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E L S E V I E R European Journal of Operational Research 93 (1996) 346-357

EUROPEANJ O U R N A LOF O PERATIONAL

RESEARCH

C o m p a r a tiv e p er fo r ma n c e o f t h e F S C L n e u ra l n e t an d K - m e a n sa lgor ithm for m arket segm entat ion

P . V . ( S u n d a r ) B a l a k r i s h n a n a , M a r t h a C . C o o p e r b , V a r g h e s e S . J a c o b c , . , P h i l l i p A . L e w i s da Bus iness Adminis tra t ion Program, Univers i ty o f Washington, Bothe l l , WA 98024, USA

b Depar tm ent o f Marke t ing , The Ohio S ta te Univers ity , Columbus , OH, USAc Depar tm ent o f Accounting a nd MIS, The Ohio S ta te Univers ity , 1775 C ollege Road, Columbus , OH 43210-1399, USAa Depar tme nt o f Marke t ing , Row an College o f New Jersey , Glassboro, N J , USA

Received 1 July 1995; revised 1 Aug ust 1995

A b s t r a c t

Given the success o f neural ne tworks in a va r ie ty o f app l ica t ions in eng inee r ing , such a s speech and im age quan tiza t ion ,i t is na tura l to consider i ts applica tion to s imilar problems in other domains. A re la ted problem that ar ises in business ismarke t segmenta t ion fo r which Clus te r ing techn iques a re used . In th i s pape r , we exp lore the ab i l i ty o f a spec i f ic neura lne twork , na me ly the Frequency-Sens i t ive Com pe t i t ive Lea rn ing A lgor i thm (FSC L) , to c lus te r da ta fo r deve lop ing s tr ategicmarke t ing dec is ions . To th is end , we invest iga te the com para t ive pe r formance o f FS CL v is -h-v is the K -means c lus te ringtechn ique . A c lus te r ana lys is conduc ted on brand cho ice da ta fo r the cof fee ca tegory r evea led tha t the two me thodolog ie sresulted in widely dif fer ing c luster solutions. In an effor t to address the dispute over the appropr ia te methodology, acom parativ e perform ance investigation was under taken using simula ted data with know n c luster solutions in a fa ir ly largeexpe r imenta l des ign to m imic va ry ing da ta qua l i ty to r e f lec t da ta co l lec t ion and m easureme nt e rro r. Based on the r e su l t s o fthese s tud ie s , i t is obse rved tha t a com bina t ion o f the two me thodo log ie s , whe re in the r e su l ts o f the FSC L ne two rk a re inpu ta s seeds to the K-m eans , seems to p rov ide more m anage r ia l ly insigh t fu l segmenta t ion schemes .Ke y wo r d s : K-means; Neural networks; Segm entation; Comparative performance; Brand choice; D ata q uality; Marketing

1 . I n t r o d u c t i o n

A c o m m o n r e s e a rc h o b j e c t iv e a c r o s s a v a r i e t y o fd i s c i p l i n e s i s t o g r o u p i t e m s t h a t a r e s i m i l a r . T h i sg r o u p i n g c a n b e o f p e o p l e f o r i d e n t if y i n g m a r k e ts e g m e n t s ; o r o f c i t i e s a n d r e g i o n s f o r s e l e c t i n g t e s tm a r k e t s i te s ; o r o f s p e c i es o f p l a n t s a n d a n i m a l s f o rc l a s si f ic a t i o n ; o r o f s p e e c h a n d s i g n al s f o r d a t a c o m -p r e s s i o n . T h e m e t h o d o l o g i e s f o r s u c h g r o u p i n g e f -f o r t s i n c l u d e m u l t i v a r i a t e a n a l y s e s a n d , m o r e r e -

* Corresponding author.

