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The weibull function as a diameter distribution model for mixed stands of Douglas-fir and Western hemlock 1980

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THE WEIBULL FUNCTION AS A DIAMETER DISTRIBUTION MODEL FOR MIXED STANDS OF DOUGLAS-FIR AND WESTERN HEMLOCK by SUSAN NANCY L I T T L E FOR. S C I . , PENNSYLVANIA STATE U N I V E R S I T Y , 1976 A T H E S I S SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n t h e D e p a r t m e n t o f FORESTRY We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF B R I T I S H COLUMBIA 1980 © S u s a n N a n c y L i t t l e , 1980 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i I m e n t o f the r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree that t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s , f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f f> C - g ^ V f The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 tt ABSTRACT The t h r e e - p a r a m e t e r W e i b u l l f u n c t i o n i s a s a t i s f a c t o r y m o d e l o f t h e d i a m e t e r d i s t r i b u t i o n s o f m i x e d s t a n d s o f w e s t e r n h e m l o c k and D o u g l a s - f i r . W e i b u l l d i s t r i b u t i o n s e s t i m a t e d by maximum l i k e l i h o o d (MLE) f i t e i g h t y of e i g h t y t h r e e o b s e r v e d d i a m e t e r d i s t r i b u t i o n s a t t h e ct = . 2 0 l e v e l o f s i g n i f i c a n c e ( K o l m o g o r o v - S m i r n o v t e s t ) . W e i b u l l p a r a m e t e r p r e d i c t o r s a r e d e r i v e d by r e g r e s s i n g s t a n d c h a r a c t e r i s t i c s o f 42 s t a n d s a g a i n s t t h e i r MLE p a r a m e t e r s . The W e i b u l l d i a m e t e r d i s t r i b u t i o n s p r e d i c t e d f r o m s t a n d a g e , mean d i a m e t e r , mean h e i g h t , s i t e i n d e x and t r e e s p e r a c r e f i t 39 o f 41 o b s e r v e d d i s t r i b u t i o n s i n t h e t e s t g r o u p a t t h e oc = . 2 0 l e v e l o f s i g n i f i c a n c e . The r e s u l t s shown h e r e c o m p a r e f a v o r a b l y w i t h t h o s e o f o t h e r a u t h o r s . The m o d e l s g i v e n r e l a t i n g s t a n d a t t r i b u t e s t o d i a m e t e r d i s t r i b u t i o n w i l l p r o v e u s e f u l i n s t a n d m o d e l i n g and i n u p d a t i n g f o r e s t i n v e n t o r i e s . TABLE OF CONTENTS P a g e T I T L E PAGE i ABSTRACT i i TABLE OF CONTENTS i i i L I S T OF TABLES i v L I S T OF FIGURES v ACKNOWLEDGEMENTS v i INTRODUCTION 1 LITERATURE REVIEW 4 WEIBULL PARAMETER ESTIMATION 9 DATA BASE 13 METHODS 22 RESULTS 24 D I S C U S S I O N , APPLICATIONS AND FURTHER RESEARCH 4'4 LITERATURE CITED 4 8 APPENDIX I 5 5 APPENDIX I I 5 8 APPENDIX I I I 5^ L I S T OF TABLES T a b l e P a g e I D e s c r i p t i v e S t a t i s t i c s o f E s t i m a t i o n P l o t s 14 I I D e s c r i p t i v e S t a t i s t i c s o f P r e d i c t i o n P l o t s 18 I I I C o m p a r i s o n o f Maximum L i k e l i h o o d E s t i m a t e d W e i b u l l D i s t r i b u t i o n s w i t h O r i g i n a l D i a m e t e r D i s t r i b u t i o n s 25 IV R e g r e s s e d a E q u a t i o n s 27 V R e g r e s s e d b E q u a t i o n s 28 VI R e g r e s s e d c E q u a t i o n s 29 V I I R e g r e s s i o n S t a t i s t i c s 31 V I I I P r e d i c t i o n M o d e l s w h i c h f i t a l l MLE D i s t r i b u t i o n s 32 IX MLE v s P r e d i c t e d D i s t r i b u t i o n 35 L I S T OF FIGURES V F i g u r e P a g e 1 The r e l a t i o n s h i p o f g o o d n e s s o f f i t t o s t a n d d i a m e t e r 37 2 The r e l a t i o n s h i p o f g o o d n e s s o f f i t t o s t a n d h e i g h t 38 3 The r e l a t i o n s h i p o f g o o d n e s s o f f i t t o s t a n d age 39 4 The r e l a t i o n s h i p o f g o o d n e s s o f f i t t o t r e e s p e r a c r e 4 0 5 The r e l a t i o n s h i p o f g o o d n e s s o f f i t t o s i t e i n d e x 41 6 O b s e r v e d and p r e d i c t e d t r e e s p e r a c r e by d i a m e t e r c l a s s f o r p l o t 71041 42 7 O b s e r v e d and p r e d i c t e d t r e e s p e r a c r e by d i a m e t e r c l a s s f o r p l o t 3 6 1 2 1 6 4 2 8 O b s e r v e d and p r e d i c t e d t r e e s p e r a c r e by d i a m e t e r c l a s s f o r p l o t 4 5 1 2 6 6 4 3 9 O b s e r v e d and p r e d i c t e d t r e e s p e r a c r e by d i a m e t e r c l a s s f o r p l o t 5 9 1 3 5 4 4 3 ACKNOWLEDGMENTS The a u t h o r i s i n d e b t e d t o a l l i n d i v i d u a l s and a g e n c i e s c o n c e r n e d w i t h h e r s t u d i e s and r e s e a r c h . The a u t h o r t h a n k s D r . D o n a l d D. Munro who s u g g e s t e d t h e p r o b l e m and whose c o n s t a n t a d v i c e , c r i t i c i s m , and e n t h u s i a s m w e r e e s s e n t i a l t o t h e c o m p l e t i o n o f t h i s p r o j e c t . D a t a f r o m t h e R e g i o n a l F o r e s t N u t r i t i o n R e s e a r c h P r o j e c t w e r e p r o v i d e d by D r . W i l l i a m A t k i n s o n and M r . A l B e c k e r o f t h e U n i v e r s i t y o f W a s h i n g t o n . W e i b u l l p a r a m e t e r e s t i m a t i n g s u b r o u t i n e s w e r e p r o v i d e d b y D r . R o b e r t L . B a i l e y , W e y e r h a e u s e r C o . , and D r . W i l l i a m G . W a r r e n , C a n a d i a n F o r e s t S e r v i c e . F i n a n c i a l and c o m p u t e r s u p p o r t w e r e s u p p l i e d by t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a and t h e U n i t e d S t a t e s F o r e s t S e r v i c e . The M a c M i l l a n B l o e d e l L t d . F e l l o w s h i p a l s o g a v e f i n a n c i a l a s s i s t a n c e . The a u t h o r e x t e n d s h e r t h a n k s t o D r s . J u l e s P . D e m a e r s c h a l k , James A r n e y , W i l l i a m G . W a r r e n and M r . M i c h a e l S t a l e y f o r t h e i r h e l p and u s e f u l f e e d b a c k i n t h e e a r l y s t a g e s o f r e s e a r c h . 1 INTRODUCTION The d e s c r i p t i o n o f d i a m e t e r ( d b h ) d i s t r i b u t i o n s p o s e s a p r o b l e m f o r t h e f o r e s t s t a n d m o d e l e r . The d i s t r i b u t i o n a l c h a r a c t e r i s t i c s o f t h e s t a n d a t a n y p o i n t i n t i m e may be r e q u i r e d f o r t h e c r e a t i o n o f s t a n d t a b l e s , w h i c h d e l i n e a t e s t a n d a t t r i b u t e s s u c h a s v o l u m e o r b a s a l a r e a p e r a c r e by d i a m e t e r c l a s s e s . Y e t t h e c o m p u t e r s t o r a g e o f t r e e d i a m e t e r s may be cumbersome and c o s t l y i n an a l r e a d y c o m p l i c a t e d m o d e l . D i a m e t e r d i s t r i b u t i o n s may be h a n d l e d i n e i t h e r o f two w a y s . The m o d e l may d e s c r i b e t h e i n i t i a l s t a n d i n i t s e n t i r e t y and " g r o w " i n d i v i d u a l t r e e s o r g r o u p s o f t r e e s o v e r t i m e ( M i t c h e l l , 1976; L e a r y , e t a l . , 1 9 7 7 ) . I n t h i s c a s e s t a n d t a b l e s a r e r e a d i l y a v a i l a b l e , as e a c h t r e e o r g r o u p o f t r e e s i s s t o r e d w i t h i t s d i a m e t e r , h e i g h t , c r o w n r a t i o , e t c . . The a l t e r n a t i v e a p p r o a c h i s t o g e n e r a t e t h e d i a m e t e r d i s t r i b u t i o n o f t h e s t a n d a t t h e t i m e i t i s n e e d e d v i a a m a t h e m a t i c a l f u n c t i o n ( D e p t a , 1 9 7 4 ) . The p a r a m e t e r s o f t h e f u n c t i o n c a n be e s t i m a t e d f r o m s t a n d a t t r i b u t e s , s u c h a s a v e r a g e d i a m e t e r , t r e e s p e r a c r e , and a v e r a g e h e i g h t . T h u s no i n d i v i d u a l t r e e d a t a n e e d be s t o r e d . H a n d l i n g t i m e i s r e d u c e d , as s t a n d t a b l e s c a n be g e n e r a t e d d i r e c t l y by f u n c t i o n s r e l a t e d t o d i a m e t e r c l a s s e s . A n o t h e r u s e f o r f u n c t i o n a l d i a m e t e r r e l a t i o n s h i p s i s i n t h e a r e a o f a e r i a l p h o t o i n t e r p r e t a t i o n . A v e r a g e h e i g h t and number o f s t e m s p e r a c r e c a n be i n t e r p r e t e d f r o m a e r i a l p h o t o g r a p h s o f f o r e s t s t a n d s . S t a n d age and s i t e i n d e x may be o b t a i n e d f r o m p a s t r e c o r d s . Once a f u n c t i o n a l r e l a t i o n s h i p h a s b e e n d e v e l o p e d b e t w e e n t h e s e s t a n d c h a r a c t e r i s t i c s and d i a m e t e r 2 d i s t r i b u t i o n , t h e p h o t o i n t e r p r e t e r c a n r e a d i l y g e n e r a t e s t a n d t a b l e s . T h i s c o u l d be a v a l u a b l e t o o l f o r p e r i o d i c a l l y u p d a t i n g i n v e n t o r i e s on a l a r g e s c a l e due t o t h e s i m p l i c i t y o f t h e m e t h o d . I n t h i s t h e s i s , t h e W e i b u l l d i s t r i b u t i o n w i l l be i n v e s t i g a t e d as a m o d e l f o r t h e d i a m e t e r d i s t r i b u t i o n s o f m i x e d s t a n d s o f s e c o n d g r o w t h P s e u d o t s u g a m e n z i e s i i ( M i r b e l ) F r a n c o ( D o u g l a s - f i r ) and T s u g a h e t e r o p h y l l a ( R a f i n e s q u e ) S a r g e n t ( w e s t e r n h e m l o c k ) . The p a r a m e t e r s , a , b , and c , o f t h e W e i b u l l f u n c t i o n : w h e r e d r e p r e s e n t s d i a m e t e r a t b r e a s t h e i g h t i n i n c h e s ; W ( d ) , number o f t r e e s p e r a c r e h a v i n g d i a m e t e r d w i l l be e s t i m a t e d f r o m t h e o b s e r v e d d i a m e t e r d i s t r i b u t i o n o f e a c h p l o t , 0 ( d ) . The n u l l h y p o t h e s i s : w i l l be r e j e c t e d i f f o r one p l o t m a x l F ( d ) - 0 ( d ) J e x c e e d s t h e K o l m o l g o r o v - S m i r n o v c r i t i c a l v a l u e d o c ( N ) . The a c c e p t a b l e p r o b a b i l i t y l e v e l o f c o m m i t t i n g a T y p e I e r r o r wi11 be = .20 t h r o u g h o u t t h i s t h e s i s . I f we c a n n o t r e j e c t HQ , t h e p a r a m e t e r s o f F ( d ) w i l l be r e g r e s s e d a g a i n s t t h e s t a n d c h a r a c t e r i s t i c s : a r e t h m e t i c mean d i a m e t e r , mean d o m i n a n t and c o d o m i n a n t h e i g h t , s t a n d a g e , number o f t r e e s p e r a c r e , and s i t e i n d e x . F ( d ) = 1 - e x p \ - ( d - a ) c H N: F ( d ) = 0 ( d ) 3 The distributions predicted from the resulting equations w i l l be compared with F(d) and 0(d) for goodness of f i t . LITERATURE REVIEW S t a n d t a b l e s , w h i c h l i s t s t a n d a t t r i b u t e s ( s u c h as number o f t r e e s p e r a c r e ) by d i a m e t e r c l a s s , h a v e p r o v e n J t o be v a l u a b l e f o r f o r e s t m a n a g e r s . The c o n s t r u c t i o n o f t h e s e t a b l e s h a s e v o l v e d a r o u n d t h e b i o m e t r i c i a n ' s a b i l i t y t o s u m m a r i z e f r e q u e n c y d i s t r i b u t i o n s i n m a t h e m a t i c a l t e r m s . M e y e r ( 1 9 3 0 ) u s e d C h a r l i e r c u r v e s t o d e s c r i b e d i a m e t e r d i s t r i b u t i o n s . B l i s s and R e i n k e r ( 1 9 6 4 ) f i t t e d M e y e r ' s d i a m e t e r d i s t r i b u t i o n s o f e v e n - a g e d s t a n d s o f D o u g l a s - f i r w i t h l o g n o r m a l c u r v e s . T h e y f o u n d t h a t t h e s e l o g n o r m a l c u r v e s c o u l d be a d e q u a t e l y d e f i n e d by t h e mean d i a m e t e r and t h e v a r i a n c e o f t h e d i a m e t e r s o f e a c h d i s t r i b u t i o n . N e l s o n ( 1 9 6 4 ) c h a r a c t e r i z e d d i a m e