h i n g the a n a l y t i c a l zone. The presence of e a s i l y i o n i z a b l e elements (such as the a l k a l i m e t a l s ) c a n c a u s e s h i f t s i n t h e i o n i z a t i o n e q u i l i b r i u m i n f l a m e s s i n c e t h e y i n c r e a s e t h e e l e c t r o n p o p u l a t i o n s i g n i f i c a n t l y . The h i g h e l e c t r o n d e n s i t y a l r e a d y p r e s e n t i n t h e a r g o n p l a s m a t e n d s t o b u f f e r t h e e f f e c t s of e a s i l y i o n i z a b l e e l e m e n t s i n t h e a n a l y t e a e r o s o l , so i o n i z a t i o n i n t e r f e r e n c e s are g r e a t l y reduced i n the ICP. The h i g h t e m p e r a t u r e of t h e p l a s m a a l s o a l l o w s many e n e r g y l e v e l s t o be p o p u l a t e d i n each e l e m e n t , l e a d i n g t o l i n e r i c h e m i s s i o n s p e c t r a . W i t h many s p e c t r a l l i n e s a v a i l a b l e f o r a n a l y s i s , t h e r e a r e of c o u r s e p r o b l e m s of s p e c t r a l o v e r l a p . One method to r e s o l v e t h i s problem tends towards h i g h e r r e s o l u t i o n i n hopes of i s o l a t i n g o v e r l a p p e d l i n e s . T h i s a p p r o a c h r e q u i r e s much e x p e n s i v e e q u i p m e n t i n c l u d i n g l a r g e s t a t e o f t h e a r t m o n o c h r o m a t o r s . The a p p r o a c h t a k e n i n t h i s t h e s i s i s t o i n s t e a d l o o k a t t h e w e a l t h of s p e c t r a l i n f o r m a t i o n c o n t a i n e d i n the spectrum as a w h o l e , and t o use p a t t e r n r e c o g n i t i o n methods t o e x t r a c t q u a l i t a t i v e and q u a n t i t a t i v e i n f o r m a t i o n f r o m c o m p l e x multicomponent s p e c t r a . T h e r e a r e s e v e r a l m o n o c h r o m a t o r and p o l y c h r o m a t o r 10 c o n f i g u r a t i o n s p r e s e n t l y u s e d f o r m u l t i e l e m e n t d e t e r m i n a t i o n s . S e q u e n t i a l s c a n n i n g s p e c t r o m e t e r s u s e a s i n g l e monochromator which has o n l y one e x i t s l i t ( F i g u r e 2a). The l i g h t f r o m t h e p l a s m a p a s s e s t h r o u g h t h e e n t r a n c e s l i t , i s d i s p e r s e d by a g r a t i n g , and t h e n p a s s e s t h r o u g h t h e e x i t s l i t t o a d e t e c t o r , which i s u s u a l l y a p h o t o m u l t i p l i e r tube, but may a l s o be a s o l i d s t a t e d e t e c t i n g d e v i c e s u c h as a p h o t o d i o d e . T h i s s y s t e m i s c a p a b l e of m e a s u r i n g s p e c t r a l i n t e n s i t y a t o n l y one w a v e l e n g t h a t a t i m e . The g r a t i n g i s r o t a t e d t o s e l e c t t h e w a v e l e n g t h o f i n t e r e s t . F o r a m u l t i e l e m e n t d e t e r m i n a t i o n , i n t e n s i t i e s a t s e v e r a l w avelengths must be measured s e q u e n t i a l l y , u s u a l l y w i t h o n l y one l i n e f o r each e l e m e n t . T h i s s y s t e m can be a c c u r a t e l y w a v e l e n g t h c a l i b r a t e d u s i n g a m e r c u r y lamp t o s u p p l y a s t a n d a r d spectrum. The need to measure s e q u e n t i a l l y , w i t h the a s s o c i a t e d w a i t i n g t i m e w h i l e the g r a t i n g i s r o t a t e d to each new w a v e l e n g t h , l i m i t s the speed w i t h which s p e c t r a may be o b t a i n e d . O n l y v e r y r a r e l y i s an e n t i r e s p e c t r u m c o l l e c t e d by slow c o n t i n u o u s s c a n n i n g of the monochromator b e c a u s e of t h i s t i m e l i m i t a t i o n . Thus, t h e a n a l y s t must a l r e a d y know where t o l o o k f o r s p e c t r a l l i n e s , s i n c e he i s l i m i t e d to spot checks of d i s c r e t e wavelengths. Erroneous r e a d i n g s can e a s i l y r e s u l t i f a s p e c t r a l i n t e r f e r e n c e i s not a c c o u n t e d f o r , or a b a c k g r o u n d c o r r e c t i o n i s not p r o p e r l y 11 a ) r diepereiv* element. ->0 detector source Figure 2. Comparison of common Spectroscopic Configurations a) slew scanning monochromator b) direct reader c) photodiode array 12 done. A v a r i a t i o n on t h i s monochromator i n v o l v e s the use of a Paschen-Runge mount, a l l o w i n g the g r a t i n g to d i s p e r s e a l a r g e p o r t i o n of the spectrum onto the i n s i d e s u r f a c e of the monochromator. T h i s i s c a l l e d a d i r e c t reading spectrometer ( F i g u r e 2 b ) . T h i s i s f a s t e r than the s e q u e n t i a l s c a n n i n g s p e c t r o m e t e r by a f a c t o r of the number of m u l t i p l e e x i t s l i t s used. The g r a t i n g i s not r o t a t e d , but remains f i x e d . The d e t e c t o r s thus r e c o r d i n t e n s i t i e s a t s e t w a v e l e n g t h s , and i t may not e a s i l y be s l e w e d to measure d i f f e r e n t w a v e l e n g t h s s h o u l d the need a r i s e . C a r e f u l c o n s i d e r a t i o n must be t a k e n to choose the p o s i t i o n of the e x i t s l i t s to measure a l l the e l e m e n t s l i k e l y to be p r e s e n t , and a t the same t i m e a t t e m p t to a v o i d l i n e s which a r e s p e c t r a l l y o v e r l a p p e d . There i s a l s o a l i m i t a t i o n on the minimum d i s t a n c e between e x i t s l i t s because of the p h y s i c a l s i z e of a p h o t o m u l t i p l i e r tube. The d i r e c t reader s u f f e r s from the same s u s c e p t i b i l i t y to s p e c t r a l i n t e r f e r e n c e s and background as the s e q u e n t i a l scanner s i n c e only spot checks of d i s c r e t e wavelengths are done, and the m a j o r i t y of the spectrum and i t s i n f o r m a t i o n i s ignored. A newer and more v e r s a t i l e method uses a c o n t i n u o u s array of photodiodes i n the e x i t f o c a l plane, r a t h e r than a s e r i e s o f d i s c r e t e d e t e c t o r s ( F i g u r e 2 c ) . An e n t i r e continuous window of spectrum can be c o l l e c t e d and s t o r e d i n computer memory, on d i s k , or p l o t t e d i n a matter of seconds. 13 Thus, i t i s as f a s t as the d i r e c t r e a d e r s , but c o l l e c t s much more i n f o r m a t i o n . The p o s s i b i l i t y of s p e c t r a l i n t e r f e r e n c e s can be c h e c k e d by u s i n g n o t one but s e v e r a l w a v e l e n g t h s t o d e t e r m i n e each e l e m e n t . The b a c k g r o u n d can a l s o be more a c c u r a t e l y c o r r e c t e d f o r . The i d e a o f u s i n g s e v e r a l w a v e l e n g t h s f o r t h e d e t e r m i n a t i o n o f one e l e m e n t can be t a k e n one s t e p f u r t h e r i f p a t t e r n r e c o g n i t i o n and f a c t o r a n a l y s i s methods of d a t a a n a l y s i s a r e e m p l o y e d . I f t h e s e methods a r e u s e d , e v e r y w a v e l e n g t h i n t e n s i t y measurement c o n t r i b u t e s t o a q u a n t i t a t i v e multicomponent d e t e r m i n a t i o n . The i n f o r m a t i o n i n each s p e c t r u m i s f u l l y u t i l i z e d . The s u i t a b i l i t y o f p h o t o d i o d e a r r a y s as a m u l t i c o m p o n e n t s i m u l t a n e o u s d e t e c t o r w i l l now be d i s c u s s e d i n more d e t a i l . 14 1.3 EVALUATION OF A PHOTODIODE ARRAY AS A MULTICHANNEL SIMULTANEOUS DETECTOR T a l m i and S i m p s o n <20> e v a l u a t e d t h e p e r f o r m a n c e of sca n n i n g photodiode a r r a y s i n s p e c t r o s c o p i c a p p l i c a t i o n s . A R e t i c o n S - s e r i e s d e v i c e was e x a m i n e d . The a s p e c t r a t i o ( h e i g h t t o w i d t h ) f o r each e l e m e n t i s a p p r o x i m a t e l y 100:1, i d e a l f o r measuring the i n t e n s i t y of an image from a narrow monochromator s l i t . The di o d e s a r e spaced 25 um a p a r t f o r a t o t a l o f 1024 d i o d e s , g i v i n g an e n t i r e a r r a y o n l y 25 mm i n l e n g t h . The p h o t o d i o d e a r r a y was p l a c e d i n t h e e x i t f o c a l p l a n e o f a number o f raonochromators f o r t h e e v a l u a t i o n of performance. A common p r o b l e m a s s o c i a t e d w i t h vacuum t u b e based d e v i c e s i s r e a d o u t l a g , r e s u l t i n g i n an i n a b i l i t y t o c o m p l e t e l y r e a d o u t a l a t e n t s i g n a l w i t h a s i n g l e s c a n . A s o l i d s t a t e d e v i c e s u c h as t h e p h o t o d i o d e a r r a y does n o t s u f f e r f r o m t h i s l a g . Even a f t e r s a t u r a t i o n , a r e s i d u a l s i g n a l ( l a g ) of o n l y a p p r o x i m a t e l y 1% i s observed. A problem w i t h many im a g i n g a r r a y s i s t h a t of bloo m i n g , or c h a r g e s p i l l o v e r t o a d j a c e n t p i x e l s . T h i s l e a d s t o an a p p a r e n t w i d e n i n g o f s a t u r a t e d p e a k s , and a l o s s o f weak l i n e s i n t h e w i n g s of i n t e n s e l i n e s . A g a i n , p h o t o d i o d e a r r a y s p e r f o r m w e l l , w i t h very l i t t l e b looming. W i t h a peak e x c e e d i n g s a t u r a t i o n by a f a c t o r of 20, and a s t r o n g / w e a k peak r a t i o of 1000:1, p i x e l c r o s s t a l k was below 1% as l i t t l e as 15 d i o d e s away, and was not d e t e c t a b l e when the i n t e n s i t y 15 r a t i o was 500:1. A n o t h e r c o n c e r n when d e t e c t i n g a r r a y s a r e used i n t h e e x i t f o c a l p l a n e i s t h a t o f g e o m e t r i c a c c u r a c y and w a v e l e n g t h r e g i s t r a t i o n . The a b s o l u t e p o s i t i o n of two l i n e s on t h e a r r a y was f o u n d t o r e m a i n c o n s t a n t o v e r a p e r i o d e x c e e d i n g one h o u r . A f t e r r o t a t i o n o f t h e g r a t i n g , t h e s e p a r a t i o n of the two l i n e s on the a r r a y remained c o n s t a n t w i t h i n one d i o d e . A l g o r i t h m s f o r c a l i b r a t i o n u s i n g c e n t r o i d f i n d i n g t e c h n i q u e s would reduce the u n c e r t a i n t y i n p o s i t i o n to a s m a l l f r a c t i o n of a diode. N o i s e i s another i m p o r t a n t parameter. A l l a r r a y s have a f i x e d p a t t e r n due t o s t r a y c a p a c i t i v e c o u p l i n g w h i c h causes v a r i a t i o n s of up to 1% of the s a t u r a t e d l e v e l . T h i s f i x e d p a t t e r n i s e a s i l y removed by d i g i t a l s u b t r a c t i o n . The f i x e d p a t t e r n n o i s e i s t e m p e r a t u r e dependent, and appears to have a 1/f c h a r a c t e r ( f l i c k e r n o i s e ) . T e m p e r a t u r e c o n t r o l can reduce the v a r i a t i o n s due to t e m p e r a t u r e f l u c t u a t i o n s . D ark c u r r e n t i s a l s o t e m p e r a t u r e d e p e n d e n t , and i s reduced g r e a t l y by r e d u c i n g the t e m p e r a t u r e . Dark c u r r e n t l i m i t s the dynamic range by r a i s i n g the n o i s e f l o o r towards the s a t u r a t i o n c e i l i n g d u r i n g l o n g i n t e g r a t i o n t i m e s . The d a r k c u r r e n t however i s r e d u c e d by a f a c t o r of 2 f o r e a ch d e c r e a s e of 6.7 d e g r e e s C. T e m p e r a t u r e r e d u c t i o n and s t a b i l i z a t i o n may be a c h i e v e d u s i n g a P e l t i e r c o o l e r i n c l o s e c o n t a c t w i t h the back of the a r r a y . 16 The dynamic range of a photodiode a r r a y may be d e f i n e d i n t e r m s o f a s i m u l t a n e o u s i n t r a s p e c t r a l d y n a m i c r a n g e , w h i c h i s t h e r a t i o o f t h e most i n t e n s e and l e a s t i n t e n s e s p e c t r a l f e a t u r e s measureable s i m u l t a n e o u s l y . T h i s can be g r e a t l y d e g r a d e d by v e i l i n g g l a r e , w h i c h i s c a u s e d by i n t e r n a l r e f l e c t i o n s w i t h i n the photodiode a r r a y e n c l o s u r e . A s i m u l t a n e o u s i n t r a s p e c t r a l dynamic range of g r e a t e r than 1000 can be a t t a i n e d i f v e i l i n g g l a r e i s kept to a minimum. T h i s may be done by r e m o v i n g t h e p h o t o d i o d e a r r a y ' s p r o t e c t i v e q u a r t z window, w h i c h a l s o i m p r o v e s r e s p o n s e i n the u l t r a v i o l e t . The l i n e a r i t y o f t h e r e s p o n s e o f a s i n g l e e l e m e n t o f t h e a r r a y was t e s t e d u s i n g a 21 g r a d i e n t n e u t r a l d e n s i t y f i l t e r . The a r r a y r e s p o n s e was l i n e a r o v e r a r a n g e o f 10,000. M i l a n o and Kim <21> used a r a p i d s c a n n i n g p h o t o d i o d e a r r a y s p e c t r o m e t e r ( R e t i c o n 256 e l e m e n t ) t o o b t a i n w h o l e b l o o d s p e c t r a i n t h e r a n g e 512 t o 621 nm. The r a p i d s c a n n i n g a b i l i t y of the a r r a y a l l o w s f o r many s p e c t r a to be c o l l e c t e d and c o - a d d e d t o i n c r e a s e t h e s i g n a l t o n o i s e r a t i o . The a p p a r e n t i n c r e a s e i n s i g n a l t o n o i s e r a t i o i s very c l o s e to the t h e o r e t i c a l l i m i t of the square r o o t of N, where N i s the number of scans. A l e a s t squares f i t a c r o s s t h e e n t i r e s p e c t r u m c o l l e c t e d ( 2 0 0 p o i n t s ) a l l o w s d e t e r m i n a t i o n of each of the hemoglobin s p e c i e s HHb, Hb02» and HbCO w i t h i m p r o v e d p r e c i s i o n o v e r a 3 w a v e l e n g t h f i t 17 w h i c h i s commonly us e d . W a v e l e n g t h r e g i s t r a t i o n can be a problem i f the s p e c t r a are taken a f t e r the p o l y c h r o m a t o r has been r e p o s i t i o n e d . T h i s may be a c c o u n t e d f o r by r e f e r e n c e t o any s t a n d a r d s p e c t r u m t a k e n b o t h b e f o r e and a f t e r r e p o s i t i o n i n g . An a p p r o p r i a t e w a v e l e n g t h s h i f t may be a p p l i e d t o keep a l l s p e c t r a p r o p e r l y r e g i s t e r e d p r i o r t o permanent d i g i t a l s t o r a g e . S h i f t s as s m a l l as a f r a c t i o n of a d i o d e can be c a r r i e d o u t , i f r e q u i r e d , by i n t e r p o l a t i o n . Thus, r a p i d d e t e r m i n a t i o n ( a p p r o x i m a t e l y 30 seconds) can be c a r r i e d o ut u s i n g a s c a n n i n g s p e c t r o m e t e r and a l e a s t squares f i t , w i t h no l o s s i n a c c u r a c y or p r e c i s i o n . The d a t a o b t a i n e d f r o m a p h o t o d i o d e a r r a y i s a l s o s u i t a b l e f o r t r e a t m e n t by p a t t e r n r e c o g n i t i o n . I n f a c t i t can v e r y e a s i l y be put i n t h e f o r m o f a d a t a m a t r i x of w a v e l e n g t h i n t e n s i t i e s ( T a b l e I ) f o r e a c h o f s e v e r a l s a m p l e s , a l l o w i n g s i m u l t a n e o u s m u l t i e l e m e n t m u l t i s a m p l e d e t e r m i n a t i o n s . 18 TABLE I Example of a_ Data M a t r i x Data v e c t o r s (pure elements and m i x t u r e s ) F e a t u r e s ( w a v e l e n g t h i n t e n s i t i e s ) Cu U V W X Fe N i 19 CHAPTER I I THEORY The t r e n d i n s p e c t r o s c o p y i n t h e p a s t few y e a r s has been t o w a r d h i g h e r r e s o l u t i o n t o a v o i d p r o b l e m s w i t h background and s p e c t r a l o v e r l a p . For example, Boumans and V r a k k i n g <22> i n v e s t i g a t e d t h e e f f e c t o f l i n e w i n g s on t h e l e v e l o f b a c k g r o u n d i n l i n e r i c h ICP s p e c t r a . They f o u n d t h a t the wings extend beyond 1-2 nm from i n t e n s e l i n e s . An a v e r a g e w i n g i n t e n s i t y f u n c t i o n c o u l d be used t o q u a n t i f y t h e e n h a n c e m e n t o f t h e b a c k g r o u n d , s i n c e w i n g s f r o m d i f f e r e n t l i n e s and d i f f e r e n t elements had s i m i l a r shapes. T h i s approach adopted u l t r a - h i g h r e s o l u t i o n as a method t o s e p a r a t e l i n e s , compute l i n e w i d t h s and l i n e wings, and thus b e t t e r c h a r a c t e r i z e the background and i t s dependence on the presence of o t h e r elements. P a t t e r n r e c o g n i t i o n approaches t h i s i n an e n t i r e l y d i f f e r e n t way. The p r e s e n c e o f i n c r e a s e d background i n r i c h s p e c t r a i s accounted f o r i n the t r a i n i n g s e t , s i n c e t h e s e l i n e w i n g s a r e p r e s e n t i n b o t h s t a n d a r d and sample. There i s no need t o ask f o r u l t r a - h i g h r e s o l u t i o n , and i n f a c t g o i n g t o l o w e r r e s o l u t i o n a c t u a l l y speeds up the a n a l y s i s , because the m a t h e m a t i c a l m a t r i c e s to be d e a l t w i t h a re s m a l l e r , w i t h l i t t l e or no adverse e f f e c t s on a c c u r a c y . The u n i v a r i a t e one l i n e a p p r o a c h r e q u i r e s i n t i m a t e knowledge of the c o n t r i b u t i o n s t o the background a t the w a v e l e n g t h of i n t e r e s t . W i t h p a t t e r n r e c o g n i t i o n , a l l 20 c o m p o n e n t s a r e d e t e r m i n e d s i m u l t a n e o u s l y , w i t h t h e i n t e r a c t i o n of background l e v e l s being accounted f o r i n the process. A number of e x c e l l e n t reviews and summaries of pa t t e r n r e c o g n i t i o n and f a c t o r a n a l y s i s are a v a i l a b l e <23,24,25,26, 27,28,29,30,31,32,33>. E s p e c i a l l y recommended are Wold et a l i n Kowalski <33> and Malinowski and Howery <27>. B a s i c t h e o r y w i l l now be o u t l i n e d w i t h r e f e r e n c e to a p p l i c a t i o n s i n emission s p e c t r a . E a c h d i g i t i z e d s p e c t r u m c o n s i s t s o f a s e t of i n t e n s i t i e s over a range of wavelengths. The spectrum can be r e p r e s e n t e d by a data v e c t o r , w i t h each element of the d a t a v e c t o r b e i n g a m e a s u r e d i n t e n s i t y a t a s p e c i f i c wavelength. A data vector composed of n measurements may be p l o t t e d i n an n-dimensional space. In p a t t e r n r e c o g n i t i o n terminology, each dimension may be r e f e r r e d to as a f e a t u r e , and the n-dimensional space as pa t t e r n space. Each spectrum taken w i l l r e duce to a s i n g l e p o i n t i n p a t t e r n space. D i s p l a c e m e n t s a l o n g the c o o r d i n a t e axes d e f i n i n g t h i s p a t t e r n space correspond to measured i n t e n s i t i e s at each of the wavelengths considered. These co o r d i n a t e axes w i l l be c a l l e d w a v e l e n g t h axes. T h i s l a b e l emphasizes t h a t each a x i s i s r e l a t e d to a d i f f e r e n t w a v e l e n g t h . The u n i t s of displacement along the a x i s are i n t e n s i t y . When a number of s p e c t r a are p l o t t e d i n p a t t e r n space, a number of p o i n t s are produced t h r o u g h o u t p a t t e r n space. 21 The r e l a t i o n s h i p s between the s e p o i n t s can y i e l d v a l u a b l e i n f o r m a t i o n about the s p e c t r a being considered. The n a t u r a l c l u s t e r i n g of p o i n t s i n p a t t e r n space s h o u l d be e x p e c t e d to show which s p e c t r a are very s i m i l a r and which are q u i t e d i f f e r e n t . Massart et a l <34> have used a c l u s t e r i n g program, MASLOC, to a t t e m p t to c l a s s i f y and i d e n t i f y i r o n m e t o e r i t e s . The d a t a v e c t o r s they used c o n t a i n e d the t r a c e e l e m e n t a l c o n c e n t r a t i o n s f o r each m e t e o r i t e . The c l a s s i f i c a t i o n s o b t a i n e d agreed very w e l l with the assignment known by other methods. C l u s t e r i n g of s t r u c t u r a l l y s i m i l a r o r g a n i c compounds has been seen when dat a v e c t o r s c o n t a i n i n g d i g i t i z e d i n f r a r e d s p e c t r a were p l o t t e d i n p a t t e r n space <12>. Two methods of c l u s t e r a n a l y s i s <25> are commonly used. The f i r s t i s h i e r a r c h i c a l c l u s t e r i n g . The r o u t i n e HIER i n ARTHUR accomplishes t h i s by generating a matrix of E u c l i d e a n d i s t a n c e s between a l l p o s s i b l e p a i r s of p o i n t s i n p a t t e r n space. T h i s i n f o r m a t i o n i s used to g e n e r a t e a c o n n e c t i o n dendrogram, a h i e r a r c h y of p o i n t s o r d e r e d by the r e l a t i v e d i s t a n c e s between p o i n t s ( F i g u r e 3). The v e r t i c a l a x i s i s l a b e l l e d the s i m i l a r i t y l e v e l , with l e s s s i m i l a r groupings h a v i n g a l o w e r l e v e l of s i m i l a r i t y . L a r g e jumps i n s i m i l a r i t y l e v e l then i n d i c a t e the b o u n d a r i e s of the c l u s t e r s . As an example, f o r 42 pure s p e c t r a of any one of 8 e l e m e n t s , the jumps i n s i m i l a r i t y are c l e a r , and the s e t of s p e c t r a f o r each element form d i s t i n c t c l u s t e r s . 22 0 0.2 C O L l J 0.4 5 0.6 h 0 0 0.8 .0 Cd Zn Cu Sr 0 0.2 0.4 0.6 - 0.8 .0 Cr Mg Fe Ni F i g u r e 3. Dendrogram showing 8 d i s t i n c t e l e m e n t a l c l u s t e r s , 23 The second method i s the g e n e r a t i o n of a m i n i m a l s p a n n i n g t r e e . The aim of t h i s method i s to m i n i m i z e the sum of the d i s t a n c e s of l i n e s connecting a l l p o i n t s together i n a s i n g l e " t r e e " (Figure 4). The longest branches of t h i s tree then correspond to l i n k s between c l u s t e r s . A maximum d i s t a n c e i s s e t f o r the b r a n c h e s , and i f t h i s d i s t a n c e i s exceeded, the branch i s "pruned" and a s e p a r a t e c l u s t e r i s generated. R e f e r r i n g back to our example, f o r an 8 element system, 8 d i s t i n c t c l u s t e r s are again found ( t a b l e I I ) . T h u s , s p e c t r a of d i f f e r e n t e l e m e n t s a p p e a r i n d i s t i n c t l y d i f f e r e n t c l u s t e r s i n p a t t e r n space. Any unknown sp e c t r u m may be i d e n t i f i e d by c l u s t e r a n a l y s i s as l o n g as two r e s t r i c t i o n s have been met. The element must have a p r e v i o u s l y c h a r a c t e r i z e d p o s i t i o n f o r i t s c l u s t e r . T h i s can be done by p r e v i o u s l y p l o t t i n g the p o s i t i o n s i n p a t t e r n space f o r s p e c t r a of any of the p o s s i b l e elements. T h i s can be c a l l e d a t r a i n i n g s e t . A l s o , the unknown must be of the same c o n c e n t r a t i o n as the known sample, i n order to maintain the same e u c l i d e a n d i s t a n c e s i n p a t t e r n space. These are r a t h e r severe r e s t r i c t i o n s s i n c e the unknown i s r e q u i r e d to be a pure s o l u t i o n of standard c o n c e n t r a t i o n . F o r t u n a t e l y , a l l these r e s t r i c t i o n s can be r e l e a s e d by the a p p r o p r i a t e use of other p a t t e r n r e c o g n i t i o n and f a c t o r a n a l y s i s methods. Dendrograms and m i n i m a l s p a n n i n g t r e e s a l l o w the a n a l y s t to d e t e c t c l u s t e r s i n p a t t e r n space. However, one 24 F E A T U R E 2 Figure A. Example of a Minimal Spanning Tree with the longest branches "pruned". 