c e n f l y , n e u ra l n e t w o r k t e c h n i q u es .M a n a g e m e n t s c i e n t i s t s a r e i n t e r e s t e d i n n e u r a ln e t w o r k s p a r t i c u l a rl y f o r a n a l y z i n g n o i s y d a t a , d e a l -

i n g w i t h p r o b l e m s t h a t h a v e n o c l e a r c u t s o lu t i o nsa n d t h e i r a b i l i t y t o l e a r n . N e u r a l n e t w o r k s h a v e b e e nd e s i g n e d f o r v a r i o u s p u r p o s e s , i n c l u d i n g m a n a g e r i a lp r o b l e m s s u c h a s h a n d l i n g c e n s u s d a t a ( O p e n s h a wa n d W y m e r , 1 9 9 1) , f o r e c a s t in g (M u r t a g h , 1 9 9 1 a ),t i m e - v a r y i n g d a t a ( R o w h e r , 1 9 9 1) , b a n k r u p t c y p r e -d i c t i o n ( W i l s o n a n d S h a r d a , 1 9 9 4 ) , t h e t r a v e l i n gs a l e s m a n p r o b l e m ( F o r t , 1 9 9 1 ) , a n d g r o u p i n g o rc l u s t e r i n g ( A h a l t e t a l . , 1 9 9 0 ; G a t h a n d G e v a , 1 9 8 8 ;K r i s h n a m u r t h y e t a l ., 1 9 9 0 ; O s i p e n k o , 1 9 8 8 ).

0377-2217/96/$15.00 Copyright 1996 Elsevier Science B.V. All fights reserved.P l l S 0 3 7 7 - 2 2 1 7 ( 9 6 ) 0 0 0 4 6 - X


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P.V. Balakrishnan et al ./ European Journal of Operational Research 93 (1996) 346-357 347H o w e v e r f o r t h e s e n e w e r t e c h n o l o g ie s , s u ch a sn e u r a l n e t w o r k s , t o b e c o m e a c c e p t e d i n p r a c ti c e t h e ys h o u l d d e m o n s t r a t e a t l e a s t p a r i t y i f n o t s u p e r i o r i t y