t e r d i s t r i b u t i o n s o f l o b l o l l y p i n e p l a n t a t i o n s w i t h t h e gamma d i s t r i b u t i o n . L e a k ( 1 9 6 5 ) u s e d n e g a t i v e e x p o n e n t i a l c u r v e s t o d e s c r i b e d i a m e t e r d i s t r i b u t i o n s o f u n e v e n - a g e d s t a n d s . A summary o f g r o w t h f u n c t i o n s a p p l i e d i n f o r e s t b i o m e t r y was g i v e n b y P r o d a n ( 1 9 6 8 ) . S u m m a r i z i n g i n f o r m a t i o n by e q u a t i o n s g r e a t l y r e d u c e s c o m p u t e r s t o r a g e and c a l c u l a t i o n r e q u i r e m e n t s . D e p t a ( 1 9 7 4 ) d e v e l o p e d a s t a n d t a b l e g e n e r a t o r f o r W e y e r h a e u s e r C o r p . G i v e n s a m p l e d a t a on b a s a l a r e a p e r a c r e , number o f t r e e s p e r a c r e , m i n i m u m , a v e r a g e , and maximum d . b . h . , and a v e r a g e t r e e h e i g h t , h i s m o d e l p r o d u c e d s t a n d t a b l e s w i t h d i a m e t e r c l a s s e s , s t e m s p e r a c r e , b a s a l a r e a p e r a c r e , a v e r a g e t o t a l h e i g h t , and c u b i c f o o t v o l u m e p e r a c r e f o r e v e n - and u n e v e n - a g e d s t a n d s . E a c h s t a n d was s u m m a r i z e d by a s e r i e s o f c o e f f i c i e n t s r e l a t i n g t o e q u a t i o n s f o r e a c h s t a n d a t t r i b u t e . I n d i v i d u a l t r e e and s i z e c l a s s s t a t i s t i c s w e r e n o t s t o r e d i n t h e c o m p u t e r , b u t w e r e g e n e r a t e d o n l y when t h e y w e r e n e e d e d f o r f i n a l o u t p u t . The •in- c a p a c i t y t o d e s c r i b e d i s t r i b u t i o n s v i a m a t h e m a t i c a l f u n c t i o n s e n a b l e d t h i s p r o g r a m t o h a n d l e l a r g e i n v e n t o r i e s and t o f i l l g a p s i n i n f o r m a t i o n . The W e i b u l l d i s t r i b u t i o n was i n t r o d u c e d t o f o r e s t b i o m e t r y by B a i l e y and D e l l ( 1 9 7 3 ) . T h i s f u n c t i o n h a s s p e c i a l a p p e a l b e c a u s e o f i t s a b i l i t y t o t a k e on - a v a r i e t y o f s h a p e s and a m o u n t s o f s k e w n e s s . S i n c e t h e p a r a m e t e r s a r e d i r e c t l y r e l a t e d t o s h a p e ( c ) and s c a l e ( b ) , t h e y s h o u l d v a r y i n a c o n s i s t e n t m a n n e r w i t h s t a n d c h a r a c t e r i s t i c s when a p p l i e d t o d i a m e t e r d i s t r i b u t i o n s . B a i l e y and D e l l g a v e an o v e r v i e w o f p a r a m e t e r e s t i m a t i o n m e t h o d s and f i t t e d f o u r d i f f e r e n t d i a m e t e r d i s t r i b u t i o n s w i t h t h e W e i b u l l . S t a u f f e r ( 1 9 7 7 ) g a v e a m a t h e m a t i c a l d e r i v a t i o n o f t h e W e i b u l l f u n c t i o n i n o r d e r t o g i v e a p r e c i s e i n t e r p r e t a t i o n f o r t h e p a r a m e t e r s . The t h r e e - p a r a m e t e r W e i b u l l f u n c t i o n i s : a , t h e l o c a t i o n p a r a m e t e r , c o r r e s p o n d s t o t h e minimum d i a m e t e r ; b , t h e s c a l e p a r a m e t e r , i s i n v e r s e l y p r o p o r t i o n a l t o t h e number o f a l l o c a t i o n s ( i . e . , number o f t r e e s ) ; and c , t h e s h a p e p a r a m e t e r , i s an i n d i c a t i o n o f t h e d e g r e e o f n o n r a n d o m n e s s . B a i l e y ( 1 9 7 3 ) u s e d maximum l i k e l i h o o d p r o c e d u r e s t o f i t e q u a t i o n s f o r 1095 P i n u s r a d i a t a D. Don d i a m e t e r d i s t r i b u t i o n s . N i n e t y p e r c e n t o f t h e s e e q u a t i o n s had an a d e q u a t e f i t a t t h e ©c = . 1 0 l e v e l o f s i g n i f i c a n c e b y t h e K o l m o g o r o v - S m i r n o v t e s t f o r g o o d n e s s o f f i t . He t h e n a t t e m p t e d t o r e g r e s s a g e , number o f s t e m s p e r a c r e , and a v e r a g e h e i g h t on t h e c h a r a c t e r i s t i c s o f t h e p e r c e n t i l e s , X p - . x p = b ( - l n ( l - p ) ) 1 / c w h e r e x i s t h e d . b . h . c l a s s o f t h e p - t h p e r c e n t i l e . He came up w i t h t h e s e r e l a t i o n s h i p s f o r t h e 2 4 - t h and 9 3 - d p e r c e n t i l e s : x . 2 4 = B 0 + B 1 A + B 2 / N + B 3 l o 9 l O ( H ) x . 2 4 = C 0 + C 1 a 2 + C 2 / N + C 3 l o 9 1 0 ( H ) From t h e s e he c a l c u l a t e d t h e p a r a m e t e r s v i a D u b e y ' s ( 1 9 6 7 ) p e r c e n t i l e e s t i m a t i o n m e t h o d . ( S e e b e l o w . ) A t t h e = . 0 5 l e v e l , 35% o f t h e s e e s t i m a t e d c u r v e s had a s a t i s f a c t o r y f i t . The W e i b u l l d i s t r i b u t i o n h a s s i n c e b e e n u s e d i n some g r o w t h m o d e l s . C l u t t e r and A l l i s o n ( 1 9 7 3 ) u s e d t h e W e i b u l l d i s t r i b u t i o n f o r d i a m e t e r s i n t h e i r g r o w t h and y i e l d m o d e l f o r P i n u s r a d i a t a . I f t h e u s e r d o e s n o t s p e c i f y a d i a m e t e r d i s t r i b u t i o n , t h e m o d e l w i l l a p p r o x i m a t e t h e d i s t r i b u t i o n u s i n g s t a n d age and number o f s t e m s as done by B a i l e y ( 1 9 7 3 ) . The W e i b u l l was u s e d b y S c h r e u d e r and Swank ( 1 9 7 4 ) t o s u m m a r i z e d i a m e t e r , b a s a l a r e a , s u r f a c e a r e a , b i o m a s s , and c r o w n p r o f i l e d i s t r i b u t i o n s f o r P i n u s s t r o b i s and P i n u s t a e d a . R u s t a g i ( 1 9 7 7 ) m o d e l e d b a s a l a r e a d i s t r i b u t i o n s f o r e v e n - a g e d D o u g l a s - f i r s t a n d s w i t h t h e W e i b u l l . Y a n g , K o z a k and S m i t h ( 1 9 7 8 ) d e s c r i b e d 7 p r e v i o u s a t t e m p t s t o f i t v o l u m e g r o w t h and i n c r e m e n t d i s t r i b u t i o n s w i t h m a t h e m a t i c a l f u n c t i o n s . T h e y t h e n d e v e l o p e d a m o d i f i e d W e i b u l l f u n c t i o n : - t h e p a r a m e t e r s o f w h i c h c a n be e s t i m a t e d v i a n o n l i n e a r r e g r e s s i o n . T h i s f u n c t i o n had t h e b e s t o v e r - a l l f i t f o r v o l u m e g r o w t h when c o m p a r e d w i t h f o u r o t h e r d i s t r i b u t i o n s f o u n d i n t h e l i t e r a t u r e . C l u t t e r and B e l c h e r ( 1 9 7 8 ) u s e d t h e W e i b u l l f u n c t i o n t o e s t i m a t e number o f t r e e s p e r a c r e by one i n c h c l a s s e s f o r s l a s h p i n e p l a n t a t i o n s . P a r a m e t e r s o f t h e d i a m e t e r d i s t r i b u t i o n s o f 487 p l o t s w e r e e s t i m a t e d v i a H a r t e r and M o o r e ' s ( 1 9 6 5 ) maximum l i k e l i h o o d a l g o r i t h m . S t a n d a r d m u l t i p l e r e g r e s s i o n was t h e n u s e d t o r e g r e s s s t a n d a t t r i b u t e s on t h e p a r a m e t e r e s t i m a t e s , r e s u l t i n g i n t h e f o l l o w i n g r e l a t i o n s h i p s : Y + Y A + Y H , + Y l n ( H ) , r = .107 0 1 2 d 3 d s = 1.044 a y •x b 6 + 6 A + 6 / N + 6 ln(H,) ' , r = .357 0 1 2 3 d s = 1.091 y x c p + p /A , r 2= .020 s = 1.091 ; 0 1 w h e r e A = s t a n d age = mean h e i g h t o f d o m i n a n t s and c o d o m i n a n t s i n f e e t a n d , N = number o f t r e e s p e r a c r e . The a u t h o r s i n s i s t e d t h a t a l t h o u g h t h e r - s q u a r e v a l u e s w e r e l o w , t h e s e e q u a t i o n s g a v e b e t t e r r e s u l t s t h a n u s i n g c o n s t a n t v a l u e s f o r a , b , and c . 6 O b s e r v e d mean d i a m e t e r d i f f e r e d f r o m p r e d i c t e d mean d i a m e t e r by l e s s t h a n one i n c h i n 99% o f t h e o b s e r v a t i o n s ( i . e . , o f t h e 487 p l o t s t h a t w e r e u s e d t o d e r i v e t h e e q u a t i o n s ) . O b s e r v e d b a s a l a r e a d i f f e r e d by l e s s t h a n 2 25 f t f r o m p r e d i c t e d b a s a l a r e a i n 9 7 . 4 % o f t h e c a s e s . S e a r c h i n g f o r a f u n c t i o n more f l e x i b l e t h a n t h e W e i b u l l , H a f l e y and S h r e u d e r ( 1 9 7 7 ) i n v e s t i g a t e d t h e p o s s i b i l i t i e s o f J o h n s o n ' s ( 1 9 4 9 ) d i s t r i b u t i o n and i t s m o d i f i c a t i o n s . The f o u r - p a r a m e t e r f u n c t i o n : f w - ; = (*- E)Ux) ~ p i ^ +  s '-fe)i23 for 6, X>0 ; - «»<Y<OO ; - e 0 < e < x < e + X<<» f(x) = 0 elsewhere. c a n t a k e on a g r e a t e r v a r i e t y o f s h a p e s and i s a p p l i c a b l e t o more c a s e s t h a n t h e W e i b u l l . S c h r e u d e r and H a f l e y ( 1 9 7 7 ) a p p l i e d t h i s c u r v e t o d i a m e t e r - h e i g h t r e l a t i o n s h i p s . T h e y e s t i m a t e d p a r a m e t e r s w i t h maximum l i k e l i h o o d t e c h n i q u e s , and a c h i e v e d a s a t i s f a c t o r y f i t a t t h e £* = . 0 5 l e v e l w i t h t h e K o l m o g o r o v - S m i r n o v t e s t . T h e y d i d n o t c o m p a r e t h e S b d i r e c t l y w i t h t h e W e i b u l l f u n c t i o n a s a m o d e l o f t r e e d i a m e t e r d i s t r i b u t i o n s . WEIBULL PARAMETER ESTIMATION W e i b u l l ( 1 9 5 5 ) d e v e l o p e d h i s e q u a t i o n i n c o n j u n c t i o n w i t h h i s a n a l y s i s o f b r e a k i n g s t r e n g t h s . I t has b e e n w i d e l y u s e d i n r e l i a b i l i t y and l i f e t e s t i n g a n a l y s i s e v e r s i n c e . I n l i f e t e s t i n g i t i s u s u a l l y n o t f e a s i b l e t o t e s t a l a r g e number o f i t e m s , n o r i s i t n e c e s s a r y t o w a i t f o r a l l i t e m s t o f a i l . S i n c e m o s t o f t h e t h e o r e t i c a l and a p p l i e d w o r k on W e i b u l l p a r a m e t e r e s t i m a t i o n h a s b e e n done i n t h e l i f e t e s t i n g f i e l d , t h e b u l k o f t h e a l g o r i t h m s f o r e s t i m a t i o n a r e b a s e d on s m a l l , h e a v i l y c e n s o r e d s a m p l e s . G r a p h i c a l e s t i m a t i o n p r o c e d u r e s w e r e d e v e l o p e d by K a o ( 1 9 5 9 ) ? • Cohen ( 1 9 6 5 ) d e f i n e d t h e l i k e l i h o o d f u n c t i o n f o r t h e W e i b u l l d i s t r i b u t i o n : H a r t e r and M o o r e ( 1 9 6 5 ) p r o d u c e d an i t e r a t i v e p r o c e d u r e f o r c a l c u l a t i n g maximum l i k e l i h o o d e s t i m a t e s . W a r r e n ( 1 9 7 6 ) , i n c o n j u n c t i o n w i t h h i s s t u d i e s on b r e a k i n g p o i n t s o f wood p r o d u c t s , h a s f u r t h e r r e f i n e d t h e maximum l i k e l i h o o d p r o c e d u r e . S e e g r i s t and A r n e r ( 1 9 7 8 ) d i s c u s s e d p r o b l e m s o f maximum l i k e l i h o o d e s t i m a t e s when e r r o r c o m p o n e n t s a r e c o r r e l a t e d due t o h a v i n g r e p e a t e d m e a s u r e m e n t s on p l o t s . Maximum l i k e l i h o o d e s t i m a t i o n c a n be t i m e c o n s u m i n g and c o m p l i c a t e d . P o i n t e s t i m a t i o n , a l t h o u g h o f t e n n o t a s a c c u r a t e , i s e a s i e r t o u s e t h a n maximum l i k e l i h o o d . Dubey ( 1 9 6 7 ) d e v i s e d a p e r c e n t i l e e s t i m a t o r f o r t h e t w o p a r a m e t e r W e i b u l l d i s t r i b u t i o n , n a m e l y : M-lnO-p,)) -ln(-1n(l-p2)) C * l n ( y p l ) - l n i y p 2 ) b * =(y p^ C* / In(l-P,) w h e r e i s t h e i - t h p e r c e n t i l e and y ^ i s t h e maximum v a l u e a t t r i b u t e d t o t h e p e r c e n t i l e . He f o u n d t h a t t h e p e r c e n t i l e s w h i c h m i n i m i z e t h e v a r i a n c e o f t h e e s t i m a t e s a r e t h e 1 7 - t h and 9 7 - t h f o r c * and t h e 4 0 - t h and 8 2 - d f o r b * . The p e r c e n t i l e a p p r o a c h i s a p p e a l i n g b e c a u s e i t d o e s n o t i n v o l v e i t e r a t i v e c a l c u l a t i o n s and i s e a s i l y i m p l e m e n t e d . E n g l e h a r d t and B a i n w r o t e p r o f u s e l y on p o i n t e s t i m a t e s o f W e i b u l l p a r a m e t e r s : ( B a i n and A n t l e ( 1 9 6 7 ) , E n g l e h a r d t and B a i n ( 1 9 7 3 , 1 9 7 4 ) , B a i n ( 1 9 7 2 ) , and E n g l e h a r d t ( 1 9 7 5 ) ) . I n 1 9 7 7 , t h e y p u b l i s h e d s i m p l i f i e d e s t i m a t o r s o f t h e p a r a m e t e r s f o r c o m p l e t e s a m p l e s and t o l e r a n c e b o u n d s f o r t h o s e e s t i m a t e s . S p e c i f i c a l l y : f o r z = 1 / c : w h e r e s = [ , 8 4 n ^ and k n i s a c o n s t a n t t o r e m o v e b i a s . Ii f o r u = l n b : n u =2 Yj/n + Y z w h e r e "f i s E u l e r ' s c o n s t a n t , 0 . 