25 TABLE II R e s u l t s of C l u s t e r A n a l y s i s by a M i n i m a l Spanning Tree 8 C l u s t e r s d e t e c t e d C l u s t e r 1 — NIB,NIE,NID,NIC,NIG,NIF,NIH . C l u s t e r 2 — CRB,CRD,CRC,CRA,CRE C l u s t e r 3 — MGA,MGE,MGC,MGB,MGD C l u s t e r 4 — ZNE,ZND,ZNB,ZNC,ZNA C l u s t e r 5 — CUE,CUB,CUC,CUA,CUD C l u s t e r 6 — CDB,CDE,CDD,CDB,CDA C l u s t e r 7 — SRA,SRB,SRE,SRC,SRD C l u s t e r 8 — FEB,FEE,FEA,FED,FEC 26 encounters problems when t r y i n g to v i s u a l i z e p a t t e r n space, s i n c e p a t t e r n space i s N - d i m e n s i o n a l , w h i l e humans can v i s u a l i z e a maximum of 3 dimensions at a time. Two methods fo r v i s u a l i z i n g p a t t e r n space are w e l l developed <32>. Eigenvector p r o j e c t i o n s are e s s e n t i a l l y a 2-dimensional window i n t o a m u l t i d i m e n s i o n a l space. p a t t e r n space can be i n i t i a l l y d e fined i n terms of a coor d i n a t e system where the coo r d i n a t e axes represent wavelengths. Eigenvector a n a l y s i s of p a t t e r n space r e s u l t s i n a new c o o r d i n a t e system where the c o o r d i n a t e axes are e i g e n v e c t o r s . These e i g e n v e c t o r s are s e l e c t e d on the b a s i s of the amount of variance spanned i n the d i m e n s i o n d e f i n e d by the e i g e n v e c t o r . The f i r s t e i g e n v e c t o r spans the maximum p o s s i b l e v a r i a n c e i n one d i m e n s i o n . The second e i g e n v e c t o r i s s e l e c t e d to span the maximum r e m a i n i n g v a r i a n c e o r t h o g o n a l t o t h e f i r s t e i g e n v e c t o r . The p r o c e s s c o n t i n u e s , w i t h each subsequent eigenvector being p e r p e n d i c u l a r to a l l those already chosen u n t i l the e n t i r e p a t t e r n space i s spanned. N e i g e n v e c t o r s are r e q u i r e d to span N dimensional space. The f i r s t two e i g e n v e c t o r s d e f i n e a pl a n e c o n t a i n i n g the maximum variance i n two dimensions. A p r o j e c t i o n of a l l p o i n t s onto t h i s plane then gives a two dimensional window i n t o p a t t e r n space where the m a j o r i t y of the po i n t s w i l l be spread out, s i n c e the variance i s maximized. Other windows i n t o p a t t e r n space may be viewed by i n s t e a d s e l e c t i n g the f i r s t and t h i r d or second and t h i r d e i g e n v e c t o r s , or a 27 c o m b i n a t i o n of any two e i g e n v e c t o r s . The N d i m e n s i o n a l p a t t e r n space d e f i n e d by w avelength a x e s i s t r a n s f o r m e d t o an e i g e n v e c t o r s p a c e d e f i n e d by e i g e n v e c t o r axes. E i g e n v e c t o r p r o j e c t i o n s of c o m b i n a t i o n s of t h e f i r s t f o u r e i g e n v e c t o r s f r o m our e x a m p l e ( F i g u r e s l l to 16) g i v e s e v e r a l d i s t i n c t windows through which to view the N d i m e n s i o n a l space. These e i g e n v e c t o r p r o j e c t i o n s w i l l be c o n s i d e r e d i n g r e a t e r d e t a i l i n the d i s c u s s i o n . A second method f o r v i e w i n g p a t t e r n space i s non l i n e a r mapping. A non l i n e a r map i s a c o m p r e s s i o n of p a t t e r n space i n t o t w o d i m e n s i o n s b r o u g h t a b o u t w i t h a m i n i m u m o f d i s t o r t i o n i n t h e i n t e r p o i n t d i s t a n c e s . T h i s amounts t o m i n i m i z i n g an e r r o r f u n c t i o n r e s u l t i n g from the d i s t o r t i o n of i n t e r p o i n t d i s t a n c e s . T h i s i s most e a s i l y u n derstood by c o n s i d e r i n g a l l data p o i n t s to be connected by s p r i n g s . As p a t t e r n s p a c e i s s l o w l y c o m p r e s s e d , some s p r i n g s w i l l s t r e t c h and o t h e r s p r i n g s w i l l compress. When p a t t e r n space i s f i n a l l y c o m p r e s s e d t o two d i m e n s i o n s , t h e s p r i n g s w i l l a l l have c o m p r e s s e d or s t r e t c h e d so as t o m i n i m i z e t h e e n e r g y s t o r e d i n t h e e n t i r e s y s t e m o f s p r i n g s . A s i d e e f f e c t o f t h i s c o m p r e s s i o n p r o c e s s i s t h a t a s t r a i g h t l i n e i n p a t t e r n s p a c e may a p p e a r c u r v e d i n a non l i n e a r map ( F i g u r e 17). T h i s non l i n e a r map w i l l a l s o be c o n s i d e r e d i n g r e a t e r d e t a i l i n the d i s c u s s i o n . Once a c l e a r i d e a of the s t r u c t u r e of p a t t e r n space i s a t t a i n e d , f u r t h e r f a c t o r a n a l y s i s <23>, i n v o l v i n g a more 28 d e t a i l e d e x a m i n a t i o n of the amount of v a r i a n c e r e p r e s e n t e d by each e i g e n v e c t o r , can g i v e a d d i t i o n a l u s e f u l i n f o r m a t i o n . E s s e n t i a l l y 100% o f t h e v a r i a n c e i s c o n t a i n e d i n t h e f i r s t few e i g e n v e c t o r s . The number of e i g e n v e c t o r s c o n t a i n i n g s i g n i f i c a n t v a r i a n c e r e p r e s e n t t h e number o f f a c t o r s p r e s e n t , F. T h i s method of d e t e r m i n i n g t h e number of f a c t o r s from the number of s i g n i f i c a n t e i g e n v e c t o r s i s w e l l e s t a b l i s h e d . The e i g e n v e c t o r a n a l y s i s i t s e l f g i v e s t h e number of e l e m e n t s p r e s e n t i n an unknown d i r e c t l y . There are numerous methods f o r d e t e r m i n i n g the c u t o f f between the s i g n i f i c a n t e i g e n v e c t o r s ( i n t o t a l r e p r e s e n t i n g the number of e lements) and the r e s i d u a l e i g e n v e c t o r s ( w h i c h r e p r e s e n t random n o i s e i n t h e s p e c t r a ) . These w i l l n o t be d i s c u s s e d h e r e i n d e t a i l . The i n t e r e s t e d r e a d e r i s r e f e r r e d t o a number of u s e f u l r e f e r e n c e s <27,35,36>. The r e s i d u a l e i g e n v e c t o r s can be d i s c a r d e d , r e s u l t i n g i n a r e d u c t i o n i n d i m e n s i o n a l i t y , and an a c c o m p a n y i n g r e d u c t i o n i n random n o i s e . The e r r o r removed by t h i s procedure i s termed e x t r a c t e d e r r o r . Some n o i s e i s s t i l l p r e s e n t i n the F r e m a i n i n g e i g e n v e c t o r s , and t h e e r r o r a s s o c i a t e d w i t h t h i s n o i s e i s t e r m e d imbedded e r r o r <35>. The r e m a i n i n g e i g e n v e c t o r s , a l s o c a l l e d a b s t r a c t f a c t o r s , span t h e same s p a c e spanned by t h e r e a l f a c t o r s ( t h e e l e m e n t s p r e s e n t i n t h e s o l u t i o n s ) , b u t do n o t 29 correspond with them. They are i n s t e a d l i n e a r combinations of the r e a l f a c t o r s . A c o o r d i n a t e t r a n s f o r m a t i o n from a b s t r a c t f a c t o r space (eigenvector space), to a r e a l f a c t o r space (element space) where each a x i s corresponds d i r e c t l y t o t h e c o n c e n t r a t i o n of one and o n l y one e l e m e n t , i s d e s i r e d . Both the r e d u c t i o n of d i m e n s i o n a l i t y ( e x t r a c t i o n of e r r o r ) and the t r a n s f o r m a t i o n to r e a l f a c t o r space (element space) can be c a r r i e d out by t a r g e t f a c t o r a n a l y s i s . A t a r g e t f a c t o r a n a l y s i s program, QUICKFACTOR, was w r i t t e n i n APL. The e x t e n s i v e m a t r i x m a n i p u l a t i o n a b i l i t i e s of t h i s computer language a l l o w the e n t i r e program to be contained i n j u s t 140 l i n e s . The m a t h e m a t i c s of t a r g e t f a c t o r a n a l y s i s are o u t l i n e d i n a book by M a l i n o w s k i and Howery <27>. In t a r g e t f a c t o r a n a l y s i s , a s e t of p o s s i b l e f a c t o r s ( t e s t v e c t o r s r e p r e s e n t i n g t h e p u r e s p e c t r a f o r e a c h element) are t e s t e d i n d i v i d u a l l y f o r t h e i r presence i n the d a t a m a t r i x . The s u c c e s s f u l t e s t v e c t o r s then c o n s t i t u t e a t r a i n i n g s e t which d e f i n e s a t r a n s f o r m a t i o n to element space. When a l l F f a c t o r s have been found, a t r a n s f o r m a t i o n of a l l data p o i n t s from the reduced (F-dimensional ) a b s t r a c t f a c t o r space to the element space i s done. In f a c t o r a n a l y s i s t e r m i n o l o g y t h i s i s termed the c o m b i n a t i o n s t e p . F i g u r e 5 shows s c h e m a t i c a l l y the e n t i r e t r a n s f o r m a t i o n 30 r o MIXTURE FEATURE1 r o or o I-o UJ > 2 UJ CO r > — LU Cu <-> I E i Si Ol m i MIXTURE EIGENVECTOR 1 Ni ^ . -j> 30 ppm Fe^ov" •Fe Figure 5. Transformation of the Pattern Space Coordinate System (wavelength space to element space) 31 p r o c e s s f r o m w a v e l e n g t h s p a c e t o e l e m e n t s p a c e . A r e c o n s t r u c t e d c o l u m n m a t r i x i s o b t a i n e d c o n t a i n i n g t h e d i s p l a c e m e n t s a l o n g e a c h r e a l f a c t o r a x i s f o r each d a t a p o i n t . These a r e r e f e r r e d to as f a c t o r l o a d i n g s . A f a c t o r l o a d i n g c o r r e s p o n d s t o t h e c o n c e n t r a t i o n o f a s p e c i f i e d e l e m e n t i n each s o l u t i o n . An e v a l u a t i o n o f t h e e r r o r s i n the f a c t o r l o a d i n g s i s a l s o done. E l e m e n t a l c o n c e n t r a t i o n s and t h e i r e s t i m a t e d e r r o r a r e o b t a i n e d s i m u l t a n e o u s l y f o r a l l s o l u t i o n s ( d a t a p o i n t s i n p a t t e r n space) of i n t e r e s t . 32 CHAPTER III LITERATURE REVIEW 3.1 DETAILED REVIEW The development of m u l t i v a r i a t e techniques as trac e d i n the l i t e r a t u r e g i v e s i n s i g h t i n t o the problems p r e v i o u s l y encountered and the s o l u t i o n s found. Although the f i e l d s i n w h i c h t h e s e a p p l i c a t i o n s a r e f o u n d a r e d i v e r s e , t h e u n i v e r s a l i t y of p a t t e r n r e c o g n i t i o n methods imply that a l l these v a r i a t i o n s are a l s o a p p l i c a b l e to emission s p e c t r a . In most c a s e s , the t h e o r y has e v o l v e d w i t h the need f o r m o d i f i e d methods to d e a l w i t h new s i t u a t i o n s . Theory and a p p l i c a t i o n are thus i n t i m a t e l y entwined, and the c l e a r e s t p i c t u r e i s o b t a i n e d by t r a c i n g the development of both c o n c u r r e n t l y . In 1965, B l a c k b u r n <37> s u g g e s t e d a l e a s t s q u a r e s method f o r determining the components of a complex gamma ray spec t r u m . At t h a t t i m e , d i r e c t s o l u t i o n of s i m u l t a n e o u s equations using matrices was d i f f i c u l t because of the la r g e amount of computer t i m e and memory r e q u i r e d f o r m a t r i x i n v e r s i o n . A number of s i m p l i f i c a t i o n s were made, m o s t l y i n v o l v i n g the e v a l u a t i o n of determinants. 400 channels from a s e t of gamma r a y s p e c t r a were u s e d i n t h e f i t . An important f a c t i l l u s t r a t e d by t h i s paper was that the s i g n a l i n each channel was j u s t the l i n e a r sum of the s i g n a l s from each component, so l i n e a r m a t r i x a l g e b r a c o u l d be used to 33 o b t a i n a l e a s t s q u a r e s f i t . Good f i t s were o b t a i n e d o n l y when t h e number of s t a n d a r d s (and t h e c omponents t h e y r e p r e s e n t e d ) matched or e x c e e d e d t h e number of e l e m e n t s a c t u a l l y p r e s e n t i n the unknowns. I f too few s t a n d a r d s were u s e d , t h e f i t s were u n a c c e p t a b l e . I t was n o t e d t h a t v e r y s m a l l peaks c o u l d be d e t e c t e d (e.g. a weak 600 count Cr peak under a 60,000 c o u n t Hg p e a k ) , c o n f i r m i n g t h e p r e s e n c e o f c e r t a i n e l e m e n t s even when s e v e r e o v e r l a p s e x i s t e d . The g e n e r a l a p p l i c a b i l i t y o f t h i s l e a s t s q u a r e s method t o any t y p e o f c o m p l e x s p e c t r a was r e c o g n i z e d , but so were t h e l i m i t a t i o n s o f c o r e memory a v a i l a b l e i n c o m p u t e r s a t t h a t t i m e . I n 1970, K a n k a r e <38> f u r t h e r d e v e l o p e d a m a t r i x a p p r o a c h t o a l e a s t s q u a r e s f i t . A l s o d i s c u s s e d was t h e p o s s i b i l i t y o f s m o o t h i n g t h e d a t a u s i n g f a c t o r a n a l y s i s p r i o r t o a l e a s t s q u a r e s d e t e r m i n a t i o n . A b s o r b a n c e s a t a t o t a l of 27 wavelengths were measured f o r 17 s o l u t i o n s a t 5 nm i n t e r v a l s from 230 to 360 nm. The components p r e s e n t i n s o l u t i o n were t h e v a r i o u s b i s m u t h c h l o r i d e c o m p l e x e s f r o m B i ^ + t h r u B i C l g - ^ * The c o n c e n t r a t i o n s f o r each o f t h e s e components were then used to d e t e r m i n e f o r m a t i o n c o n s t a n t s . F a c t o r a n a l y s i s was found to be u s e f u l f o r p r e l i m i n a r y data a n a l y s i s , and g a v e r e s u l t s f o r f o r m a t i o n c o n s t a n t s c o n s i s t e n t w i t h o t h e r w o r k e r s . I t was s u g g e s t e d t h a t a l a r g e number of w a v e l e n g t h s be u s e d , p r e f e r a b l y s p a c e d a t e q u a l i n t e r v a l s . T h i s i n t u i t i v e l y makes s e n s e , s i n c e a 34 l a r g e r number o f w a v e l e n g t h s g i v e more i n f o r m a t i o n . W i t h more i n f o r m a t i o n , a more a c c u r a t e r e s u l t can be expected. I n 1 9 7 1 , Hugus and E l - A w a d y <39> d i s c u s s e d t h e d e t e r m i n a t i o n of the number of s p e c i e s p r e s e n t i n a system through the use of m a t r i x rank a n a l y s i s . At any w a v e l e n g t h , t h e a b s o r b a n c e was t h e sum o f t h e a b s o r b a n c e s o f each s p e c i e s i n s o l u t i o n . A m a t r i x c o n s i s t e d o f t h e measured a b s o r b a n c e s a t i w a v e l e n g t h s f o r j s o l u t i o n s . T h i s g e n e r a l l y l e a d t o a r e c t a n g u l a r m a t r i x . M u l t i p l i c a t i o n of t h i s d ata m a t r i x by i t s t r a n s p o s e gave a square m a t r i x . In t h e a b s e n c e o f e x p e r i m e n t a l e r r o r s t h e r a n k o f t h e m a t r i x w o u l d be e q u a l t o t h e number o f i n d e p e n d e n t l y a b s o r b i n g s p e c i e s i n s o l u t i o n . I f the data were e n t i r e l y e r r o r f r e e , n e i g e n v e c t o r s and e i g e n v a l u e s would be found, c o r r e s p o n d i n g t o t h e n a b s o r b i n g s p e c i e s . E r r o r s a l w a y s e x i s t , and t h i s l e a d t o i e i g e n v e c t o r s ( o r j , i f j was l e s s t h a n i ) . The r a n k o f t h e m a t r i x w o u l d t h e n be t h e same as i t s o r d e r . Only n e i g e n v a l u e s a re s t a t i s t i c a l l y s i g n i f i c a n t , r e f l e c t i n g r e a l components, and the r e m a i n i n g i - n e i g e n v a l u e s r e f l e c t o n l y n o i s e . A problem a r o s e i n d e t e r m i n i n g the proper v a l u e f o r n. C o m p a r i s o n of each e i g e n v a l u e w i t h t h e s q u a r e r o o t of i t s v a r i a n c e was i n v e s t i g a t e d as one p o s s i b l e c r i t e r i o n . T h i s i n d i c a t o r was ambiguous, i n d i c a t i n g e i t h e r two or t h r e e components f o r a system of 38 s o l u t i o n s a t 9 wavelengths. A r e s i d u a l f u n c t i o n ( o b t a i n e d from the r e s i d u a l m a t r i x a f t e r 35 t h e e x t r a c t i o n of each e i g e n v e c t o r ) was f o u n d t o g i v e much b e t t e r r e s u l t s , i n d i c a t i n g c l e a r l y t h a t o n l y 2 components were p r e s e n t i n the t e s t s e t . The v a l u e of c h i - s q u a r e d was a l s o computed, and p r o v i d e d another r e l i a b l e i n d i c a t o r when used w i t h a s y s t e m o f 8 s o l u t i o n s a t 8 w a v e l e n g t h s . I t i s a p p a r e n t f r o m t h i s t h a t t h e r e i s no one i n d i c a t o r w h i c h i s r e l i a b l e i n a l l c a s e s . I n s t e a d , many i n d i c a t o r f u n c t i o n s s h o u l d be e v a l u a t e d t o p r o v i d e a b e t t e r i d e a o f t h e number of components i n a system u n t i l an unambiguous i n d i c a t o r f o r a p a r t i c u l a r case i s found. I n 1972, K o w a l s k i e t a l <3> used p a t t e r n r e c o g n i t i o n methods t o i n v e s t i g a t e t h e p o s s i b l e i d e n t i f i c a t i o n of a r c h e o l o g i c a l a r t i f a c t s u s i n g t r a c e e l e m e n t d a t a . The c o n c e n t r a t i o n s of each of 10 e l e m e n t s were d e t e r m i n e d by c o n v e n t i o n a l X ray F l u o r e s c e n c e s p e c t r o s c o p i c methods f o r 45 s o u r c e ( s a m p l e s o f c o n f i r m e d o r i g i n ) and 29 a r t i f a c t ( u n k n o w n ) s a m p l e s . A l l s a m p l e s w e r e p l o t t e d i n a 10 d i m e n s i o n a l s p a c e ( d e f i n e d by t h e t e n e l e m e n t a l c o n c e n t r a t i o n s ) . The d i f f i c u l t i e s o f v i s u a l i z i n g 10 d i m e n s i o n a l space, and methods of r e d u c i n g the graphs t o two d i m e n s i o n s w e r e d i s c u s s e d . The v i s u a l d i s p l a y o f m u l t i d i m e n s i o n a l data s e t s by r e d u c t i o n to a t w o - d i m e n s i o n a l non l i n e a r map was shown f o r t h e s a m p l e s . The n a t u r a l c l u s t e r i n g o f s i m i l a r s a m p l e s was e v i d e n t . F o u r d i s t i n c t c l u s t e r s were f o u n d , c o r r e s p o n d i n g t o t h e f o u r g e o g r a p h i c r e g i o n s f r o m w h i c h t h e s o u r c e s came. U n a m b i g u o u s 36 i d e n t i f i c a t i o n of most a r t i f a c t samples was p o s s i b l e . Many methods of d e t e r m i n a t i o n of c l a s s were used. H y p e r p l a n e s e p a r a t i o n s were used to d e f i n e the d i f f e r e n t volumes of m u l t i d i m e n s i o n a l space corresponding to the four c l a s s e s . A K nearest neighbour r u l e could a l s o be used. O v e r a l l , four c l a s s i f i c a t i o n methods were used, with no one method showing o b v i o u s s u p e r i o r i t y over the o t h e r s . The p o s s i b i l i t y of a u t o m a t i c c l a s s i f i c a t i o n u s i n g r u l e s r i g i d l y s e t by a knowledgeable s c i e n t i s t was c l e a r l y demonstrated. As new t e c h n i q u e s of o b t a i n i n g d a t a were d e v e l o p e d , methods of a n a l y s i s were adapted to t h e s e t e c h n i q u e s . In 1977, Antoon et a l <A0>, used l e a s t s q u a r e s c u r v e f i t t i n g with a set of F o u r i e r transform i n f r a r e d s p e c t r a of polymer systems. The advent of computer systems which c o u l d r o u t i n e l y and e f f i c i e n t l y g e n e r a t e d i f f e r e n c e s p e c t r a g r e a t l y a s s i s t e d i n the development of these techniques. A s t r a i g h t f o r w a r d l e a s t squares f i t of standard s p e c t r a of p-o- and m-xylene s o l u t i o n s to complex m i x t u r e s p e c t r a was done. The bes t s p e c t r a l r e g i o n i n terms of minimum e r r o r was a l s o i n v e s t i g a t e d . G r e a t e r a c c u r a c y i n t h e de t e r m i n a t i o n of s u b t r a c t i o n c o e f f i c i e n t s was p o s s i b l e with a l e a s t s q u a r e s f i t over an e n t i r e r e g i o n of sp e c t r u m as opposed to t h a t o b t a i n e d from the a b s o r p t i o n of a s i n g l e band or wavelength. I t was suggested that the i n c l u s i o n of " i n t e r a c t i o n s p e c t r a " , based on s p e c t r a l m o d i f i c a t i o n s of 37 one component by t h e p r e s e n c e o f a n o t h e r , i n a s t a n d a r d s p e c t r a l l i b r a r y f o r cu r v e f i t t i n g would a l l o w the e x t e n t of i n t e r a c t i o n t o be q u a n t i f i e d . As of 1977, the a p p l i c a b i l i t y of l e a s t squares methods was w e l l e s t a b l i s h e d . The u s e o f f a c t o r a n a l y s i s i n c h e m i s t r y was s t i l l b e i n g d e v e l o p e d , w i t h t h e m a j o r f o c u s s h i f t i n g t o a d e s i r e t o d e v e l o p more s t r i n g e n t r u l e s f o r d e t e r m i n i n g t h e number o f components i n t h e p r e s e n c e o f random e r r o r o r n o i s e . The g e n e r a l e f f e c t o f e r r o r s on f a c t o r a n a l y s e s <35> and t h e e f f e c t o f e r r o r on t h e d e t e r m i n a t i o n of the number of f a c t o r s <36> was d i s c u s s e d by M a l i n o w s k i . I n r e a l data c o n t a i n i n g e x p e r i m e n t a l e r r o r , the number of f a c t o r s ( e i g e n v a l u e s and e i g e n v e c t o r s ) i s al w a y s g r e a t e r t h a n t h e number o f r e a l f a c t o r s , n. The f i r s t n e i g e n v e c t o r s c o n t a i n t h e t r u e f a c t o r s a l o n g w i t h some imb e d d e d e r r o r . The r e m a i n i n g e i g e n v e c t o r s r e p r e s e n t e x p e r i m e n t a l e r r o r o n l y . The e r r o r i s p a r t i t i o n e d i n t o i m b edded e r r o r , i n t i m a t e l y a s s o c i a t e d w i t h t h e f i r s t n e i g e n v e c t o r s , and t h u s unremoveab1e, and t h e e x t r a c t e d e r r o r , a s s o c i a t e d e n t i r e l y w i t h the r e m a i n i n g e i g e n v e c t o r s . The d i m e n s i o n a l i t y o f t h e d a t a can be r e d u c e d t o t h a t r e p r e s e n t e d by t h e f i r s t n e i g e n v e c t o r s , w i t h an accompanying removal of the e x t r a c t e d e r r o r c o n t a i n e d i n the r e m a i n i n g e i g e n v e c t o r s . The r e a l e r r o r i s a Pythagorean sum of t h e imbedded and e x t r a c t e d e r r o r . The r e a l e r r o r i s e v a l u a t e d t h rough i t r e l a t i o n s h i p to the r e s i d u a l s t a n d a r d 38 d e v i a t i o n (they are e q u i v a l e n t ) f o r the s u c c e s s i v e r e m o v a l of