o v e r e x i s t i n g t e c h n i q u e s . T o t h i s e n d , i n t h is p a p e rwe an a ly ze th e cap ab i l i t i e s o f a sp ec i f i c n eu r a l n e t -w o r k i m p l e m e n t a t i o n a g a i n s t a m o r e t r a d i ti o n a l te c h -n i q u e , n a m e l y K - m e a n s , f o r t h e g e n e r a l p r o b l e m o fc lu s t e r in g .A l t h o u g h s e v e r a l n e t w o r k s h a v e b e e n p r o p o s e d ,l i t t l e co mp ar i so n wi th t r ad i t io n a l t e ch n iq u es i s av a i l -ab le . Fu r th e r , d i f f e r en t ap p l i ca t io n s o f t h e same b as i ct e c h n i q u e m a y p r o d u c e d i f f e r e n t r e s u l ts l e a d i n g t o al ack o f s t ab i l i t y in th e c lu s t e r so lu t io n s a s i l l u s t r a t edi n S e c t i o n 3 . T h e r e s e a r c h e r m u s t d e t e r m i n e w h i c hd a t a r ed u c t i o n m e t h o d t o u s e f o r h i s / h e r p u r po s e s .W h i c h m e t h o d i s b e s t f o r th e d a t a c h a r a c te r i s ti c s , t h es a m p l e s i z e , t h e a m o u n t o f d a t a r e d u c t i o n n e e d e d ?S o m e o f t h e m o r e c o m m o n t e ch n i q ue s , s uc h a sf a c t o r a n a l y s is a n d c l u s t e r i n g , h a v e e x t e n s i v e r e -s e a r c h b a s e s b e h i n d t h e m t o s u g g e s t w h e r e p r o b l e m sm a y a r i s e a n d w h i c h m e t h o d s m i g h t b e b e s t f o rw h i c h d a t a c h a r a c t e ri s t ic s a n d p u r p o s e s ( e . g . , B e z d e ka n d H a t h a w a y , 1 9 9 0 ; M i l l i g a n , 1 9 8 0 ; M i l l i g a n a n dC o o p e r , 1 9 8 5 ; C o o p e r a n d M i l l i g a n , 1 9 8 8 ; D u b e s ,1987) .T h e r e c e n t i n t e r e s t in n e u r a l n e t w o r k a n a l y s is h a sr e s u l t e d i n n u m e r o u s t e c h n i q u e s s u g g e s t e d f o r m a n yd i f f e r e n t a p p l ic a t i o n s ( e . g ., F o r t , 1 9 9 1 ; H e c h t - N i e l -s e n , 1 9 9 0 ; K o h o n e n , 1 9 8 2 a , b , 1 9 8 9 ) . T h i s r e l a t i v e l yn e w m e t h o d o l o g i c a l a r e a h a s h a d l i t tl e o f t h e r i g o r-o u s t e s t in g o f t h e s e n e w l y s u g g e s t e d m e t h o d s t o h e l pt h e r e s e a r c h e r a n s w e r c o m p r e h e n s i v e l y t h e q u e s t io n sp o sed in th e p r eced in g p a r ag r ap h . A sma l l l i t e r a tu r e ,h o w e v e r , i s b e g i n n i n g t o d e v e l o p ( e .g . , B a l a k r i sh n a ne t a l. , 1 9 9 2 , 1 9 9 4 ; B e z d e k , 1 9 9 1 ; N a y l o r a n d L i ,1988) .T h i s p a p e r r e p o r t s o n t h e p e r f o r m a n c e o f o n e k i n do f n e u r a l n e t w o r k , t h e F r e q u e n c y - S e n s i t i v e C o m p e t i -t i v e l_ ~a rn in g ( F S C L ) n e t w o r k d e v e l o p e d b y K r i s h -n a m u r t h y e t a l. ( 1 9 9 0 ) f o r V e c t o r q u a n t i z a t i o n . T h i sm e t h o d i s c o m p a r e d w i t h K - m e a n s n o n - h i e r a r c h i c a lc l u st e ri n g . B o t h o f t h es e m e t h o d s a t t e m p t t o a c h i e v et h e s a m e o b j e c t i v e , n a m e l y , a s s o c ia t i n g a s e t o fo b jec t s w i th a co r r e s p o n d in g r ep r e sen ta t iv e o b jec t . I nv e c t o r q u a n t i z a t i o n , t h e g o a l i s t o c l a s s i f y v e c t o r sin to c l a sse s . I n en g in ee r in g ap p l i ca t io n s th e se v ec to r sr e p r e s e n t i n p u t f r o m s p e e c h o r i m a g e s . I n m a n a g e -m e n t a p p l i c a t io n s , s u c h a s m a r k e t s e g m e n t a t i o n , d a t a

a r e c o l l e c t e d i n t h e f o r m o f a t t r i b u t e ' s v a l u e s a n dc l u s t e r i n g t e c h n i q u e s a r e u s e d t o c l a s s i f y e a c h d a t ap o in t . I n e i th e r ca se , t h e g o a l i s t o c l a s s i f y th e in p u ti n t o o n e o f t h e p r e - s p e c i f i e d n u m b e r o f o u t p u t g ro u p s .O n e o f t h e p r o b l e m s i n a p p l y i n g t h e s e t e c h n i q u e sd i r e c t l y t o r e a l d a t a f o r m a r k e t s e g m e n t a t i o n i s t h a tt h e r e s u l t s o b t a i n e d f r o m t h e t w o t e c h n i q u e s d o n o tt y p i c a l l y m a t c h . S i n c e t h e r e i s n o a b s o l u t e c r i t e ri o nt o u n e q u i v o c a l l y sa y t h e r e s u l t s o f o n e t e c h n i q u e a r er i g h t a n d t h e o t h e r w r o n g i t b e c o m e s i n c u m b e n t o nt h e r e s e a r c h e r to t e s t t h e s e t e c h n i q u e s u s i n g s i m u -l a t e d d a t a w h o s e p r o p e r t i e s a r e k n o w n ( M i l l i g a n a n dC o o p e r , 1 9 8 7) . T h i s p a p e r , t h e r e f o r e , e x a m i n e s b o t hs i m u l a t e d d a t a a n d b r a n d c h o i c e p r o b a b i l i ti e s d a t a t oc o m p a r e t h e F S C L n e u r al n e t w o r k w i th t h e K - m e a n sa lg o r i th m.