5 7 7 2 . T h i s p r o c e d u r e , l i k e t h e p e r c e n t i l e a p p r o a c h , i s e a s y t o u s e . A l t h o u g h t h e d e b i a s i n g c o n s t a n t , k , i s d i f f i c u l t t o c a l c u l a t e , E n g l e h a r d t and B a i n h a v e s o l v e d f o r v a l u e s f o r k o f a l l s a m p l e s i z e s u n d e r 60 and f o r i n f i n i t e l y l a r g e s a m p l e s . A c o m p l e t e d i s c u s s i o n o f t h e W e i b u l l f u n c t i o n , i t s h i s t o r y and a p p l i c a t i o n s , was g i v e n by Mann ( 1 9 6 8 ) . She d i s c u s s e d t h e a p p l i c a t i o n s and e f f i c i e n c y o f s e v e r a l e s t i m a t i o n p r o c e d u r e s f o r c e n s o r e d and u n c e n s o r e d s a m p l e s , as w e l l a s some m e t h o d s o f i m p o s i n g c o n f i d e n c e b o u n d s on t h e v a r i o u s e s t i m a t e s . M a n n , S h a f e r , and S i n g p u r w a l l a ( 1 9 7 4 ) p u b l i s h e d a t e x t on s t a t i s t i c a l a n a l y s i s o f r e l i a b i l i t y and l i f e t e s t i n g d a t a . T h e y d i s c u s s e d t h e t h e o r y and m a t h e m a t i c a l d e r i v a t i o n o f e s t i m a t i o n p r o c e d u r e s and t h e a p p l i c a t i o n s t h e r e o f . T h e y c o n d e n s e d t h e work o f s e v e r a l p r e v i o u s a u t h o r s , i n c l u d i n g d ' A g o s t i n o ( 1 9 7 1 ) ( l i n e a r e s t i m a t i o n ) and Thomas and W i l s o n ( 1 9 7 2 ) ( p o i n t e s t i m a t i o n ) . T h e t o l e r a n c e b o u n d s and c o n f i d e n c e l i m i t s d e f i n e d by J o h n s 12 and L e i b e r m a n ( 1 9 6 6 ) , T h o m a n , B a i n , and A n t l e ( 1 9 6 9 ) , B o g d a n o f f and P i e r c e ( 1 9 7 3 ) , and L a w l e s s ( 1 9 7 5 ) w e r e a l s o d i s c u s s e d . The t e x t i s an e x c e l l e n t r e f e r e n c e f o r a n y o n e i n t e r e s t e d i n t h e u s e o f t h e W e i b u l l d i s t r i b u t i o n . 13 DATA BASE The d a t a f o r t h i s t h e s i s came f r o m t h e R e g i o n a l F o r e s t N u t r i t i o n R e s e a r c h P r o j e c t ( U n i v . W a s h . , 1 9 7 6 ) ( a c o o p e r a t i v e f e r t i l i z a t i o n s t u d y w i t h f o r e s t i n d u s t r y and s e v e r a l g o v e r n m e n t r e s o u r c e a g e n c i e s ) . The s t u d y was c o o r d i n a t e d by t h e U n i v e r s i t y o f W a s h i n g t o n u n d e r D r . W i l l i a m A t k i n s o n . E i g h t y t h r e e o f t h e c o n t r o l p l o t s i n n a t u r a l s t a n d s w e r e s e l e c t e d f o r my s t u d y b a s e d on t h e f o l l o w i n g c r i t e r i a : 1 . P l o t e l e v a t i o n b e t w e e n 0 and 2 , 5 0 0 f t . - ' ' a b o v e s e a l e v e l . 2 . D o u g l a s - f i r s i t e i n d e x ( K i n g , 50 ) t o t a l h e i g h t o f s i t e t r e e s 7 7 - 1 4 3 f e e t a t 50 y e a r s f r o m s e e d . 3 . Age a t b r e a s t h e i g h t 1 2 - 4 2 y r s . 4 . A l l p l o t s l o c a t e d w e s t o f t h e C a s c a d e s . 5 . M i x e d s t a n d s o f w e s t e r n h e m l o c k and D o u g l a s - f i r . P l o t s i z e r a n g e d f r o m 1/10 a c r e t o 1 / 5 a c r e . T r e e h e i g h t was m e a s u r e d t o t h e n e a r e s t f o o t and dbh t o t h e 0 . 1 i n c h . T a b l e s I and I I l i s t t h e s e p l o t s a l o n g w i t h some o f t h e i r s t a n d s t a t i s t i c s . 1 / B e c a u s e t h e d a t a w e r e a l l r e p o r t e d i n E n g l i s h u n i t s , a l l m e a s u r e m e n t s and r e s u l t s r e p o r t e d h e r e a r e l i k e w i s e i n E n g l i s h u n i t s . iH TABLE Is ST^KD DATA - ESTIMATION SET P l o t Species 11006 71037 81048 131076 141082 141084 TOTAL Douglas-fir W. Hemlock TOTAL Douglas-fir W. Hemlock TPA 460 447 13 8.3 8.5 2.0 205 14.3 200 14.3 5 14.1 8 D 3.0 3.4 161095 201118 211123 211125 % in 1.8 7.6 TOTAL Douglas-fir Big l e a f Maple 327 320 7 11.2 11.4 3.5 4.1 3.5 Dmax 19.8 21.8 TOTAL 950 5.8 2.2 1.7 10.4 Douglas-fir 950 5.8 TOTAL 1460 4.1 1.8 1.7 10.0 Douglas-fir 1360 4.1 W. Hemlock 50 4.2 W. Redcedar 10 1.8 Other 40 2.5 TOTAL 470 8.8 3.6 2.0 16.1 Douglas-fir 340 9.9 W. Hemlock 110 6.2 W. Redcedar 15 2.6 Red Alder 5 8.5 TOTAL 480 8.4 5.5 1.7 25.8 Douglas-fir 215 12.7 W. Hemlock 145 5.4 W. Redcedar 95 2.9 Sitka Spruce 5 2.7 Red Alder 20 10.2 TOTAL 640 7.8 4.2 2.0 18.3 Douglas-fir 334 10.7 W. Hemlock 280 4.9 W. Redcedar 13 4.2 Other 13 2.5 TOTAL 400 8.4 3.3 2.8 15.5 Douglas-fir 393 8.5 Other 7 5.0 TOTAL 380 11.4 3.3 5.6 18.2 Douglas-fir 373 11.4 Other 7 6.9 21.1 Stand age 35 39 21 36 52 52 35 26 39 39 S i t e index 113 133 130 80 118 118 129 143 130 130 Average height 82 112 67 62 108 114 98 85 110 102 D — Mean d i a m e t e r ( i n c h e s ) s D - - S t a n d a r d d e v i a t i o n o f d i a m e t e r s ( i n c h e s ) D . ~ M i n i m u m d i a m e t e r ( i n c h e s ) m m D — Maximum d i a m e t e r ( i n c h e s ) max TPA — T r e e s p e r a c r e S t a n d Age — ( y e a r s ) S i t e I n d e x — K i n g , D o u g l a s - f i r ( f e e t a t 50 y r s ) A v e r a g e H e i g h t — Mean d o m i n a n t , c o d o m i n a n t h t . ( f e e t ] \5 TABLE I: STAND DATA - ESTIMATION SET Plo t Species TPA Dmin Dma Stand age S i t e index Average height 211128 TOTAL 407 9.6 3.8 2.1 18.8 45 124 105 Douglas-fir 354 9.9 W. Hemlock 40 6.1 Red Alder 13 10.0 221132 TOTAL 420 10.3 5.0 2.3 20.7 45 124 113 Douglas-fir 293 12.5 W. Hemlock 107 5.6 Red Alder 20 3.6 33119 3 TOTAL 880 4.2 2.3 1.6 12.2 19 123 51 Douglas-fir 450 5.4 W. Hemlock 430 2.8 341202 TOTAL 610 8.0 3.1 3.8 18.1 27 137 87 Douglas-fir 580 8.1 W. Hemlock 30 7.2 361215 TOTAL 305 10.2 5.7 1.7 21.9 39 138 110 Douglas-fir 135 14.8 W. Hemlock 145 6.4 Red Alder 15 11.2 Other 10 1.7 371221 TOTAL 320 11.3 3.7 3.0 19.7 42 127 109 Douglas-fir 180 13.4 W. Hemlock 130 8.7 Red Alder 10 7.3 411244 TOTAL 387 8.9 3.3 1.7 17.2 28 140 92 Douglas-fir 354 9.6 W. Hemlock 13 2.3 Other 20 2.6 431253 TOTAL 1610 5.1 2.2 1.7 11.2 37 98 67 Douglas-fir 580 5.7 W. Hemlock 640 5.2 W. Redcedar 390 4.1 451270 TOTAL 620 7.5 4.2 2.0 23.1 33 131 94 Douglas-fir 167 12.3 W. Hemlock 313 6.2 W. Redcedar 80 4.7 Big l e a f Maple 20 5.7 Red Alder 7 6.2 Other 33 4.7 16 TABLE I J STAND DATA - ESTIMATION SET P l o t Species TPA X> 8 D Pnin °max Stand S i t e Average age index height 531313 TOTAL 493 7.5 2.0 3.8 13.0 21 125 65 Douglas-fir 493 7.5 531317 TOTAL 527 7.0 3.4 1.5 15.0 21 125 58 Douglas-fir 447 7.9 Other 80 1.7 541319 TOTAL 1100 4.6 2.3 1.6 12.0 35 106 72 Douglas-fir 1100 4.6 551327 TOTAL 640 7.4 2.9 2.0 13.4 37 94 77 Douglas-f i r 640 7.4 571341 TOTAL 400 9.9 3.5 2.1 17.4 29 137 88 Douglas-fir 333 10.6 W. Hemlock 20 6.7 H. Redcedar 27 4.5 Red Alder 13 7.4 Other 7 7.0 601355 TOTAL 500 7.6 4.0 1.6 17.1 42 105 97 Douglas-fir 400 9.0 Other 100 1.9 601360 TOTAL 840 6.4 2.2 1.6 15.3 42 105 77 Douglas-fir 820 6.5 Other 20 1.7 681404 TOTAL 570 6.7 4.7 1.6 15.3 30 135 91 Douglas-fir 340 9.9 Other 230 1.9 761453 TOTAL 420 6.7 3.6 1.5 16.4 32 98 73 Douglas-fir 360 7.0 Other 60 4.9 771462 TOTAL 1280 3.7 1.8 1.6 8.9 15 77 43 Douglas-fir 1270 3.7 Sitk a Spruce 10 2.1 791473 TOTAL 630 7.3 3.3 1.5 15.4 30 121 87 Douglas-fir 420 8.5 W. Hemlock 40 3.6 W. Redcedar 50 1.9 Big l e a f Maple 20 2.5 Red Alder 100 7.3 811485 TOTAL 600 8.2 2.4 4.3 14.6 31 109 69 Douglas-fir 600 8.2 iT TABLE Is STAND DATA - ESTIMATION SET Plot Species TPA Dmln Dmax s t a n d s i t e Average age index height 821490 TOTAL 1110 5.2 2.6 Douglas-fir 220 8.0 W. Hemlock 720 4.9 Red Alder 10 8.1 Other 160 2.4 831494 TOTAL 1560 5.1 2.4 Douglas-fir 520 6.5 W. Hemlock 950 4.5 Other 90 2.9 951568 TOTAL 1340 4.5 1.7 Douglas-fir 1340 4.5 961576 TOTAL 280 11.3 . 3.4 Douglas-fir 280 11.3 971582 TOTAL 520 7.2 3.6 Douglas-fir 420 8.3 Other 100 2.8 981588 TOTAL 374 10.2 3.4 Douglas-fir 347 10.7 Big leaf Maple 20 3.7 Other 7 2.5 991591 TOTAL 380 6.8 5.4 Douglas-fir 313 7.7 W. Hemlock 47 3.0 Other 20 1.9 1011602 TOTAL 730 6.2 2.4 Douglas-fir 730 6.2 1011605 TOTAL 1060 4.9 2.0 Douglas-fir 1060 4.9 1101660 TOTAL 830 3.7 2.0 Douglas-fir 400 4.2 H. Hemlock 340 2.3 W. Redcedar 10 1.7 Other 80 2.8 1.7 12.8 1.5 12.2 21 122 60 3.0 18.6 1.5 14.6 2.5 18.4 1.6 28.5 2.8 12.7 1.2 9.4 1.7 6.8 22 1.5 9.1 19 39 30 38 34 24 24 12 130 100 112 108 125 121 118 118 132 68 44 94 72 100 79 71 59 36 1131675 TOTAL 1300 4.0 Douglas-fir 1080 4.0 Other 220 4.0 2.0 1.6 9.9 23 117 59 I S TABLE II: STAND DATA - PREDICTION SET Plot Species TPA D 8D °fliin Dm ax Stand age Site index Average height 11004 TOTAL 407 9.3 2.9 4.3 15.2 35 113 81 Douglas-fir 407 9.3 51025 TOTAL 593 6.4 2.4 1.7 11.7 30 101 67 Douglas-fir 553 6.6 W. Hemlock 40 4.1 51027 TOTAL 626 7.0 2.4 2.2 17.0 30 101 65 Douglas-f i r 613 7.1 W. Hemlock 13 4.0 71041 TOTAL 220 13.0 4.0 5.8 20.2 39 133 106 Douglas-fir 205 13.4 W. Hemlock 15 7.4 B1043 TOTAL. 1180 5.3 2.0 2.0 11.4 21 130 66 Douglas-fir 1170 5.3 Other 10 2.4 131074 TOTAL 1600 4.4 2.1 1.6 9.4 36 80 55 Douglas-fir 1580 4.3 Other 20 . 5.1 161091 TOTAL 659 7.7 3.7 2.2 16.8 35 129 98 Douglas-fir 373 9.9 W. Hemlock 253 4.9 W. Redcedar 20 4.0 Other 13 5.3 171098 TOTAL 1530 4.9 2.0 1.8 12.3 20 122 63 Douglas-fir 980 5.4 W. Hemlock 550 4.1 171100 TOTAL 2280 3.8 1.7 1.6 9.9 20 122 59 Douglas-fir 1550 4.1 W. Hemlock 730 3.1 191111 TOTAL 314 10.7 4.2 1.5 18.5 45 126 117 Douglas-fir 280 11.7 W. Hemlock 7 2.3 W. Redcedar 27 2.2 191114 TOTAL 267 11.7 3.5 7.0 20.0 45 126 117 Douglas-fir 267 11.7 201119 TOTAL 687 6.4 3.3 1.8 17.9 26 143 84 Douglas-fir 680 6.4 Other 7 2.6 X - Mean d i a m e t e r ( i n c h e s ) - - Standard d e v i a t i o n o f d i a m e t e r s ( i n c h e s ) — Minimum^djameter ( i n c h e s ) Maximum d i a m e t e r ( i n c h e s ) TPA — Trees per a c r e Stand Age — ( y e a r s ) S i t e Index — K i n g , D o u g l a s - f i r ( f e e t a t 50 y r s ) Average H e i g h t — Mean dominant, codosrinant h t . ( f e e t ) TABLE l i s STAND DATA - PREDICTION SET P l o t Species TPA 8D Drain ^max s t a n d S i t e Average age index height 331194 341204 361216 371222 411246 431254 451266 541324 551330 571342 591353 TOTAL Douglas-fir W. Hemlock TOTAL Douglas-fir W. Hemlock Other TOTAL Douglas-fir W. Hemlock TOTAL Douglas-fir W. Hemlock TOTAL Douglas-fir Other TOTAL Douglas-fir W. Hemlock W. Redcedar TOTAL Douglas-fir W. Hemlock W. Redcedar Other TOTAL Douglas-fir TOTAL Douglas-fir TOTAL Douglas-fir W. Hemlock TOTAL Douglas-fir W. Hemlock W. Redcedar 1150 900 250 680 480 160 40 340 185 155 430 240 190 373 360 13 1050 340 480 230 473 193 100 73 107 790 790 930 930 360 327 33 813 520 173 120 4.7 5.1 3.2 6.8 7.9 3.7 5.5 10.9 14.2 7.0 10.3 12.9 7.0 9.8 10.0 4.8 6.0 7.2 5.8 4.5 7.0 12.8 4.2 3.0 2.0 6.0 6.0 5.8 5.8 9.7 9.8 9.3 1.9 3.2 5.3 4.4 3.0 3.0 5.4 3.0 2.2 3.8 3.5 1.8 8.9 19 1.8 15.0 27 2.0 3.1 3.1 1.5 1.5 23.3 18.9 16.8 13.5 19.6 39 42 28 37 33 1.6 14.6 35 2.4 13.3 37 2.1 17.2 29 1.4 17.1 34 123 137 138 127 140 98 131 106 94 137 99 52 86 109 108 91 83 95 80 73 82 72 20 T A B L E I I : STAND DATA - P R E D I C T I O N S E T Plot Species TPA D 8D Pmin Dmax Stand age Site index Average height 591354 TOTAL 993 5.2 3.5 1.4 23.0 34 99 71 Douglas-fir 633 6.2 W. Hemlock 73 2.7 \ W. Redcedar 280 3.6 Other 7 3.6 681407 TOTAL 420 8.5 3.9 1.5 16.8 30 135 90 Douglas-fir 350 9.9 Other 70 1.7 691410 TOTAL 260 9.3 4.5 2.0 17.5 29 130 84 Douglas-fir 247 9.6 Other 13 3.8 691412 TOTAL 327 8.4 5.3 1.7 20.2 29 130 87 Douglas-fir 220 11.4 W. Redcedar 20 2.2 Red Alder 7 9.7 Other 80 1.9 7614 54 TOTAL 1774 4.0 2.0 1.6 11.4 32 98 61 Douglas-fir 1687 3.8 Other 87 6.8 771458 TOTAL 860 3.5 1.7 1.4 9.2 15 77 39 Douglas-fir 790 3.6 Other 70 2.5 781468 TOTAL 1013 5.7 2.4 2.2 16.7 42 108 85 Douglas-fir 960 5.7 W. Hemlock 13 8.2 Red Alder 33 7.0 Other 7 2.3 791472 TOTAL 690 6.8 3.1 1.6 13.7 30 121 85 Douglas-fir 350 8.5 W. Hemlock 10 1.6 W. Redcedar 100 2.5 Big leaf Maple 140 . 