e i g e n v e c t o r s . When t o o many e i g e n v e c t o r s have been removed, the c a l c u l a t e d r e a l e r r o r w i l l r i s e , s i n c e not o n l y i s n o i s e b e i n g removed, but a l s o i n f o r m a t i o n r e l a t i n g t o the r e a l f a c t o r s p r e s e n t . Thus, a jump i n t h e c a l c u l a t e d r e a l e r r o r i n d i c a t e s the border between r e a l f a c t o r s and f a c t o r s due t o n o i s e ( e x p e r i m e n t a l e r r o r ) . A number o f f u n c t i o n s can e x p l i c i t l y i n d i c a t e t h e number of r e a l f a c t o r s p r e s e n t i n a data s e t . The imbedded e r r o r f u n c t i o n was shown to c o r r e c t l y i n d i c a t e f o u r f a c t o r s i n an a r t i f i c i a l d a t a s e t w i t h a r t i f i c i a l n o i s e . I n r e a l d a t a , the imbedded e r r o r f u n c t i o n s s h o u l d show a l e v e l l i n g o f f o r a minimum a t t h e c o r r e c t number o f f a c t o r s . The f a c t o r i n d i c a t o r f u n c t i o n ( I N D , s e e a p p e n d i x B) was g e n e r a l l y more s e n s i t i v e , g i v i n g a d e f i n i t e minimum a t the c o r r e c t number of f a c t o r s . The v a l u e of the c a l c u l a t e d r e a l e r r o r a t the c o r r e c t number of f a c t o r s agreed w e l l w i t h the magnitude of the e r r o r known to be p r e s e n t i n an a r t i f i c i a l d a t a s e t . I t a l s o gave an u n b i a s e d e s t i m a t e o f t h e e r r o r p r e s e n t i n r e a l d a t a . T h e s e i n d i c a t o r s w e r e u s e d by M a l i n o w s k i on t h e d a t a c o n t a i n e d i n t h e p a p e r by Hugus and E l - A w a d y <39>, and t h e r e s u l t s g e n e r a l l y a g r e e d , w i t h one e x c e p t i o n . They r e p o r t e d 3 f a c t o r s f o r a m a t r i x c o n s i s t i n g o f 8 m e t h y l r e d s o l u t i o n s measured a t 8 w a v e l e n g t h s . The p o s s i b i l i t y of a f o u r t h component was not su p p o r t e d by t h e i r 39 i n d i c a t o r s (the number of r e s i d u a l s g r e a t e r than t h r e e t i m e s the e s t i m a t e d v a r i a n c e , s t a n d a r d e r r o r i n the e i g e n v a l u e , or t h e c h i - s q u a r e d t e s t ) w h i c h w e r e a l l d e p e n d e n t on a p r e l i m i n a r y e s t i m a t e of t h e r e a l e r r o r . I n c o n t r a s t , t h e f a c t o r i n d i c a t o r f u n c t i o n , IND, an e m p i r i c a l f u n c t i o n independent of any e s t i m a t e of the e r r o r , showed a minimum a t 4 f a c t o r s . The p r e d i c t e d r e a l e r r o r f o r t h i s number of f a c t o r s a g r e e d w e l l w i t h an e x p e c t e d e r r o r o f 0.0005 t o 0.0015 absorbance u n i t s . T h i s f u r t h e r i l l u s t r a t e s t h e i m p o r t a n c e of u s i n g s e v e r a l i n d i c a t o r f u n c t i o n s , as where one may be ambiguous or m i s l e a d i n g , a consensus among a l l f u n c t i o n s may r e v e a l a c l e a r e r e s t i m a t e of the number of f a c t o r s . I n y e t a n o t h e r e x a m p l e o f a u s e a b l e i n d i c a t o r , W i n d i g et a l <15> used the SPSS ( S t a t i s t i c a l Package f o r the S o c i a l S c i e n c e s ) p a c k a g e f o r f a c t o r and d i s c r i m i n a n t a n a l y s i s o f p y r o l y s i s mass s p e c t r a of a number of complex samples. The samples, i n c l u d i n g f u n g i and b a c t e r i a , were a n a l y z e d t o f i n d r e l a t i o n s h i p s b e t w e e n s t r u c t u r a l f e a t u r e s and s p e c t r a l f e a t u r e s . These s t r u c t u r a l f e a t u r e s c o u l d then be used f o r f u r t h e r c l a s s i f i c a t i o n . The component axes found c o u l d be r e l a t e d t o s t r u c t u r a l f e a t u r e s i n c l u d i n g a p e n t o s e a x i s , a 6-deoxyhexose a x i s , and a p r o t e i n a x i s . A g e n e r a l s e l e c t i o n of o n l y t h o s e e i g e n v e c t o r s w i t h e i g e n v a l u e s g r e a t e r t h a n u n i t y ( u s i n g n o r m a l i z e d s p e c t r a f o r t h e f a c t o r a n a l y s i s ) gave good c l a s s i f i c a t i o n of unknowns. 40 K o r m o s and Waugh <14> u s e d f a c t o r a n a l y s i s t o i n v e s t i g a t e t h e com p o n e n t s p r e s e n t i n s o l i d s t a t e n u c l e a r magnetic resonance e x p e r i m e n t s . U s i n g the IND f u n c t i o n as an i n d i c a t o r , t h r e e c o mponents were f o u n d i n a s y s t e m c o n s i s t i n g o f p h o s p h o r u s 31 s p e c t r a o f o c t a - c a l c i u m p h o s p h a t e . A p l o t o f l o g ( e i g e n v a l u e ) vs t h e number of components a l s o i n d i c a t e d t h r e e components. The a p p l i c a t i o n of f a c t o r a n a l y s i s t o many d i f f e r e n t c h e m i c a l and b i o l o g i c a l p r o b l e m s was shown i n a r e v i e w by W e i n e r <'2 3>. He n o t e d t h a t many g r a p h s of one de p e n d e n t v a r i a b l e as a f u n c t i o n o f an i n d e p e n d e n t v a r i a b l e a r e se e n i n t h e l i t e r a t u r e , and i n many c a s e s a s t r a i g h t l i n e i s drawn t h r o u g h t h e d a t a p o i n t s . An a s s u m p t i o n i s made i n th e s e cases which may not be v a l i d . The dependent v a r i a b l e i s assumed t o be a f u n c t i o n o f o n l y one f a c t o r , o r i f i t i s r e c o g n i z e d t o be a f u n c t i o n of more t h a n one f a c t o r , i t i s assumed t h a t the f i r s t f a c t o r i s o v e r w h e l m i n g l y dominant. A u n i v a r i a t e s o l u t i o n i s f o r c e d upon what may v e r y w e l l be a m u l t i v a r i a t e problem. W i t h f a c t o r a n a l y s i s , as many f a c t o r s a r e i n c o r p o r a t e d i n t o a f i t as a r e r e q u i r e d t o a d e q u a t e l y r e p r e s e n t t h e d a t a . The d i m e n s i o n a l i t y o f t h e d a t a i s n o t f o r c e d down t o one d i m e n s i o n . I n many c a s e s a l i n e a r c o m b i n a t i o n of many v a r i a b l e s w i l l g i v e a f a r b e t t e r model f o r the system under i n v e s t i g a t i o n . I n f a c t o r a n a l y s i s , the d a t a s e t w i l l n o t be p r o p e r l y r e p r o d u c e d u n l e s s t h e model i n c o r p o r a t e s a l l t h e r e l e v a n t f a c t o r s . As an e x a m p l e , a 41 m a t r i x o f gas l i q u i d c h r o m a t o g r a p h y r e t e n t i o n i n d i c e s was f a c t o r a n a l y z e d . F i v e f a c t o r s w e r e f o u n d , w h i c h c o r r e s p o n d e d to a l i n e a r c o m b i n a t i o n of carbon number, heat of v a p o r i z a t i o n , Van der Wa a l s c o n s t a n t s , s q u a r e o f t h e d i p o l e moment, and molar p o l a r i z a t i o n . From these f a c t o r s , i t was then p o s s i b l e t o generate e q u a t i o n s which p r e d i c t e d t hese q u a n t i t i e s f o r g i v e n r e t e n t i o n d a t a . S o l v e n t e f f e c t s i n NMR, and c l i n i c a l d e t e r m i n a t i o n s o f t h e e f f e c t o f d r u g s on e l e c t r o e n c e p h a l o g r a m s were g i v e n as o t h e r examples where f a c t o r a n a l y s i s p r o v i d e d new i n f o r m a t i o n . A g e n e r a l d i s c u s s i o n o f t h e many l e v e l s o f c o m p l e x i t y i n the a p p l i c a t i o n of p a t t e r n r e c o g n i t i o n appeared i n 1978 <41>. Wold and co-workers d e f i n e d f o u r l e v e l s of i n c r e a s i n g c o m p l e x i t y . The l o w e s t l e v e l c a r r i e d out assignment of an unknown i n t o any one o f a number o f p r e d e f i n e d c l a s s e s . C l a s s i f i c a t i o n was a c h i e v e d by g e n e r a t i n g d i s c r i m i n a n t f u n c t i o n s which changed s i g n as a border between c l a s s e s was c r o s s e d (e.g. a h y p e r p l a n e s e p a r a t i o n ) . A n o t h e r common c l a s s i f i c a t i o n scheme compared t h e d i s t a n c e s t o K n e a r e s t n e i g h b o u r s . The number of n e a r e s t n e i g h b o u r s c o n s i d e r e d (K) was v a r i e d , and i f more than one neighbour was c o n s i d e r e d , a wei g h t e d average or a v o t i n g scheme d e t e r m i n e d the c l a s s . The s e c o n d l e v e l was s i m i l a r t o t h e f i r s t e x c e p t t h a t the s e p a r a t i o n was e f f e c t e d by a c l o s e d s u r f a c e r a t h e r than a p l a n e t o a l l o w f o r t h e p o s s i b i l i t y o f o u t l i e r s . I n an 42 e n t r o p y miniraax method, one or s e v e r a l p a r a l l e l o p i p e d s were c o n s t r u c t e d t o e n c l o s e each c l a s s , d e t e r m i n e d by m i n i m i z i n g t h e e n t r o p y w i t h i n t h e c l a s s . I n SIMCA ( S o f t I n d e p e n d e n t M o d e l l i n g o f C l a s s A n a l o g y ) , a s l a b was c o n s t r u c t e d c o n t a i n i n g as many p o i n t s as p o s s i b l e . The t h i c k n e s s of the s l a b was r e l a t e d t o a c o n f i d e n c e l e v e l f o r c l a s s i f i c a t i o n . P o i n t s l y i n g o u t s i d e t h e s e s u r f a c e s were i d e n t i f i e d as o u t l i e r s . At the t h i r d l e v e l , a r e l a t i o n s h i p was d e s i r e d between t h e p o s i t i o n o f a d a t a p o i n t i n t h e h y p e r p l a n e used i n SIMCA, and an " e x t e r n a l e f f e c t " . T h i s a l l o w e d f o r example, t h e p r e d i c t i o n of a l e v e l of a c t i v i t y ( o r i n a c t i v i t y ) o f a d r u g . C l a s s i f i c a t i o n (e.g. a c t i v e o r i n a c t i v e ) f i r s t t o o k p l a c e , f o l l o w e d by comparison of i t s p o s i t i o n i n the SIMCA plane w i t h t h a t of knowns. A m u l t i p l e r e g r e s s i o n f i t t o the axes of the h y p e r p l a n e was sometimes done. T h i s seems to be analogous to the p r e d i c t i o n s made u s i n g a b s t r a c t f a c t o r axes i n f a c t o r a n a l y s i s . The a b s t r a c t f a c t o r s t h e m s e l v e s do not c o r r e s p o n d t o i n d i v i d u a l p r o p e r t i e s ( i n s t e a d they are l i n e a r c o m b i n a t i o n s of them). The o v e r a l l p o s i t i o n of an unknown i n a b s t r a c t f a c t o r s p a c e t h e n g i v e s g e n e r a l i n f o r m a t i o n about i t s c l a s s i f i c a t i o n or g e n e r a l p r o p e r t i e s . At t h e f o u r t h l e v e l , i t was d e s i r e d t o r e l a t e t h e p o s i t i o n of a data p o i n t not o n l y t o g e n e r a l p r o p e r t i e s but t o q u a n t i f y s p e c i f i c p r o p e r t i e s . " E f f e c t v a r i a b l e s " were f o u n d , a n d i r r e l e v a n t v a r i a b l e s w e r e d e l e t e d . 43 C h a r a c t e r i z i n g v a r i a b l e s were then sought f o r the s i m p l i f i e d d a t a . The magnitude of e x t e r n a l p r o p e r t i e s (e.g. s t r e n g t h , c o r r o s i v i t y ) c o u l d t h e n be f o u n d . T h i s seems s i m i l a r i n t h e o r y t o t h e f a c t o r a n a l y s i s a p p r o a c h used by M a l i n o w s k i <27>. Wold r e f e r r e d t o "good" and "bad" v a r i a b l e s , w i t h the "good" r e l a t i n g t o e x t e r n a l p r o p e r t i e s , and the "bad" bei n g i r r e l e v a n t . I f one i n s t e a d c a l l s t h e s e "good" v a r i a b l e s " a b s t r a c t f a c t o r s " or s i g n i f i c a n t e i g e n v e c t o r s , and the bad v a r i a b l e s " r e s i d u a l e i g e n v e c t o r s " o r n o i s e , t h e a n a l o g y i s c l e a r . The s e a r c h f o r c h a r a c t e r i z i n g v a r i a b l e s among t h e "good" v a r i a b l e s i s analogous to t a r g e t t e s t i n g the a b s t r a c t f a c t o r space to o b t a i n r e a l , p h y s i c a l l y s i g n i f i c a n t f a c t o r s . Thus a p r o c e s s o f f a c t o r a n a l y s i s f o l l o w e d by r e d u c t i o n t o th e s i g n i f i c a n t e i g e n v e c t o r s g i v e s " e f f e c t v a r i a b l e s " o r a b s t r a c t f a c t o r s . T r a n s f o r m a t i o n of the c o o r d i n a t e system, t h r o u g h t a r g e t t e s t i n g , t o one d e f i n e d by r e a l f a c t o r a x e s g i v e s a s p a c e where " e x t e r n a l p r o p e r t y v a r i a b l e s " e x t e n d a l o n g each a x i s . The i n t i m a t e r e l a t i o n s h i p between p a t t e r n r e c o g n i t i o n and f a c t o r a n a l y s i s s h o u l d be o b v i o u s , y e t some papers seem to c o n s i d e r them s e p a r a t e t e c h n i q u e s . Indeed, the new c a t c h p h r a s e " e x p e r t s y s t e m s " i s i n r e a l i t y j u s t a f u r t h e r e x t e n s i o n o f t h e s e t e c h n i q u e s . A l l t h e s e m a t h e m a t i c a l t e c h n i q u e s and t h e i r c o m p u t e r p r o g r a m o f f s p r i n g s h o u l d be c o n s i d e r e d as m e r e l y d i f f e r e n t f a c e t s o f a common f i e l d . 44 That f i e l d i n v o l v e s the e x t r a c t i o n of u s e f u l , q u a n t i t a t i v e a n d / o r q u a l i t a t i v e i n f o r m a t i o n f r o m a s y s t e m where many v a r i a b l e s a r e p r e s e n t . Thus, o p t i m i z a t i o n , e x p e r t systems, p a t t e r n r e c o g n i t i o n , f a c t o r a n a l y s i s a nd a r t i f i c i a l i n t e l l i g e n c e a l l have t h e same r o o t s . Even w i t h i n one of the above a r e a s , many a l t e r n a t e names are used. L i n d b e r g et a l <42> p o i n t e d o u t t h a t f a c t o r a n a l y s i s goes under many names. I t i s a l s o c a l l e d s i n g u l a r v a l u e d e c o m p o s i t i o n ( n u m e r i c a l a n a l y s i s ) , p r i n c i p a l c o m p o n e n t s a n a l y s i s ( s t a t i s t i c s ) , t h e K a r h u n e n - L o e v e e x p a n s i o n ( e l e c t r o n i c s ) , e i g e n v e c t o r p r o j e c t i o n , and r a n k a n n i h i l a t i o n ( c h e m i s t r y ) . C a r e must be t a k e n t o a v o i d c o n s i d e r i n g t h a t d i f f e r e n t l a b e l s r e p r e s e n t d i f f e r e n t methods. As r e s e a r c h c o n t i n u e d , new f u n c t i o n s t o i n d i c a t e t h e number o f f a c t o r s p r e s e n t were t r i e d , a l o n g w i t h many v a r i a t i o n s on those a l r e a d y e s t a b l i s h e d . Wold <43> p r o p o s e d c r o s s v a l i d a t o r y e s t i m a t i o n as a means of d e t e r m i n i n g the number of components p r e s e n t i n a s y s t e m . The p e r f o r m a n c e o f t h e method w i t h a r t i f i c i a l l y c r e a t e d d a t a was i m p r e s s i v e , b u t i t s p e r f o r m a n c e on r e a l d a t a was n o t f u l l y i n v e s t i g a t e d . When t h e number o f r e a l f a c t o r s approached w i t h i n 3 or 4 of the s m a l l e s t d i m e n s i o n of the data m a t r i x , the number of f a c t o r s was o v e r e s t i m a t e d by one. Thus, t h i s method c o u l d be added t o t h o s e a l r e a d y u s e d , b u t t h i s l i m i t a t i o n s h o u l d be k e p t i n m i n d . A consensus of methods i s a l w a y s best f o r the d e t e r m i n a t i o n of 45 the number of f a c t o r s , n. A n t o o n e t a l <9> a p p l i e d f a c t o r a n a l y s i s t o F o u r i e r t r a n s f o r m e d i n f r a r e d s p e c t r a o f o-, m- and p - x y l e n e s . S i m i l a r s p e c t r a had been p r e v i o u s l y d e a l t w i t h by a s i m p l e l e a s t squares f i t by Antoon et a l <40>. They concl u d e d t h a t a s u i t a b l e i n d i c a t o r f o r the d e t e r m i n a t i o n of the number of f a c t o r s was a drop of up t o s e v e r a l o r d e r s o f m a g n i t u d e i n t h e e i g e n v a l u e s a t t h e c o r r e c t number of f a c t o r s , w i t h t h e r e m a i n i n g e i g e n v a l u e s e x h i b i t i n g s i m i l a r m a g n i t u d e s a f t e r the drop. A p l o t of the l o g of the e i g e n v a l u e vs the number of components showed a d i s t i n c t drop a f t e r t h r e e components, c o n f i r m i n g t h a t the t h r e e d i f f e r e n t x y l e n e s were a l l p r e s e n t i n a t e s t d a t a s e t . A r e s t r i c t i o n t h a t a l l a b s o r b a n c e s be l e s s t h a n 1.2 a b s o r b a n c e u n i t s was i m p o s e d t o e n s u r e t h e l i n e a r i t y o f Beer's l a w . A s p e c t r a l window f r o m 3200 t o 2700 cm-1 a t a r e s o l u t i o n o f 4 cm-1 was t a k e n . N a r r o w i n g t h e s p e c t r a l window w h i l e r e t a i n i n g t h e same r e s o l u t i o n showed a s m a l l but steady decrease i n the prominence of the drop i n e i g e n v a l u e . T h i s was to be e x p e c t e d , s i n c e r e g i o n s of t h e s p e c t r u m c o n t a i n i n g d i s t i n c t i v e p eaks w o u l d be e x c l u d e d i f a n a r r o w e r window were c h o s e n . F o r t h e same s p e c t r a l window, a d e c r e a s e i n r e s o l u t i o n f r o m 4 cm-1 t o 8 cm-1 had no s i g n i f i c a n t e f f e c t on t h e d i s t i n c t i v e d r o p i n e i g e n v a l u e . Thus, h i g h r e s o l u t i o n was n o t a r e q u i r e m e n t . When a l a r g e w i n d o w was u s e d i n i t s e n t i r e t y , t h e 46 i n f o r m a t i o n c o n t e n t was not i n c r e a s e d s i g n i f i c a n t l y f o r f a c t o r a n a l y s i s t e c h n i q u e s by u s i n g h i g h e r r e s o l u t i o n . U l t i m a t e l y , a d e c r e a s e i n r e s o l u t i o n w i l l a f f e c t t h e d e t e r m i n a t i o n of the number of components, but o n l y when the r e s o l u t i o n i s so l o w t h a t t h e t o t a l number of a b s o r p t i o n m e a s u r e m e n t s a p p r o a c h e s t h e same o r d e r as t h e number of components. In t h i s case t h i s would be a r e s o l u t i o n of the o r d e r of 100 cm-1! V a r i a t i o n s i n a c o n s t a n t background due t o s c a t t e r i n g were f o u n d t o have no s i g n i f i c a n t e f f e c t . Even a v a r y i n g s t r u c t u r e d b a c k g r o u n d w o u l d be e a s i l y a c c o u n t e d f o r by an a d d i t i o n a l f a c t o r , so any t y p e o f background as l o n g as i t i s c o n s t a n t i s no problem. The r e g i o n f r o m 1300 t o 1700 cra-1 was a c t u a l l y f o u n d t o be s u p e r i o r t o t h e 3200 t o 2700 cm-1 r e g i o n , f o r some of t h e more d i l u t e s o l u t i o n s . T h i s i s presumably due to the h i g h e r d e n s i t y o f i n f o r m a t i o n f o u n d i n t h e f i n g e r p r i n t r e g i o n of the spectrum. I t was m e n t i o n e d t h a t a number o f n o n - l i n e a r e f f e c t s r e l e v a n t p a r t i c u l a r l y t o i n f r a r e d s p e c t r o s c o p y c o u l d be p r e s e n t , i n c l u d i n g s c a t t e r i n g , r e f l e c t i o n , and i n t e r f e r e n c e f r i n g e s (from the c e l l windows), but i f proper p r e c a u t i o n s i n sample p r e p a r a t i o n and h a n d l i n g were t a k e n , these e f f e c t s s h o u l d be s m a l l enough t o be l o s t i n t h e n o i s e , and were thus s w a l l o w e d up i n the r e s i d u a l e i g e n v e c t o r s . Analogous n o n - l i n e a r e f f e c t s i n e m i s s i o n s p e c t r a , such as s t r a y l i g h t w i t h i n t h e m o n o c h r o m a t o r , w o u l d a l s o be e x p e c t e d t o be 47 p r e s e n t , but s h o u l d be s m a l l . Ho e t a l <7> a l s o n o t e d t h e e f f e c t o f r e d u c i n g r e s o l u t i o n on the r e s u l t s of an a n a l y s i s . They a p p l i e d the method of rank a n n i h i l a t i o n t o the q u a n t i t a t i v e a n a l y s i s of m u 1 t i c o m p o n e n t f l u o r e s c e n c e d a t a . R e d u c t i o n o f t h e r e s o l u t i o n f r o m 60 c h a n n e l s , t o 30 f e a t u r e s of two summed c h a n n e l s e a c h , h e r e a l s o had a n e g l i g i b l e e f f e c t on t h e r e s u l t s . H a a l a n d and E a s t e r l i n g < 4 4 > d e m o n s t r a t e d t h a t , even w i t h a s i m p l e l e a s t s q u a r e s a p p r o a c h , l i m i t s f o r t h e d e t e c t i o n o f t r a c e g a s e s were i m p r o v e d by a f a c t o r o f 5 t o 7 over c o n v e n t i o n a l methods. To a c h i e v e the improvement, they used a r e s o l u t i o n of 0.063 cm-1, which a l l o w e d the s p e c t r a l r o t a t i o n a l bands f o r each gas to be r e s o l v e d . A window 200 cm-1 w i d e was u s e d , and up t o 200 i n t e r f e r o g r a m s were averaged f o r an improvement i n s i g n a l t o n o i s e r a t i o . Four m e t h o d s o f b a s e l i n e c o r r e c t i o n w e r e c o m p a r e d . The i m p r o v e m e n t i n d e t e c t i o n l i m i t s a r o s e p r i m a r i l y f r o m t h e f a c t t h a t a l l the i n f o r m a t i o n i n a l a r g e p o r t i o n of spectrum was used i n the a n a l y s i s , a l l o w i n g d e t e r m i n a t i o n s even when i n d i v i d u a l peaks a p p e a r e d l o s t i n t h e n o i s e . The f u r t h e r c o n v e n i e n c e o f l e a s t s q u a r e s methods f o r d e a l i n g w i t h s p e c t r a l o v e r l a p s was not however mentioned i n t h i s paper, and would be expected to a t l e a s t m a i n t a i n t h e s e d e c t e c t i o n l i m i t s i n more complex m i x t u r e s of gaseous components. 48 S e v e r a l examples of a p p l i c a t i o n s of p a t t e r n r e c o g n i t i o n t e c h n i q u e s i n d i v e r s e a r e a s were g i v e n by E r i c k s o n e t a l <45>. ARTHUR, a pa c k a g e o f p a t t e r n r e c o g n i t i o n p r o g r a m s , was u s e d . C o u n t e r f e i t w h i s k e y was i d e n t i f i e d u s i n g t h e r e s u l t s o f g a s c h r o m a t o g r a p h i c a n a l y s e s . P a t t e r n r e c o g n i t i o n i d e n t i f i e d t h e few compounds i n t h e a n a l y s e s w i t h the g r e a t e s t v a r i a n c e between c l a s s e s , and on the b a s i s of t h e s e few f e a t u r e s , p e r f e c t s e p a r a t i o n was a c h i e v e d w i t h a r e d u c t i o n i n c o s t of the study as an added bonus. The i d e n t i f i c a t i o n of t h e s o u r c e o f o i l s p i l l s c an be d i f f i c u l t , p a r t i c u l a r l y s i n c e the o i l undergoes w e a t h e r i n g o n c e r e l e a s e d t o t h e n a t u r a l e n v i r o n m e n t . N e u t r o n a c t i v a t i o n a n a l y s i s s p e c t r a were c o l l e c t e d , and even i n the c a s e o f t h e w e a t h e r e d s p e c t r a , a p a r t i c u l a r o i l was s t i l l c h a r a c t e r i z e d i n a r e l a t i v e l y s m a l l r e g i o n i n m u l t i d i m e n s i o n a l space. The SIMCA method was found t o g i v e the b e s t c l a s s i f i c a t i o n r e s u l t s , w i t h 40 d i f f e r e n t t y p e s of o i l b e i n g i d e n t i f i a b l e w i t h r e a s o n a b l e r e l i a b i l i t y . A i r p o l l u t i o n , p r o d u c t q u a l i t y and w i n e q u a l i t y were a l s o e v a l u a t e d w i t h s u c c e s s f u l r e s u l t s . O i l s were a l s o i d e n t i f i e d by t h e v e c t o r p o s i t i o n o f t h e i r f l u o r e s c e n c e s p e c t r a i n m u l t i d i m e n s i o n a l space <46>. T h i s g e n e r a t e d a c u r v e d s u r f a c e i n s p a c e c o r r e s p o n d i n g t o the c o n c e n t r a t i o n and degree of w e a t h e r i n g . Thus the degree of w e a t h e r i n g c o u l d a l s o be q u a n t i f i e d i n a d d i t i o n t o t h e p o s s i b l e s o u r c e of the o i l . 