T h e r e m a i n d e r o f t h e p a p e r i s o r g a n i z e d i n t o ab r i e f e x p la n a t io n o f t h e t w o a p p r o a c h e s, t h e m e t h o d -o l o g y f o r t h e s t u d y , t h e r e s u l ts o f t h e s t u d y , l i m i t a-t i o n s o f t h e s t u d y , a n d c o n c l u s i o n s a n d f u t u r e r e -sea r ch d i r ec t io n s .

2 . N e u r a l n e t w o r k s

A n e u r a l n e t w o r k c o n s i s t s o f t w o b a s i c c o m p o -n en t s : processing elements ( o r n o d e s ) a n d t h e i r in-terconnections. The p r o c e s s i n g e l e m e n t r e c e i v e s an u m b e r o f i n p u t s i g n a ls a n d t h e n g e n e r a t e s a s i n g l eo u t p u t s i g n a l w h i c h i s t h e n " s e n t " t o o n e o r m o r ep r o c e s s i n g e l e m e n t s v i a t h e i n t e r c o n n e c t i o n s . H o wt h e n o d e s a r e c o n n e c t e d t o e a c h o t h e r b e t w e e n t h ev a r i o u s l a y e r s c o n s t i t u t e s t h e s y s t e m ' s k n o w l e d g ea n d d e t e r m i n e s h o w t h e n e u r a l n e t w o r k w i l l r e s p on dt o a n y a r b i t r a r y i n p u t . A c o n n e c t i o n b e t w e e n t w on o d e s c o u l d h a v e p o s i t i v e o r n e g a t i v e w e i g h t s . T h ew e i g h t w i y i s p o s i t iv e i f n o d e j e x c i t e s n o d e i a n dn e g a t i v e i f j i n h i b i t s i . T h e s t r e n g t h o f t h e c o n n e c -t i o n i s g i v e n b y t h e a b s o l u t e v a l u e o f t h e w e i g h t w i j.T h e p a t te r n o f c o n n e c t i v i t y c a n b e r e p r e s en t e d b y aw e i g h t m a t r i x W i n w h i c h w i j re p r e s e n t s th e s t r e n g tha n d s e n s e ( i . e . , e x c i t a t o r y o r i n h i b i t o r y ) o f t h e c o n -n e c t i o n b e t w e e n : j a n d i . D u r i n g t h e l e a rn i n g p r o c e s st h e se : w e i g ht s a r e m o d i f i e d b a s e d o n t h e l e a rn i n ga l g o r i t h m u s e d a n d t h e d e s i r e d o u t p u t ( s u p e r v i s e dl e a r n i n g ) f o r a g i v e n i n p u t . S i n c e s e v e r a l s o u r c e sd i s c u s s t h e c o n c e p t o f n e u r a l n e t w o r k s i n d e t a i l ( s e ef o r e x a m p l e , R u m e l h a r t e t a l . , 1 9 8 7 ; N e l s o n a n d


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348 P.V. Balakrishnan et aL / European Journal of Operational Research 93 (1996) 346-357I l l i n g w o r t h , 1 9 9 1 ; H e c h t - N i e l s e n , 1 9 9 0 ) , h e r e w ew i l l o n l y d is c u s s t h e s p e c i f i c a p p r o a c h w e h a v e u s e d .2 .1 . T h e F S C L n e t w o r k