5.5 Red Alder 90 7.1 811481 TOTAL 550 8.5 3.1 1.8 17.5 31 109 75 Douglas-fir 530 8.8 Other 20 2.0 21 TABLE lis STAND DATA - PREDICTION SET Plot Species TPA Dfliin °max Stand Site Average age index height 821492 TOTAL Douglas-fir W. Hemlock Red Alder Other 880 400 420 30 30 6.7 B.l 5.4 8.0 2.7 2.7 2.3 12.1 21 122 59 831498 TOTAL Douglas-fir W. Hemlock 1270 610 660 5.7 7.2 4.4 2.4 1.7 12.8 22 130 62 951565 TOTAL Douglas-fir Red Alder Other 1930 1850 20 60 1.7 1.5 9.0 19 100 49 961572 TOTAL Douglas-fir W. Hemlock Other 380 340 27 13 9.8 10.5 4.5 3.1 3.3 1.9 15.4 39 112 93 971580 981584 TOTAL Douglas-fir Big leaf Maple TOTAL Douglas-fir Other 470 440 30 327 320 7 8.5 8.7 5.5 11.3 11.5 2.3 3.0 2.9 3.4 2.3 14.8 17.0 30 38 108 125 74 94 991592 TOTAL Douglas-fir W. Hemlock Sitka Spruce Other 599 253 153 33 160 6.9 10.4 6.1 6.0 2.4 4.4 1.5 16.5 34 121 79 1101658 TOTAL 670 3.6 Douglas-fir 410 4.2 W. Hemlock 110 1.8 Red Alder 20 4.6 Other 130 1.9 1.9 1.7 7.7 12 132 34 1131678 TOTAL 850 5.2 Douglas-fir 550 5.7 W. Hemlock 20 3.5 Red Alder 100 4.9 Other 180 3.8 2.7 1.6 11.5 23 117 65 22 METHODS The d a t a w e r e s p l i t by DUPLEX, ,an a l g o r i t h m d e v e l o p e d by R. W. K e n n a r d ( S n e e , 1 9 7 7 ) , i n t o an e s t i m a t i o n s e t and a p r e d i c t i o n s e t . E a c h s e t c o n t a i n e d a c o m p a r a b l e r a n g e o f s i t e s , d e n s i t i e s , and age g r o u p s . The e s t i m a t i o n s e t was u s e d t o f i n d a s u i t a b l e r e l a t i o n s h i p b e t w e e n s t a n d c h a r a c t e r i s t i c s and t h e W e i b u l l d i a m e t e r p a r a m e t e r s a , b and c . The p r e d i c t i o n s e t was u s e d t o t e s t t h a t r e l a t i o n s h i p . The p a r a m e t e r s o f t h e W e i b u l l d i s t r i b u t i o n f o r e a c h p l o t w e r e e s t i m a t e d f r o m t h e g i v e n i n d i v i d u a l t r e e d i a m e t e r d a t a . T h i s e s t i m a t i o n r e l i e d on t h e a v a i l a b i l i t y o f an a d e q u a t e e s t i m a t i n g r o u t i n e . T h r e e a l g o r i t h m s f o u n d i n t h e l i t e r a t u r e w e r e t e s t e d : D u b e y ' s ( 1 9 6 7 ) p e r c e n t i l e m e t h o d , and two maximum l i k e l i h o o d r o u t i n e s , W a r r e n ( 1 9 7 6 ) and B a i l e y ( 1 9 7 3 ) ( s e e A p p e n d i x I ) . B a i l e y ' s FITTER r o u t i n e was s e l e c t e d b e c a u s e i t g a v e a b e t t e r f i t t h a n t h e p e r c e n t i l e m e t h o d a n d , u n l i k e W a r r e n ' s r o u t i n e , i t g a v e u n b i a s e d e s t i m a t e s . A f t e r t h e W e i b u l l p a r a m e t e r s w e r e e s t i m a t e d f o r t h e d i a m e t e r d i s t r i b u t i o n s o f a l l p l o t s , s t a n d c h a r a c t e r i s t i c s f r o m t h e d a t a w e r e r e g r e s s e d * on t h e e s t i m a t e d p a r a m e t e r s o f t h e e s t i m a t i o n p l o t s . The c h a r a c t e r i s t i c s c o n s i d e r e d w e r e : s t a n d t o t a l a g e , mean d o m i n a n t and c o d o m i n a n t h e i g h t , 1 / R e g r e s s i o n d o n e v i a BMD:2R r o u t i n e . £ 3 a r i t h m e t i c mean d i a m e t e r , s i t e i n d e x , number o f s t e m s ( > 0 . 1 i n c h d . b . h . ) p e r a c r e , and t h e i n v e r s e s and s q u a r e s t h e r e o f . The b e s t o v e r a l l m o d e l s w e r e t h e n u s e d t o p r e d i c t t h e W e i b u l l p a r a m e t e r s f o r t h e p r e d i c t i o n p l o t s . The r e s u l t i n g d i s t r i b u t i o n s w e r e c o m p a r e d w i t h t h e c o r r e s p o n d i n g e s t i m a t e s f r o m t h e o b s e r v e d d i a m e t e r d a t a . ( S e e A p p e n d i x I ) . To be a u s e f u l m o d e l , t h e W e i b u l l d i s t r i b u t i o n s h o u l d d e s c r i b e t h e d i a m e t e r d i s t r i b u t i o n s o f m i x e d s p e c i e s s t a n d s as w e l l a s , i f n o t b e t t e r t h a n s i m p l e r m o d e l s s u c h a s t h e n o r m a l d i s t r i b u t i o n . H e r e , a r e g r e s s e d r e l a t i o n s h i p was deemed s a t i s f a c t o r y i f a l l o f t h e p r e d i c t e d d i s t r i b u t i o n s s a t i s f i e d t h e K o l m o g o r o v - S m i r n o v ( K - S ) g o o d n e s s o f f i t t e s t a t t h e « - = 0 . 2 0 l e v e l ( M a s s e y , 1 9 5 1 ) . T h i s l e v e l o f s i g n i f i c a n c e was t h o u g h t t o be an a c c e p t a b l e c o m p r o m i s e b e t w e e n t h e g o o d n e s s o f f i t o b t a i n e d f o r W e i b u l l d i s t r i b u t i o n s on e v e n - a g e d , s i n g l e s p e c i e s s t a n d s ( B l i s s & R i e n k e r , 1 9 6 4 ; • < * = 0 . 1 5 ; B a i l e y , 1 9 7 3 . 35% f i t a t oc = 0 . 0 5 ) and t h e f i t s o b t a i n a b l e w i t h s i m p l e r o n e - and t w o - p a r a m e t e r f u n c t i o n s . RESULTS The W e i b u l l p a r a m e t e r s a , b , and c w e r e e s t i m a t e d b y B a i l e y ' s FITTER r o u t i n e f o r a l l 83 s t a n d s . The c u m u l a t i v e d i s t r i b u t i o n s d e f i n e d by t h e maximum l i k e l i h o o d e s t i m a t e d p a r a m e t e r s w e r e c o m p a r e d w i t h t h e o b s e r v e d d i a m e t e r d i s t r i b u t i o n s . A l l o f t h e e s t i m a t e d d i s t r i b u t i o n s p a s s e d t h e K o l m o g o r o v - S m i r n o v ( K - S ) t e s t a t t h e <*.= 0 . 2 0 l e v e l o f s i g n i f i c a n c e ; d 2 ( 3 0 ) = . 1 3 1 - / ( T a b l e I I I ) . The s t a n d c h a r a c t e r i s t i c s mean d i a m e t e r , mean d o m i n a n t and c o d o m i n a n t h e i g h t , s t a n d a g e , t r e e s p e r a c r e , and s i t e i n d e x w e r e r e g r e s s e d a g a i n s t t h e e s t i m a t e d p a r a m e t e r s o f t h e W e i b u l l d i a m e t e r d i s t r i b u t i o n s o f t h e e s t i m a t i o n s e t . The b e s t r e s u l t i n g p r e d i c t o r s f o r t h e p a r a m e t e r s b a s e d on c o r r e l a t i o n c o e f f i c i e n t s and F r a t i o s a r e l i s t e d i n T a b l e s I V , V , and V I . The d i a m e t e r d i s t r i b u t i o n s o f t h e p r e d i c t i o n s e t as d e f i n e d by a l l c o m b i n a t i o n s o f a , b , and c p r e d i c t o r s w e r e c o m p a r e d w i t h t h e d i s t r i b u t i o n s w h i c h w e r e e s t i m a t e d f r o m t h e o b s e r v e d d i a m e t e r d i s t r i b u t i o n s v i a maximum l i k e l i h o o d . The o n l y c o m b i n a t i o n s t o f a i l t h e K - S t e s t a t t h e <*-= 0 . 2 0 l e v e l w e r e t h o s e i n v o l v i n g p r e d i c t o r a ( 8 ) o r b ( 8 ) . 1 / L i l l i e f o r s ( 1 9 6 7 ) c r i t i c a l v a l u e f o r d 2 ( 3 0 ) was u s e d h e r e b e c a u s e t h e p a r a m e t e r s o f t h e d i s t r i b u t i o n w e r e e s t i m a t e d f r o m "the s a m p l e . X 1? T A B L E I I I : C o m p a r i s o n o f M a x i m u m L i k e l i h o o d E s t i m a t e d W e i b u l l D i s t r i b u t i o n w i t h O r i g i n a l D i a m e t e r D i s t r i b u t i o n > ( N ) i / P l o t *w b * TW 5 . 9 2 1 6 1 . 8 6 1 7 . 0 6 8 . 1 5 9 8 . 1 9 5 2 2 . 4 9 6 9 . 0 5 6 . 1 5 3 6 . 0 0 7 1 2 . 3 7 1 6 . 0 6 2 . 1 3 1 6 . 1 9 3 5 2 . 4 0 9 8 . 0 6 3 . 1 3 0 8 . 7 0 1 4 2 . 3 6 9 0 . 0 5 3 . 1 8 3 1 1 . 2 2 4 1 2 . 6 9 9 2 . 0 7 2 . 1 7 7 4 . 2 6 9 6 1 . 9 5 4 9 . 0 4 1 . 1 6 6 5 . 4 4 4 0 2 . 3 9 0 2 . 0 6 1 . 1 3 0 3 . 7 7 2 6 1 . 6 7 0 9 . 0 9 5 . 1 0 0 3 . 4 6 6 1 1 . 7 8 6 9 . 0 9 3 . 1 0 5 9 . 2 8 4 9 2 . 4 7 0 1 . 0 5 8 . 1 3 0 8 . 0 0 5 4 1 . 3 3 5 5 . 0 7 1 . 1 3 0 6 . 9 5 1 4 1 . 7 1 9 2 . 0 5 4 . 1 2 6 6 . 9 6 1 3 1 . 4 7 5 2 . 0 4 1 . 1 3 0 3 . 8 2 3 5 1 . 7 4 7 9 . 0 4 6 . 1 0 3 3 . 1 7 2 8 1 . 8 1 5 2 . 0 8 0 . 0 8 4 1 1 . 9 0 2 1 2 . 7 5 3 5 . 0 9 4 . 1 7 3 5 . 7 0 8 2 1 . 4 4 9 4 . 0 8 7 . 1 8 3 7 . 1 9 3 2 1 . 9 8 2 7 . 0 5 1 . 1 6 0 5 . 4 7 0 6 1 . 5 6 4 5 . 0 4 6 . 1 2 4 7 . 1 3 9 6 1 . 9 9 2 2 . 0 6 3 . 1 6 3 9 . 8 3 1 0 2 . 2 6 8 1 . 0 9 2 . 1 6 9 9 . 0 5 5 0 2 . 2 2 6 4 . 0 5 5 . 1 5 9 1 0 . 4 2 7 5 1 . 9 1 7 5 . 0 7 6 . 1 5 7 3 . 4 8 6 0 1 . 4 4 3 1 . 0 8 1 . 1 3 1 3 . 5 6 3 3 1 . 7 2 1 8 . 0 8 9 . 1 1 8 4 . 9 7 6 0 1 . 4 9 0 0 . 0 4 9 . 1 5 9 5 . 8 8 8 0 1 . 6 4 7 0 . 0 5 1 . 1 5 3 1 0 . 1 4 2 5 1 . 5 6 5 1 . 0 6 3 . 1 5 9 1 2 . 2 8 8 0 2 . 1 6 8 3 . 0 7 5 . 1 5 3 1 2 . 5 5 5 8 3 . 2 5 7 9 . 0 5 6 . 2 0 6 9 . 3 3 2 8 1 . 9 5 5 4 . 0 5 1 . 1 7 7 9 . 9 6 7 8 2 . 8 6 3 5 . 0 6 6 . 1 6 0 1 0 . 8 7 8 2 3 . 6 0 3 8 . 0 5 7 . 1 6 3 4 . 6 3 2 7 1 . 9 4 4 2 . 0 4 6 . 1 0 0 5 . 5 1 9 4 1 . 6 6 8 4 . 0 4 7 . 1 2 4 6 . 1 2 4 3 1 . 0 3 8 1 . 0 9 8 . 1 5 0 6 . 7 2 1 1 1 . 5 0 6 0 . 0 9 5 . 1 3 2 4 . 4 5 7 2 2 . 0 1 7 7 . 0 7 5 . 1 4 9 7 . 8 7 3 2 2 . 1 9 1 8 . 0 8 3 . 1 4 2 4 . 0 3 4 9 1 . 5 9 6 2 . 0 5 5 . 1 2 1 5 . 5 9 7 8 1 . 7 0 2 8 . 0 4 3 . 1 4 2 1 1 0 0 4 1 1 0 0 6 5 1 0 2 5 5 1 0 2 7 7 1 0 3 7 7 1 0 4 1 8 1 0 4 3 8 1 0 4 8 1 3 1 0 7 4 1 3 1 0 7 6 1 4 1 0 8 2 1 4 1 0 8 4 1 6 1 0 9 1 1 6 1 0 9 5 1 7 1 0 9 8 1 7 1 1 0 0 1 9 1 1 1 1 1 9 1 1 1 4 2 0 1 1 1 8 2 0 1 1 1 9 2 1 1 1 2 3 2 1 1 1 2 5 2 2 1 1 2 8 2 2 1 1 3 2 3 3 1 1 9 3 3 3 1 1 9 4 3 4 1 2 0 2 3 4 1 2 0 4 3 6 1 2 1 5 3 6 1 2 1 6 3 7 1 2 2 1 3 7 1 2 2 2 4 1 1 2 4 4 4 1 1 2 4 6 4 3 1 2 5 3 4 3 1 2 5 4 4 5 1 2 6 6 4 5 1 2 7 0 5 3 1 3 1 3 5 3 1 3 1 7 5 4 1 3 1 9 5 4 1 3 2 4 4 . 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 5 0 0 0 . 0 0 0 0 5 0 0 0 1 . 0 0 0 0 1 . 0 0 0 0 1 . 0 0 0 0 0 . 5 0 0 0 1 . 0 0 0 0 1 . 5 0 0 0 1 . 5 0 0 0 1 . 5 0 0 0 1 . 0 0 0 0 0 . 0 0 0 0 6 . 5 0 0 0 2 . 0 0 0 0 1 . 5 0 0 0 5 . 0 0 0 0 2 . 5 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 5 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 1 . 0 0 0 0 1 . 0 0 0 0 1 . 0 0 0 0 1 . 5 0 0 0 3 . 5 0 0 0 0 . 0 0 0 0 1 . 0 0 0 0 1 . 0 0 0 0 1 . 1 . 1 . 1 . 3. 1 . 1 . 0 . 0 . y a * , b * , c * a r e t h e e s t i m a t e d W e i b u l l p a r a m e t e r s V d - m a x / S n ( x ) - F D ( x ) 3 / a . 