49 Wines were i n v e s t i g a t e d by Van der G r e e f e t a l <47> who used non l i n e a r mapping, K n e a r e s t n e i g hbour, and p r i n c i p a l component a n a l y s i s t o d i s c r i m i n a t e on t h e b a s i s of mass s p e c t r a . Bordeaux and Rhone wines were c l a s s i f i e d w i t h o u t a m b i g u i t y . They a l s o t e s t e d u r i n e s a m p l e s and f o u n d t h a t u r i n e s a m p l e s f r o m men and women c o u l d be e a s i l y d i s t i n g u i s h e d i n a non l i n e a r map when proper a v e r a g i n g and w e i g h t i n g was done. They c o n c l u d e d t h a t non l i n e a r maps gave good g r a p h i c a l r e p r e s e n t a t i o n s of t h e s e p a r a t i o n of c l a s s e s , b u t p r i n c i p a l c o m p o n e n t a n a l y s i s g a v e more i n f o r m a t i o n a b o u t t h e f e a t u r e s t h a t b e s t c h a r a c t e r i z e c l a s s e s . K o e n i g e t a l <10> a p p l i e d f a c t o r a n a l y s i s t o p o l y m e r systems produced from p o l y p h e n y l e n e o x i d e and p o l y s t y r e n e . F o u r i e r t r a n s f o r m i n f r a r e d s p e c t r a o f t h e p o l y m e r s were shown t o have t h r e e c o m p o n e n t s r a t h e r t h a n j u s t t h e two e x p e c t e d f r o m t h e i n d i v i d u a l c o m p o n e n t s . T h i s t h i r d component was a t t r i b u t e d t o c o n f o r m a t i o n a l c h a nges w h i c h were d e p e n d e n t upon t e m p e r a t u r e and t h e r a t i o o f t h e two m a t e r i a l s . The i n t e r a c t i o n s p e c t r a , r e p r e s e n t i n g t h e c o n t r i b u t i o n t o the s p e c t r a l change due to the i n t e r a c t i o n was p l o t t e d , and a maximum was f o u n d . Thus, i n f o r m a t i o n c o n c e r n i n g phase c h a n g e s i n t h e p o l y m e r s y s t e m c o u l d be o b t a i n e d by l o o k i n g a t t h e s m a l l b u t s i g n i f i c a n t i n t e r a c t i o n s r e m a i n i n g a f t e r t h e m a j o r c o m p o n e n t s were 50 a c c o u n t e d f o r . P r e s u m a b l y s m a l l e f f e c t s i n any t y p e o f s p e c t r a , such as m a t r i x e f f e c t s i n e m i s s i o n s p e c t r a , c o u l d be q u a n t i f i e d i n t h i s manner. S h a r a f and K o w a l s k i <13> used mass s p e c t r a o b t a i n e d t h r o u g h o u t t h e e l u t i o n o f c o m p o n e n t s f r o m a g a s c h r o m a t o g r a p h y c o l u m n t o d e v e l o p a method of q u a n t i t a t i v e a n a l y s i s and the g e n e r a t i o n of i n d i v i d u a l mass s p e c t r a f o r each component r e g a r d l e s s of e l u t i o n o v e r l a p . I n many cases o v e r l a p , sometimes c o m p l e t e , of e l u t i n g components c o u l d not be a v o i d e d . I n t h e s e s i t u a t i o n s l i b r a r y mass s p e c t r a l s e a r c h e s became i n a c c u r a t e or t o t a l l y u n r e l i a b l e . However i f f a c t o r a n a l y s i s was a p p l i e d to the system of s p e c t r a , the number of components c o u l d be d e t e r m i n e d u s i n g many of the p r e v i o u s l y mentioned i n d i c a t o r s . P a r e n t mass s p e c t r a were o b t a i n e d f r o m t h e d a t a w i t h an a v e r a g e d e v i a t i o n o f l e s s than 3%. When r e s o l u t i o n was degraded by a f a c t o r of 5, the r e s u l t s r e m a i n e d s t a b l e when f a c t o r a n a l y s i s was u s e d . A s e c o n d , more common, a p p r o a c h u s e d o n l y t h e s p e c t r a c o l l e c t e d d u r i n g e l u t i o n peaks, and f a i l e d c o m p l e t e l y w i t h the l o w e r r e s o l u t i o n . Thus, r e s o l u t i o n c o u l d be s a c r i f i c e d w i t h l i t t l e e f f e c t on the outcome of a f a c t o r a n a l y s i s , but t h e same c a n n o t be s a i d f o r t h e s e c o n d t e c h n i q u e . R e g e n e r a t e d p u r e s p e c t r a p r o d u c e d t h r o u g h f a c t o r a n a l y s i s c o u l d then be l i b r a r y s earched w i t h much g r e a t e r s u c c e s s . I t i s i m p o r t a n t t h a t a s u f f i c i e n t number of f a c t o r s be i n c l u d e d when r e c o n s t r u c t i n g a d a t a s e t . E l i a s s o n e t a l 51 <48> i n v e s t i g a t e d t h e i n f l u e n c e of s o l v e n t s on c h e m i c a l s h i f t s i n n u c l e a r magnetic resonance. A two component model reproduced 80-90% of the observed e x p e r i m e n t a l s h i f t s , w h i l e any one parameter model c o u l d reproduce no more than 58% of t h e t h e e f f e c t . T h u s , f a c t o r a n a l y s i s i n c r e a s e d t h e r e l i a b i l i t y of a model and i t s p r e d i c t i o n s over a u n i v a r i a t e model. F a c t o r a n a l y s i s has been a p p l i e d t o t h e p r o b l e m s o f o v e r l a p p i n g s p e c t r a i n o t h e r a r e a s as w e l l . G i l b e r t e t a l <49> were a b l e t o r e p r o d u c e r e s o l v e d X r a y p h o t o e l e c t r o n s p e c t r a of p l a t i n u m complexes. P r i o r knowledge of the data s e t o r c u r v e f i t t i n g p a r a m e t e r s were n o t r e q u i r e d . They c h o s e t h e number o f p r i n c i p a l c omponents so t h a t 99.9% of the v a r i a n c e was r e t a i n e d . T h i s r e q u i r e d o n l y two a b s t r a c t f a c t o r s . R o t a t i o n o f f a c t o r s p a c e t h e n g e n e r a t e d t h e pu r e s p e c t r a as r e a l f a c t o r s . From t h e r e p r o d u c e d s p e c t r a , e v i d e n c e o f two o x i d a t i o n s t a t e s f o r P t (2+ and 4+) was f o u n d . T h i s i n f o r m a t i o n was s t a t e d t o be d i f f i c u l t t o o b t a i n except by c a r r y i n g out a multicomponent a n a l y s i s . G i l l e t t e and K o e n i g <11> e x p l o r e d t h e p o s s i b i l i t y o f u s i n g f a c t o r a n a l y s i s t o a c h i e v e a r e d u c t i o n i n n o i s e i n F o u r i e r t r a n s f o r m i n f r a r e d s p e c t r a . They r e f e r r e d t o M a l i n o w s k i ' s d e f i n i t i o n s of r e a l , e x t r a c t e d and imbedded e r r o r i n o u t l i n i n g t h e t h e o r y b e h i n d n o i s e r e d u c t i o n . A number o f i n d i c a t o r s were used t o d e t e r m i n e t h e number of 52 f a c t o r s . T h e y g e n e r a t e d g r a p h s o f l o g ( e i g e n v a l u e ) , v a r i a n c e , c u m u l a t i v e % v a r i a n c e , r e a l , i m b e d d e d , and e x t r a c t e d e r r o r , and t h e IND f u n c t i o n vs t h e number of com p o n e n t s . Of a l l t h e s e i n d i c a t o r s , t h e l o g ( e i g e n v a l u e ) g r a p h showed a v e r y d i s t i n c t i v e d rop i n d i c a t i n g 2 f a c t o r s . The r e a l e r r o r , v a r i a n c e , and c u m u l a t i v e % v a r i a n c e s u g g e s t e d 2 f a c t o r s a l s o , but were n o t as c l e a r . The IND f u n c t i o n showed no d i s t i n c t minimum, so was u n a b l e t o i n d i c a t e t h e number of f a c t o r s i n t h i s i n s t a n c e . Weak s p e c t r a c o n t a i n i n g t w o L o r e n t z i a n p e a k s w e r e f a c t o r a n a l y z e d , and t h e n r e c o n s t r u c t e d f r o m t h e 2 f a c t o r s i n d i c a t e d . A r e c o n s t r u c t e d spectrum c o n t a i n i n g v i s i b l y l e s s n o i s e was o b t a i n e d , c o n f i r m i n g t h e i m p r o v e m e n t i n n o i s e which was t h e o r e t i c a l l y p r e d i c t e d . G i l l e t t e e t a l <50> f u r t h e r a p p l i e d f a c t o r a n a l y s i s to p e r m i t q u a n t i t a t i v e a n a l y s i s u s i n g a r e f e r e n c e s p e c t r a l l i b r a r y . 254 p o i n t s were used f o r each s p e c t r u m i n t h e r e g i o n 3200 to 800 cm-1. The e i g e n v e c t o r s c o r r e s p o n d i n g to a b s t r a c t f a c t o r s were s e p a r a t e d f r o m t h o s e due t o n o i s e . T h i s l e f t a s e r i e s of a b s t r a c t f a c t o r s which were d i s p l a y e d as " e i g e n s p e c t r a " . These e i g e n s p e c t r a r e p r e s e n t e d l i n e a r c o m b i n a t i o n s of the pure component s p e c t r a . I t was p o s s i b l e t o g e n e r a t e t h e pu r e component s p e c t r a f r o m t h e a b s t r a c t s p e c t r a i f an a p p r o p r i a t e l i n e a r c o m b i n a t i o n was atte m p t e d . A l e a s t s q u a r e s f i t c o u l d be c a r r i e d o u t t o f i t a c o m b i n a t i o n o f e i g e n s p e c t r a t o e a c h p u r e s p e c t r u m i n t h e 53 s p e c t r a l l i b r a r y . I n t h e e x a m p l e g i v e n , a h e x a n e / c a r b o n t e t r a c h l o r i d e m i x t u r e , f o u r r e f e r e n c e s p e c t r a were w e l l f i t by the a b s t r a c t s p e c t r a . Only two components were i n d i c a t e d by t h e d r o p i n e i g e n v a l u e s . D i f f e r e n c e s p e c t r a w e r e o b t a i n e d between the f i t s and the s t a n d a r d s . The d i f f e r e n c e s p e c t r a o f hexane and c a r b o n t e t r a c h l o r i d e showed o n l y random f l u c t u a t i o n s , w h i l e t h e d i f f e r e n c e s p e c t r a o f a m y l d i s u l f i d e and pentane showed d i f f e r e n c e s i n s t r u c t u r e beyond the n o i s e l e v e l . The main d i s a d v a n t a g e of t h i s method i s t h a t a l e a s t s q u a r e s f i t must be done f o r e v e r y r e f e r e n c e s p e c t r u m , and i n t h e i n f r a r e d , o v e r 2300 have a l r e a d y been c o m p i l e d by S a d t l e r L a b o r a t o r i e s . T h i s i s s i m i l a r i n p r i n c i p l e t o M a l i n o w s k i ' s t a r g e t t e s t i n g , which a l l o w s r e f e r e n c e s p e c t r a to be t e s t e d f o r t h e i r presence i n the a b s t r a c t f a c t o r s . S o g l i e r o e t a l <8> d i s c u s s e d t h e e s t a b l i s h m e n t o f a l i b r a r y o f l o w t e m p e r a t u r e l u m i n e s c e n c e ( L T L ) and room t e m p e r a t u r e f l u o r e s c e n c e (RTF) s p e c t r a f o r use w i t h p a t t e r n r e c o g n i t i o n methods t o i d e n t i f y h a z a r d o u s c h e m i c a l s . An unknown was matched t o a l i b r a r y member by t h r e e measures of s i m i l a r i t y , e u c l i d e a n d i s t a n c e , a n g l e b e t w e e n t h e two v e c t o r s , and c o r r e l a t i o n c o e f f i c i e n t . The r e s u l t s f r o m s e v e r a l r e p l i c a t e i d e n t i f i c a t i o n s of unknowns was t h a t RTF s p e c t r a were more r e p e a t a b l e , but LTL s p e c t r a c o u l d p r o v i d e more s u b t l e d i f f e r e n c e s b e t w e e n s i m i l a r s p e c t r a and t h u s 54 a s s i s t i n an a u t o m a t e d c l a s s i f i c a t i o n p r o c e s s . F u r t h e r s t e p s need to be taken t o e v a l u a t e the most d i s c r i m i n a t o r y s e t o f f e a t u r e s ( b e s t s p e c t r a l window) f o r b o t h o f t h e s e types of s p e c t r a . F r a n k e l <12> a p p l i e d two p a t t e r n r e c o g n i t i o n t e c h n i q u e s t o t h e a n a l y s i s o f t h e U. S. EPA l i b r a r y o f F o u r i e r t r a n s f o r m i n f r a r e d s p e c t r a . T h i s c o n s i s t e d of 2300 s p e c t r a a t 4 cm-1 r e s o l u t i o n c o v e r i n g t h e r a n g e 450-4000 cm-1. I n a l l c a l c u l a t i o n s the r e s o l u t i o n was reduced n u m e r i c a l l y t o 16 cm-1. The f i r s t t e c h n i q u e used was c l u s t e r i n g t o y i e l d d e n d r o g r a m s . The s e c o n d t e c h n i q u e g e n e r a t e d e i g e n v e c t o r p l o t s . B o t h m e t h o d s s h o w e d c h e m i c a l l y s i g n i f i c a n t g r o u p i n g s . He c o n c l u d e d t h a t t h e EPA l i b r a r y was a u s e f u l database t o f o r m u l a t e a t e c h n i q u e f o r the c l a s s a n a l y s i s of complex m i n t u r e s u s i n g p a t t e r n r e c o g n i t i o n . Haaland and E a s t e r l i n g <51> f u r t h e r r e f i n e d t h e i r l e a s t squares approach, u s i n g the same x y l e n e s as b e f o r e <44>, and t h e same b a s e l i n e c o r r e c t i o n s . As a means o f c o n s e r v i n g c o m p u t e r memory, a t h r e s h o l d was d e f i n e d , b e l o w w h i c h s p e c t r a l i n f o r m a t i o n was not c o n s i d e r e d s i g n i f i c a n t . T h i s r e s u l t e d i n windows f o r each s p e c t r u m o n l y i n t h e r e g i o n s where pea k s were p r e s e n t i n a p a r t i c u l a r s p e c t r u m . The s i m p l i c i t y o f t r e a t i n g t h e e n t i r e s p e c t r u m w i t h o u t peak s e l e c t i o n p r o c e d u r e s a p p e a r e d l o s t when t h i s more c o m p l i c a t e d approach was taken. The r e s u l t s were very good, w i t h a p p r o x i m a t e l y 0.1% a v e r a g e r e l a t i v e e r r o r when t h e 55 s i g n a l t o n o i s e r a t i o i s 250. W i t h t h e s i g n a l t o n o i s e r a t i o a s l o w as 2.5, t h e r e l a t i v e e r r o r was o n l y a p p r o x i m a t e l y 15%. A l l c o m p o n e n t s p r e s e n t must be p r e s e n t i n t h e s e t of s t a n d a r d s h o w e v e r , o r l a r g e e r r o r s may e c i s t . I n d e e d , i f t h e w r o n g s t a n d a r d s a r e c h o s e n , and a f i t i s f o r c e d , a n omolously h i g h r e s u l t s may be found f o r components which a r e n o t even p r e s e n t . The e f f e c t s o f f o r c i n g of a f i t , w h i c h i s i n h e r e n t i n l e a s t s q u a r e s f i t s s h o u l d be k e p t i n mind when e v a l u a t i n g the r e s u l t s of a sample which i s not a t l e a s t q u a l i t a t i v e l y w e l l c h a r a c t e r i z e d . I f t h e s y s t e m i s w e l l c h a r a c t e r i z e d , v e r y good r e s u l t s can be o b t a i n e d . Hangac e t a l <52> used t h e i n f o r m a t i o n c o n t a i n e d i n a 2300 compound i n f r a r e d s p e c t r a l l i b r a r y w i t h a b s o r b a n c e m e a s u r e m e n t s a t 185 d i s c r e t e f r e q u e n c i e s . By f a c t o r a n a l y s i s of t h i s d ata s e t , the number of f a c t o r s r e q u i r e d t o c o r r e c t l y i d e n t i f y an unknown was c o n s i d e r a b l y reduced. No one i n d i c a t o r was a b l e t o p i n p o i n t a c o n s i s t e n t number o f f a c t o r s . T a r g e t t r a n s f o r m e d r e c o n s t r u c t e d s p e c t r a were w i t h i n e x p e r i m e n t a l e r r o r when 75 e i g e n v e c t o r s were u s e d . S i g n i f i c a n t d i s c r e p a n c i e s were not n o t e d u n t i l o n l y 55 e i g e n v e c t o r s were used. M a l i n o w s k i ' s IND f u n c t i o n e x h i b i t e d a minimum a t 37 e i g e n v e c t o r s . Hoping f o r the maximum amount o f s i m p l i f i c a t i o n f o r t h e l e a s t l o s s i n a c c u r a c y , 37 e i g e n v e c t o r s were chosen f o r the reduced s p e c t r a l l i b r a r y . 56 A l l unknowns were then t r a n s f o r m e d to 37 d i m e n s i o n a l space. C o n v e n t i o n a l l i b r a r y s e a r c h t e c h n i q u e s were t h e n used t o a r r i v e a t the c l o s e s t matches i n the l i b r a r y . The r e d u c t i o n i n l i b r a r y s i z e d i d not a d v e r s e l y a f f e c t i d e n t i f i c a t i o n of s e v e r a l t e s t unknowns. The r e d u c t i o n i n l i b r a r y s i z e g r e a t l y d e c r e a s e d t h e t i m e r e q u i r e d f o r a s e a r c h , and a p p e a r e d t o be c o m p a r a b l e i n a c c u r a c y t o f u l l l i b r a r y s e a r c h e s . I n t h i s c a s e h o w e v e r , t h e IND f u n c t i o n was p r o b a b l y n o t a r e l i a b l e i n d i c a t o r , as t h e o t h e r i n d i c a t o r s suggested more f a c t o r s . A more r e a s o n a b l e number of f a c t o r s t o s e l e c t w o u l d p r o b a b l y have been 55, i f n o t more. Can must be taken to not throw out u s e f u l i n f o r m a t i o n when the d e s i r e t o g r e a t l y s i m p l i f y a d a t a s e t s t r i k e s . I n r e c o n s t r u c t i n g a d a t a s e t w i t h r e d u c e d n o i s e , i t i s f a r b e t t e r t o e r r on t h e s i d e o f t o o many f a c t o r s r a t h e r t h a n t o o f e w , s i n c e t o o many w i l l j u s t add s l i g h t l y more n o i s e , w h i l e too few can s e r i o u s l y d i s t o r t the data. S i n c e f a c t o r a n a l y s i s l o o k s a t t h e v a r i a n c e i n a l l m e a s u r e m e n t s , i t h a s no b u i l t i n b i a s a b o u t w h i c h m e a s u r e m e n t s s h o u l d be i m p o r t a n t . T h i s c a n l e a d t o t h e d i s c o v e r y of d e p e n d e n c i e s w h i c h w o u l d n o t o t h e r w i s e be p r e d i c t e d . As an e x a m p l e o f t h i s , w a t e r q u a l i t y was i n v e s t i g a t e d by m o n i t o r i n g t h e 5 day b i o c h e m i c a l oxygen demand (BOD) <53>. ARTHUR was used t o g e n e r a t e a r e s p o n s e s u r f a c e f o r t h e BOD f r o m t h e two l a r g e s t e i g e n v e c t o r s r e p r e s e n t i n g 19 v a r i a b l e s . A r e g i o n of optimum e f f i c i e n c y 57 w i t h a minimum i n t h e BOD was f o u n d . The most h i g h l y c o r r e l a t e d v a r i a b l e s were t h e n i d e n t i f i e d . C o n t r o l o v e r t h e s e v a r i a b l e s would be s u f f i c i e n t t o b r i n g the BOD down to n e a r optimum v a l u e s . A number o f r e l a t i o n s h i p s w o u l d n ot o t h e r w i s e have been found, such as the i m p o r t a n c e of c a l c i u m and magnesium and the d e t r i m e n t a l e f f e c t of i r o n on the BOD. M o r e r e c e n t l y , A r u n a c h a l a m and G a n g a d h a r a n <54> i n t e r p r e t e d carbon 13 NMR and mass s p e c t r a u s i n g p r i n c i p a l components a n a l y s i s and d i s c r i m i n a n t f u n c t i o n s . The a b i l i t y o f p r i n c i p a l c o m p o n e n t s a n a l y s i s t o e x t r a c t u s e f u l i n f o r m a t i o n was demonstrated when m i x t u r e s of c y c l o h e x a n e and n-hexane were a n a l y z e d . I n c l u s i o n o f t h e i n t e n s i t y i n f o r m a t i o n f o r the peak a t 28 m/z meant t h a t t h r e e f a c t o r s r a t h e r than two were r e q u i r e d to a d e q u a t e l y account f o r the v a r i a n c e o f t h e d a t a . T h i s was s t r o n g e v i d e n c e f o r t h e presence of n i t r o g e n as an i m p u r i t y i n the samples. A p l o t of the e i g e n v e c t o r c o e f f i c i e n t s showed t h a t peaks a t 57 and 86 were c o l l i n e a r , and were a s s o c i a t e d s o l e l y w i t h n-hexane. P e a k s a t 84, 85 and 54 were a l s o c o l l i n e a r , and a s s o c i a t e d s o l e l y w i t h c y c l o h e x a n e . O t h e r p e a k s l a y b e t w e e n t h e s e a x e s , i n d i c a t i n g c o n t r i b u t i o n s f r o m b o t h compounds. The mole f r a c t i o n s o f e a c h compound were e a s i l y o b t a i n a b l e by c o m p a r i s o n o f t h e r e l a t i v e i n t e n s i t i e s o f t h e s e l i n e s . A q u a n t i t a t i v e d e t e r m i n a t i o n was a l s o done u s i n g o n l y t h e l i n e s known t o have c o n t r i b u t i o n s f r o m b o t h ( c o m p l e t e 58 s p e c t r a l o v e r l a p ) . The v a l u e of mole f r a c t i o n s d e t e r m i n e d w i t h t h e s e peaks were i n good a g r e e m e n t w i t h t h e a c t u a l v a l u e s . They s u g g e s t e d t h a t t h e mass f r a g m e n t s o b t a i n e d d u r i n g c o - e l u t i o n o f s e v e r a l c o m p o n e n t s i n g a s c h r o m a t o g r a p h y - m a s s s p e c t r o s c o p y c o u l d be a n a l y z e d t o d e t e r m i n e the number of components. They d i d not seem to be aware o f t h e work done by S h a r a f and K o w a l s k i <13> i n t h i s same s u b j e c t . H o o g e r b r u g g e e t a l <55> used t h e e i g e n v e c t o r a n a l y s i s r o u t i n e s i n ARTHUR t o g e n e r a t e d i s c r i m i n a n t f u n c t i o n s . G r a p h s i n v o l v i n g t h e f i r s t two f u n c t i o n s gave r e a s o n a b l y good s e p a r a t i o n of p y r o l y s i s mass s p e c t r a of c a r b o h y d r a t e s , y e a s t s , and m y c o b a c t e r i a . The method i n v o l v e d s c a l i n g the d a t a ( S C A L E ) . P r i n c i p a l c o m p o n e n t a n a l y s i s ( K A P R I N ) f o l l o w e d by t r a n s f o r m a t i o n t o t h e new e i g e n v e c t o r s p a c e (KATRAN) gave p r i n c i p a l component p l o t s (VARVAR). T h i s new d a t a m a t r i x was t h e n s u b j e c t e d t o t h e e n t i r e t r e a t m e n t a s e c o n d t i m e w i t h t h e s c a l e d d a t a a d d i t i o n a l l y r e p l a c e d by c a t e g o r y means (MEAN). The e n t i r e s e q u e n c e was SCALE, KAPRIN, KATRAN, VARVAR t o g e n e r a t e p r i n c i p a l component p l o t s , then SCALE, MEAN, KAPRIN, KATRAN, VARVAR t o generate f i n a l d i s c r i m i n a n t f u n c t i o n s . Good d i s c r i m i n a t i o n o f t h e f i n a l c a t e g o r i e s was o b t a i n e d . The r o u t i n e MEAN was not i n t h e s t a n d a r d ARTHUR p a c k a g e , but was a s i m p l e a d d i t i o n t o the program. A u t o s c a l i n g d u r i n g the SCALE r o u t i n e was found t o g i v e t h e h i g h e s t a c c u r a c y . A c o m p a r i s o n w i t h t h e more 59 s o p h i s t i c a t e d " S t a t i s t i c a l P a c k a g e f o r S o c i a l S c i e n c e ( S P S S ) " showed t h a t ARTHUR was a b l e t o g i v e d i s c r i m i n a n t p l o t s r i v a l i n g the SPSS package f o r g r a p h i c a l p r e s e n t a t i o n . L i n d b e r g e t a l <42> a p p l i e d