K r i s h n a m u r t h y e t a l. ( 1 9 9 0 ) a n d A h a l t e t al . ( 1 9 9 0 )r e p o r t o n a r e v i s e d m e t h o d o f v e c t o r q u a n t i z a t i o nu s i n g f r e q u e n c y s e n s i t i v e c o m p e t i t i v e l e a r n i n g( F S C L ) . V e c t o r q u a n t i z a t i o n i s d e s c r i b e d a s " a s t a -t is t ic a l m e t h o d o f e n c o d i n g d a t a f o r t r a n s m i s s i o n t o ar e c e i v e r " ( K r i s h n a m u r t h y e t a l . , 1 9 9 0 ) . N e u r a l n e t -w o r k s a r e s u g g e s t e d a s p a r t i c u l a r l y s u i t e d t o t h is t a s ks i n c e th e y a r e a d a p t i v e i n t w o s e n s e s . F i rs t , e a c h n e wt r a i n i n g v e c t o r c a n m o d i f y t h e c o d e b o o k v e c t o r s s ot h e r e i s o n - l i n e l e a r n i n g . T h i s m a y b e n e e d e d w h e nt h e s o u r c e s t a t i s t i c s c h a n g e o v e r t i m e , f o r e x a m p l e ,w h e n c o m m u n i c a t i o n c h a n n e l s a r e c h a n g e d . S e c o n d ,n e u r a l n e t w o r k s c a n p r o c e s s o n e v e c t o r o f d a t a a t at i m e i n s t e a d o f r e q u i r in g t h e e n t i r e t r a i n i n g s e t t o b ep r o c e s s e d a t o n c e i n a b a t c h m o d e , a s i n t h e a p -p r o a c h b y L i n d e e t a l . ( 1 9 8 0 ) . T h e l a t t e r i s p a r t i c u -l a r l y c o m p u t a t i o n a l l y i n t e n s i v e ( K r i s h n a m u r t h y e ta l . , 1990) .T h e f r e q u e n c y s e n s i t i v e c o m p e t i t i v e l e a r n i n g a l -g o r i th m w o r k s a s f o l l o w s . I n f o r m a t i o n o r v o i c e s c a nb e r e p r e s e n t e d a s a v e c t o r x c o n s i s t i n g o f k d i m e n -s i o n s. U s u a l l y , k d i m e n s i o n s a r e t o o l a r g e f o r tr a n s -m i s s i o n o v e r c u r r e n t l in e s o r f i b e r o p t i c s. T h e r e f o r e ,t h e k d i m e n s i o n s m u s t b e r e d u c e d t o a s m a l l e rn u m b e r f o r t r a n sm i s s io n .T o c o m p r e s s t h e d a t a o r v o i c e f o r t r a n s m i s s i o n ,t h e r e n e e d s t o b e a s y s t e m f o r e n c o d i n g t h e d a t ab e f o r e t r a n s m i s s i o n a n d t h e n d e c o d i n g i t a t t h e r e -c e i v i n g l o c a t i o n . A c o d e b o o k i s u s e d f o r t h i s p u r -p o s e . T h e c o d e b o o k C c o n s i st s o f M c o d e w o r d s c ,.,e a c h o f k d i m e n s io n s . O n e o f th e M c o d e w o r d s isu s e d t o r e p r e s e n t t h e k - d i m e n s i o n a l v e c t o r x . T h i s i sb a s i c a l l y a d a t a r e d u c t i o n a p p r o a c h . T h e c o d e b o o k i sd e v e l o p e d u s i n g a l a r g e s e t o f t ra i n i n g d a t a .

W h e n a v e c t o r i s t o b e e n c o d e d , i t i s m a t c h e dw i t h t h e c l o s e s t c o d e w o r d b y c a l c u l a t i n g t h e d i s t o r -t ion d ( x , c i) , i = 1 . . . . M b e t w e e n t h e v e c t o r a n de a c h c o d e w o r d . I f th e j t h c o d e w o r d i s t h e c lo s e s t ,t he v e c t o r i s t h e n e n c o d e d a s c o d e w o r d c j. , U p o nr e c e i p t o f t h e s i g n a l , t h e r e c e i v e r d e c o d e s u s i n g t h ea p p r o p r i a t e c o d e w o r d .