2 ( N ) - K o l m o g o r o r - S m i r n o v l i m i t f o r d f o r f i t a t 2 0 % l e v e l o f s i g n i f i c a n c e Z.6 TABLE III (Continued) Plot b* c* d V d. 2(N)i/ 551327 1.0000 7.1695 2.3550 .056 .157 551330 2.0000 4.2236 1.7714 .047 .132 571341 0.0000 10.9818 3.0761 .038 .160 571342 0.0000 10.9145 2.8211 .073 .166 591353 1.0000 5.8602 1.5572 .049 .155 591354 1.0000 4.5779 1.3143 .047 .104 601355 0.0000 8.4775 1.9472 .124 .169 601360 1.0000 6.0191 2.5392 .062 .139 681404 1.0000 5.8259 1.0608 .202 .163 681407 0.0000 9.5229 2.2623 .138 .183 691410 1.0000 9.2792 1.8733 .084 .190 691412 1.0000 7.8292 1.1820 .162 .169 761453 1.0000 6.2550 1.5107 .076 .159 761454 1.0000 3.3434 1.6222 .080 .078 771458 1.0000 2.7473 1.5137 .099 .137 771462 1.0000 3.0129 1.6326 .078 .112 781468 1.5000 4.7896 1.9342 .067 .103 791472 0.5000 7.0273 2.0676 .060 .153 791473 0.0000 8.1590 2.3065 .055 .159 811481 0.0000 9.4805 2.9009 .079 .163 8114 85 4.0000 4.7198 1.8031 .033 .160 821490 1.0000 4.6869 1.6929 .064 .121 821492 2.0000 5.1969 1.7361 .063 .131 831494 1.0000 4.6034 1.7963 .040 .102 831498 1.0000 5.3542 2.0770 .056 .112 951565 1.0000 2.9076 1.6261 .098 .091 951568 1.0000 3.9121 2.1489 .067 .110 961572 0.0000 10.9154 3.4388 .084 .163 961576 0.0000 12.4302 3.6758 .056 .183 971580 2.0000 7.3454 2.2804 .097 .173 971582 0.5000 7.5120 1.9103 .086 .166 981584 0.0000 12.3532 4.5658 .071 .169 981588 0.0000 11.3259 3.3332 .054 .163 991591 1.0000 6.1411 1.1413 .061 .163 991592 1.0000 6.3467 1.2426 .123 .131 1011602 2.5000 4.1552 1.6311 .093 .150 1011605 0.5000 4.9292 2.3247 .061 .123 1101658 1.0000 2.9.53 1.5847 .089 .157 1101660 1.0000 2.9956 1.7325 .107 .149 1131675 1.0000 3.3690 1.6392 .081 .111 1131678 1.0000 4.6670 1.6160 .113 .137 TABLE IV; a Equations a(l) • 0.4737 + 448.4/TPA a(2) = 3.086 - 0.6655(D) + 0.08021(D2) - 27.58(A)/TPA a(3) =• -413.3 - 43.22(D) + 640.1/D - 574.1/TPA + 1.032(D2) + 296.6(ln D) - 21.83(A)/TPA a(4) = -444.1 - 46.31(D) + 689.5/D - 1275.0/TPA + 1.093(D2) + 318.9(ln D) - 2.872(A)/SI a(5) = -410.7 - 43.33(D) + 635.0/D +0.00005556(SI2) + 295.3(ln D) + 1.0 4 8 ( 02) - 14.86(HT)/TPA a(6) = -466.0 - 47.99(D) + 724.6/D - 1347.0/TPA + 1.126(D2) + 333.3(ln D) - 2.544(HT)/SI a(7) = -325.8 - 35.84(D) +496.8/D + 0.8543(D2) + 0.00003266(SI2) + 238.6(ln D) + 0.01514(TPA)/A a(8) = 372.5 -39.37(D) + 0.008466(SI) + 567.8/D - 0.0002378(A2) + 0.91931D2) + 268.2(ln D) + 0.06373(TPA)/HT a(9) = -332.4 - 35.63(D) + 509.5/D - 0.0004968(A2) + 0.8411(D2) + 241.2(ln D) + 0.0726(TPA)/SI ,a(10) " -386.3 - 40.60(D) + 598.1/D - 0.0006213 (A2) + 0.965KD2) + 278.1(ln D) - 7.318 (SI)/TPA A = D = HT stand age (total years) SI = site index (King, Douglas f i r , feet at 50 yrs) arithmetic mean diameter (inches) TPA = trees per acre (dbh 0.1 in.) = ave. dominant & codominant height (feet) TABLE V; b Equations b(l) = 2.648 + 0.06581(HT) - 0.04242(TPA)/A b(2) = 6.297 + 0.001277(A2) + 0.00007376(SI2) - 0.1870(TPA)/HT b(3) = -0.2065 + 0.05548{HT) + 1.159(In A) - 0.2231(TPA)/SI b(4) = 17.75 + 0.05619(HT) - 2.412(ln TPA) b(5) - 13.10 + 3.914(ln A) - 2.600(ln TPA) - 7.488(A)/HT b(6) = 3.497 + 3.188(ln HT) - 1.791(ln TPA) + 14.72 (A)/TPA b(7) = 6.364 + 0.03952(A) - 29.30/A - 236.9/SI + 13.84(HT)/TPA b(8) " 23.20 + 5.665(HT)/SI - 3.156(ln TPA) b(9) = 18.90 + 2.617(ln A) - 3.276(ln TPA) b(10) - 4.011 + 0.0007134(A2) + 4.451(ln D) - 1.038(ln TPA) 00' TABLE VI: c Equations c(l) - 1.562 + 255.6/TPA c(2) • 0.8321 + 0.2908(D) - 0.0004183(A2) - 0.0002045(HT2) c(3) = 0.6100 + 0.4136(D) - 311.65/TPA + 0.0004033(A2) - 0.0002179(HT2) c(4) » 165.4 + 16.77(D) - 253.8/D - 0.3649(E)2) - 0.0001321 (HT2) - 0.00003068 (SI 2) - 115.9(ln D) c(5) = 38.96 - 0.1473(A) + 0.3883(D) + 0.4969(HT) + 0.002466(A2) - 0.002059(HT2) - 14.72(ln HT) + 0.0674(TPA)/SI c(6) » -0.1859 + 0.6313(D) + 0.1174(HT) + 10.28/D + 0.001338(A2) - 0.0008732(HT2) - 2.415(ln A) - 2.940 (SD/TPA 3o S e v e r a l c o m b i n a t i o n s of p r e d i c t i n g e q u a t i o n s passed the K-S t e s t at t h e <* = 0 . 2 0 l e v e l w i t h 100% f i t o f a l l p l o t s . These s u p e r i o r c o m b i n a t i o n s , ( o r d i s t r i b u t i o n m o d e l s ) , can be found i n T a b l e V I I I w i t h t h e i r r e s p e c t i v e d s t a t i s t i c s . As a c l a s s , t h e c o m b i n a t i o n o f e q u a t i o n s a(2) and b(10) was a s u p e r i o r m o d e l , h a v i n g t h e l o w e s t average d s t a t i s t i c and t h e l e a s t amount o f v a r i a n c e i n f i t r e g a r d l e s s of t h e c e q u a t i o n u s e d . The b e s t model was: a(2)=3.086 - 0.6655(D) + 0.08021 (D 2 ) - 27 .58 (A)/TPA b(10) = 4.011 + 0.0007134 ( A 2 ) + 4 . 4 5 1 ( l n D) - 1 . 0 3 8 ( l n TPA) c ( 4 ) = 165.4 + 16.77(D) - 253.8/D - .3649 (D 2 ) - .0001321 (HT 2 ) - .00003068 ( S I 2 ) - 115.9 ( l n D) where D = mean s t a n d d i a m e t e r ( i n c h e s ) A = s t a n d age ( t o t a l y e a r s ) SI = s i t e index ( K i n g , D o u g l a s - f i r , h e i g h t at 50 y r s ) HT = mean dominant and codominant h e i g h t ( f e e t ) TPA = t r e e s per a c r e w i t h an average d = . 0 5 8 7 , s H = . 0 3 1 7 , and max d = .1441 . 31 TABLE VII: Regression Stat is t ics Equation F(n,v) R S E E (inches) a(l) 3.896(1,40) .2979 1.336 a(2) 6.717(3,38) .5887 1.161 a(3) 5.737(6,35) .7042 1.063 a(4) 5.480 (6,35) .6960 , 1.075 a(5) 5.674(6,35) .7022 1.066 a(6) 5.736(6,35) .7041 1.063 a(7) 3.852(6,35) .6306 1.162 a(8) 3.432(7,34) .6434 1.163 a(9) 4.088(6,35) .6419 1.148 a(10) 4.820 (6,35) .6727 1.108 b(l) 37.58(2,39) .8114 1.597 b(2) 22.62(3,38) .8006 1.659 b(3) 26.95(3,38) .8248 1.565 b(4) 51.29(2,39) .8512 1.434 b(5) 33.97(3,38) .8535 1.443 b(6) 34.45(3,38) .8551 1.445 b(7) 20.57(4,37) .8306 1.562 b(8) 48.64(2,39) .8449 1.462 b(9) 48.42(2,39) .8442 1.464 b(10) 39.18(3,38) .8693 1.368 c(l) 7.719(1,40) .4022 .5412 c(2) 8.863 (3,38) .6416 .4652 c(3) 7.485(4,37) .6588 .4570 c(4) 5.827(6,35) .7069 .4470 c(5) 5.068(7,34) .7146 .4485 c(6) 5.039(7,34) .7136 .4492 3£ Table VIII — Prediction Models which f i t a l l MLE Distributions EQUATION # d s d max d a b c 2 5 2 .0852 .0408 .1625 2 5 3 .0830 .0405 .1746 2 5 4 .0843 .0371 .1694 2 5 6 .0839 .0426 .1890 2 6 2 .0780 .0407 .1732 2 6 3 .0750 .0408 .1663 2 6 4 .0650 .0389 .1672 2 6 5 .0827 .0439 .1868 2 6 6 .0767 .0421 .1868 2 11 1 .0735 .0572 .1705 2 11 2 .0591 .0346 .1549 2 11 3 .0563 .0346 .1739 2 11 4 .0587 .0317 .1441 2 11 6 .0566 .0384 .1883 7 11 2 .08 21 .0459 .1666 7 11 3 .0795 .0465 .1775 7 11 4 .0789 .0422 .1733 7 11 6 .0813 .0446 .1794 1. d = average K-S s t a t i s t i c over a l l prediction plots s^ = standard deviation of d max d = maximum value of d found over a l l plots 3 3 A p l o t by p l o t c o m p a r i s o n o f t h e d i s t r i b u t i o n s p r e d i c t e d by t h e m o d e l : w i t h t h e o b s e r v e d s t a n d d i a m e t e r d i s t r i b u t i o n s c a n be f o u n d i n T a b l e I X . The K - S s t a t i s t i c d was l e s s t h a n d ^(N) f o r 9 5 % o f t h e p l o t s . A c c o r d i n g t o t h e K o l m o g o r o v - S m i r n o v t e s t , t h e p r e d i c t e d W e i b u l l d i a m e t e r d i s t r i b u t i o n s a r e s a t i s f a c t o r y m o d e l s o f t h e o b s e r v e d d i s t r i b u t i o n s . The q u a l i t y o f f i t , a s r e f l e c t e d by t h e K - S s t a t i s t i c d , was n o t d e p e n d e n t on t h e v a l u e o f a n y one s t a n d a t t r i b u t e ( F i g u r e s 1 - 5 ) . The m o d e l p r e d i c t s c o n s i s t a n t l y o v e r t h e r a n g e o f mean d i a m e t e r , h e i g h t , s i t e , s t o c k i n g , and s t a n d age e x h i b i t e d i n t h e p r e d i c t i o n s e t . The f i g u r e s i n A p p e n d i x I I I show a l l t h r e e d i s t r i b u t i o n s ( M L E , p r e d i c t e d , and o b s e r v e d ) f o r 18 o f t h e p r e d i c t i o n s e t The p r e d i c t e d d i s t r i b u t i o n s d i d n o t f i t t h e o b s e r v e d d i a m e t e r d i s t r i b u t i o n s a t t h e oc = . 2 0 l e v e l o f s i g n i f i c a n c e f o r p l o t s 6 9 1 4 1 2 and 9 9 1 5 9 2 . The t r e e s i n t h e s e p l o t s d i v i d e d e a s i l y i n t o d i s t i n c t s p e c i e s - s i z e g r o u p s . S t a n d 6 9 1 4 1 2 ( f i t a t d = . 0 1 ) was 67% D o u g l a s - f i r (mean d i a m e t e r 1 1 . 4 i n c h e s ) and 33% c e d a r and o t h e r s p e c i e s (mean d i a m e t e r 1 . 9 i n c h e s ) . S t a n d 9 9 1 5 9 2 ( f i t a t oc = . 1 0 ) was 42% D o u g l a s - f i r ( 1 0 . 4 i n . d b h ) 31% h e m l o c k and s p r u c e ( 6 . 0 i n . d b h ) and 27% o t h e r s p e c i e s ( 2 . 4 i n . d b h ) . The p o l y m o d a l n a t u r e o f t h e s e two d i a m e t e r d e n s i t y d i s t r i b u t i o n s a c c o u n t s f o r t h e l a c k o f f i t w i t h t h e u n i m o d a l W e i b u l l m o d e l . F ( d ) = 1 . 0 - e x p p l o t s . 34 A l t h o u g h t h e d i f f e r e n c e b e t w e e n t h e o b s e r v e d and p r e d i c t e d d i s t r i b u t i o n s was n o t s i g n i f i c a n t a t t h e oc = . 2 0 p r o b a b i l i t y l e v e l , p l o t s 7 1 0 4 1 , 3 6 1 2 1 6 , 4 5 1 2 6 6 , and 5 9 1 3 5 4 showed a t e n d e n c y i n t h e m o d e l t o w a r d s b i a s . The p r e d i c t e d d i s t r i b u t i o n s f o r t h e s e p l o t s u n d e r e s t i m a t e d t h e number o f t r e e s i n t h e s m a l l d i a m e t e r c l a s s e s and o v e r e s t i m a t e d t h e number o f t r e e s i n t h e m i d d l e d i a m e t e r c l a s s e s ( F i g u r e s 6 - 9 ) . TABLE IX: MLE vs Predicted Distribution (a(2) b(10) c(4)) d .20OO) = .190 Plot a y b y c y dMLE y d<)3 11004 4.0000 5.9126 1.8617 .0470 .0683 1.4432 8.5643 2.7774 51025 1.0000 6.0771 2.3716 .0594 .0651 0.7155 6.2829 2.0695 51027 1.5000 6.1935 2.4098 .0441 .1027 1.0467 6.6419 2.2585 71041 3.0000 11.2241 2.6992 .0301 .1006 3.1164 10.9179 2.9267 81043 1.5000 4.2696 1.9549 .0495 .0580 1.3184 4.3939 1.7620 131074 1.0000 3.7726 1.6709 .0881 .1825 1.0847 3.8379 2.1205 161091 1.5000 6.9514 1.7192 .0379 .0469 1.2519 7.2286 1.5712 171098 1.5000 3.8235 1.7479 .0327 .0388 1.3898 3.7538 1.8461 171100 1.0000 3.1728 1.8152 .1082 .0792 1.4731 2.2186 1.4335 191111 0.0000 11.9021 2.7535 .0578 .1295 1.2053 10.0470 2.3373 191114 6.5000 5.7082 1.4494 .1226 .1355 1.6340 10.6077 2.6085 201119 1.5000 5.4706 1.5645 .0580 .0949 1.0683 5.9763 1.4188 33119 4 1.5000 3.5633 1.7218 .0190 .0821 1.2734 3.8310 1.9692 341204 1.5000 5.8884 1.6467 .0441 .0800 1.1693 6.2855 1.5151 361216 0.0000 12.2880 2.1683 .1061 .1588 2.2319 9.6907 2.5561 371222 2.0000 9.3328 1.9554 .0700 .0988 2.0727 9.3668 2.4634 411246 0.0000 10.8782 3.6038 .0199 .0682 2.2281 8.5962 2.6055 1/ The f i r s t l ine contains the maximum likel ihood estimate, the second l ine contains the predicted parameters. 2/ K-S s t a t i s t i c for predicted distr ibution against MLE. 3/ K~S s ta t i s t i c for predicted distr ibution against observed. TABLE IX: continued. Plot a b c dMLE a 0 431254 1.0000 5.5479 1.6959 .0220 .0580 1.0048 5.7322 1.7233 451266 1.0000 6.1243 1.0381 .0637 .1600 0.4506 7.0768 1.4097 541324 1.0000 5.5978 1.7028 .0138 .0475 0.7587 5.9345 1.7206 551330 2.0000 4.2236 1.7714 .0727 .1000 0.8161 5.6846 1.9221 571342 0.0000 10.9145 2.8211 .0573 .1335 1.9970 8.6355 2.7846 5913 53 1.0000 5.8602 1.5572 .0628 .1090 0.9106 6.0446 1.9707 591354 1.0000 4.5779 1.3143 .1010 .1552 0.8491 5.0046 1.8804 681407 0.0000 9.5229 2.2623 .0332 .1571 1.2825 7.9296 2.0860 691410 1.0000 9.2792 1.8733 .1067 .1883 0.7497 8.7600 2.5967 691412 1.0000 7.8292 1.1820 .1399 .2809 0.7406 8.1007 2.1373 761454 1.0000 3.3434 1.6222 .0428 .1099 1.2103 3.1180 1.7182 771458 1.0000 2.7473 1.5137 .0578 .1485 1.2616 2.6943 1.4734 781468 1.5000 4.7896 1.9342 .0942 .1407 0.7623 5.8533 1.5856 791472 0.5000 7.0273 2.0676 .0361 .0810 1.0497 6.3680 1.6580 811481 0.0000 9.4805 2.9009 .0364 .0907 1.6804 7.6802 2.5801 821492 2.0000 5.1969 1.7361 .0305 .0946 1.5498 5.7217 2.1237 831498 1.0000 5.3542 2.0770 .0320 .0483 1.4287 4.7089 1.8574 951565 1.0000 2.9076 1.6261 .0806 .0686 1.4584 2.1003 1.4339 961572 0.0000 10.9154 3.4388 .0422 .1058 1.4 815 9.1113 2.7657 971580 2.0000 7.3454 2.2804 .019 2 .1235 1.4864 7.8087 2.6171 981584 0.0000 12.3532 4.5658 .0389 .0956 2.6398 9.8388 3.1718 991592 1.0000 6.3467 1.2426 .1000 .2172 0.7664 6.8167 1.8280 1101658 1.0000 2.9122 1.5748 .0780 .1499 1.2042 3.1270 1.3974 1131678 1.0000 4.6665 1.6158 .0462 .1632 1.0407 4.6872 1.8621 3T Figure 1. The r e l a t i o n s h i p of goodness of f i t to stand diameter. 