a p a r t i a l l e a s t s q u a r e s method to the a n a l y s i s of f l u o r e s c e n c e s p e c t r a of humic a c i d and l i g n i n s u l f o n a t e . The p a r t i a l l e a s t squares method was a v a r i a t i o n o f f a c t o r a n a l y s i s w h i c h f o u n d n o t o n l y t h e s m a l l e s t number o f e i g e n v e c t o r s w h i c h c o u l d a d e q u a t e l y d e s c r i b e the data space ( r e p r e s e n t e d as a m a t r i x T) but a l s o c o r r e l a t e d a component c o n c e n t r a t i o n m a t r i x Y, which a l l o w e d a s i m u l t a n e o u s l e a s t s q u a r e s s o l u t i o n t o be f o u n d f o r any c o m p l e x m i x t u r e . They t o o k o v e r 20 s p e c t r a , c o n s i s t i n g of pur e s p e c t r a o f t h e components o f i n t e r e s t ( h u m i c a c i d and l i g n i n s u l f o n a t e ) and m i x t u r e s o f t h e co m p o n e n t s . Each spectrum c o n s i s t e d of 27 e q u a l l y spaced wavelength i n t e n s i t y measurements. T h i s formed the data m a t r i x . The p r e d i c t e d c o n c e n t r a t i o n s agreed w e l l w i t h the a c t u a l c o n c e n t r a t i o n s . However, more than t h r e e p r i n c i p a l components were used i n the a n a l y s i s , s i n c e a d d i t i o n a l f a c t o r s seemed s i g n i f i c a n t to the a c c u r a c y of the method. These a d d i t i o n a l f a c t o r s were a t t r i b u t e d t o i n t e r a c t i o n s b e t w e e n c o m p o n e n t s i n t h e s o l u t i o n m i x t u r e s . Thus, a d d i t i o n a l f a c t o r s w h i c h may a f f e c t t h e a c c u r a c y o f t h e method can be a c c o u n t e d f o r by p r i n c i p a l components methods. McCue and M a l i n o w s k i <56> used t a r g e t f a c t o r a n a l y s i s 60 on t h e u l t r a v i o l e t s p e c t r a o f u n r e s o l v e d l i q u i d c h r o m a t o g r a p h i c f r a c t i o n s t o v e r i f y t h e p r e s e n c e of components and i d e n t i f y those f r a c t i o n s c o n t a i n i n g o t h e r i m p u r i t e s . The parameter SPOIL was defined to d e s c r i b e the s u c c e s s or f a i l u r e of a p a r t i c u l a r t a r g e t t e s t . A v a l u e below 3 confirmed the t e s t vector was a true f a c t o r , between 3 and 6 i t was u n c e r t a i n but p r o b a b l e , and above 6 i t was not a f a c t o r . M i x t u r e s of o - x y l e n e , ethy1 benzene, and p-xylene, which a l l mutually overlap, were prepared. I n i t i a l f a c t o r a n a l y s i s with r e a l e r r o r as an i n d i c a t o r p r e d i c t e d 3 f a c t o r s . An i m p u r i t y ( f o u r t h component) was seen o n l y i n f r a c t i o n 2, a t t r i b u t e d to an u n c l e a n c o l l e c t i o n v e s s e l . Thus, an u n s u s p e c t e d i n p u r i t y p r e s e n t o n l y i n one of the many f r a c t i o n s was detected. T h i s demonstrates the power of f a c t o r a n a l y s i s methods to i d e n t i f y a d d i t i o n a l unsuspected components i n an unknown sample. The errant f r a c t i o n 2 was omitted from the data matrix f o r subsequent t a r g e t t e s t i n g a g a i n s t s e v e r a l standard s o l u t i o n s . The values of SPOIL f o r o - x y l e n e , ethy1 benzene, and p - x y l e n e as t e s t v e c t o r s were 0.81, 0.39 and 5.5 r e s p e c t i v e l y , i n d i c a t i n g the p r e s e n c e of a l l three i n the data matrix. Benzene, toluene and m-xylene gave SPOIL values of 47, 19 and 56 r e s p e c t i v e l y , c o n f i r m i n g the l a c k of t h e s e components. The s l i g h t l y h i g h v a l u e of SPOIL f o r p - x y l e n e was b e l i e v e d to be due to a s l i g h t wavelength s h i f t e r r o r , i n d i c a t i n g the importance of proper wavelength r e g i s t r a t i o n . T h i s e r r o r i n the t e s t vector was 61 n o t r e a l i z e d u n t i l t h e t a r g e t f a c t o r a n a l y s i s was c a r r i e d out, i n d i c a t i n g the s e n s i t i v i t y of t h e s e t e c h n i q u e s t o s m a l l but d i s t i n c t i v e f e a t u r e s i n s p e c t r a . They f u r t h e r a p p l i e d rank a n n i h i l a t i o n f a c t o r a n a l y s i s t o t h e same s y s t e m t o g i v e a c t u a l c o n c e n t r a t i o n s of e a ch component i n each f r a c t i o n <5>. The e r r o r i n the d e t e r m i n e d c o n c e n t r a t i o n s was o n l y v e r y s l i g h t l y more t h a n t h e e r r o r a s s o c i a t e d w i t h the p r e p a r a t i o n of the samples t h e m s e l v e s . Thus q u a n t i t a t i v e a n a l y s e s r i v a l l i n g c o n v e n t i o n a l methods were p o s s i b l e , w i t h the added a b i l i t y to handle more complex m i x t u r e s than can be handled by o t h e r methods. W h i t n e y <57> p o i n t e d o u t t h a t p r i n c i p a l component a n a l y s i s d e r i v e s an i n t e r m e d i a t e s e t of m u t u a l l y o r t h o g o n a l l i n e a r c o m b i n a t i o n s of the o r i g i n a l v a r i a b l e s to g i v e a new s p a c e d e f i n e d by l a t e n t v a r i a b l e s , or e i g e n v e c t o r s . T h i s l e a d to a n a t u r a l r e d u c t i o n i n d i m e n s i o n a l i t y to the minimum r e q u i r e d t o c h a r a c t e r i z e t h e s y s t e m . I n a t t e m p t i n g t o c l a s s i f y s t e l l a r s p e c t r a l t y p e s , 3 d i m e n s i o n s appeared to be s u f f i c i e n t t o a c h i e v e c l a s s i f i c a t i o n . The t h i r d v a r i a b l e was a l m o s t l o s t i n t h e n o i s e h o w e v e r , and s o l u t i o n s w i t h j u s t 2 d i m e n s i o n s a l s o worked w e l l . P r e d i c t i o n s were then b a s e d on 6 s p e c t r a l i n d i c e s , w h i c h r e d u c e d t o a 2 d i m e n s i o n a l space f o r the purposes of c l a s s i f i c a t i o n . The p r i n c i p a l component a n a l y s i s method gave the best r e s u l t s of any c l a s s i f i c a t i o n method used i n a s t r o n o m y . I t s s u c c e s s 62 was a t t r i b u t e d t o the a b i l i t y t o s y s t e m a t i c a l l y s e l e c t the most i m p o r t a n t d a t a p o i n t s and e l i m i n a t e t h e i n f l u e n c e o f e r r o r s i n t h e d a t a . T h i s a p p l i c a t i o n e m p h a s i z e s t h e m u l t i d i s c i p l i n a r y n a t u r e of t h e s e methods. T h i s a l s o e m p h a s i z e s t h e u n f o r t u n a t e f a c t t h a t t h e s e methods a r e commonly i n use i n many d i v e r s e f i e l d s , and t h o s e i n one f i e l d (e.g. astronomy) may not be aware of the advances made i n t h e s e methods i n a n o t h e r f i e l d (e.g. c h e m i s t r y ) , even though the r e s u l t s a re ve r y r e l e v a n t . C a r e must be t a k e n when c o n s t r u c t i n g a d a t a m a t r i x t o s e l e c t r e p r e s e n t a t i v e s a m p l e s w i t h s u f f i c i e n t v a r i a n c e . Wold and Dunn <58> d i s c u s s e d t h e c o n d i t i o n s r e q u i r e d f o r r e l a t i o n s b e t w e e n s t r u c t u r e and a c t i v i t y t o be f o u n d by v a r i o u s p a t t e r n r e c o g n i t i o n methods. The a b i l i t y f o r e x a m p l e t o p r e d i c t b i o l o g i c a l a c t i v i t y f r o m s t r u c t u r a l f e a t u r e s was a p r i m a r y g o a l . They warned of the p o s s i b i l i t y o f s a m p l i n g a r t i f a c t s and t h e need t o t a k e a p p r o p r i a t e p r e c a u t i o n s t o i n s u r e t h a t s u p e r f l u o u s c o r r e l a t i o n s w o u l d not c o m p r o m i s e t h e p r e d i c t i v e a b i l i t y of a mo d e l . I f t h e t r a i n i n g s e t was b i a s e d w i t h r e s p e c t t o too many occura n c e s of an uncommon f e a t u r e r e l a t i v e t o t h e more common on e 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 p r e d i c t i o n s c o u l d be made w h i c h w o u l d have no r e a l s i g n i f i c a n c e . An e x a m p l e was g i v e n c o n c e r n i n g the N-N=0 group i n n i t r o s a m i n e s . As a r e s u l t of the d i s c o v e r y t h a t many n i t r o s a m i n e s a re c a r c i n o g e n s , many more were s y n t h e s i z e d and t e s t e d f o r t h e i r a c t i v i t y t h a n 63 w o u l d n o r m a l l y be t h e c a s e . The model was t h e n b i a s e d t o w a r d s t h e p r e s e n c e o f N-N=0 i n c a r c i n o g e n i c s u b s t a n c e s , and a s t r o n g s t r u c t u r e a c t i v i t y r e l a t i o n s h i p between N-N=0 and c a r c i n o g e n e i t y a r o s e . Many r e l a t e d n i t r o s a m i n e s w i t h o u t a c t i v i t y were n ot f u r t h e r i n v e s t i g a t e d , so t h e model was b i a s e d towards those w i t h h i g h a c t i v i t y . C a r p e n t e r and T i l l <16> were one o f t h e few g r o u p s t o use p a t t e r n r e c o g n i t i o n methods on d a t a o b t a i n e d f r o m i n d u c t i v e l y c o u p l e d p l a s m a o p t i c a l e m i s s i o n s p e c t r o m e t r y . They d i d n o t d e a l d i r e c t l y w i t h t h e s p e c t r a l d a t a however. I n s t e a d t h e y d e t e r m i n e d 8 e l e m e n t a l c o n c e n t r a t i o n s by t r a d i t i o n a l u n i v a r i a t e t e c h n i q u e s . The r e s u l t s f o r 8 d i f f e r e n t t y p e s o f b r a s s e s were t h e n a n a l y z e d f i r s t by s i m p l y l o o k i n g f o r t h e p r e s e n c e o r a b s e n c e of a p a r t i c u l a r element as a s o l e d i s t i n g u i s h i n g f e a t u r e . T h i s was a g a i n a u n i v a r i a t e a p p r o a c h , and p r o v e d s a t i s f a c t o r y o n l y f o r p a r t i a l c l a s s i f i c a t i o n . P a t t e r n r e c o g n i t i o n methods were t h e n a p p l i e d . A h i e r a r c h i c a l c l u s t e r i n g method gave 6 g r o u p s , c o m p a r i n g w e l l w i t h t h e 8 a c t u a l l y p r e s e n t e x c e p t f o r 3 w h i c h were c l u s t e r e d as one. The most s u c c e s s f u l method o f d i s p l a y was non l i n e a r m a pping, w h i c h showed 8 g r o u p s . O n l y 2 o f ,the g r o u p s were q u e s t i o n a b l y c l o s e t o g e t h e r . A l l o t h e r s were w e l l s e p a r a t e d . The f o r e n s i c a p p l i c a t i o n s of these c l a s s i f i c a t i o n s of b r a s s samples were s t r e s s e d . 64 D e v i a t i o n s from l i n e a r i t y can sometimes be a d j u s t e d to a l l o w t h e s y s t e m t o be t r e a t e d as c o m p l e t e l y l i n e a r . H a a l a n d e t a l <59> a p p l i e d w e i g h t e d m u l t i v a r i a t e l e a s t s q u a r e s m e t h o d s . N o n l i n e a r i t i e s i n B e e r ' s l a w w e r e a p p r o x i m a t e d by a l l o w i n g the r e f e r e n c e s p e c t r a to be known m i x t u r e s . W i t h s e v e r e l y o v e r l a p p i n g s p e c t r a l bands, t h e average r e l a t i v e e r r o r i n e s t i m a t e d c o n c e n t r a t i o n s was l e s s t h a n 1%. V e r y r e c e n t l y , a c o r r e l a t i o n a p p r o a c h t o i n f o r m a t i o n e x t r a c t i o n from complex i n d u c t i v e l y c o u p l e d plasma s p e c t r a was c a r r i e d o u t . Ng and H o r l i c k <60,61> u s e d t h e i n t e r f e r o g r a m s from a F o u r i e r t r a n s f o r m s p e c t r o m e t e r system a s t h e i r d a t a b a s e . S y n t h e t i c s i n g l e f r e q u e n c y i n t e r f e r o g r a m s ( s i n e waves) were p r o d u c e d t o g i v e a one f r e q u e n c y mask f o r each e l e m e n t s e a r c h e d f o r by s p e c t r a l c o r r e l a t i o n . A peak a t z e r o c o r r e l a t i o n s h i f t i d e n t i f i e d t h e p r e s e n c e o f an e l e m e n t . T h i s was e q u i v a l e n t t o d e t e r m i n i n g the presence of an element by s p e c t r a l i n t e n s i t y at one p a r t i c u l a r c h a r a c t e r i s t i c w a v e l e n g t h , and shows a l l t h e same l i m i t a t i o n s as any o t h e r u n i v a r i a t e t e c h n i q u e , e s p e c i a l l y s u s c e p t i b i l i t y t o s p e c t r a l i n t e r f e r e n c e s . To b r i n g about an improvement i n a c c u r a c y , and to p r o v i d e some method f o r d e t e r m i n i n g w h i c h i n t e r f e r e n c e s were p r e s e n t , t h r e e f r e q u e n c i e s were taken f o r each element g i v i n g t h r e e m a s k s t o be r u n f o r e a c h e l e m e n t . T h u s , w h e r e an a n o m o l o u s l y h i g h response might be seen a t one f r e q u e n c y , i f 65 c o r r e s p o n d i n g r e s p o n s e s w e r e n o t s e e n a t t h e o t h e r f r e q u e n c i e s , t h e i n i t i a l r e s p o n s e c o u l d be due t o an i n t e r f e r i n g l i n e r a t h e r t h a n t h e p r e s e n c e o f t h e e l e m e n t . T h i s worked w e l l i n t h e e x a m p l e s g i v e n i n t h e p a p e r , but q u a n t i t a t i v e r e s u l t s were s t i l l more d i f f i c u l t t o o b t a i n when i n t e r f e r e n c e s were p r e s e n t , h o w e v e r , t h e s e r e s u l t s w o u l d be i m p o s s i b l e t o o b t a i n u s i n g j u s t one l i n e . T h i s f u r t h e r i l l u s t r a t e s t h a t t h e a p p l i c a t i o n o f m u l t i v a r i a t e methods g i v e s s u p e r i o r r e s u l t s t o a u n i v a r i a t e approach. A f u r t h e r i m p r o v e m e n t t o t h e c o r r e l a t i o n masks c o u l d be s u g g e s t e d . I f t h e mask were a s i n g l e mask composed o f s e v e r a l f r e q u e n c i e s f o r a l l s t r o n g and weak l i n e s , a very-s t r o n g r e s p o n s e w o u l d be e x p e c t e d i f t h e e l e m e n t were p r e s e n t . I f t h e e l e m e n t were n o t p r e s e n t , l i t t l e o r no response would be seen, s i n c e s p u r i o u s c o r r e l a t i o n s brought about by s p e c t r a l i n t e r f e r e n c e s would be g r e a t l y outweighed by t h e i n f o r m a t i o n f r o m t h e u n o v e r l a p p e d r e g i o n s of t h e s p e c t r u m . Thus, t h e b e s t c o r r e l a t i o n mask t o use w o u l d be one c o m p r i s i n g t h e e n t i r e s p e c t r u m o f t h e e l e m e n t o f i n t e r e s t . S i m i l a r l y , i n any o t h e r m u l t i v a r i a t e a p p r o a c h , t h e u s e o f t h e e n t i r e s p e c t r u m a l l o w s t h e maximum i n f o r m a t i o n to be used f o r e l e m e n t a l i d e n t i f i c a t i o n . T h i s r e p r e s e n t s t h e e x t e n t o f t h e r e s e a r c h b e i n g p r e s e n t l y c a r r i e d o u t i n p a t t e r n r e c o g n i t i o n i n t h e s c i e n t i f i c , and e s p e c i a l l y c h e m i c a l f i e l d . These methods 66 have been adapted to the a r e a of e m i s s i o n s p e c t r o s c o p y w i t h an e m p h a s i s on t h e a n a l y t i c a l a p p l i c a t i o n s t o i n d u c t i v e l y c o u p l e d plasmas. 67 3.2 SUMMARY OF CONCEPTS FROM THE LITERATURE A t t h i s p o i n t , a s h o r t s u m m a r y o f p r e v i o u s w o r k , w i t h a n e m p h a s i s o n t h e u s e f u l c o n c e p t s d e v e l o p e d , i s i n o r d e r b e f o r e p r o c e e d i n g t o t h e w o r k done i n t h i s t h e s i s . L e a s t s q u a r e s m a t r i x a l g e b r a c a n s o l v e c o m p l e x m a t r i c e s r e p r e s e n t i n g s p e c t r a a n d g i v e c o m p o n e n t c o n c e n t r a t i o n s , s i n c e e a c h i n t e n s i t y m e a s u r e m e n t i s a l i n e a r sum o f s i g n a l s f r o m e a c h c o m p o n e n t . A c o m p u t e r i s r e q u i r e d t o a c c o m p l i s h t h i s e f f i c i e n t l y . Good q u a n t i t a t i v e f i t s a r e o b t a i n e d when a l l c o m p o n e n t s i d e n t i f i e d b e f o r e t h e l e a s t s q u a r e s f i t . M e a s u r e m e n t s a t m a n y w a v e l e n g t h s ( m u l t i v a r i a t e ) g i v e g e n e r a l l y b e t t e r r e s u l t s t h a n o n e l i n e ( u n i v a r i a t e ) m e a s u r e m e n t s . C a r e s h o u l d b e t a k e n t o a v o i d f o r c i n g u n i v a r i a t e s o l u t i o n s on m u l t i v a r i a t e p r o b l e m s . F a c t o r a n a l y s i s c a n i m p r o v e u p o n a l e a s t s q u a r e s f i t and r e d u c e t h e c o m p l e x i t y o f a s y s t e m by s p e c i f y i n g t h e m i n i m u m n u m b e r o f f a c t o r s ( F ) r e q u i r e d t o r e p r o d u c e t h e d a t a . F a c t o r a n a l y s i s c a n r e m o v e n o i s e f r o m d a t a s e t s by r e m o v i n g t h e n o n -s i g n i f i c a n t e i g e n v e c t o r s a f t e r F . Many i n d i c a t o r s o f F a r e u s e d , e a c h w i t h i t s own s t r e n g t h s a r e w e a k n e s s e s . I t i s c o n v e n i e n t t o e x p r e s s m u l t i v a r i a t e d a t a s e t s i n a m u l t i d i m e n s i o n a l p a t t e r n s p a c e . W h e n t h i s i s d o n e , t h i s s p a c e c a n be v i e w e d t h r o u g h t w o d i m e n s i o n a l w i n d o w s f o r i n t e r p r e t a t i o n u s i n g e i g e n v e c t o r p r o j e c t i o n s and n o n l i n e a r m a p s . T h e r e l a t i o n s h i p s b e t w e e n s i m i l a r g r o u p s , a n d t h e c o m p l e x i n f l u e n c e s o f many s i m u l t a n e o u s l y v a r y i n g p a r a m e t e r s 68 can be seen . T a r g e t f a c t o r a n a l y s i s and p a t t e r n r e c o g n i t i o n a r e r e l a t e d . S u c c e s s f u l t e s t v e c t o r s a r e s o u g h t t o i d e n t i f y components i n the f o r m e r , and " e f f e c t v a r i a b l e s " a re sought to i d e n t i f y components i n the l a t t e r . The s i z e of a s p e c t r a l window c o l l e c t e d f o r a n a l y s i s i s i m p o r t a n t i n m u l t i v a r i a t e methods. N a r r o w i n g a s p e c t r a l window r e d u c e s t h e i n f o r m a t i o n c o n t e n t o f t h e s p e c t r u m d r a s t i c a l l y and a d v e r s e l y a f f e c t s t h e a c c u r a c y o f t h e r e s u l t s . I n c o n t r a s t , l o w e r i n g the r e s o l u t i o n r e t a i n s much of t h e i n t e n s i t y i n f o r m a t i o n i n t h e s p e c t r u m , and does n o t d r a m a t i c a l l y a f f e c t m u l t i v a r i a t e a n a l y s i s r e s u l t s . The s u i t a b i l i t y of a s p e c t r a l window depends on the d e n s i t y of i n f o r m a t i o n i t c o n t a i n s , w i t h many, many l i n e s b e i n g b e s t . D e t e c t i o n l i m i t s can be i m p r o v e d by t a k i n g an e n t i r e s p e c t r a l window f o r a n a l y s i s r a t h e r t h a n j u s t one or two l i n e s . V a r i a t i o n s i n s t r u c t u r e d b a c k g r o u n d can be a c c o u n t e d f o r by i n c l u d i n g b a c k g r o u n d as a f a c t o r i f n e c c e s s a r y . U n s u s p e c t e d c o m p o n e n t s may be d e t e c t e d and s u b e s e q u e n t l y i d e n t i f i e d i n a m u l t i v a r i a t e a n a l y s i s e v e n when t h e component pea k s a r e c o m p l e t e l y o v e r l a p p e d w i t h o t h e r components. T a r g e t t e s t i n g can q u a l i t a t i v e l y i d e n t i f y i n d i v i d u a l components i n complex m i x t u r e s . S m a l l d e v i a t i o n s i n m i x t u r e 69 s p e c t r a can be t r e a t e d l i n e a r l y , and p o s s i b l y be r e l a t e d to p h y s i c a l p arameters a f f e c t i n g the m i x t u r e (such as changes i n s t r u c t u r a l c o n f o r m a t i o n s i n p o l y m e r s ) . F i n a l l y , the c o o r d i n a t e system of p a t t e r n space can be r o t a t e d so t h a t the axes c o r r e s p o n d t o p h y s i c a l p r o p e r t i e s , such as c o n c e n t r a t i o n s . I n t h e f o l l o w i n g c h a p t e r , t h e s e p r i n c i p l e s w i l l be a p p l i e d t o t h e i n v e s t i g a t i o n and i n t e r p r e t a t i o n o f i n d u c t i v e l y c o u p l e d plasma s p e c t r a . 70 CHAPTER IV EXPERIMENTAL An i n d u c t i v e l y c o u p l e d p l a s m a u n i t m a n u f a c t u r e d by Plasma Therm Inc., K r e s s e n N. J . was used. A PT-2500 plasma t o r c h a s s e m b l y c o n t a i n e d t h e p l a s m a i t s e l f . An HFP-2500E r a d i o f r e q u e n c y g e n e r a t o r , c a p a b l e o f up t o 2.5 KW o f RF power a t 27.18 Mhz s u s t a i n e d the plasma w i t h the a s s i s t a n c e of an AMN-2500E a u t o m a t i c i m p e d e n c e m a t c h i n g n e t w o r k . A c o n c e n t r i c g l a s s n e b u l i z e r ( m o d e l GN-5601) and a c o n v e n t i o n a l c o n c e n t r i c b a r r e l type spray chamber (model SC-50 3 7 ) , p r o v i d e d t h e s a m p l e a e r o s o l . The p l a s m a f l o w r a t e was 11 1 / r a i n , and t h e a u x i l i a r y f l o w r a t e was 0.9 1/min. The n e b u l i z e r f l o w r a t e was 1.5 l / m i n . An image o f t h e p l a s m a was f o r m e d on t h e e n t r a n c e s l i t by a f u s e d s i l i c a l e n s o f 150 mm f o c a l l e n g t h . O p t i c a l a l i g n m e n t of t h e p l a s m a , l e n s , and m o n o c h r o m a t o r was m a i n t a i n e d by p l a c i n g the e n t i r e a p p a r a t u s on an o p t i c a l r a i l . A S c h o e f f e l - M c P h e r s o n M o d e l 2 7 0 , 0.35 m e t e r monochromator w i t h a 1200 line/mm g r a t i n g b l a z e d a t 500 nm was u s e d . The e n t r a n c e s l i t was s e t a t 100 um, w i t h a 5 mm a p e r a t u r e . R a t h e r t h a n a s i n g l e e x i t s l i t , a 1024 e l e m e n t p h o t o d i o d e a r r a y was p l a c e d i n t h e e x i t f o c a l p l a n e . T h i s a p p a r a t u s has been d e s c r i b e d p r e v i o u s l y by B l a d e s <62>. A b l o c k d i a g r a m i l l u s t r a t e s t h e b a s i c c o n f i g u r a t i o n o f t h e ap p a r a t u s ( F i g u r e 6). 71 RF Iipedence Power Hatching I Generator Network A lens •j load . T coils . 0 r c L h i Liquid Argon sain flow regulator nebulizer flow regulator nebulizer spray chaiber analyte solution Bonochronatcr photodiode array PDA readout board ADC Coapupro Systea 8/16 printer l^ plotterj •onitor floppy disk and hard disk F i g u r e 6. B l o c k Diagram of the I n s t r u m e n t a t i o n . 