T h e b a s i c n e u r a l n e t w o r k w i l l h a v e M n e u r a lu n i t s , e a c h w i t h a n a s s o c i a t e d w e i g h t v e c t o r t h a t i st h e i t h co d ew o rd , i . e . , w i = cg. A l l k d i m en s i o n s o f

t h e i n p u t v ec t o r x a r e f ed i n p a ra l l e l t o t h e n eu ra ln e t w o r k . D i s t o r t i o n s d( x , w i ) a re c a l c u la t e d b e t w e e nt h e i n p u t v e c t o r a n d e a c h o f th e M u n i ts . T h ed i s t o r t i o n s m a y b e m e a s u r e d u s i n g s o m e d i s t a n c em e a s u r e , t h e E u c l i d e a n d i s t a n c e b e i n g a c o m m o na p p r o a c h f o r c l u s t e r in g t e c h n i q u e s . T h e u n i t w i t h th em i n i m u m d i st o rt io n ' w i n s ' a n d t h e in p u t v e c t o r i sa s s i g n e d t o t h e c o d e w o r d c i .

I n t h is p r o c e s s, a f e w c o d e w o r d s m a y a c c o u n t fo ra d i s p r o p o r t i o n a t e s h a r e o f t h e i n p u t v e c t o r s . G r o s s -b e r g ( 1 9 7 6 ) a n d H e c h t - N i e l s e n (1 9 8 8 ) s u g g e s t m e a n so f m i n i m i z i n g t h e w i n n e r - t a k e - a l l p r o b l e m . L i p p -m a n n ( 1 9 8 7 ) a n d W i n t e r s a n d R o s e ( 1 9 8 9 ) i n d i c a t em e a n s o f f in d i n g t h e w i n n e r . C o m p e t i t i v e le a r n i n ga l g o r i t h m s a d j u s t t h e w e i g h t v e c t o r s o f n e u r a l u n i t st h a t w i n t o o f r e q u e n t l y , o r a re i n t h e n e i g h b o r h o o d o ft h e w i n n e r i n th e c a s e o f t h e K o h o n e n s e l f - o r g a n i z -i n g m a p ( K o h o n e n , 1 9 8 9 ). C o m p e t i t i v e l e a r n i n g isu s e f u l i f a c r i t e r i o n o f t h e e f f o r t i s t o m a x i m i z ee n t r o p y , t h a t i s , t o h a v e c o d e w o r d s c h o s e n w i t hr e l a t i v e l y e q u a l f r e q u e n c y .

T h e F r e q u e n c y - S e n s i t i v e C o m p e t i t i v e L e a r n i n g( F S C L ) T r a i n i n g A l g o r i t h m c o u n t s t h e n u m b e r o ft i m es , u i , a p a r t i cu l a r u n i t w i n s an d i n c reas es t h ed i s t o r t i o n m e a s u r e u s i n g t h e n o n - d e c r e a s i n g ' f a i r -n e s s ' f u n c t i o n F ( u ) . T h e f a i rn es s fu n c t i o n i n i t i a l l yf a v o r s u n i f o r m c o d e w o r d u s a g e b u t m i n i m i z e s d i s -t o r t io n a s t r a i n in g p r o g r e s s e s . T h e a l g o r i t h m i s a ni m p l e m e n t a t i o n o f t h e G r o s s b e r g c o n s c i e n c e p r i n c i -p l e a n d w o r k s a s f o l l o w s ( K r i s h n a m u r t h y e t a l . ,1990) :1 . C o n s i d e r a n i n p u t v e c t o r x .2 . F i n d t h e d i s t o r t i o n D i = F ( u i ) d ( x , w i ) fo r a l l

o u t p u t u n i t s .3 . Se l ec t th e o u t p u t u n i t i * w i t h t h e s m a l l e s t d i s t o r -

t i o n a n d l a b e l i t a s th e w i n n i n g u n i t a n d i n c r e m e n tg i * "4 . A d j u s t t h e s e l e c t e d w e i g h t v e c t o r w i . ( n + 1 ) =w i , ( n ) + E ( n) [ x ( n ) - w i , ( n )] w h ere n i s t h et r a in i n g t i m e a n d 0 < E ( n ) < 1 .5 . R e p e a t s t e p s 1 - 4 f o r a ll t r a i n in g v e c t o r s .