4.10 5.35 6.60 7.85 9.11 10.36 11.61 12.37 14.11 . + i + « + + + + 1 + + e (• + 4 + 1  .25 • 20 .17 -o .15 o •r— -t-> to •r— +-> <0 +-> oo CO 1 .10 .02 * • 0 + . -t + + 1——+ + + — - + — - + • + •( + , + + + - ^ + _ + .3.47 _ 5.98 7.23 8.48 9.73 10.99 12.24 <3.49 14.75 16.00 Diameter (inches) STATISTICS.. R SQUARED .00848 . SLOPE (B) - .00160 STD FRRGfi OF P - Figure 2. The relationship of goodness of f i t to stand height. 39.00 49.00 59.00 69.00 79.00 89.00 99.00 109.00 119.00 129.00 . + r + r + + H (- + + + + 1 r + -t r- + + + T .25 + .22 .20 .17 ,15 .12 +•> «/> T- •I » rO 4-> CO CO ^ .10 .07 .05 .02 it * 4 ft * * * + ^ + + + i + t + + + + + + <—-+ +-—+ 1-77—+- .00 44.00 54.00 64.00 74.00 84.00 94.00 104.00 "114.00 124.00 1 Height (feet) STATISTICS.. R SQUARED .00360 SLOPE (B) .00013 STD ERROR OF B - 33 Figure 3. The relationship of goodness of f i t to stand age. 14.00 13.00 22.00 26.00 30.00 34.00 3B.00 42.00 46.00 50.00 . + -f + 1 (• + i (• + + + r 1 + r r + + + + +. .25 + • * .22 .20 .17 • ° .15 4-> «/» +-> fO +J i/"> CO i .12 .10 .07 .05 0 + « * . + + + + + r 1- 1 +: 1 * + + + + 1—--•t- + + +- 12.00 . 16.00 20.00 24.00 23.00 32.00 36.00 40.00 44.00 48.00 'STATISTICS.. R SQUARED Stand Age (total years) .02950 SLOPE lb) - .00092 STD ERROR OF B -+, 2.00 .OOuf; F i g u r e 4 . T h e r e l a t i o n s h i p o f g o o d n e s s o f f i t t o t r e e s p e r a c r e . C " 323.00 529.00 735.00 941.00 1147.00 1353.00 1559.00 1765.00 1971.00 2177.00 * 1- 1 i i 1 i i 1 1 •+ ^ <• + * + < + +—-+ +• . 2 5 *** .22 + .20 . 1 7 u 5 -12 -t-> » J O . 0 7 .05 . 0 2 , » • * * * * « * * * 0 ' * ft + t -+ 4 - • -+ - / /22o!« ' 4.26.00 632.00 + 838.00 1044.00 ~T250.00 ^1456.00 ,1662.00 1868T.0O 2074.00 2288.00 •STATISTICS.. R SQUARED T r e e s p e r A c r e .07018 SLOPE <B) - -.00002 " STD ERROR OF B - .00001 Figure 5. The relationship of goodness of f i t to site index. 81.00 89.00 97.00 105.00 , 113.00 121.00 129.00 137.00 ' 145.00 153.00 • + < < ' 1 < + 1- i i r- H 1 1 + ( 1 1 , e— .22 .20 ,17 .15 4-> t/> •r~ % .12 +-> CO CO .10 .07 .05 0 + .+-- 77.00 i i « ~H f + e f. + + H i < + + ^ -+ + +-85.00 93.00 101.00 109.00 117.00 125.00 133.'00 ' 141.'00 ' 14y.'oo""+~^LOO •STATISTICS.. R SQUARED Site Index (feet at 50 years),'• .00299 SLOPE (B) - -.00015 STD ERROR OF B .00043 Figure 6. Observed and predicted trees per acre by diameter class for plot 710^1. 80- 60 H HQ] Zo *f-8 8-12. 12.-16 Diameter Class Observed I—I Predicted Figure 7. Observed and predicted trees per acre by diameter class for plot 361216. IbOA 12,0- &• SO 5 5 " - ' o 10-/5 Diameter Glass l^>l Observed I J Predicted Figure 8. Observed and predicted trees per acre by diameter class for plot 451266 Diameter Glass Figure 9. Observed and predicted trees per acre by diameter class for plot 591354. 5 Hoo < 6 Zoo- zoo l o o i \^ZA. Observed [ f Predicted 2 O - M N - 8 8 M 2 Diameter Glass D I S C U S S I O N , A P P L I C A T I O N S , and FUTURE RESEARCH A l l o f t h e 8 3 MLE d i a m e t e r d i s t r i b u t i o n s f i t t h e o b s e r v e d d i s t r i b u t i o n a t t h e p r o b a b i l i t y l e v e l <X = . 2 0 . The n u l l h y p o t h e s i s , F ( d ) = 0 ( d ) , c a n n o t be r e j e c t e d . The m o d e l c u m u l a t i v e d i a m e t e r d i s t r i b u t i o n d = dbh F ( d ) = c u m u l a t i v e f r e q u e n c y a = f ( m e a n d i a m e t e r , s t a n d a g e , t r e e s p e r a c r e ) b = f ( m e a n d i a m e t e r , s t a n d a g e , t r e e s p e r a c r e ) c = f ( m e a n d i a m e t e r , h e i g h t , s i t e i n d e x ) s u c c e s s f u l l y d e s c r i b e s t h e MLE d i a m e t e r d i s t r i b u t i o n s o f t h e 41 p r e d i c t i o n p l o t s . T h i s d o e s n o t g u a r e n t e e t h a t t h e m o d e l i s s o u n d . When t e s t e d a g a i n s t t h e o b s e r v e d d i s t r i b u t i o n s , t h e m o d e l d i d n o t f i t f o r two p l o t s . I n o t h e r c a s e s , b i a s was o b s e r v e d t o w a r d s u n d e r e s t i m a t i o n o f t h e number o f s m a l l t r e e s . P r e v i o u s a p p l i c a t i o n s o f t h e W e i b u l l t o d i a m e t e r d i s t r i b u t i o n s a r e d i f f i c u l t t o c o m p a r e due t o t h e d i f f e r e n t s t a t i s t i c a l t e s t s and t h e c r i t e r i a f o r f i t c h o s e n . S h r e u d e r and Swank ( 1 9 7 4 ) c o m p a r e d f o u r d i s t r i b u t i o n s by l o g l i k e l i h o o d ( l n L ) s t a t i s t i c s . A l t h o u g h t h e W e i b u l l h a s l a r g e r l n L t h a n t h e o t h e r m o d e l s f o r s i x o f s e v e n s a m p l e s a l l o f t h e l n L a r e e x t r e m e l y s m a l l ( l n L — e " ^ ^ ) . No c r i t e r i a o f f i t was g i v e n by t h e a u t h o r s t o t e s t t h e s t r e n g t h o f t h e W e i b u l l a s a m o d e l f o r d i a m e t e r d i s t r i b u t i o n s . C l u t t e r and c F ( d ) = 1 . 0 - e x p i - H5 B e l c h e r (1978) g a v e t h e c o e f f i c i e n t s o f d e t e r m i n a t i o n f o r t h e i r p r e d i c t i o n e q u a t i o n s ( a = f ( a g e , h e i g h t ) , r = .107; b = f ( a g e , h e i g h t , t r e e s p e r a c r e ) , 2 2 r = .357; c = f ( a g e ) , r = .200). T h e y c o m p a r e d p r e d i c t e d and o b s e r v e d mean d i a m e t e r and b a s a l a r e a p e r a c r e . T h i s c h o i c e o f t e s t shows a c o n c e r n f o r a v e r a g e s t a n d d e s c r i p t o r s r a t h e r t h a n an i n t e r e s t i n d i s t r i b u t i o n a l q u a l i t i e s . B a i l e y (1973) p r e d i c t e d t h e p e r c e n t i l e s o f t h e two p a r a m e t e r W e i b u l l f r o m a g e , h e i g h t , and t r e e s p e r a c r e ( r > .95) and t h e n o b t a i n e d t h e W e i b u l l p a r a m e t e r s f r o m t h e p e r c e n t i l e s . S i x t y f i v e p e r c e n t o f t h e p r e d i c t e d d i s t r i b u t i o n s f i t t h e o b s e r v e d d i a m e t e r d i s t r i b u t i o n s a t t h e c< = .05 l e v e l ( K - S ) . The l e v e l o f s i g n i f i c a n c e c h o s e n by B a i l e y i s n o t a s e x a c t i n g a s t h e one u s e d i n t h e p r e s e n t s t u d y ( d Q^(N) = 1.36/NTIT ; d 20^) = 1«07//~N). A l t h o u g h t h e r e s u l t s o f t h i s s t u d y a r e n o t d i r e c t l y c o m p a r a b l e w i t h t h o s e o f o t h e r a u t h o r s , t h e f i t s d e m o n s t r a t e d h e r e f o r m i x e d s p e c i e s s t a n d s a p p e a r as good i f n o t b e t t e r t h a n t h o s e f o u n d f o r p l a n t a t i o n s o f p i n e . The p r i m a r y a p p l i c a t i o n f o r t h e W e i b u l l d i a m e t e r d i s t r i b u t i o n m o d e l w i l l be i n c o m p u t e r s i m u l a t i o n o f f o r e s t s t a n d s . I t i s n o t n e c e s s a r y f o r t h e s i m u l a t o r t o " g r o w " i n d i v i d u a l t r e e s i n o r d e r t o m a i n t a i n d i s t r i b u t i o n a l i n f o r m a t i o n a t e a c h t i m e i n t e r v a l . The m o d e l p r e s e n t e d h e r e t r e a t s d i a m e t e r d i s t r i b u t i o n a s i n d e p e n d e n t o f s t a n d h i s t o r y . I f t h e s i m u l a t o r p r e d i c t s t r e e s p e r a c r e , mean d i a m e t e r , and mean h e i g h t f o r a s t a n d g i v e n s i t e i n d e x , t h e d i a m e t e r d i s t r i b u t i o n o f t h e s t a n d c a n be g e n e r a t e d a t a n y a g e . T h i s g r e a t l y r e d u c e s t h e c o m p u t a t i o n t i m e and s t o r a g e r e q u i r e m e n t s o f t h e s i m u l a t o r . HQ> The m o d e l d e v e l o p e d i n t h i s t h e s i s d e s c r i b e s t h e d i a m e t e r d i s t r i b u t i o n o f t h e e n t i r e s t a n d . M o d e l i n g t h e d i a m e t e r d i s t r i b u t i o n o f i n d i v i d u a l s p e c i e s w i t h i n a m i x e d s p e c i e s s t a n d may e l i m i n a t e t h e b i a s e v i d e n t i n t h e m o d e l p r e s e n t e d i n t h i s t h e s i s . Mean d i a m e t e r , h e i g h t , and t r e e s p e r a c r e may be a l l t h a t i s n e e d e d t o p r e d i c t i n d i v i d u a l d i s t r i b u t i o n s . H o w e v e r , i t may p r o v e n e c e s s a r y t o t r a c k t h e p a r a m e t e r s o f t h e W e i b u l l o v e r t i m e f o r e a c h s p e c i e s i n t h e s t a n d . T h e s e q u e s t i o n s s h o u l d be a n s w e r e d t h r o u g h f u t u r e r e s e a r c h . The r e s u l t s r e p o r t e d h e r e a r e b a s e d on d a t a f r o m u n t r e a t e d s e c o n d g r o w t h s t a n d s . S i l v i c u l t u r a l t r e a t m e n t i s i n t e n d e d t o h a v e a p o s i t i v e i m p a c t on t h e d i a m e t e r d i s t r i b u t i o n o f t h e s t a n d . F e r t i l i z a t i o n may i n c r e a s e d i a m e t e r g r o w t h t h r o u g h an i n c r e a s e i n s i t e q u a l i t y . T h i n n i n g d i r e c t l y a l t e r s t h e d i s t r i b u t i o n t h r o u g h s e l e c t i v e r e m o v a l o f t r e e s . F e r t i l i z a t i o n may i n d u c e a s h i f t i n t h e p o p u l a t i o n mean d i a m e t e r ; t h i n n i n g w i l l t e n d t o skew t h e d i a m e t e r d i s t r i b u t i o n . The W e i b u l l f u n c t i o n i s f l e x i b l e e n o u g h t o h a n d l e s u c h v a r i a t i o n . W h e t h e r o r n o t t h e m o d e l p r e s e n t e d h e r e c a n r e f l e c t t h e c h a n g e s i n d u c e d t h r o u g h s i l v i c u l t u r e , e i t h e r by c h a n g i n g i n p u t v a l u e s ( s i t e i n d e x , mean d i a m e t e r , t r e e s p e r a c r e ) , o r by c a l i b r a t i n g t h e c o e f f i c i e n t s w i l l h a v e t o be a n s w e r e d by f u r t h e r r e s e a r c h . HI The d i a m e t e r d i s t r i b u t i o n m o d e l may be u s e f u l i n i n v e n t o r y u p d a t i n g . T h r o u g h a e r i a l p h o t o g r a p h i n t e r p r e t a t i o n , mean h e i g h t , mean d i a m e t e r , and t r e e s p e r a c r e c a n be e s t i m a t e d . S i t e i n d e x and age may be o b t a i n e d f r o m p a s t r e c o r d s . W i t h t h e s e d a t a , d i a m e t e r d i s t r i b u t i o n s c a n be p r e d i c t e d f o r e a c h s t a n d . T h i s w i l l s p e e d t h e p r o c e s s o f i n v e n t o r y u p d a t i n g and r e d u c e t h e number o f c o s t l y f i e l d p l o t s n e e d e d . The W e i b u l l f u n c t i o n c a n m o d e l t h e d i a m e t e r d i s t r i b u t i o n s o f m i x e d s t a n d s o f D o u g l a s - f i r and w e s t e r n h e m l o c k . I t s f u t u r e w i l l d e p e n d on t h e d e v e l o p m e n t o f m o d e l s w h i c h a c c u r a t e l y p r e d i c t s m a l l d i a m e t e r c l a s s e s and w h i c h p r e d i c t s e p a r a t e d i s t r i b u t i o n s f o r i n d i v i d u a l s p e c i e s g r o u p s w i t h i n a m i x e d s t a n d . HQ LITERATURE CITED B a i l e y , R o b e r t L . , 1 9 7 3 . W e i b u l l m o d e l f o r P i n u s r a d i a t a d i a m e t e r d i s t r i b u t i o n . I n S t a t i s t i c s i n F o r e s t r y R e s e a r c h , I . U . F . R . O . P r o c . s u b g r o u p S 6 . 0 2 ; V a n c o u v e r , B . C . p p . 