72 The p r e s e n c e o f a 1024 e l e m e n t p h o t o d i o d e a r r a y ( R e t i c o n RL-1024S) i n the e x i t f o c a l p l a n e a l l o w e d an e n t i r e 50 nm w i n d o w o f t h e I C P e m i s s i o n s p e c t r u m t o be s i m u l t a n e o u s l y a c q u i r e d . The i n t e g r a t i o n t i m e of the a r r a y was s e t t o a v o i d s a t u r a t i o n w i t h a l l s o l u t i o n s . D a t a a c q u i s i t i o n and a r r a y t i m i n g was s u p e r v i s e d by a Compupro S y s t e m 8/16, u s i n g c u s t o m s o f t w a r e w r i t t e n i n F o r t r a n and d e v e l o p e d i n o u r l a b o r a t o r y . F r o m e a c h 1024 e l e m e n t photodiode spectrum, a segment of the spectrum c o n s i s t i n g of 380 a d j a c e n t p h o t o d i o d e s was e x t r a c t e d and used f o r f u r t h e r d a t a a n a l y s i s ( F i g u r e s 7 and 8 ) . T h i s s p e c t r u m was d i v i d e d i n t o 38 s e t s o f 10 d i o d e s e a c h . The measured i n t e n s i t i e s a c r o s s each s e t of t e n d i o d e s were summed. T h i s g a v e a r e d u c e d s p e c t r u m o f 38 i n t e g r a t e d i n t e n s i t y measurements w i t h an e f f e c t i v e s p e c t r a l r e s o l u t i o n of 4 to 5 A ( F i g u r e 9 ) . S t a n d a r d s o l u t i o n s o f 500 ppm were p r e p a r e d f o r t h e f o l l o w i n g e l e m e n t s f r o m a n a l y t i c a l q u a l i t y r e a g e n t s . Cadmium (CdCl2'2 1/2 H2O, F i s h e r S c i e n t i f i c ) , c h r o m i u m ( C r ( N 0 3 ) 3 * 9 H 2 0 , Ma 1 1 i n k r o d t ) , c o p p e r (Cu(NO3)2 * 3 H 2 0 • A n a l a r ) , m a g n e s i u m ( M g C l 2 ' 6 H 2 0 , Ma 1 1 i n k r o d t ) , n i c k e l ( N i(N03)2'6H20, BDH C h e m i c a l s ) , s t r o n t i u m (SrCl2*6H2°» F i s h e r S c i e n t i f i c ) , z i n c ( Z n C l 2 , a n h y d r o u s , A l d r i c h ) , and barium (BaCl2*2H2^) were prepa r e d by d i s s o l v i n g i n d e i o n i z e d w a t e r , and d i l u t i n g t o t h e mark i n a 250 ml v o l u m e t r i c f l a s k . I r o n (Fe w i r e ) was d i s s o l v e d i n d i l u t e n i t r i c a c i d 73 ure 7. Spectral Window (380 photodiodes) for Cd, Cr, Cu, and Fe. 74 Zn Figure 8. Spectral Window (380 photodiodes) for Mg, Ni, Sr, and Zn. 75 F i g u r e 9 . Reduced S p e c t r a of Fe 76 and t h e n d i l u t e d t o t h e mark i n a 250 ml v o l u m e t r i c f l a s k . S p e c t r a of the 500 ppm s t a n d a r d aqueous s o l u t i o n s were taken w i t h t h e s e l e c t e d s p e c t r a l window a t 453 t o 471 nm. F i v e r e p l i c a t e s p e c t r a (7 f o r N i ) w e r e c o l l e c t e d f o r e a c h e l e m e n t . Each s p e c t r u m c o n s i s t e d o f 25 s c a n s c o - a d d e d t o improve s i g n a l to n o i s e r a t i o and the background s u b t r a c t e d to remove non a n a l y t e l i n e s and continuum. S p e c t r a of t h r e e m i x t u r e s (250 ppm Fe/250 ppm N i , 250 ppm Ni/250 ppm Cu, and a m i x t u r e o f f i v e e l e m e n t s (Cr , M g , N i , Sr , Zn) a l l a t 250 ppm were o b t a i n e d i n t h e same manner. The d i g i t i z e d s p e c t r a were s t o r e d on 8 i n c h f l o p p y d i s k s as s t a n d a r d CPM f i l e s , t h e n t r a n s f e r r e d t o t h e UBC campus c o m p u t e r s y s t e m (an MTS based system) where programs w r i t t e n i n F o r t r a n and APL were used t o c a r r y o u t t h e s u b s e q u e n t m u l t i v a r i a t e a n a l y s i s . ARTHUR <63>, a p a t t e r n r e c o g n i t i o n p r o g r a m w r i t t e n i n F o r t r a n , was used t o g e n e r a t e e i g e n v e c t o r p r o j e c t i o n s and non l i n e a r maps. QUICKFACT0R, a p r o g r a m w r i t t e n i n APL i n our l a b and b a s e d on t h e p r i n c i p l e s o u t l i n e d i n M a l i n o w s k i and Howery <27> was used f o r a l l f a c t o r a n a l y s i s r o u t i n e s . 77 CHAPTER V RESULTS AND DISCUSSION Each o f t h e s p e c t r a c o l l e c t e d were r e d u c e d t o 38 f e a t u r e s , r e p r e s e n t i n g the i n t e n s i t y over 10 a d j a c e n t d i o d e s f o r each f e a t u r e . Each spectrum i s thus r e p r e s e n t e d by a 38 d i m e n s i o n a l d a t a v e c t o r . The s p e c t r u m can be p l o t t e d i n p a t t e r n s p a c e , and w i l l a p p e a r as a s i n g l e p o i n t . S i n c e s p e c t r a f o r d i f f e r e n t elements each c o n t a i n t h e i r own unique c o m b i n a t i o n o f s p e c t r a l i n t e n s i t i e s , t h e s p e c t r a o f d i f f e r e n t e l e m e n t s s h o u l d a p p e a r i n d i s t i n c t l y d i f f e r e n t p o s i t i o n s i n p a t t e r n s p a c e . E i g h t d i s t i n c t c l u s t e r s a r e c l e a r l y s een i n a non l i n e a r map of p a t t e r n s p a c e ( F i g u r e 10). A n a t u r a l c l u s t e r i n g o f l i k e s p e c t r a o c c u r s , w i t h s p e c t r a of u n l i k e elements b e i n g e a s i l y d i s t i n g u i s h e d . The n e x t s t e p i n t h i s i n v e s t i g a t i o n of t h e a p p e a r a n c e of s p e c t r a i n p a t t e r n s p a c e i s t o probe t h e e f f e c t o f c o n c e n t r a t i o n on the p o s i t i o n of a pure e l e m e n t a l spectrum i n p a t t e r n s p a c e . The i n t e n s i t i e s a t a l l w a v e l e n g t h s f o r one a v e r a g e d s p e c t r u m f o r each e l e m e n t were s c a l e d t o g e n e r a t e s p e c t r a c o r r e s p o n d i n g t o 500, 400, 300, 200, and 100 ppm. T h i s s c a l i n g p r o c e d u r e i s c o n s i s t e n t w i t h t h e p r a c t i c e of g e n e r a t i n g w o r k i n g c u r v e s from two p o i n t s . The f i r s t p o i n t i s a r e a d i n g o r a v e r a g e o f r e a d i n g s f o r a s e t c o n c e n t r a t i o n , w i t h t h e s e c o n d p o i n t on t h e w o r k i n g c u r v e b e i n g t h e r e a d i n g f o r a b l a n k s o l u t i o n . T h i s c a n be 78 79 r e l i a b l y done when i t i s known t h a t response i s l i n e a r over t h e r a n g e c o n s i d e r e d . The r e s p o n s e o f t h e I C P t o c o n c e n t r a t i o n changes i s g e n e r a l l y l i n e a r over 4 to 5 o r d e r s of magnitude. The range from 500 to 100 ppm i s w e l l w i t h i n these l i m i t s . These s p e c t r a f o r 8 elements (Cd, Cr, Cu, Fe, Mg, N i , Sr and Zn) were t h e n p l o t t e d i n p a t t e r n s p a c e ( F i g u r e s 11 to 16). The barium spectrum was saved f o r l a t e r t a r g e t t e s t i n g , to r e p r e s e n t an unknown not i n c l u d e d i n the o r i g i n a l s e t of 8 elements. A number of e i g e n v e c t o r p r o j e c t i o n s were p r o d u c e d t o o b t a i n s e v e r a l v i e w s of p a t t e r n space. A f t e r e x a m i n a t i o n of t h e s e p r o j e c t i o n s , s e v e r a l o b s e r v a t i o n s may be made. 1) Each s e r i e s of c o n c e n t r a t i o n s d e f i n e s an a x i s i n p a t t e r n s p a c e and each a x i s i s i n a u n i q u e d i r e c t i o n . These a x e s a r e t h e e l e m e n t a x e s . 2) Each a x i s r a d i a t e s f r o m a common p o i n t , w h i c h i s t h e p o s i t i o n o f a b a c k g r o u n d s p e c t r u m i n p a t t e r n s p a c e . 3) E x a m i n a t i o n o f t h e s c a l e s on each a x i s s h o w s t h a t d i s p l a c e m e n t a l o n g e a c h a x i s i s d i r e c t l y p r o p o r t i o n a l to the c o n c e n t r a t i o n of the element r e p r e s e n t e d by t h a t a x i s . A l t h o u g h t h e a x e s a r e w e l l s e p a r a t e d , i n t h e e i g e n v e c t o r p r o j e c t i o n s i t appears t h a t some axes are r a t h e r c l o s e t o each o t h e r . I f t h i s were t h e c a s e , i t m i g h t make d i s c r i m i n a t i o n of one element from another more d i f f i c u l t . I t s h o u l d be k e p t i n mind t h a t t h e v a r i a n c e i n o n l y two d i m e n s i o n s i s b e i n g v i e w e d a t any one t i m e , and t h u s w o u l d CM cr o i— O U J Sr Ni Zn Cr Mg o UJ Fe E I G E N V E C T O R 1 Figure 11. Eigenvector Projection of Working Curves (eigenvector 1 vs eigenvector 2). 81 E I G E N V E C T O R I e 12. Eigenvector Projection of Working Curves (eigenvector 1 vs eigenvector 3). 82 Cd Figure 13. Eigenvector Projection of Working Curve (eigenvector 2 vs eigenvector 3). 83 Figure 14. Eigenvector Projection of Working Curves (eigenvector 1 vs eigenvector 4). 84 o i— o UJ UJ CD UJ Fe Cd Ni Zn E I G E N V E C T O R 2 Figure 15. Eigenvector Projection of Working Curves (eigenvector 2 vs eigenvector 4). 85 a: o i— o U J UJ UJ E I G E N V E C T O R 3 Figure 16. Eigenvector Projection of Working Curves (eigenvector 3 vs eigenvector 4). 86 be e x p e c t e d t o g i v e b e s t r e s u l t s i f o n l y two m a j o r f a c t o r s a f f e c t e d the spread of the data p o i n t s . In the case of an 8 e l e m e n t s y s t e m , a t l e a s t 8 f a c t o r s (one f o r each e l e m e n t ) a r e a t work. An e i g e n v e c t o r p r o j e c t i o n may be u n a b l e t o show a l l t h e s e p a r a t i o n i n h e r e n t i n p a t t e r n s p a c e i n o n l y t w o d i m e n s i o n s . The n e e d t o v e r i f y t h e n e a r e q u a l s e p a r a b i l i t y o f a l l e l e m e n t a x e s l e a d s t o t h e use o f a non l i n e a r map. A non l i n e a r map was p r o d u c e d w i t h t h e 8 e l e m e n t s p e c t r a l d a t a ( F i g u r e 17). I t can be c l e a r l y seen t h a t a l l axes are w e l l s e p a r a t e d . B o t h o f t h e s e t y p e s o f g r a p h s a c t u a l l y r e p r e s e n t w o r k i n g c u r v e s f o r each of the elements s t u d i e d . These m u l t i d i m e n s i o n a l w o r k i n g c u r v e s can be b e t t e r u n d e r s t o o d i f a one d i m e n s i o n a l w o r k i n g c u r v e i s f i r s t c o n s i d e r e d and then the concepts developed a re extended to h i g h e r d i m e n s i o n s . A c o n v e n t i o n a l (one d i m e n s i o n a l ) working curve i s one where the c o n c e n t r a t i o n of some component (e.g. an e l e m e n t ) i s d i r e c t l y p r o p o r t i o n a l t o some measured p a r a m e t e r (e.g. t h e i n t e n s i t y of a s i n g l e s p e c t r a l l i n e ) . F i g u r e 18a shows t h e i n t e n s i t y of a s p e c t r a l l i n e as a f u n c t i o n o f t h e c o n c e n t r a t i o n o f i r o n . The i n t e n s i t y a x i s may be c a l i b r a t e d u s i n g t h i s g r a p h so t h a t a s p e c i f i c i n t e n s i t y r e p r e s e n t s a s p e c i f i c c o n c e n t r a t i o n . I n F i g u r e 18b, a s a m p l e c o n t a i n i n g two e l e m e n t s , c o p p e r and i r o n , i s 87 Figure 17. Non Linear Map of Working Curves. 88 a) /[Ni] Figure 18. Examples of Working Curves in 1, 2 and 3 Dimensions. a) a conventional working curve of intensity vs concentration. b) determination of the relative concentrations of two elements in a mixture. c) extension to three elements. d) eigenvector projection of eigenvectors 1 and 2 for a system with 3 elemental working curves. e) eigenvector projection of eigenvectors 1 and 3 for a system with 3 elemental working curves. 89 r e p r e s e n t e d by a p o i n t . The r e l a t i v e c o n c e n t r a t i o n s of two elements can be d e t e r m i n e d by l o o k i n g at the magnitudes of t h e r e l a t i v e i n t e n s i t i e s of a n a l y t i c a l l i n e s f o r each element. A b s o l u t e c o n c e n t r a t i o n s can be o b t a i n e d by proper s c a l i n g a l o n g each of the two axes. The d i s p l a c e m e n t a l o n g each a x i s then r e p r e s e n t s the c o n c e n t r a t i o n of t h a t element. A 3 e l e m e n t c a s e can be r e p r e s e n t e d by u s i n g a 3 d i m e n s i o n a l c a r t e s i a n c o o r d i n a t e s y s t e m ( F i g u r e 1 8 c ) . The l o g i c a l e x t e n s i o n o f t h i s c o n c e p t t o 8 d i m e n s i o n s g i v e s an 8 d i m e n s i o n a l p a t t e r n space, and 8 independent element axes. By d rawing upon knowledge of 3 d i m e n s i o n a l c o o r d i n a t e systems, one can b e t t e r i n t e r p r e t the 2 d i m e n s i o n a l images of 8 d i m e n s i o n a l p a t t e r n s p a c e o b t a i n e d by e i g e n v e c t o r p r o j e c t i o n s and non l i n e a r maps. A 3 d i m e n s i o n a l c a r t e s i a n c o o r d i n a t e system appears as i n F i g u r e 18c. An e i g e n v e c t o r p r o j e c t i o n of the f i r s t two e i g e n v a l u e s from an e i g e n v e c t o r a n a l y s i s o f t h e 3 d i m e n s i o n a l c a r t e s i a n c o o r d i n a t e s y s t e m w i l l g i v e F i g u r e 18d. T h i s v i e w c o r r e s p o n d s t o one t a k e n down an a x i s d e f i n e d by t h e v e c t o r (1,1,1) a l o n g a cube d i a g o n a l t o t h e c o o r d i n a t e a x e s . I f i n s t e a d 1 and 3 a r e p l o t t e d , one o b t a i n s F i g u r e 18e. B e i n g f a m i l i a r w i t h 2 d i m e n s i o n a l r e p r e s e n t a t i o n s of 3 d i m e n s i o n a l space, t h i s can be seen to be a view of the 3 c o o r d i n a t e axes from the s i d e . Note t h a t two o f t h e ax e s a p p e a r t o a l m o s t o v e r l a p i n t h i s p r o j e c t i o n , even t h o u g h i t i s known t h a t t h e y a r e a l l e q u a l l y w e l l s e p a r a t e d . 90 U n d e r s t a n d a b l y , i n an 8 d i m e n s i o n a l c a s e r e d u c e d t o 2 d i m e n s i o n s , t h e r e i s much more p o s s i b i l i t y o f two axe s a p p e a r i n g o v e r l a p p e d , even t h o u g h t h e y a r e s t i l l w e l l s e p a r a t e d i n p a t t e r n space. An e x c e l l e n t example of t h i s i s seen i n t h e 4 vs 3 p r o j e c t i o n ( F i g u r e 1 6 ) , where Fe and Cd a x e s a p p e a r t o be v e r y c l o s e t o g e t h e r . I n t h e 4 vs 2 p r o j e c t i o n ( F i g u r e 15) they appear the best s e p a r a t e d o f a l l axes. E x a m i n a t i o n of many e i g e n v e c t o r p r o j e c t i o n s l e a d s to the c o n c l u s i o n t h a t a l l e l e m e n t a l axes are w e l l s e p a r a t e d . The i n d e p e n d e n c e of each e l e m e n t a l a x i s may a l s o be a r g u e d f r o m a c o n s i d e r a t i o n of t h e p r o p e r t i e s o f a t o m i c e m i s s i o n s p e c t r a . Each e l e m e n t has many c h a r a c t e r i s t i c l i n e s . Many o f t h e s e may a l s o o v e r l a p w i t h l i n e s o f o t h e r elements. However, each element has a unique c o m b i n a t i o n of l i n e s . T h i s c l e a r l y u n i q u e p a t t e r n s e r v e s t o d i s t i n g u i s h the spectrum from o t h e r s . In p a t t e r n space, the data v e c t o r f o r the e l e m e n t a l spectrum i s p l a c e d i n a unique d i r e c t i o n where no o t h e r p o i n t i s found. In low r e s o l u t i o n s p e c t r a and c o m p l i c a t e d m i x t u r e s i t becomes more d i f f i c u l t t o f i n d a l i n e f r e e f r o m s p e c t r a l i n t e r f e r e n c e . I n the worst case, t h e r e may be no c l e a r l i n e p r e s e n t . T h i s i s not a problem when m u l t i v a r i a t e a n a l y s i s i s used. S i n c e a l a r g e segment of spectrum i s bei n g viewed, the absence of a l i n e can be j u s t as u s e f u l q u a l i t a t i v e l y as t h e p r e s e n c e o f one. The a b s e n c e o f an e l e m e n t a l l i n e 91 p r o v i d e s s t r o n g e v i d e n c e t h a t t h e e l e m e n t i s not p r e s e n t . A l l the i n f o r m a t i o n c o n t a i n e d i n the presence or absence of l i n e s and t h e i r i n t e n s i t y l e a d s to a d i s p l a c e m e n t a l o n g one o r more o f t h e e l e m e n t a l a x e s , b u t t h e r e w i l l be no d i s p l a c e m e n t a l o n g an e l e m e n t a x i s i f t h e e n t i r e s p e c t r u m c h a r a c t e r i s t i c of t h a t element i s not p r e s e n t . The number o f d i f f e r e n t e l e m e n t s p r e s e n t i n a s e t of s p e c t r a may be d e t e r m i n e d by e i g e n v e c t o r a n a l y s i s . I n d i c a t o r f u n c t i o n s a s s i s t i n d e t e r m i n i n g the c o r r e c t number of f a c t o r s , w h i c h i n t u r n r e f l e c t s t h e number of e l e m e n t s p r e s e n t . Q u a l i t a t i v e i d e n t i f i c a t i o n of these e lements may be done w i t h t a r g e t f a c t o r a n a l y s i s , by t a r g e t t e s t i n g f o r each e l e m e n t s e p a r a t e l y u s i n g a t e s t v e c t o r composed of a pure element spectrum. When a l l elements p r e s e n t have been i d e n t i f i e d , a c o m b i n a t i o n s t e p i s c a r r i e d o u t , and t h e c o n c e n t r a t i o n s of each element are found i n each sample. T a r g e t f a c t o r a n a l y s i s was c a r r i e d o u t on a s y s t e m of 42 s p e c t r a . Each s p e c t r u m i n t h e s e t was e i t h e r t h a t o f a pure element or the m i x t u r e s s p e c i f i e d i n the e x p e r i m e n t a l s e c t i o n . E x a m i n a t i o n o f a t y p i c a l r u n o f QUICKFACTOR b e s t i l l u s t r a t e s t h e s e t e c h n i q u e s . The d a t a m a t r i x ( T a b l e I ) c o n s i s t s of p u r e a v e r a g e d s p e c t r a of each of the 8 el e m e n t s , 4 r e p l i c a t e Ni/Cu m i x t u r e s p e c t r a , 3 r e p l i c a t e F e/Ni m i x t u r e s p e c t r a , and 2 s p e c t r a of a m i x t u r e of 5 e l e m e n t s . The e i g e n v a l u e s o b t a i n e d a r e 92 l i s t e d i n T a b l e I I I . F o l l o w i n g t h i s , t h e r e a l , i m b e d d e d , and e x t r a c t e d e r r o r s f o r a l l p o s s i b l e numbers of f a c t o r s are o b t a i n e d . These f u n c t i o n s s h o u l d a l l l e v e l o f f a t t h e a p p r o p r i a t e number of f a c t o r s . A d d i t i o n a l l y , t h e IND f u n c t i o n <36> i s g i v e n , which s h o u l d show a minimum at the c o r r e c t number of f a c t o r s . T a b l e IV c o m p a r e s t h e many i n d i c a t o r s used and t h e i r c h a r a c t e r i s t i c b e h a v i o r a t t h e c o r r e c t number of f a c t o r s . W i t h t h i s d a t a , none of t h e s e i n d i c a t o r s a r e e n t i r e l y s a t i s f a c t o r y by t h e m s e l v e s f o r i n d i c a t i n g the c o r r e c t number of s i g n i f i c a n t f a c t o r s , F. In th e c o u r s e of t h i s work, a n o t h e r measure has been f o u n d w h i c h a p p e a r s t o be f a i r l y r e l i a b l e . S u c c e s s i v e r a t i o s of t h e m a g n i t u d e s o f one e i g e n v a l u e t o t h e n e x t a r e p r o d u c e d (e.g. e i g e n v a l u e l / e i g e n v a l u e 2 , or e i g e n v a l u e 2 / e i g e n v a 1 u e 3 , e t c . ) . T a b l e l l l s u m m a r i z e s t h e i n f o r m a t i o n o b t a i n e d f r o m t h e i n d i c a t o r s . The r a t i o f u n c t i o n i s p l o t t e d i n F i g u r e l 9 . T h i s s i m p l e f u n c t i o n shows maxima f o r s e v e r a l d i f f e r e n t p o s s i b l e v a l u e s of F. With some knowledge of the na t u r e of t h e s p e c t r a , t h e s e maxima can be i n t e r p r e t e d . The f i r s t o c c u r s b e tween t h e f o u r t h and f i f t h e i g e n v a l u e . T h i s i n d i c a t e s f o u r m a j o r f a c t o r s a r e p r e s e n t . E x a m i n a t i o n of t h e s p e c t r a ( F i g u r e s 7 and 8) s h o w s t h a t , o f t h e 8 e l e m e n t a l s p e c t r a u s e d , o n l y A (Cd, S r , C r , Z n ) have s t r o n g l i n e s i n the r e g i o n of i n t e r e s t . A second, l e s s pronounced maximum i s seen at 8 f a c t o r s . These f o u r a d d i t i o n a l f a c t o r s 93 TABLE III E i g e n v a l u e s and I n d i c a t o r s o b t a i n e d from QUICKFACTOR f o r 42 s p e c t r a R a t i o t o Ei g e n E i g e n v a l u e RE IE XE IND next lower v e c t o r e i g e n v a l u e 1 98299599 473 .4 114.8 459. 2 1. 849 1 .21 2 81114630 311 .0 106.7 292. 1 1. 382 2 .02 3 40149439 167 .9 70.5 152. 3 0. 856 3 .00 4 13367989 57 .3 27.8 50. 1 0. 339 15 .86 5 843269 41 .3 22.4 34. 7 0. 287 1 .80 6 467197 27 .6 16.2 21. 9 0. 225 2 .16 7 216136 15 .8 10.1 12. 1 0. 158 3 .12 8 69231 8 .57 5.88 6. 23 0. 106 4 .38 9 15797 5 .55 4.03 3. 80 0. 087 3 .25 10 4857. o - 4 .11 3.15 2. 64 0. 084 1 .50 11 3238. 9 2 .34 1.88 1. 39 0. 065 3 .94 12 822. 19 1 .50 1.26 0. 81 0. 060 4 .92 13 167. 41 1 .31 1.15 0. 64 0. 082 1 .60 14 104. 81 1 .18 1.07 0. 50 0. 130 1 .61 15 65. 146 1 .10 1.04 0. 38 0. 276 1 .19 16 54. 769 1 .00 0.97 0. 24 1. 000 1 .44 17 37. 973 — — _ _ — 94 TABLE IV B e h a v i o r of I n d i c a t o r s near the C o r r e c t Number of F a c t o r s , F_ I n d i c a t o r Standard e r r o r i n e i g e n v a l u e R e s i d u a l s g r e a t e r than 3 times the v a r i a n c e Chi-Squared Imbedded e r r o r ( I E ) E x t r a c t e d e r r o r (XE) Real e r r o r (RE) B e h a v i o r e r r o r becomes l a r g e r than the e i g e n v a l u e a f t e r F approaches z e r o at F l e v e l s o f f a f t e r F l e v e l s o f f or minimum at F l e v e l s o f f at F r i s e s a f t e r F I n d i c a t o r f u n c t i o n (IND) minimum a t F U n i t y L o g ( e i g e n v a l u e ) E i g e n v a l u e s C u mulative % V a r i a n c e o n l y e i g e n v e c t o r s >1 from n o r m a l i z e d data s e t s are s i g n i f i c a n t sharp drop a f t e r F drops at F and l e v e l s o f f a f t e r w a r d s 99.9% of v a r i a n c e r e t a i n e d f o r F f a c t o r s Ref. 39 39 39 36 36 36 36 15 14 9 49 R a t i o maximum at F t h i s work 95 T 1 1 1 1 1 1 1 1 1 1 I ' 1 1 ' 2 4 6 8 10 12 14 16 No. of Factors ure 19. Ratio Function as an Indicator of the Number of Significant Eigenvectors, F. 96 r e f l e c t the presence of 4 e lements w i t h weaker l i n e s i n the r e g i o n of i n t e r e s t . T h i s a l s o r e p r e s e n t s the c u t o f f between the s i g n i f i c a n t and n o n - s i g n i f i c a n t e i g e n v e c t o r s . The r e m a i n i n g e i g e n v e c t o r s may be i n t e r p r e t e d to be due o n l y to n o i s e . When a d a t a m a t r i x c o n t a i n i n g o n l y pure s p e c t r a i s f a c t o r a n a l y z e d , t h i s c u t o f f i s v e r y c l e a r , and a f t e r t h e c u t o f f t h e e i g e n v a l u e s s l o w l y d e c r e a s e , w i t h t h e r a t i o f u n c t i o n f l u c t u a t i n g around about 2. When s p e c t r a l m i x t u r e s a r e i n c l u d e d i n t h e d a t a m a t r i x , t h e c u t o f f i s s t i l l p r e s e n t , but i s l e s s pronounced. A t h i r d maximum i s seen a t 12 f a c t o r s , and a l t h o u g h n o t v e r y h i g h , may be p h y s i c a l l y s i g n i f i c a n t . F a c t o r s 9 t o 12 may r e f l e c t t h e v e r y s m a l l i n f l u e n c e s t h e p r e s e n c e of one element may have on the s p e c t r a of a second element. Thus, f a c t o r s 9 to 12, which are b a r e l y r e s o l v a b l e from the n o i s e may r e p r e s e n t i n t e r e l e m e n t e f f e c t s , s i n c e i t i s not observed when o n l y pure s p e c t r a are used. I t may indeed be p o s s i b l e to q u a n t i f y i n t e r e l e m e n t and m a t r i x e f f e c t s ( f o r e x a m p l e i o n i z a t i o n i n t e r f e r e n c e s ) i n c a s e s where p e r t u r b a t i o n s of the spectrum of an element are l a r g e enough. The f o r m a t i o n of r e f r a c t o r y compounds or o x i d e s , and t h e p r e s e n c e of e a s i l y i o n i z a b l e e l e m e n t s , l e a d i n g t o s h i f t s i n t h e i o n i z a t i o n e q u i l i b r i u m , a r e n o t as m a j o r a p r o b l e m i n plasmas as i n f l a m e s as a r e s u l t of the h i g h t e m p e r a t u r e of the plasma, but may s t i l l be l a r g e enough to be d e t e c t a b l e . T h i s a r e a c e r t a i n l y d e s e r v e s f u r t h e r a t t e n t i o n . 