2 .2 . K-means c lu s t e r ingC l u s t e r i n g a l g o r i t h m s a r e a c l a s s o f d a t a r e d u c t io n

t e c h n i q u e s . H i e r a r c h i c a l c l u s t e r i n g a l g o r i t h m s s t a r tw i t h n c l u s t e r s, w h e r e n i s t h e n u m b e r o f o b s e r v a -t i o n s , s u c h a s p e o p l e , e a c h h a v i n g i n f o r m a t i o n o n k


4/12

P . V . B a la k r is h n a n e t a l . / E u r o p e a n J o u r n a l o f O p e r a tio n a l R e s e a r ch 9 3 (19 9 6) 3 4 6 -3 5 7 34 9d i m e n s i o n s , s u c h a s d e m o g r a p h i c d a t a . T h e d i s t a n c eb e t w e e n o b s e r v a t i o n s i s c a l c u l a t e d in k - d i m e n s i o n a ls p a c e . T h e t w o c l o s e s t p o i n t s a r e m e r g e d i n t o ac l u s t e r . T h i s p r o c e s s c o n t i n u e s u n t i l a l l o b s e r v a t i o n sa r e i n o n e c l u s t e r . O n e t h e n h a s t o d e t e r m i n e t h en u m b e r o f c lu s t e r s u s in g a d e c i s i o n ru l e ( M i l l i g a na n d C o o p e r , 1 9 8 5 ) .

I n n o n - h i e r a r c h i c a l c l u s t e r in g , t h e r e s e a r c h e r s p e c -i f i e s t h e n u m b er o f c l u s t e r s i n t h e d a t a s e t a p r i o r i .S i n c e t h i s a p p r o a c h m a t c h e s t h e v e c t o r q u a n t i z a t i o np r o b l e m , a n o n - h i e r a r c h i c a l m e t h o d i s u s e d h e r e f o rc o m p a r i s o n . T h e K - m e a n s p r o c e d u r e , a s u s e d b yF A S T C L U S i n S A S ( S A S , 1 9 9 0 ) , s e le c t s M r a n d o mp o i n t s f r o m t h e d a t a s e t . T h e s e a r e u s e d a s c l u s t e rs e e d s a n d a l l o t h e r p o i n t s a r e a s s i g n e d t o t h e n e a r e s tc l u s t e r s e e d . S u c c e s s i v e i t e r a t i o n s i n v o l v e r e p l a c i n gt h e c u r r e n t c l u s t e r s e e d b y t h e c l u s t e r m e a n a n d t h e nr e a s s i g n i n g a l l p o i n ts t o t h e n e a r e s t n e w c l u s t e r s e e d .T h e p r o c e s s c o n t i n u e s u n t i l t h e r e i s n o c h a n g e i nc l u s t e r m e a n s f r o m t h e p r e v i o u s i t e r a t i o n o r t h ed i f f e r e n c e i s v e r y s m a l l .

3 . A n a p p l i c a ti o n o f t h e F S C L n e t w o r k a n d K -m e a n s m e t h o d o lo g y

T h e a c t u a l d a t a e m p l o y e d i n t h i s s t u d y r e p r e s e n tt h e b r a n d s w i t c h i n g p r o b a b i l i t i e s o n 1 8 d i f f e r e n tc o f f e e b r a n d s d e r i v e d f r o m s c a n n e r d a t a . T h e d a t ac o l l e c t i o n p r o c e d u r e a n d m e t h o d o f c o n s t r u c t i n g t h e s es w i t c h i n g p r o b a b i l i t i e s a r e d e s c r i b e d i n B u c k l i n a n dS r i n i v a s a n ( 1 9 9 1 ). T h e i n p u t t o t h e K - m e a n s c l u s te r -i n g a l g o r i t h m a n d F S C L n e t w o r k w a s a n a v e r a g ep r o b a b i l i t y s w i t c h i n g v e c t o r ( 1 8 - d i m e n s i o n a l ) a t th es u b - h o u s e h o l d l e v e l . T h e r e w e r e a t o ta l o f 2 0 7 s u b -h o u s e h o l d s w h i c h w e r e t o b e c l u s t e r e d i n t o s i x d i s -t i n c t s e g m e n t s a s r e c o m m e n d e d b y B u c k l i n a n dS r i n i v a s a n ( 1 9 9 1 ) . E a c h o f t h e s e 2 0 7 o b j e c t s t o b eg r o u p e d w a s r e p r e s e n t e d b y a v e c t o r s p e c i f y i n gc h o i c e p r o b a b i l it i e s f o r 1 8 b r a n d s o f c o f f e e .