5 1 - 1 9 . B a i l e y , R o b e r t L . and T . R. D e l l , 1 9 7 3 . Q u a n t i f y i n g d i a m e t e r d i s t r i b u t i o n s w i t h t h e W e i b u l l f u n c t i o n . F o r e s t S c i e n c e 1 9 : 9 7 - 1 0 4 . B a i n , L e e J . and C h a r l e s E. A n t l e , 1 9 6 7 . E s t i m a t i o n o f p a r a m e t e r s i n t h e W e i b u l l d i s t r i b u t i o n . T e c h n o m e t r i c s 9 : 6 2 1 - 6 2 7 . B l i s s , C . I . and K. A . R e i n k e r , 1 9 6 4 . A l o g n o r m a l a p p r o a c h t o d i a m e t e r d i s t r i b u t i o n s i n e v e n - a g e d s t a n d s . F o r e s t S c i e n c e 1 0 : 3 5 0 - 3 6 0 . B o g d a n o f f , D a v i d A . and D o n a l d A . P i e r c e , 1 9 7 3 . B a y e s f i d u c i a l i n f e r e n c e f o r t h e W e i b u l l d i s t r i b u t i o n . J o u r n a l o f A m e r i c a n S t a t i s t i c a l A s s o c i a t i o n 6 8 : 6 5 9 - 6 6 4 . C l u t t e r , J e r o m e L . and B . J . A l l i s o n , 1 9 7 3 . A g r o w t h and y i e l d m o d e l f o r P i n u s r a d i a t a i n New Z e a l a n d . I n G r o w t h M o d e l s f o r T r e e and S t a n d S i m u l a t i o n . P r o c . I . U . F . R . O . w o r k i n g p a r t y S 4 , 0 1 - 4 ; E d . J . F r i e s . I n s t S k o g s p r o d . , S t o c k h o l m . R a p p , U p p s a t s e r N o . 3 0 , p p . 1 3 6 - 1 6 0 . 19 C l u t t e r , J e r o m e L . and D a v i d M . B e l c h e r , 1 9 7 8 . Y i e l d o f s i t e p r e p a r e d s l a s h p i n e p l a n t a t i o n s i n t h e l o w e r c o a s e p l a i n o f G e o r g i a and F l o r i d a . I n G r o w t h M o d e l s f o r L o n g Term F o r c a s t i n g o f T i m b e r Y i e l d s , E d . J o r a n F r i e s , e t a l . P u b l . F W S - 1 - 7 8 S c h o o l o f F o r e s t and W i l d l i f e R e s o u r c e s , VPI and S t a t e U n i v . ; p p . 5 3 - 7 0 . C o h e n , C l i f f o r d A . , 1 9 6 5 . Maximum l i k e l i h o o d e s t i m a t i o n i n t h e W e i b u l l d i s t r i b u t i o n b a s e d on c o m p l e t e and on c e n s o r e d s a m p l e s . T e c h n o m e t r i c s 1 3 : 1 7 1 - 1 8 2 . D ' A g o s t i n o , R a l p h B . , 1 9 7 1 . L i n e a r e s t i m a t i o n o f t h e W e i b u l l p a r a m e t e r s . T e c h n o m e t r i c s 1 3 : 1 7 1 - 1 8 2 . D e p t a , D a v i d J . , 1 9 7 4 . L a r g e i n - p l a c e i n v e n t o r i e s b a s e d upon s t a n d t a b l e s I n I n v e n t o r y D e s i g n and A n a l y s i s , E d . W. E. F r a y e r , G . B . H a r t m a n , D. R. B o w e r . P r o c . W o r k s h o p o f I n v e n t o r y W o r k i n g G r o u p , S . A . F . p p . 2 7 5 - 2 8 8 . D o w n t o n , F . , 1 9 6 5 . L i n e a r e s t i m a t e s o f p a r a m e t e r s i n t h e e x t r e m e v a l u e d i s t r i b u t i o n . T e c h n o m e t r i c s 8 : 3 - 1 7 . D u b e y , S a t y a D . , 1 9 6 7 . Some p e r c e n t i l e e s t i m a t e s f o r W e i b u l l p a r a m e t e r s . T e c h n o m e t r i c s 9 : 1 1 9 - 1 2 9 . E k , A . R . , J . N . I s s o s and R. L . B a i l e y , 1 9 7 5 . S o l v i n g f o r W e i b u l l d i a m e t e r d i s t r i b u t i o n p a r a m e t e r s t o o b t a i n s p e c i f i e d mean d i a m e t e r s . F o r e s t S c i e n c e 2 1 : 2 9 0 - 2 9 2 . 50 E n g l e h a r d t , Max and L e e J . B a i n , 1 9 7 4 . Some r e s u l t s on p o i n t e s t i m a t i o n f o r t h e two p a r a m e t e r W e i b u l l o r e x t r e m e v a l u e d i s t r i b u t i o n . T e c h n o m e t r i c s 1 6 : 4 9 - 5 6 . E n g l e h a r d t , Max and L e e J . B a i n , 1 9 7 7 . S i m p l i f i e d s t a t i s t i c a l p r o c e d u r e s f o r t h e W e i b u l l o r e x t r e m e v a l u e d i s t r i b u t i o n . T e c h n o m e t r i c s 1 9 : 3 2 3 - 3 3 1 . F i n k e l s t e i n , J . M . , 1 9 7 6 . C o n f i d e n c e b o u n d s on t h e p a r a m e t e r s o f t h e W e i b u l l p r o c e s s . T e c h n o m e t r i c s 1 8 : 1 1 5 - 1 1 7 . F r e u n d , J o h n E . , 1 9 7 1 . M a t h e m a t i c a l S t a t i s t i c s , P r e n t i c e H a l l , I n c . E n g l e w o o d C l i f f s , N . J . p p . 125+ H a f l e y , W. L . and H . T . S c h r e u d e r , 1977. S t a t i s t i c a l d i s t r i b u t i o n s f o r f i t t i n g d i a m e t e r and h e i g h t d a t a i n e v e n - a g e d s t a n d s . C a n a d i a n J o u r n a l o f F o r e s t R e s e a r c h 7 : 4 8 1 - 4 8 7 . H a r t e r , H . L e o n and A l b e r t H. M o o r e , 1 9 6 5 . Maximum l i k e l i h o o d e s t i m a t i o n o f t h e p a r a m e t e r s o f Gamma and W e i b u l l p o p u l a t i o n s f r o m c o m p l e t e and f r o m c e n s o r e d s a m p l e s . T e c h n o m e t r i c s 7 : 6 3 9 - 6 4 3 . J o h n s , M . V . and G. J . L i e b e r m a n , 1 9 6 6 . An e x a c t a s y m p t o t i c a l e f f i c i e n t c o n f i d e n c e b o u n d f o r r e l i a b i l i t y i n t h e c a s e o f t h e W e i b u l l d i s t r i b u t i o n . T e c h n o m e t r i c s 8 : 1 3 5 - 1 7 5 . 51 K a o , J . H . K . , 1 9 5 9 . A g r a p h i c a l e s t i m a t i o n o f m i x e d W e i b u l l p a r a m e t e r s i n l i f e t e s t i n g o f e l e c t r o n t u b e s . T e c h n o m e t r i c s 1 : 3 8 9 - 4 0 7 . L a w l e s s , J . F . , 1 9 7 5 . C o n s t r u c t i o n o f t o l e r a n c e b o u n d s f o r t h e e x t r e m e - v a l u e and W e i b u l l d i s t r i b u t i o n s . T e c h n o m e t r i c s 1 7 : 2 5 5 - 2 6 1 . L e a k , W i l l i a m B . , 1 9 6 5 . The J - s h a p e d p r o b a b i l i t y d i s t r i b u t i o n . F o r e s t S c i e n c e 1 1 : 4 0 5 - 4 1 1 . L e a r y , R o l p h A . , J e r o l d T . H a h n , and R o l a n d G . B u c h m a n , 1 9 7 7 . A g e n e r a l i z e d f o r e s t g r o w t h p r o j e c t i o n s y s t e m - p r e l i m i n a r y r e s e a r c h f i n d i n g s . P h a s e I o f F o r e s t R e s o u r c e s E v a l u a t i o n P r o g r a m ( F R E P ) . M a n u s c r i p t , N o r t h C e n t r a l F o r e s t E x p e r i m e n t S t a t i o n , USDA ( 3 0 0 p p . ) L i l l i e f o r s , H u b e r t W . , 1 9 6 7 . On t h e K o l m o g o r o v - S m i r n o v t e s t f o r n o r m a l i t y w i t h mean and v a r i a n c e u n k n o w n . A m e r i c a n S t a t i s t i c a l A s s o c . J o u r n a l 6 2 : 3 9 9 - 4 0 2 . M a n n , N a n c y R . , 1 9 6 8 . P o i n t and i n t e r v a l e s t i m a t i o n p r o c e d u r e s f o r t h e t w o - p a r a m e t e r W e i b u l l a n d e x t r e m e - v a l u e d i s t r i b u t i o n s . T e c h n o m e t r i c s 1 0 ( 2 ) : 2 3 1 - 2 5 6 . M a n n , N a n c y R . , S c h a f e r , R a y E . , and S i n g p u r w a l l a , N o z e r D . , 1 9 7 4 . M e t h o d s o f S t a t i s t i c a l A n a l y s i s f o r L i f e T e s t i n g . J o h n W i l e y , New Y o r k . M a s s e y , F r a n k J . , J r . , 1 9 5 1 . The K o l m o g o r o v - S m i r n o v t e s t f o r g o o d n e s s o f f i t . A m e r i c a n S t a t i s t i c a l A s s o c . J o u r n a l 4 6 : 6 8 - 7 8 . 5 2 M e y e r , W a l t e r H . , 1 9 3 0 . D i a m e t e r d i s t r i b u t i o n s e r i e s i n e v e n - a g e d f o r e s t s t a n d s . Y a l e U n i v . S c h . F o r . B u l l . 2 8 , 105 p p . M i t c h e l l , K e n n e t h J . , 1 9 7 5 . D y n a m i c s and s i m u l a t e d y i e l d o f D o u g l a s - f i r . F o r e s t S c i e n c e M o n o g r a p o h N o . 17, 39 p p . N e l s o n , Thomas C , 1 9 6 4 . D i a m e t e r d i s t r i b u t i o n and g r o w t h o f l o b l o l l y p i n e . F o r e s t S c i e n c e 1 0 : 1 0 5 - 1 1 4 . P r o d a n , M i c h a i l , 1 9 6 8 . F o r e s t B i o m e t r i c s . P e r g a m o n P r e s s . N . Y . pp 3 9 - 5 6 . R u s t a g i , K r i s h n a P . , 1 9 7 8 . P r e d i c t i n g s t a n d s t r u c t u r e i n e v e n - a g e d s t a n d s . I n G r o w t h M o d e l s f o r L o n g Term F o r e c a s t i n g o f T i m b e r Y i e l d s , E d . J o r a n F r i e s , e t a j . P u b l . F W S - 1 - 7 8 . S c h o o l o f F o r e s t and W i l d l i f e R e s o u r c e s , VPI and S t a t e U n i v . ; p p . 1 9 3 - 2 0 8 . S c h m i d t , J . W. and R. E. T a y l o r , 1 9 7 0 . S i m u l a t i o n and A n a l y s i s o f I n d u s t r i a l S y s t e m s . R i c h a r d D. I r w i n , I n c . , Homewood, 1 1 1 . 644 p p . S c h r e u d e r , Hans T . and W. L . H a f l e y , 1 9 7 7 . A u s e f u l b i v a r i a t e d i s t r i b u t i o n f o r d e s c r i b i n g s t a n d s t r u c t u r e o f t r e e h e i g h t s and d i a m e t e r s . B i o m e t r i c s 3 3 : 4 7 1 - 4 7 8 . S c h r e u d e r , Hans T . and Wayne T . S w a n k , 1 9 7 4 . C o n i f e r o u s s t a n d s c h a r a c t e r i z e d w i t h t h e W e i b u l l d i s t r i b u t i o n . C a n a d i a n J o u r n a l o f F o r e s t R e s e a r c h 4 : 5 1 8 - 5 2 3 . 53 S e e g r i s t , D o n a l d W. and S t a n f o r d L . A r n e r , 1 9 7 8 . S t a t i s t i c a l a n a l y s i s o f l i n e a r g r o w t h and y i e l d m o d e l s w i t h c o r r e l a t e d o b s e r v a t i o n s f r o m p e r m a n e n t p l o t s r e m e a s u r e d a t f i x e d i n t e r v a l s . I n G r o w t h M o d e l s f o r L o n g Term F o r e c a s t i n g o f T i m b e r Y i e l d s . E d . J o r a n F r i e s , e t j H . Pub 1 . F W S - 1 - 7 8 S c h o o l o f F o r e s t and W i l d l i f e R e s o u r c e s , V P I ; p p . 2 0 9 - 2 2 3 . S n e e , R o n a l d D . , 1 9 7 7 . V a l i d a t i o n o f r e g r e s s i o n m o d e l s : m e t h o d s and e x a m p l e s . T e c h n o m e t r i c s 1 9 : 4 1 5 - 4 2 8 . S t a u f f e r , 1977. A d e r i v a t i o n f o r t h e W e i b u l l D i s t r i b u t i o n . U n i v . o f B . C . , F a c u l t y o f F o r e s t r y . U n p u b l i s h e d m a n u s c r i p t . T h o m a n , D . R . , L . J . B a i n and C . E . A n t l e , 1 9 6 9 . I n f e r e n c e s on t h e p a r a m e t e r s o f t h e W e i b u l l d i s t r i b u t i o n . T e c h n o m e t r i c s 1 1 : 4 4 5 - 4 6 0 . T h o m a s , D a v i d R. and Wanda M . W i l s o n , 1 9 7 2 . L i n e a r o r d e r s t a t i s t i c e s t i m a t i o n f o r t h e t w o - p a r a m e t e r W e i b u l l and e x t r e m e - v a l u e d i s t r i b u t i o n s f r o m T y p e I I p r o g r e s s i v e l y c e n s o r e d s a m p l e s . T e c h n o m e t r i c s 1 4 : 6 7 9 - 6 9 2 . U n i v e r s i t y o f W a s h i n g t o n , C o l l e g e o f F o r e s t R e s o u r c e s , 1 9 7 6 . R e g i o n a l F o r e s t N u t r i t i o n R e s e a r c h P r o j e c t . B i e n n i a l R e p o r t 1 9 7 4 - 1 9 7 6 . 67 p p . W a r r e n , W i l l i a m G . , 1 9 7 6 . Maximum l i k e l i h o o d e s t i m a t i o n o f t h e t h r e e - p a r a m e t e r W e i b u l l . F o r i n t e k , V a n c o u v e r , B . C . , u n p u b l i s h e d . W e i b u l l W . , 1 9 5 5 . A s t a t i s t i c a l d i s t r i b u t i o n f u n c t i o n o f w i d e a p p l i c a b i l i t y . J o u r n a l o f A p p l i e d M e c h a n i c s 1 8 : 2 9 3 - 2 9 7 . 