97 A f t e r an e v a l u a t i o n o f t h e number of f a c t o r s p r e s e n t , QUICKFACTOR c o n t i n u e s w i t h a r o u t i n e to i n d i v i d u a l l y t a r g e t t e s t f o r l i k e l y f a c t o r s . Pure e l e m e n t a l s p e c t r a are used as the t a r g e t v e c t o r s . V a l u e s of r e a l and apparent e r r o r , and the SPOIL c r i t e r i o n , e v a l u a t e the s u c c e s s of each t e s t . The v a l u e o f S P O I L i s e s p e c i a l l y u s e f u l . A c c o r d i n g t o M a l i n o w s k i <27>, a v a l u e of 0-3 i n d i c a t e s a good f a c t o r , 3-6 i n d i c a t e s the t e s t v e c t o r may be a f a c t o r , and v a l u e s above 6 a l m o s t c e r t a i n l y i n d i c a t e t h a t i t i s n o t a f a c t o r . S p e c t r a o f a l l 8 e l e m e n t s known t o be i n t h e d a t a m a t r i x s u c c e e d , w i t h a v a l u e o f SPOIL n e a r 1 i n a l l c a s e s ( T a b l e V). I f a s p e c t r u m of an e l e m e n t not p r e s e n t i n t h e d a t a m a t r i x i s t e s t e d , t h e v a l u e o f SPOIL becomes u n a c c e p t a b l y l a r g e , and the t e s t v e c t o r i s r e j e c t e d . T h i s i s shown when a Ba s p e c t r u m i s t e s t e d f o r , w i t h a r e s u l t i n g v a l u e o f SPOIL of 34 i n d i c a t i n g t h a t i t i s not p r e s e n t . I n t h i s e x a m p l e , a l l 8 e l e m e n t s were p r e s e n t as p u r e s p e c t r a i n t h e d a t a m a t r i x . The e l e m e n t s need n o t be r e p r e s e n t e d by pure s p e c t r a i n the data m a t r i x , and i n f a c t a l l o f t h e s p e c t r a i n t h e d a t a m a t r i x can be m i x t u r e s , as l o n g as t h e r e i s a v a r i e t y of m i x t u r e s c o n t a i n i n g d i f f e r e n t e l e m e n t s . For example, a Ni/Cu m i x t u r e and a Cd/Cu m i x t u r e i n t h e d a t a m a t r i x w o u l d be s u f f i c i e n t t o c h a r a c t e r i z e a p u r e Cu s p e c t r u m . T h e r e i s an a d v a n t a g e t o h a v i n g a l l t h e s u s p e c t e d e l e m e n t s p r e s e n t as p u r e s p e c t r a i n t h e d a t a 98 TABLE V R e s u l t s of Target T e s t i n g f o r the Presence of Elements Data v e c t o r (element) Cadmium Chromium Copper I r o n Barium Magnesium N i c k e l S t r o n t i u m Z i n c V a l u e of SPOIL 0.9999 0.8835 0.9500 0.7391 34.3298 0.9485 1.7014 0.7915 0.9688 P r e s e n t i n da t a s e t yes yes yes yes no yes yes yes yes 99 m a t r i x . The d i m e n s i o n a l i t y o f t h e d a t a w o u l d t h e n be e x p e c t e d t o be t h e same as t h e number of pure s p e c t r a p r e s e n t i n the data m a t r i x , no m a t t e r how many m i x t u r e s of the e lements may be p r e s e n t . I f an a d d i t i o n a l d i m e n s i o n i s seen when t h e d a t a m a t r i x i s a n a l y z e d , t h i s i n d i c a t e s t h a t a n o ther element i s p r e s e n t which has not been accounted f o r . T h i s a d d i t i o n a l e l e m e n t can t h e n be d e t e r m i n e d by a d d i n g s p e c t r a of more elements to the data s e t u n t i l the expected and p r e d i c t e d number of f a c t o r s a g r e e . T a r g e t t e s t i n g can be c a r r i e d o u t on t h e o r i g i n a l d a t a m a t r i x u s i n g t h e p ure e l e m e n t a l s p e c t r a as t a r g e t v e c t o r s to c o n f i r m the presence of the a d d i t i o n a l element. So f a r t h i s e v a l u a t i o n o f t h e d a t a m a t r i x has been q u a l i t a t i v e , i n d i c a t i n g o n l y t h e i d e n t i t y o f t h e e l e m e n t s p r e s e n t . These q u a l i t a t i v e r e s u l t s c o u l d then be used by a p r o g r a m m a b l e s l e w s c a n s y s t e m t o measure t h e b e s t l i n e s ( c o n s i d e r i n g l i n e i n t e n s i t i e s and a v o i d i n g s p e c t r a l o v e r l a p s ) f o r each element, g i v i n g q u a n t i t a t i v e r e s u l t s i n a c o n v e n t i o n a l manner. However, q u a n t i t a t i v e i n f o r m a t i o n can be o b t a i n e d d i r e c t l y from the data m a t r i x , e l i m i n a t i n g the need f o r c o n v e n t i o n a l m e a s u r e m e n t s of i n d i v i d u a l l i n e s e n t i r e l y . Once t a r g e t t e s t i n g i s c o m p l e t e , i t i s a s i m p l e m a t t e r t o o b t a i n c o n c e n t r a t i o n s f o r e a c h e l e m e n t . A c o m b i n a t i o n s t e p i s c a r r i e d o u t u s i n g t h e s u c c e s s f u l t e s t v e c t o r s , and a c o l u m n m a t r i x i s o b t a i n e d c o n t a i n i n g t h e c o n c e n t r a t i o n o f each e l e m e n t ( i n u n i t s r e l a t i v e t o t h e 100 s t a n d a r d s used as t e s t v e c t o r s ) f o r e a c h s p e c t r u m i n t h e o r i g i n a l d a t a m a t r i x . The e r r o r s i n each f a c t o r l o a d i n g ( c o n c e n t r a t i o n ) a re a l s o g i v e n . The r e s u l t s are shown i n Table VI. The a c c u r a c y of the r e s u l t s i s i m p r e s s i v e when one c o n s i d e r s t h a t t h e s p e c t r a had an e f f e c t i v e r e s o l u t i o n of 4-5 A. S e v e r a l comments s h o u l d be made. Remember t h a t o n l y 4 of the 8 elements had any s t r o n g l i n e s i n the r e g i o n of i n t e r e s t . The Mg spectrum i n f a c t had o n l y a s i n g l e v e r y weak l i n e t o d i s t i n g u i s h i t f r o m t h e b a c k g r o u n d . Y e t , t h e Mg s p e c t r a c l e a r l y d i s t i n g u i s h t h e m s e l v e s i n the non l i n e a r map ( F i g u r e 17) and a l s o i n the q u a n t i t a t i v e r e s u l t s . T h i s c l e a r l y shows t h a t a q u a n t i t a t i v e m u l t i e l e m e n t a n a l y s i s c a n be c a r r i e d o u t u s i n g a l o w r e s o l u t i o n p o 1 y c h r o m a t o r . The s p e c t r a l i n f o r m a t i o n u s e d ( w e a k , i n t e r f e r e d l i n e s ) t o c a r r y out the a n a l y s i s would be u s e l e s s i f c o n v e n t i o n a l s i n g l e l i n e methods were us e d . However, thr o u g h the use of f a c t o r a n a l y s i s and p a t t e r n r e c o g n i t i o n , t h e l a r g e amount o f i n f o r m a t i o n p r e s e n t i n t h e s e l i n e s i s u t i l i z e d . A w i d e n i n g o f t h e s p e c t r a l window t o i n c l u d e more s t r o n g l i n e s , e s p e c i a l l y f o r C r , Zn, Mg, and Cu ( w h i c h had o n l y weak l i n e s i n t h i s a n a l y s i s ) , s h o u l d i n c r e a s e t h e a c c u r a c y o f t h e f i n a l r e s u l t s , and a l s o s h a r p e n t h e d i s t i n c t i v e d r o p i n e i g e n v a l u e a t t h e c o r r e c t number of 101 TABLE VI E l e m e n t a l C o n c e n t r a t i o n s o b t a i n e d d i r e c t l y  from F a c t o r L o a d i n g s C o n c e n t r a t i o n P e r c e n t a g e Element A c t u a l P r e d i c t e d E r r o r C o m p o s i t i o n I r o n / N i c k e l M i x t u r e Fe 250 ppm 254.1 ppm 1.4 52.8 Ni 250 ppm 226.8 ppm 1.2 47.2 C o p p e r / N i c k e l M i x t u r e Cu 250 ppm 247.1 ppm 7.5 53.3 N i 250 ppm 216.4 ppm 1.2 46.7 5 Element M i x t u r e Cr 250 ppm 199.5 ppm 8.0 18.4 Mg 250 ppm 242.1 ppm 31.0 22.3 Ni 250 ppm 198.7 ppm 2.3 18.3 Sr 250 ppm 223.6 ppm 0.6 20.6 Zn 250 ppm 220.4 ppm 18.8 20.3 102 f a c t o r s , F. A s p e c t r a l window c o n t a i n i n g a l a r g e r number of s t r o n g l i n e s f o r a l l e l e m e n t s w i l l a l s o i m p r o v e d e t e c t i o n l i m i t s f o r some of t h e e l e m e n t s . W i t h t h i s e x a m p l e , a s i g n a l t o n o i s e r a t i o o f t h e o r d e r o f 100 i s o b t a i n e d . The s e n s i t i v i t y f o r each element i s weighted by the number and s t r e n g t h of i t s l i n e s p r e s e n t i n the window. G e n e r a l l y , any e l e m e n t a t l e s s t h a n 1% of t h e c o n c e n t r a t i o n of t h e most c o n c e n t r a t e d c o m p o n e n t w i l l be l o s t i n t h e n o i s e e i g e n v e c t o r s . T h i s l i m i t a t i o n i s c a u s e d p r i m a r i l y by t h e l i m i t e d dynamic range of a photodiode a r r a y (approx. 1000). A l t h o u g h s l i g h t l y s l o w e r i n t h e d a t a a c q u i s i t i o n s t a g e , a f a s t scan monochromator w i t h a p h o t o m u l t i p l i e r tube (dynamic range 10E6) c o u l d p o s s i b l y improve the d e t e c t i o n l i m i t s of t h e m i n o r e l e m e n t a l c o n s t i t u e n t s i n t h e p r e s e n c e of m a j o r o n e s . A l t e r n a t e l y , a m e t h o d o f s c a l i n g a s e r i e s o f p h o t o d i o d e a r r a y s c a n s a t d i f f e r e n t i n t e g r a t i o n t i m e s t o generate a spectrum w i t h o u t s a t u r a t i o n and a g r e a t e r dynamic r a n g e c o u l d a l s o work. A t h i r d o p t i o n , w i t h a l l t h e speed of p h o todiode a r r a y d e t e c t i o n , would i n v o l v e c o l l e c t i n g the d a t a by F o u r i e r t r a n s f o r m i n d u c t i v e l y c o u p l e d p l a s m a s p e c t r o s c o p y <6A,65>. The t e s t v e c t o r s used i n f a c t o r a n a l y s i s c o n s t i t u t e a t r a i n i n g s e t by w h i c h t h e method can q u a n t i f y i n d i v i d u a l c o m p o n e n t s . The t r a i n i n g s e t f o r e m i s s i o n s p e c t r a may be very compact, c o n s i s t i n g of one scan averaged spectrum f o r 103 e a c h e l e m e n t . F o r i n d u c t i v e l y c o u p l e d p l a s m a s , some 60 elements may be a n a l y z e d , thus the t r a i n i n g s e t of s t a n d a r d s would c o n s i s t of o n l y 60 pure s p e c t r a . T h i s c o n t r a s t s w i t h t h e s p e c t r a l l i b r a r i e s used i n i n f r a r e d <52> and mass s p e c t r o s c o p y <15> where h u n d r e d s o r t h o u s a n d s o f s e p a r a t e s p e c t r a a r e r e q u i r e d , and t h e l i b r a r i e s a r e s t i l l i n c o m p l e t e . A s e t o f 60 e m i s s i o n s p e c t r a f o r e l e m e n t s i n aqueous s o l u t i o n w o u l d f o r m a c o m p l e t e s e t f o r a s p e c t r a l l i b r a r y . R e p r o d u c i b i l i t y of s p e c t r a f r o m one i n d u c t i v e l y c o u p l e d plasma s p e c t r o m e t e r t o the next i s p r e s e n t l y a problem, but g u i d e l i n e s toward a s t a n d a r d s e t of o p e r a t i n g c o n d i t i o n s are s l o w l y b e i n g f o r m u l a t e d <66>. Once a system i s s t a n d a r d i z e d i t s h o u l d be a b l e t o make u s e o f a s t a n d a r d l i b r a r y a v a i l a b l e t o a l l . In a d d i t i o n to the 60 e l e m e n t s , o t h e r f a c t o r s may a l s o be p r e s e n t i n u n u s u a l c i r c u m s t a n c e s . I n n o n - a q u e o u s s o l u t i o n s , the background w i l l be q u i t e d i f f e r e n t , but t h i s can be accounted f o r by u s i n g the background spectrum as one of the f a c t o r s i n the a n a l y s i s . I n t e r e l e m e n t e f f e c t s have a l s o been m e n t i o n e d as a d d i t i o n a l f a c t o r s . W i t h improved s i g n a l t o n o i s e r a t i o s , t h e s e f a c t o r s s h o u l d be q u a n t i f i a b l e , a t l e a s t i n the cases where they make a d e t e c t a b l e m o d i f i c a t i o n to the spectrum, and would t h e r e f o r e decrease the a c c u r a c y of an a n a l y s i s . 104 Another a r e a not yet i n v e s t i g a t e d by f a c t o r a n a l y s i s i s t h a t o f i o n t o atom r a t i o s <18>. F a c t o r a n a l y s i s s h o u l d a l l o w a pure atom and p u r e i o n s p e c t r u m t o be d e r i v e d f r o m a s e r i e s of s p e c t r a t a k e n a t d i f f e r e n t power s e t t i n g s . The i o n t o atom r a t i o depends upon t h e t e m p e r a t u r e of t h e p l a s m a , w i t h h o t t e r t e m p e r a t u r e s f a v o r i n g g r e a t e r i o n i z a t i o n . The t e m p e r a t u r e of the plasma i s dependent upon t h e power c o u p l e d i n t o t h e p l a s m a . -A v a r i a t i o n i n t h e i o n and atom p o p u l a t i o n s w i l l t h e r e f o r e o c c u r as t h e power i s changed. T h i s v a r i a t i o n s h o u l d be s u f f i c i e n t t o a l l o w f a c t o r a n a l y s i s t o i d e n t i f y two f a c t o r s , f o r t h e s e p a r a t e i o n and atom s p e c t r a . 105 CHAPTER VI CONCLUSIONS M u l t i v a r i a t e methods of data a n a l y s i s have been shown t o be p a r t i c u l a r l y s u i t a b l e t o t h e q u a l i t a t i v e and q u a n t i t a t i v e i n t e r p r e t a t i o n of i n d u c t i v e l y c o u p l e d p l a s m a s p e c t r a . The c o l l e c t i o n o f s p e c t r a l w i n d o w s u s i n g a p hotodiode a r r a y , r a t h e r than the c o l l e c t i o n of i n d i v i d u a l l i n e i n t e n s i t i e s , as i s done c o n v e n t i o n a l l y i n a n a l y t i c a l a p p l i c a t i o n s has been shown to have many advantages. The u n i q u e n a t u r e of a t o m i c e m i s s i o n s p e c t r a has been shown to l e a d to d i s t i n c t i v e c l u s t e r s f o r each element i n a m u l t i d i m e n s i o n a l p a t t e r n s p a c e . Each e l e m e n t d e f i n e s a u n i q u e d i r e c t i o n i n p a t t e r n s p a c e , and t h i s has a l l o w e d m u l t i e l e m e n t w o r k i n g c u r v e s t o be g e n e r a t e d . The d i s p l a c e m e n t a l o n g e a c h a x i s g a v e q u a n t i t a t i v e d e t e r m i n a t i o n s of c o n c e n t r a t i o n f o r a l l elements p r e s e n t i n an unknown sample. The use of f a c t o r a n a l y s i s and i n d i c a t o r f u n c t i o n s , and e s p e c i a l l y t h e r a t i o f u n c t i o n , has a l l o w e d a q u a l i t a t i v e e v a l u a t i o n of any data m a t r i x c o n t a i n i n g a s e t of d i g i t i z e d s p e c t r a . T h e s e f u n c t i o n s d e t e r m i n e t h e number o f s i g n i f i c a n t f a c t o r s r e f l e c t i n g t h e number of e l e m e n t s p r e s e n t i n a s a m p l e , and may a l s o r e v e a l t h e p r e s e n c e of unsuspected components. Q u a l i t a t i v e c h a r a c t e r i z a t i o n of each component was done 106 by t a r g e t f a c t o r a n a l y s i s , where a s u c c e s s f u l t a r g e t t e s t i n d i c a t e d t h e p r e s e n c e of a s p e c i f i c e l e m e n t i n t h e d a t a m a t r i x . The absence of o t h e r elements may a l s o be c o n f i r m e d by t a r g e t f a c t o r a n a l y s i s , s i n c e anoraolously h i g h v a l u e s of SPOIL w i l l i n d i c a t e a poor match f o r a t e s t spectrum. I t may be p o s s i b l e t o d e t e c t p e r t u r b a t i o n s t o t h e s p e c t r a as a r e s u l t o f m a t r i x e f f e c t s , and t h e s e e f f e c t s c o u l d c o n c e i v a b l y be q u a n t i f i e d w i t h f u r t h e r s t u d y . The a c c u r a c y o f t h e q u a n t i t a t i v e r e s u l t s c a n be i m p r o v e d by f u r t h e r w i d e n i n g of t h e s p e c t r a l window. The w i d t h o f t h e s p e c t r a l w i n d o w a p p e a r s t o be f a r more i m p o r t a n t t h a n t h e s p e c t r a l r e s o l u t i o n . P r e s u m a b l y t h i s r e s u l t s from the i n c l u s i o n of more l i n e s and t h e r e f o r e more i n f o r m a t i o n i n a w i d e r s p e c t r a l window. The dynamic range of t h e p h o t o d i o d e a r r a y r e s t r i c t s t h e d e t e c t i o n l i m i t s o f t h e method. The d y n a m i c r a n g e c o u l d l i k e l y be i m p r o v e d by d i g i t a l s c a l i n g of s p e c t r a w i t h d i f f e r e n t i n t e g r a t i o n t i m e s . A l t e r n a t e l y , a method such a F o u r i e r t r a n s f o r m i n d u c t i v e l y c o u p l e d p l a s m a s p e c t r o s c o p y w o u l d a l l o w a g r e a t e r d y n a m i c range and hence improve the d e t e c t i o n l i m i t s . The e s t a b l i s h m e n t o f a s t a n d a r d s e t o f o p e r a t i n g c o n d i t i o n s f o r t h e i n d u c t i v e l y c o u p l e d p l a s m a c o u l d have s e v e r a l advantages, and s h o u l d be encouraged. Among these advantages i s the e s t a b l i s h m e n t of a v ery compact s p e c t r a l l i b r a r y . T h i s l i b r a r y w o u l d c o n s i s t o f one s t a n d a r d s p e c t r u m f o r e a c h e l e m e n t and w o u l d g r e a t l y a s s i s t i n t h e 107 t a r g e t t e s t i n g s t e p s i n f a c t o r a n a l y s i s . M u l t i v a r i a t e t e c h n i q u e s s h o u l d prove to be very u s e f u l to the f u t u r e of automated and semi-automated s i m u l t a n e o u s m u l t i e l e m e n t a n a l y s e s . T h i s paper i s meant o n l y to show the e f f e c t i v e n e s s and s u i t a b i l i t y of these t e c h n i q u e s i n d e a l i n g w i t h i n d u c t i v e l y c o u p l e d plasma s p e c t r a , and much r e s e a r c h r e m a i n s t o be done. The s e l e c t i o n o f t h e b e s t o v e r a l l s p e c t r a l window, and the s t a n d a r d i z a t i o n and o p t i m i z a t i o n of i n d u c t i v e l y c o u p l e d p l a s m a s p e c t r o m e t e r s a r e two of many areas t h a t need to be f u r t h e r e x p l o r e d . 108 BIBLIOGRAPHY Boumans, P. W. J . M.; V r a k k i n g , J . J . A. M. Spectrochimica Acta 1984, 39B, 1239-1260 Boumans, P. W. J . M.; V r a k k i n g , J . J . A. M. Spectrochimica Acta 1984, 39B, 1261-1290 K o w a l s k i , B. R.; S c h a t z k i , T. F.; S t r o s s , F. H. Anal. Chem. 1972, 44, 2176-2180 S a x b e r g , B. E. H.; Duewer, D. L.; B o o k e r , J . L.; K o w a l s k i , B. R. Anal. Chim. Acta 1978, 103, 201-212 McCue, M.; M a l i n o w s k i , E. R. J. Chromatogr. Sci. 1983, 11, 229-234 Warner, I . M.; D a v i d s o n , E. R.; C h r i s t i a n , G. D. Anal. Chem. 1977, 49, 2155-2159 Ho, C. N.; C h r i s t i a n , G. D.; D a v i d s o n , E. R. Anal. Chem. 1980, 52, 1071-1079 S o g l i e r o , G.; E a s t w o o d , D.; Ehmer, R. Appl. Spectrosc. 1982, 36, 110-116 A n t o o n , M. K.; D ' e s p o s i t o , L.; K o e n i g , J . L. Appl. Spectrosc. 1979, 33, 351-357 K o e n i g , J . L.; T o v a r R o d r i q u e z , M. J . M. Appl. Spectrosc. 1981, 35., 543-548 G i l l e t t e , P. C ; K o e n i g , J . L. Appl. Spectrosc. 1982, 36, 535-539 F r a n k e l , D. S. Anal. Chem. 1984, 56, 1011-1014 S h a r a f , M. A.; K o w a l s k i , B. R. Anal. Chem. 1981 , 53, 518-522 Kormos, D. W.; Waugh, J . S. Anal. Chem. 1983, 5_5, 633-638 W i n d i g , W.; Haverkamp, J . ; K i s t e m a k e r , P. G. Anal. Chem. 1983, 55, 81-88 C a r p e n t e r , R. C ; T i l l C. Analyst 1984, 109, 881-884 Chen, F. F. " I n t r o d u c t i o n t o Plasma P h y s i c s " ; Plenum P r e s s : New York, 1977 109 C a u g h l i n , B. L.; B l a d e s , M. W. Spectrochimica Acta 1984, 39B, 1583-1602 D a h l q u i s t , R. L.; K n o l l , J . W. Appl. Spectrosc. 1978, 32., 1-29 T a l m i , Y.; S i m p s o n , R. W. Appl. Opt. 1981, 1_9, 1401-1414 M i l a n o , M. J . ; K i m , K. Y. Anal. Chem. 1977, 4_9, 555-559 Boumans, P. W. J . M.; V r a k k i n g , J . J . A. M. spectrochemica Acta 1984, 39B, 1291-1305 Weiner, P. H. Chemtech 1977, 321-328 I s e n h o u r , T. L.; K o w a l s k i , B. R.; J u r s , P. C. CRC C r i t . Rev. Anal. Chem. 1974, 4, 1-45 Ma s s a r t , D. L.; Kaufman, L. "The I n t e r p r e t a t i o n of A n a l y t i c a l C h e m i c a l Data by the use of C l u s t e r A n a l y s i s " ; W i l e y : New York, 1983 M a s s a r t , D. L.; D i j k s t r a , A.; Kaufman, L. " E v a l u a t i o n and O p t i m i z a t i o n of L a b o r a t o r y Methods and A n a l y t i c a l P r o cedures"; W i l e y : New York, 1978 M a l i n o w s k i , E. R.; Howery, D. G. " F a c t o r A n a l y s i s i n Che m i s t r y " ; W i l e y : New York, 1980 Varmuza, K. Anal. Chim. Acta 1980, 122, 227-240 K o w a l s k i , B. R. Anal. Chem. 1980, 52, 112R-122R D e l a n e y , M. F. Anal. Chem. 1984, 56, 261R-277R F r a n k , I . E.; K o w l a s k i , B. R. Anal. Ghem. 1982, 54, 232R-243R K r y g e r , L. Talanta 1981, 2j3, 871-877 K o w a l s k i , B. R. (Ed. ) , Wold, S. i n " C h e m o m e t r i c s , M a t h e m a t i c s and S t a t i s t i c s i n Che m i s t r y " ; D. R e i d e l P u b l i s h i n g : New York, 1984 M a s s a r t , D. L.; Kaufman, L.; E s b e n s e n , K. H. Anal. Chem. 1982, 5_4, 911-917 M a l i n o w s k i , E. R. Anal. Chem. 1977, 49., 606-612 M a l i n o w s k i , E. R. Anal. Chem. 1977, .49, 612-617 110 B l a c k b u r n , J . A. Anal. Chem. 1965, 3J7, 1000-1003 K a n a k a r e , J . J . Anal. Chem. 1970, 4_2, 1322-1326 Hugus, Z. Z.; E l - A w a d y , A. A. J. Phys. Chem. 1971, 75, 2954-2957 A n t o o n , M. K.; K o e n i g , J . H.; K o e n i g , J . L. Appl. Spectrosc. 1977, _31, 518-524 A l b a n o , C ; Dunn, W.; E d l u n d , U.; J o h a n s s o n , E.; Norden, B.; S j o s t r o m , M.; Wold, S. Anal. Chim. Acta 1978, 103, 429-443 L i n d b e r g , W.; P e r s s o n , J . A.; Wold, S. Anal. Chem. 1983, 55, 643-648 Wold, S. Technometrics 1978, .20, 397-405 H a a l a n d , D. M.; E a s t e r l i n g , R. G. Appl. Spectrosc. 1980, 34, 539-548 E r i c k s o n , G. A.; G e r l a c h , G. W.; Jochum, C. J . ; K o w l a s k i , B. R. Cereal Foods World 1981, 26, 383-388 K i l l e e n , T. J . ; E a s t w o o d , D.; H e n d r i c k , M. S. Talanta 1981, 28., 1-6 Van der G r e e f , J . ; Tas, A. C ; Bouman, J . ; Ten Noever de Brauw, M. C ; S c h r e u r s , W. H. P. Anal. Chim. Acta 1983, 1_50, 45-52 E l i a s s o n , B.; J o h n e l s , D.; Wold, S.; E d l u n d , U. Acta Chem. Scand. B 1982, 36, 155-164 G i l b e r t , R. A.; L L e w e l l y n , J . A.; S w a r t z , W. E.; Palmer, J . W. Appl. Spectrosc. 1982, 36., 428-430 G i l l e t t e , P. C ; Lando, J . B.; K o e n i g , J . L. Appl. Spectrosc. 