5H Y a n g , R. C , K o z a k , A . and J . H . G . S m i t h , 1 9 7 8 . The p o t e n t i a l o f W e i b u l 1 - t y p e f u n c t i o n s a s f l e x i b l e g r o w t h c u r v e s . C a n a d i a n J o u r n a l o f F o r e s t R e s e a r c h 8 : 4 2 4 - 4 3 1 . APPENDIX I The s t r e n g t h o f t h e r e g r e s s i o n m o d e l s d e p e n d e d on t h e a c c u r a c y o f t h e i n p u t s . I t was n e c e s s a r y , t h e r e f o r e , t o e s t i m a t e t h e W e i b u l l p a r a m e t e r s o f t h e i n d i v i d u a l p l o t d i a m e t e r d i s t r i b u t i o n s f r o m t h e t r e e d i a m e t e r d a t a as a c c u r a t e l y as p o s s i b l e . The p a r a m e t e r e s t i m a t i o n a l g o r i t h m s d e v e l o p e d i n t h e l i f e - t e s t i n g r e s e a r c h w e r e n o t c o n s i d e r e d , as t h e y w e r e d e v e l o p e d f o r h e a v i l y c e n s o r e d s a m p l e s , and i n m o s t c a s e s w e r e t i m e c o n s u m i n g and i n a d e q u a t e when u s i n g c o m p l e t e s a m p l e s o f o n e h u n d r e d o r more o b s e r v a t i o n s . ( F o r a m o r e t h o r o u g h d i s c u s s i o n , s e e M a n n , et a j _ . , 1 9 7 4 . ) The E n g l e h a r d t and B a i n ( 1 9 7 7 ) a l g o r i t h m f o r s i m p l i f i e d p o i n t e s t i m a t e s was a b a n d o n e d b e c a u s e t h e d e b i a s i n g c o n s t a n t , k , w o u l d h a v e t o be c a l c u l a t e d f o r e a c h s a m p l e , t h u s r e n d e r i n g t h e t e c h n i q u e more c u m b e r s o m e t h a n t h e m o r e a c c u r a t e maximum l i k e l i h o o d a l g o r i t h m s . T h r e e e s t i m a t i o n a l g o r i t h m s , D u b e y ' s ( 1 9 6 7 ) p e r c e n t i l e m e t h o d , B a i l e y ' s ( 1 9 7 3 ) maximum l i k e l i h o o d e s t i m a t o r , and W a r r e n ' s ( 1 9 7 7 ) maximum l i k e l i h o o d m e t h o d w e r e c o n s i d e r e d f o r t h e e s t i m a t i o n o f t h e d i a m e t e r d i s t r i b u t i o n s . T h e s e r o u t i n e s , DUBEY ( D u b e y ) , FITTER ( B a i l e y ) , and WINWAR ( W a r r e n ) w e r e t e s t e d and c o m p a r e d i n t h e f o l l o w i n g m a n n e r : U s i n g a r a n d o m number g e n e r a t o r , s a m p l e s w e r e t a k e n f r o m c u m u l a t i v e W e i b u l l d i s t r i b u t i o n s h a v i n g b and c p a r a m e t e r s w i t h i n t h e r a n g e o f t h o s e e x p e c t e d f o r t h e d a t a s e t . A n u m b e r , y ( 0 ^ y < l ) , c o r r e s p o n d i n g i n t h i s c a s e t o a c u m u l a t i v e f r e q u e n c y , was r a n d o m l y s e l e c t e d f r o m a u n i f o r m 5.6 d i s t r i b u t i o n . From t h e W e i b u l l f o r m u l a i t f o l l o w s t h a t x = bC-lnd-y))^. • G i v e n y , b , and c , t h e d i a m e t e r , x , was c a l c u l a t e d . T h i s was r e p e a t e d u n t i l a s a m p l e o f 100 d i a m e t e r s was o b t a i n e d . ( S e e F r e u n d ( 1 9 7 1 ) f o r more d e t a i l . ) The s a m p l e was t h e n r u n t h r o u g h e a c h o f t h e t h r e e - p a r a m e t e r e s t i m a t i n g r o u t i n e s , y i e l d i n g t h r e e e s t i m a t e d d i a m e t e r d i s t r i b u t i o n s . The K o l m o g o r o v - S m i r n o v t e s t f o r g o o d n e s s o f f i t was u s e d t o c o m p a r e t h e s e d i s t r i b u t i o n s w i t h t h e o r i g i n a l d i s t r i b u t i o n . C r i t i c a l v a l u e s , d ^ N ) w e r e g i v e n b y M a s s e y 1 9 5 1 ) s u c h t h a t P r j ™ * x | s N ( x ) - F o ( x ) | > d ( N ) j = oc w h e r e F ( X ) i s t h e t h e o r e t i c a l c u m u l a t i v e d i s t r i b u t i o n o S ^ ( X ) i s an o b s e r v e d c u m u l a t i v e d i s t r i b u t i o n f o r a s a m p l e o f N. The d i f f e r e n c e , d = | S N ( X ) - F (X)( was c a l c u l a t e d f o r t w e n t y o b s e r v a t i o n s . The a v e r a g e maximum d , d , was c a l c u l i a t e d f o r t e n s a m p l e s f r o m e a c h d i s t r i b u t i o n . The d and c o r r e s p o n d i n g s t a n d a r d d e v i a t i o n s , s ^ , a r e l i s t e d i n A p p e n d i x I I . A c c o r d i n g t o C h e b y s h e v ' s t h e o r e m : P R ( ^ x - k e r < x < v i + ko-) ^ 1 - 1 / k 2 - I f k = 1 / 7 * , P r (x < )X +(r/fai., I n t h i s c a s e , jx and cr w e r e e s t i m a t e d by d and s ^ , r e s p e c t i v e l y , so t h a t P r ( d < d + Sj -Ala) > 1 - <*• I f f o r ( l - a ) % o f t h e s a m p l e s , d < d a ( N ) , t h e e s t i m a t e d d i s t r i b u t i o n h a s a good f i t a t t h e O L l e v e l o f p r o b a b i l i t y . I n o t h e r w o r d s , i f d + s g -AR <6JH) t h e e s t i m a t e d d i s t r i b u t i o n f i t s t h e o r i g i n a l d i s t r u b u t i o n a t t h e l e v e l . A c c o r d i n g t o t h e r e s u l t s i n A p p e n d i x I I , a l l o f t h e c u r v e s e s t i m a t e d by FITTER f i t a t t h e o c = 0 . 0 1 l e v e l , ( d 0 1 ( 2 0 ) = 0 . 3 5 6 ) . WINWAR e s t i m a t e s f o r d i s t r i b u t i o n s w h e r e b = 10 w i t h 3 . 5 < c 5 . 0 f i t a t t h e oc= . 0 5 l e v e l . The r e m a i n i n g WINWAR e s t i m a t e s f i t a t t h e ot= 0 . 0 1 l e v e l . T h e DUBEY e s t i m a t e s f o r (b = 1 0 , 3 . 5 < c < 5 . 0 ) d i d n o t f i t a t the<X= 0 . 2 0 l e v e l o f p r o b a b i l i t y . The r e m a i n d e r f i t a t t h e 0 . 0 5 l e v e l . B a i l e y ' s FITTER r o u t i n e was c h o s e n f o r t h i s s t u d y b e c a u s e i t g a v e t h e b e s t o v e r a l l f i t f o r d i s t r i b u t i o n s w i t h i n t h e e x p e c t e d d a t a r a n g e , a n d , u n l i k e WINWAR, i t i s an u n b i a s e d e s t i m a t o r . APPENDIX H i Comparison of Dubey, Warren (WINWAR), and Bailey (FITTER) estimates for b and c over a range of true values for b and c. with a - 1.0. For each combination of b and c, each estimator used the same random sample (N - 100) from the given population. Twenty observations from the estimated curve were used to calculate d. The Kolmogoror-Smirnov s t a t i s t i c , d, was averaged over ten estimations. 10.0 2.0 2.5 3.0 3.5 4.0 4.5 5.0 8 d 8 d a d s d Sd 8 d 8 d DUBEY .0806 WINWAR .0334 FITTER .0335 .0337 .0232 .0229 .0906 .0330 .0333 .0392 .0233 .0230 .1034 .0330 .0333 .0424 .0234 .0230 .1179 .0394 .0324 .0440 .0324 .0231 .1334 .0390 .0324 .0450 .0319 .0231 .1497 .0457 .0330 .0458 .0340 .2300 .1664 .0451 .0327 .0456 .0336 .0229 b - 20.0 DUBEY .0640 WINWAR .0345 FITTER .0351 .0290 .0244 .0242 .0666 .0339 .0346 .0337 .0234 .0239 .0709 .0374 .0363 .0379 .0305 .0268 .0753 .0373 .0363 .0419 .0297 .0267 .0813 .0385 .0365 .0446 .0280 .0268 .0877 .0381 .0363 .0476 .0284 .0270 .0943 .0384 .0363 .0504 .0281 .0271 b - 30.0 DUBEY .0410 WINWAR .0381 FITTER .0377 .0296 .0278 .0270 .0418 .0377 .0379 .0327 .0270 .0264 .0435 .0388 .0372 .0353 .0279 .0267 .0462 .0317 .0369 .0377 .0287 .0263 .0498 .0378 .0368 .0393 .0298 .0258 .0545 .0383 .0372 .0396 .0306 .0258 .0597 .0388 .0368 .0402 .0299 .0262 b » 40.0 DUBEY .0432 WINWAR .0386 FITTER .0371 .0275 .0228 .0236 .0430 .0376 .0363 .0259 .0238 .0241 .0436 .0377 .0362 .0248 .0242 .0243 .0477 .0341 .0327 .0237 .0236 .0241 .0460 .0350 .0329 .0234 .0236 .0212 .0439 .0384 .0369 .0366 .0303 .0261 .0461 .0386 .0371 .0384 .0300 .0261 03> APPENDIX I I I G r a p h i c a l i l l u s t r a t i o n o f t h e o b s e r v e d , e s t i m a t e d , and p r e d i c t e d d i a m e t e r d i s t r i b u t i o n s o f some t y p i c a l s t a n d s i n t h e p r e d i c t i o n s e t . F i g u r e 1. O b s e r v e d , e s t i m a t e d , a n d p r e d i c t e d d i a m e t e r d i s t r i b u t i o n s o f s t a n d noo4 O • 5 . LO I S S O D i a m e t e r ( i n c h e s ) F i g u r e 2. O b s e r v e d , e s t i m a t e d , a n d p r e d i c t e d d i a m e t e r d i s t r i b u t i o n s o f s t a n d 71041 D i a m e t e r ( i n c h e s ) 1 / e s t i m a t e d f r o m o b s e r v e d d i s t r i b u t i o n by maximum l i k e l i h o o d 2 / p r e d i c t e d f r o m s t a n d c h a r a c t e r i s t i c s b y m o d e l a ( 2 ) b ( 1 0 ) c ( 4 ) F i g u r e 3 . O b s e r v e d , e s t i m a t e d , a n d p r e d i c t e d d i a m e t e r d i s t r i b u t i o n s o f s t a n d 161091 D i a m e t e r ( i n c h e s ) F i g u r e 4 . O b s e r v e d , e s t i m a t e d , a n d p r e d i c t e d d i a m e t e r d i s t r i b u t i o n s o f s t a n d 191114 D i a m e t e r ( i n c h e s ) 1 / e s t i m a t e d f r o m o b s e r v e d d i s t r i b u t i o n b y maximum l i k e l i h o o d 2 / p r e d i c t e d f r o m s t a n d c h a r a c t e r i s t i c s b y m o d e l a ( 2 ) b ( 1 0 ) c ( 4 ) Figure 5. Observed, estimated, and p r e d i c t e d diameter d i s t r i b u t i o n s of stand 331194 Diameter (inches) Figure 6. Observed, estimated, and p r e d i c t e d diameter d i s t r i b u t i o n s of stand 361216 Diameter (inches) 1/ estimated from observed d i s t r i b u t i o n by maximum l i k e l i h o o d 2/ p r e d i c t e d from stand c h a r a c t e r i s t i c s by model a(2) b(10) c(4) F i g u r e 7. O b s e r v e d , e s t i m a t e d , a n d p r e d i c t e d d i a m e t e r d i s t r i b u t i o n s o f s t a n d 431254 O S 1 CD I S 2 0 2 5 D i a m e t e r ( i n c h e s ) F i g u r e 8. O b s e r v e d , e s t i m a t e d , a n d p r e d i c t e d d i a m e t e r d i s t r i b u t i o n s o f s t a n d 451266 o s L O i s 20 D i a m e t e r ( i n c h e s ) 1/ e s t i m a t e d f r o m o b s e r v e d d i s t r i b u t i o n b y maximum l i k e l i h o o d dj p r e d i c t e d f r o m s t a n d c h a r a c t e r i s t i c s b y m o d e l a ( 2 ) b ( 1 0 ) c ( 4 ) F i g u r e 9. O b s e r v e d , e s t i m a t e d , a n d p r e d i c t e d d i a m e t e r d i s t r i b u t i o n s o f s t a n d 541324 D i a m e t e r ( i n c h e s ) F i g u r e 10. O b s e r v e d , e s t i m a t e d , a n d p r e d i c t e d d i a m e t e r d i s t r i b u t i o n s o f s t a n d 541324 i . o r — D i a m e t e r ( i n c h e s ) 1/ e s t i m a t e d f r o m o b s e r v e d d i s t r i b u t i o n b y maximum l i k e l i h o o d 2 / p r e d i c t e d f r o m s t a n d c h a r a c t e r i s t i c s b y m o d e l a ( 2 ) b ( 1 0 ) c ( 4 ) <o5 Figure 11. Observed, estimated, and p r e d i c t e d diameter d i s t r i b u t i o n s of stand 551330 Diameter (inches) Figure 12. Observed, estimated, and p r e d i c t e d diameter d i s t r i b u t i o n s of stand 591354 i . O i — o c CT 0) > o . e J— (D.S. (— o . o 1 o i s Diameter (inches) 2 0 1/ estimated from observed d i s t r i b u t i o n by ^ i ^ . J i ^ ^ o o d 1/ p r e d i c t e d from stand c h a r a c t e r i s t i c s by model a(2) b(10). c(4} &6 F i g u r e 13. O b s e r v e d , e s t i m a t e d , a n d p r e d i c t e d d i a m e t e r d i s t r i b u t i o n s o f s t a n d 691410 D i a m e t e r ( i n c h e s ) F i g u r e 14. O b s e r v e d , e s t i m a t e d , a n d p r e d i c t e d d i a m e t e r d i s t r i b u t i o n s o f s t a n d 791472 >- o o . e QJ CT <D > ro O . •<* 3 0.2 O . O o b s e r v e d , , e s t i m a t e d ^ p r e d i c t e d - • • • i i i — i — — i — i i o i s D i a m e t e r ( i n c h e s ) 1 / e s t i m a t e d f r o m o b s e r v e d d i s t r i b u t i o n by " " i 1 ^ . " 1 * } . 1 ! ! ? 0 ^ - , il p r e d i c t e d f r o m s t a n d c h a r a c t e r i s t i c s b y m o d e l a(2) b(10) d4) 67 F i g u r e 15. O b s e r v e d , e s t i m a t e d , a n d p r e d i c t e d d i a m e t e r d i s t r i b u t i o n s o f s t a n d 821492 O S . 1 O 15 20 25 D i a m e t e r ( i n c h e s ) F i g u r e 16 . O b s e r v e d , e s t i m a t e d , a n d p r e d i c t e d d i a m e t e r d i s t r i b u t i o n s o f s t a n d 831498 D i a m e t e r ( i n c h e s ) 1/ e s t i m a t e d f r o m o b s e r v e d d i s t r i b u t i o n by maximum l i k e l i h o o d J/ p r e d i c t e d f r o m s t a n d c h a r a c t e r i s t i c s b y m o d e l a ( 2 ) b ( 1 0 ) c ( 4 ) F i g u r e 17. O b s e r v e d , e s t i m a t e d , a n d p r e d i c t e d d i a m e t e r d i s t r i b u t i o n s o f s t a n d 991592 D i a m e t e r ( i n c h e s ) F i g u r e 18. O b s e r v e d , e s t i m a t e d , a n d p r e d i c t e d d i a m e t e r d i s t r i b u t i o n s o f s t a n d 1101658 o o . c at cr QJ S- o . u. > •r- 4-> _ ro O. O. 2 O.O I i i_ I O I S D i a m e t e r ( i n c h e s ) o b s e r v e d 1/ e s t i m a t e d ^ p r e d i c t e d — _i i i_ 2 0 1/ e s t i m a t e d f r o m o b s e r v e d d i s t r i b u t i o n by maximum l i k e l i h o o d 27 p r e d i c t e d f r o m s t a n d c h a r a c t e r i s t i c s b y m o d e l a(2) b(10) c(t)

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