1982, 36.. 661-665 H a a l a n d , G.; E a s t e r l i n g , R. G. Appl. Spectrosc. 1982, 36., 665-673 Hangac, G.; W i e b o l d t , R. C ; Lam, R. B.; I s e n h o u r , T. L. Appl. Spectrosc. 1982, ^ 36, 40-47 S c h u e t z l e , D.; K o s k i n e n , J . R.; H o r s f a l l , F. L. J. WCPF 1982, 54_, 457-465 111 Arunachalam, J .; Gangadharan, S. Anal. Chira. Acta 1984, 157, 245-260 H o o g e r b r u g g e , R.; W i l l i g , S. J . ; K i s t e m a k e r , P. G. Anal. Chem. 1983, 55, 1710-1712 McCue, M.; M a l i n o w s k i , E. R. Appl. Spectrosc. 1983, 3 7, 463-469 W h i t n e y , C. A. Astron. Astrophys. Suppl. Ser. 1983, 51 443-461 Wold, S.; Dunn, W. J . J. Chem. Inf. Comput. Sci. 1983, 23, 6-13 H a a l a n d , D. M.; E a s t e r l i n g , R. G.; V o p i c k a , D. A. Appl. Spectrosc. 1985, 39, 73-84 Ng, R. C. L.; H o r l i c k , G. i n p r e s s Ng, R. C. L.; H o r l i c k , G. i n p r e s s B l a d e s , M. W., Appl. Spectrosc. 1983, 3_7, 371-375 "ARTHUR", I n f o m e t r i x Inc., S e a t t l e , WA. F a i r e s , L. M., PhD d i s s e r t a t i o n , Los Alamos N a t i o n a l L a b o r a t o r y / U n i v e r s i t y .of New Mexico , 1983 H a l l , R. H., PhD d i s s e r t a t i o n , U n i v e r s i t y of A l b e r t a , 1979 R a y s o n , G. D.; M a r s h a l l , K. A.; R e z a a i y a a n , R.; H i e f t j e , G. M. P i t t s b u r g h Conference on A n a l y t i c a l C h e m i s t r y and A p p l i e d S p e c t r o s c o p y , New O r l e a n s , LA, Febr u a r y 1985, paper 487 112 APPENDIX A GENERAL TARGET FACTOR ANALYSIS The details of the mathematics involved have not yet been developed r i g o r o u s l y i n t h i s t h e s i s , since the mathematics are but a tool used in the same manner as an ohmraeter or a plotter. It is the concepts as embodied in a lo g i c a l geometric interpretation (position of spectra in pattern space) which are important. Some of the formulae used w i l l be outlined here for the sake of competeness. A more detailed discussion is found throughout reference <27>, from which many of these formulae were adapted. Data matrix Each column i s a data vector (e.g. U,V,W,X,...) of intensity measurements at equal intervals of wavelength for one particular sample. Covariance matrix Samples [D] = U Wavelength Intensities V W X [Z] = [D] T[D] 113 D i a g o n a l i z a t i o n / Column m a t r i x [ C ] [ Z ] [ C ] - 1 = [L] [C] and [ L ] a r e f o u n d s u c h t h a t [ L ] i s a d i a g o n a l m a t r i x c o n t a i n i n g e i g e n v a l u e s a l o n g t h e d i a g o n a l . [C] i s the column m a t r i x , and i s composed of the e i g e n v e c t o r s f o r each e i g e n v a l u e . Row m a t r i x [R] i s found such t h a t [R]T[R] = [ L ] The row m a t r i x i s composed of t h e p r o j e c t i o n s o f e a ch data p o i n t onto the e i g e n v e c t o r axes. W i t h t h e row and c o l u m n m a t r i c e s , t h e o r i g i n a l d a t a m a t r i x may be r e c o n s t r u c t e d d i r e c t l y . [D] = [ R ] [ C ] The data s e t however c o n t a i n s n o i s e . Because of t h i s , the number of e i g e n v e c t o r s (N 1) r e q u i r e d to span the f a c t o r s p a c e ( p a t t e r n s p a c e ) i s l e s s t h a n t h e t o t a l number of e i g e n v e c t o r s (N). Only the f i r s t N' e i g e n v e c t o r s are taken to r e c o n s t r u c t the data m a t r i x . The f i r s t N' columns of the row m a t r i x ( l a b e l l e d [ R * ] ) and t h e f i r s t N' rows of t h e c o l u m n m a t r i x ( l a b e l l e d [ C * ] ) a r e t a k e n . The f i r s t N' e i g e n v a l u e s are taken and form the d i a g o n a l m a t r i x [ L * ] . [D*] = [ R * ] [ C * ] T h i s reproduced data m a t r i x has e x t r a c t e d e r r o r ( n o i s e ) r e m o v ed, and [R*] and [ C * ] a r e c a l l e d t h e a b s t r a c t row and 114 c o l u m n m a t r i c e s r e s p e c t i v e l y . The d e t a i l s o f t h e d e t e r m i n a t i o n o f t h e number o f f a c t o r s N 1 a r e d i s c u s s e d i n appendix B. R o t a t i o n / T r a n s f o r m a t i o n The c o o r d i n a t e s y s t e m i n p a t t e r n s p a c e may be changed t o any o t h e r by use of a t r a n s f o r m a t i o n m a t r i x [T] ( t o be d i s c u s s e d i n d e t a i l l a t e r ) . A new row and column m a t r i x i s gen e r a t e d f o r t h i s new c o o r d i n a t e system. [R] = [ R * ] [ T ] [C] = [ T ] - 1 [ C * ] The new row and c o l u m n m a t r i x e s s t i l l r e p r o d u c e t h e same da t a m a t r i x w i t h i n e x p e r i m e n t a l e r r o r when m u l t i p l i e d t o g e t h e r . [D*] = [ R ] [ C ] An i n f i n i t e number o f new c o o r d i n a t e s y s t e m s may be gen e r a t e d i n t h i s manner, but o n l y one w i l l have c o o r d i n a t e s c o r r e s p o n d i n g t o p h y s i c a l l y s i g n i f i c a n t parameters. The t r a n s f o r m a t i o n m a t r i x may be found which g i v e s t h i s one s i g n i f i c a n t r o t a t i o n by t h e p r o c e s s o f t a r g e t f a c t o r a n a l y s i s . 115 A number o f p o s s i b l e d a t a v e c t o r s p r e s e n t i n t h e d a t a m a t r i x may be t e s t e d f o r i n d i v i d u a l l y . A t r a n s f o r m a t i o n v e c t o r f o r each i s produced. T i = [ L * ] " l [ R * ] T R l R]_ i s t h e t e s t v e c t o r . A p r e d i c t e d v e c t o r R_i i s c a l c u l a t e d . I i = [ R * ] T i I f R^ and R^ a r e s i m i l a r , t h e n one may c o n c l u d e t h a t t h e t e s t v e c t o r i s a f a c t o r . A number of f u n c t i o n s a r e u s e f u l i n e v a l u a t i n g t h e s u c c e s s o f t h e t e s t (e.g. SPOIL). These are d i s c u s s e d i n appendix B. When a l l N' f a c t o r s a r e f o u n d t h r o u g h t a r g e t t e s t i n g , the t o t a l t r a n s f o r m a t i o n m a t r i x [T] i s c o n s t r u c t e d from the i n d i v i d u a l t r a n s f o r m a t i o n v e c t o r s from s u c c e s s f u l t e s t s . [T] = [ T i T 2 T 3 • • • T N * ] The p h y s i c a l l y s i g n i f i c a n t c o l u m n m a t r i x r e l a t i n g t o r e a l f a c t o r s may then be c o n s t r u c t e d . [C] = [ T ] [ C * ] The e n t r i e s i n t h i s c o l u m n m a t r i x a r e t e r m e d f a c t o r l o a d i n g s , and c o r r e s p o n d t o t h e p r o j e c t i o n s of each d a t a v e c t o r o n t o t h e r e a l f a c t o r a x e s i n p a t t e r n s p a c e . D i s p l a c e m e n t a l o n g t h e s e a x e s i n t h e c a s e o f e m i s s i o n s p e c t r a p r o v i d e s d i r e c t readout of the c o n c e n t r a t i o n of each component. 116 APPENDIX B ERRORS IN THE DETERMINATION OF THE NUMBER OF FACTORS, TARGET TESTING, AND FACTOR LOADINGS D e t e r m i n i n g the number of f a c t o r s The r e a l , imbedded and e x t r a c t e d e r r o r s a r e c a l c u l a t e d as shown below. RE N' = RSD = N H — - , 1/2 i=N'+l M(N-N') where, RE = r e a l e r r o r RSD « r e s i d u a l s t a n d a r d d e v i a t i o n N ^ i i = sum of e i g e n v a l u e s assumed t o be n o i s e i=N'+l I E = RSD ( N» / N ) J / 2 XE = RSD N - N' N —i 1/2 e x t r a c t e d e r r o r ( the e r r o r removeable by f a c t o r a n a l y s i s ) The r e a l e r r o r s h o u l d approach the e s t i m a t e d e r r o r when t h e c o r r e c t number o f f a c t o r s i s u s e d . The imb e d d e d e r r o r s h o u l d show a minimum o r a l e v e l i n g o f f a t t h e c o r r e c t number o f f a c t o r s . The r e l a t i o n s h i p b e t w e e n t h e t h r e e i s Pythago r e a n . ( R E ) 2 = ( I E ) 2 + ( X E ) 2 117 The factor indicator function (IND) i s an empirical function f i r s t proposed by Malinowski <36>. IND = RE (N-N')2 This function should show a minimum at the correct number of factors. In this thesis, another empirical function was found useful. A drop in the log of the eigenvalue at the correct number of factors has been used previously <11>. This information i s better represented by taking the ratio of successive pairs of eigenvalues to evaluate the drop from one eigenvalue to the next. As an example, eigenvalue ratio 25 1.04 24 1.09 22 1.05 21 3.00 a maximum 7 1.17 6 1.20 5 The rat i o function shows a maximum at the correct number of factors. Evaluating the success of target testing Apparent error in the test vector J 1/2 ( l l , i ~ M The apparent error in the test vector contains the sum of the squares of the differences between a l l components in M AET = Yl i = l 118 the p r e d i c t e d and t e s t v e c t o r s . Root mean square e r r o r i n the p r e d i c t e d v e c t o r REP = RECTx'Tx)1^2 T]_ i s the t r a n s f o r m a t i o n v e c t o r f o r the t e s t . R e a l e r r o r i n the t a r g e t v e c t o r RET = ( ( A E T ) 2 - ( R E P ) 2 ) 1 / 2 The most i n f o r m a t i v e parameter t o e v a l u a t e the suc c e s s of the t e s t v e c t o r i s the SPOIL f u n c t i o n , a l s o i n t r o d u c e d by M a l i n o w s k i <27>. SPOIL = RET REP SPOIL < 3 - v e c t o r i s a f a c t o r 3 < SPOIL < 6 - v e c t o r i s p r o b a b l y a f a c t o r and sh o u l d be examined more c a r e f u l l y 6 < SPOIL - v e c t o r i s not a f a c t o r E r r o r s i n the f a c t o r l o a d i n g s The e r r o r s i n t h e f a c t o r l o a d i n g s can be d e t e r m i n e d e a s i l y a f t e r the t r a n s f o r m a t i o n m a t r i x has been found a t the c o m p l e t i o n o f t a r g e t f a c t o r a n a l y s i s . F i r s t [J_] i s f o u n d , d e f i n e d t o be, [T] = [ T ] " 1 ^ * ] - 1 / 2 Each row of [ T ] , l a b e l l e d Tj i s then t a k e n t o c a l c u l a t e the e r r o r f o r each f a c t o r l o a d i n g . E F L j = R E ( T j - T j ) l / 2 The e r r o r s i n t h o s e f a c t o r l o a d i n g s r e f l e c t t h e e x p e c t e d e r r o r s i n t h e o v e r a l l p r o j e c t i o n s o f t h e d a t a v e c t o r s onto the r e a l f a c t o r axes. 119 APPENDIX C QUICKFACTOR PROGRAM LISTING The f o l l o w i n g i s a l i s t i n g of t h e f a c t o r a n a l y s i s p r o g r a m w r i t t e n i n APL and run under an MTS o p e r a t i n g system on the U n i v e r s i t y of B r i t i s h Columbia campus computer. Appendix D i s a t a b l e of t r a n s l i t e r a t e d c h a r a c t e r s showing the s t a n d a r d APL c h a r a c t e r s e t , and t h e s u b s t i t u t i o n s used i n t h e MTS ( M i c h i g a n T e r m i n a l System) i m p l e m e n t a t i o n of APL. Log Output: Doug W i r s z , MWBX, 20:32:00 Thu Jun 27/85 "QUICKFACT0R2<#>" " QUICKFACT0R2 1] 'ENTER THE NAME OF YOUR DATA MATRIX' 2] DAT=# 3] MN=$,DAT 4] M=MN[1] 5] N=MN[2] 6] NC0V:'THE COVARIANCE MATRIX IS BEING CALCULATED' 7] COVAR=($TRDAT)+.*DAT 8] QEIG: 'THE EIGENVECTORS ARE NOW BEING CALCULATED' 9] EIG=EIGENR COVAR 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 'THE EIGENVALUES ARE:' EIGVAL=(1,N)$TAEIG EIGVEC=(-N,N)$TAEIG $TREIGVAL REALERR0R:1=0 RE=0 NPRIME = 0,(N-1)$TA($.N) L1:I=I+1 EIGTAKE=(0,I)$DREIGVAL EIGTAKE=(N-I)$, EIGTAKE J=0 SUM=0 INL1:J=J+1 SUM=SUM+EIGTAKE[J] $>INL1 I F ( J $ L T ( N - I ) ) REI=(SUM/(M*(N-I)))@0.5 RE=RE,REI $>L1 IF(I$LT(N-1)) I E = RE*((NPRIME/N)@0.5) XE=RE*(((N-NPRIME)/N)@0.5) IND=RE/((N-NPRIME)@2) QRE:'PRINT THE REAL, IMBEDDED AND EXTRACTED ERRORS, AND THE IND FUNCTION' 'FOR 1 THRU ',($FM(N-1)),» POSSIBLE FACTORS? Y/N' Q=$# $>PERR IF((1$TAQ)$EQ'Y') 120 35] $>SETNUMFAC I F ( ( 1 $TAQ)$EQ'N') 36] $>QRE 37] PERR: 'THE REAL ERRORS FOR 1 THRU 1, ($FM(N-1) ) ,' POSSIBLE FACTORS ARE:' 38] 1$DRRE 39] ' 1 40] 'THE IMBEDDED ERRORS ARE:' 41] 1$DRIE 42] » « 43] 'THE EXTRACTED (RMS) ERRORS ARE:' 44] 1$DRXE 45] ' ' 46] 'THE IND FUNCTION EXHIBITS A MINIMUM AT THE CORRECT NUMBER OF FACTORS' 47] 'THE IND FUNCTION VALUES ARE:' 48] 1$DRIND 49] ' ' 50] SETNUMFAC:'SELECT THE NUMBER OF FACTORS TO BE FOUND' 51] NPRIME=# 52] NUMFACFOUND=0 53] NAMEFACFOUND=' ' 54] COMPARE:'DO YOU WANT THE UNIQUENESS TEST OPTION ON? Y/N' 55] QC=$# 56] $>COMPARE I F ( ( ( 1 $TAQC) $NE'Y')&((1$TAQC)$NE'N')) 57] TOTALTRANS = (NPRIME,NPRIME)$ ,0 58] 'THE ABSTRACT COLUMN MATRIX IS NOW BEING CONSTRUCTED' 59] CDAGGER=$TR((-N ,-N)$TAEIG) 60] CDAGGER=(NPRIME,-N)$TACDAGGER 61] QAROW: 'THE ABSTRACT ROW MATRIX IS NOW BEING CONSTRUCTED' 62] RDAGGER = (DAT)+.*( $TRCD AGGER) 63] DIAG2:I=NPRIME 64] LAMDAGGER=(NPRIME,NPRIME)$,0 65] LAMDAGGERINV=LAMDAGGER 66] EIGVALINV=1/EIGVAL 67] DIAG:LAMDAGGER[I;I]=EIGVAL[1;I] 68] LAMDAGGERINV[I;I]=EIGVALINV[1;I] 69] 1=1-1 70] $>DIAG I F I$GT0 71] $>TESTS 72] TESTS:'THE NUMBER OF FACTORS TO BE FOUND IS »,($FMNPRIME),' WITH ', ( $FMNUMFACFOUND) ,' FACTORS FOUND SO FAR.' 73] 'THE FACTORS FOUND SO FAR ARE:' 74] NAMEFACFOUND 75] 'ENTER A TEST VECTOR (OR RESET TO RESET THE NUMBER OF FACTORS)' 76] '#:' 77] ROWTESTNAME=$# 78] $>SETNUMFAC IF(1$TAROWTESTNAME$EQ'R') 121 79] ROWTEST=$EXROWTESTNAME 80] R0WTEST=(M,1)$,R0WTEST 81] 'THE TRANSFORMATION VECTOR IS BEING CALCULATED' 82] TRANS VECL = ( $/LAMDAGGER)+.*($TRRDAGGER) + .*(ROWTEST) 83] PRED:'THE PREDICTED VECTOR IS BEING CALCULATED FROM THE TEST VECTOR' 84] PREDVEC=RDAGGER+.*TRANSVECL 85] 'EVALUATION OF ERRORS IN THE TEST VECTORS FOLLOWS' 86] VDIF=PREDVEC-ROWTEST 87] VDIF=M$,VDIF 88] VDIF=VDIF@2 89] SUM=VDIF+.*1 90] AET=((SUM)/M)@0.5 91] 'APPARENT ERROR IN THE TEST VECTOR' 92] AET 93] TV=NPRIME$,TRANSVECL 94] REP=(RE[NPRIME+1])*((TV+.*TV)@0.5) 95] 'RMS ERROR IN PREDICTED VECTOR' 96] REP 97 ] RET=(((AET@2)-(REP@2))@2)@0.25 98] 'REAL ERROR IN THE TARGET VECTOR' 99] RET 100] SP0IL=RET/REP 101] 'VALUE OF SPOIL (0-3 EXCELLENT, 3-6 FAIR, 6+ UNACCEPTABLE (MALINOWSKI)) 1 102] SPOIL 103] $>ENTERFAC IF((1$TAQC)$NE'Y') 104] 'THE TEST AND PREDICTED VECTORS MAY BE COMPARED' 105] 'FEATURE, TESTVECTOR, PREDICTED VECTOR' 106] LABELS=(M,1)$,$.M 107] LABELS,ROWTEST.PREDVEC 108] ENTERFAC:'ENTER THIS TEST VECTOR AS A REAL FACTOR? Y/N' 109] ENT=$# 110] $>TESTS IF((1$TAENT)$EQ'N') 111] $>ADDFAC IF((1$TAENT)$EQ'Y') 112] $>ENTERFAC 113] ADDFAC:TOTALTRANS=TOTALTRANS,TRANSVECL 114] NUMFACF0UND=NUMFACF0UND+1 115] NAMEFACFOUND=NAMEFACFOUND,' '.ROWTESTNAME 116] $>ALLFACFOUND IF((2*NPRIME)$EQ(($.TOTALTRANS) [ 2 ] ) ) 117] 'THIS TEST VECTOR HAS BEEN ADDED TO THE SET OF REAL FACTORS' 118] $>TESTS 119] ALLFACFOUND: 'THE *, ($FMNPRIME),' FACTORS YOU SPECIFIED HAVE BEEN ENTERED' 120] TOTALTRANS=(NPRIME,-NPRIME)$TATOTALTRANS 121] 'THE ROW MATRIX IS BEING CALCULATED' 122 ] ROWREAL=RDAGGER+.*TOTALTRANS 123] 'THE RECONSTRUCTED ROW MATRIX CAN BE FOUND IN VARIABLE ROWREAL .' 122 124] 'THE COLUMN MATRIX IS NOW BEING CALCULATED' 125] COLREAL=($/TOTALTRANS)+.*CDAGGER 126] 'THE RECONSTRUCTED COLUMN MATRIX IS:' 127] COLREAL 128] 'THE FACTORS FOUND ARE:' 129] NAMEFACFOUND 130] TBAR = ($/T0TALTRANS)+.*(LAMDAGGERINV@0.5) 131] EFL=(RE[NPRIME+l])*((0+.+(($TRTBAR)@2))@0.5) 132] 'THE ERROR IN EACH FACTOR LOADING IS:' 133] EFL=(NPRIME,1)$,EFL 134] EFL 135] CHANGEFAC:'TRY A DIFFERENT NUMBER OF FACTORS? Y/N' 136] Q=$# 137] $>SETNUMFAC IF((1$TAQ)$EQ'Y') 138] $>END IF((1$TAQ)$EQ'N') 139] $>CHANGEFAC 140] END: 'THATS ALL FOR NOW FOLKS' tl 123 APPENDIX D APL/MTS T r a n s l i t e r a t i o n T a b l e Transliteration Table ALPHA a $AL LESS OR EQUAL $LE ALPHABET A-Z A-Z LESS THAN < $LT A-Z. a-z A- Z LINE FEED AND A & LOCKED FUNCTION ASSIGN *• • LOGARITHM • $@ $L0 $LN BRACKETS C < MAXIMUM r $MA $CE 1 > MEMBERSHIP e $EP $ME BRANCH -»• $> $G0 MINIMUM L $MI SFL MINUS - -CAP n $CA MODULUS 1 $1 CIRCULAR FUNCTIONS o $$ SCI $PI MULTIPLY X * COLON : • $N& COMMA(CATENATION) • * NAND tl s-COMMENT A $* $C0 NEGATION - TNI COMPRESSION / % NOR V $NR COMPRESSIONS + SCI NOT $N0 CUP U $cu NOT EQUAL * $NE NULL e $: DECODE J. $DE $BA $0M DEFINITION (.DEL) V ii OMEGA 6) DELTA A $" OR V 1 W DELTAU A $U" i DIERESIS $DS PARENTHESES ( DIGITS 0-9 0-9 ) ) DIMENSION P $, $RH PERIOD DIVIDE • / PLUS + + DROP $DR $D0 QUAD 0 # ENCODE T $EN $RP QUAD - DIVIDE ffl $/ EQUAL = " $EQ QUOTE - QUAD • $# ESCAPE 0 $0UT QUOTE t « EXPANSION \ $X EXPANSIONS) \ $X1 RANDOM ? ? SRV EXPONENTIATION * § ROTATION $R0 EXECUTE $EX ROTATIONlll 0 $R1 RSUB $RS FACTORIAL t • $FA FORMAT $FM SEMICOLON » i SUB C Ssu GRADE DOWN t $GD SYSTEM I SIB SSY GRADE UP i $GU STA SUP GREATER OR EQUAL $GE TAKE + GREATER THAN > $6T TRANSPOSITION STR INDEX $. $10 $IN " UNDERSCORE - SUN 124 APPENDIX E QUICKFACTOR PROGRAM DESCRIPTION An i n t e r a c t i v e program was w r i t t e n to c a r r y out t a r g e t f a c t o r a n a l y s i s . T h i s program was c a l l e d QUICKFACTOR. I t was w r i t t e n i n APL to take advantage of the e x t e n s i v e m a t r i x m a n i p u l a t i o n a b i l i t i e s of t h i s language. T h i s f a c i l i t a t e d v e r y f a s t m o d i f i c a t i o n s of the program and ex p a n s i o n of the p r o g r a m s t e p by s t e p . D e b u g g i n g was done by r u n n i n g a s e t of s t a n d a r d d a t a t h r o u g h t h e p r o g r a m and m o n i t o r i n g i t s m a n i p u l a t i o n a t each s t a g e of t h e p r o g r a m . A d d i t i o n a l l y , use o f APL a l l o w s t h e pr o g r a m t o be v e r y c o m p a c t , w i t h t h e e n t i r e t a r g e t f a c t o r a n a l y s i s b e i n g c a r r i e d o u t on a d a t a m a t r i x of any s i z e i n o n l y 140 l i n e s . The program i s mo d e l l e d on the m a t h e m a t i c a l s t e p s f o r t a r g e t f a c t o r a n a l y s i s o u t l i n e d i n M a l i n o w s k i and Howery <27>. The r e a d e r i s r e f e r r e d t o t h i s book f o r a d e t a i l e d t r e a t m e n t . A b r i e f o u t l i n e o f each s t e p and where i t i s found i n the program f o l l o w s . L i n e s 1 t h r u 5 e n t e r t h e i n f o r m a t i o n i n t h e d a t a m a t r i x and measure t h e number o f r o w s and c o l u m n s (M and N) p r e s e n t . L i n e 7 c a l c u l a t e s t h e c o v a r i a n c e m a t r i x u s e d i n subsequent c a l c u l a t i o n s . The c o v a r i a n c e m a t r i x was chosen r a t h e r than a c o r r e l a t i o n m a t r i x s i n c e a l l measurements were of a s i m i l a r t y pe and magnitude, e m i s s i o n i n t e n s i t i e s . The n o r m a l i z a t i o n a s s o c i a t e d w i t h a c o r r e l a t i o n m a t r i x i s n o r m a l l y o n l y w a r r a n t e d when "each column of da t a i n v o l v e s a 125 d i f f e r e n t p r o p e r t y h a v i n g s i g n i f i c a n t l y d i f f e r e n t o r d e r s of magnitude" (pp. 30). L i n e 9 c a l l s on a s t a n d a r d APL l i b r a r y r o u t i n e EIGENR which f i n d s the e i g e n v a l u e s and e i g e n v e c t o r s of a m a t r i x . The v a r i a b l e s EIGVAL and EIGVEC s t o r e t h i s i n f o r m a t i o n f o r f u r t h e r r e f e r e n c e . The e i g e n v a l u e s a r e p r i n t e d out a t l i n e 13. L i n e s 14 t h r u 30 c a l c u l a t e v a l u e s of r e a l imbedded and e x t r a c t e d e r r o r s , and the IND f u n c t i o n f o r a l l p o s s i b l e numbers of f a c t o r s . The o p e r a t o r may then e x a m i n e t h e s e r e s u l t s and on t h a t b a s i s , a l o n g w i t h t h e e i g e n v a l u e s , s e l e c t t h e number of f a c t o r s t o be f o u n d . F o l l o w i n g the s e l e c t i o n of the number of f a c t o r s , a f l a g i s s e t f o r l a t e r use a t l i n e 55, t h e u n i q u e n e s s t e s t o p t i o n . I f t h i s i s e n a b l e d , th e p r e d i c t e d and t e s t v e c t o r s w i l l be d i s p l a y e d f o r comparison a f t e r the t a r g e t t e s t i n g s t e p . In l i n e s 58 t o 71, a b s t r a c t row and c o l u m n m a t r i c e s a r e c o n s t r u c t e d a c c o r d i n g to the number of f a c t o r s s e l e c t e d . At l i n e 75, t h e a c t u a l t a r g e t t e s t i n g i s e n t e r e d , and i t s s u i t a b i l i t y i s e v a l u a t e d i n t e r m s of a p p a r e n t e r r o r , RMS e r r o r i n t h e t a r g e t v e c t o r , and r e a l e r r o r i n t h e t a r g e t v e c t o r , c a l c u l a t e d i n l i n e s 81 to 99. These numbers s h o u l d r e f l e c t the a p p r o x i m a t e expected magnitude of the e r r o r f o r a s u c c e s s f u l t e s t . The v a l u e o f t h e p a r a m e t e r S P O I L c a l c u l a t e d a t l i n e 100, i s a f a i r l y good i n d i c a t o r of t h e s u c c e s s of the t e s t . V a l u e s below 3 c o n f i r m a f a c t o r , below 6 s t r o n g l y s u g g e s t a f a c t o r , and above 6 r e j e c t a f a c t o r . 126 The e v a l u a t i o n of success i s at p r e s e n t under the judgement of the o p e r a t o r , but the v a l u e s of SPOIL have proven i n t h i s r e s e a r c h t o be v e r y r e l i a b l e , so t h i s e v a l u a t i o n c o u l d e a s i l y be a u t o m a t e d , w i t h t h e o p e r a t o r s u p p l y i n g t h e t e s t v e c t o r s , and t h e p r o g r a m a c c e p t i n g or r e j e c t i n g them as a p p r o p r i a t e on the b a s i s of SPOIL v a l u e . I f t h e t e s t v e c t o r i s a c c e p t a b l e , i t i s added t o t h e t o t a l t r a n s f o r m a t i o n m a t r i x i n l i n e s 113 t o 118, and t h e program then r e t u r n s to the b e g i n n i n g of the t a r g e t t e s t i n g r o u t i n e a t l i n e 72 t o t e s t a n o t h e r v e c t o r . T h i s c y c l e of t e s t i n g c o n t i n u e s u n t i l a number of f a c t o r s e q u a l t o t h a t s e l e c t e d a t l i n e 50 i s f o u n d . The r e a l row and c o l u m n m a t r i c e s a r e t h e n c o n s t r u c t e d f r o m t h e a b s t r a c t row and c o l u m n m a t r i c e s i n l i n e s 119 t o 1 2 7 , u s i n g t h e t r a n s f o r m a t i o n m a t r i x , w h i c h i s i n t u r n composed of t h e s u c c e s s f u l t e s t v e c t o r s . The c o l u m n m a t r i x c o n t a i n s a l l t h e f a c t o r l o a d i n g s ( c o n c e n t r a t i o n s of t a r g e t t e s t e d components) f o r each of the o r i g i n a l s a m p l e s i n t h e d a t a m a t r i x . The e r r o r s i n e a ch f a c t o r l o a d i n g a r e computed and d i s p l a y e d by l i n e s 130 t o 134 a l o n g w i t h t h e f a c t o r l o a d i n g s . The u n i t s o f c o n c e n t r a t i o n depend on t h e c o n c e n t r a t i o n s r e p r e s e n t e d by the t e s t v e c t o r s , w i t h the t e s t v e c t o r c o n c e n t r a t i o n b e i n g e q u a l to one u n i t . The o p t i o n of t r y i n g a d i f f e r e n t number of f a c t o r s i n the event of u n s a t i s f a c t o r y r e s u l t s i s i n c l u d e d i n l i n e s 135 127 t o 139 w h i c h r e t u r n s t h e p r o g r a m t o t h e s t a g e where t h e number of f a c t o r s are o r i g i n a l l y s e t a t l i n e 50. 128 

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