T h e d a t a s e t c o m p r i s i n g 2 0 7 1 8 c h o i c e p r o b a -b i l i t i e s w a s t h e n c l u s t e r e d u s i n g t h e F S C L n e t w o r ka s w e l l a s t h e K - m e a n s a l g o r i t h m t o g e n e r a t e as i x - s e g m e n t s o l u t i o n . I n t h e s t u d y , t h e K - m e a n sp r o c e d u r e f o r n o n - h i e r a r c h i c a l c l u s t e r i n g w a s p e r -f o r m e d w i t h i t e r a t i o n s . I n t h i s c a s e , t h e a l g o r i t h mn e e d e d 1 1 i t e ra t i o n s t o c o n v e r g e u s i n g t h e d e f a u l tv a l u e s a n d s t o p p i n g r u l e .

T h e r e s u l t in g c l u s t e r s o l u t io n s t h a t w e r e g e n e r a t e db y t h e s e a l t e m a t e m e t h o d s w e r e t h e n c o m p a r e d f o rs i m i l a r i t y o n t h e c r i t e r i a o f ( 1 ) s e g m e n t s i z e , ( 2 )s e g m e n t m e a n s , a n d ( 3 ) t h e i r m a n a g e r i a l i n t e r p r e t a -t io n . I n T a b l e 1 , w e s h o w t h e c o m p a r a t i v e r e s u l ts o ft h e s i x - c l u s t e r s o l u t i o n s p r o d u c e d b y t h e K - m e a n sp r o c e d u r e ( F A S T C L U S ) a n d t h e F S C L n e t w o r k(A _h a lt e t a l . , 1 9 9 0 ) . E v en a cu r s o ry g l an c e a t t h ist a b l e i n d i c a t e s t h a t t h e F S C L n e t w o r k r e s u l t s a r e n o ts i m i la r to t h o s e p r o d u c e d b y K - m e a n s . B o t h m e t h o d sd i f f e r e d i n s o m e t h i n g a s f u n d a m e n t a l a s t h e s i z e o ft h e i n d i v i d u a l c l u s t e r s . Fo r i n s t an ce , t h e l a rg es t s eg -m e n t h a d 6 7 m e m b e r s u si n g K - m e a n s w h i le th el a r g e st F S C L c l u st e r h a d 4 2 m e m b e r s . I n t e r e st in g l y ,t h e F S C L p r o c e d u r e g e n e r a t e d s i x c l u s t e r s t h a t , c o n -s i st e n t w i t h o u r p r i o r e x p e c t a t i o n s , w e r e m o r e o r l e sso f e q u a l s i z e , v a r y i n g o n l y f r o m t h e l a r g e s t w i t h 4 2t o t h e s m a l l e s t w i t h 3 1 m e m b e r s . O n t h e o t h e r h a n d ,t he K - m e a n s p r o c e d u r e g e n e r a te d w i d e l y v a r y i n gs i z e d c l u s t e r s . N a t u r a l l y e n o u g h , t h e t w o p r o c e d u r e sd i d n o t p r o d u c e c l u s t e r m e a n s t h a t w e r e i d e n t i c a l .

G i v e n t h a t c l u s t e r s d i f f e r b y m e t h o d , t h e n e x ti s s u e i s t h e i r m a n a g e r i a l i n te r p r e t a b i li t y . F o r c o m p a r -a t i v e p u r p o s e s a n d t o b e c o n s i s t e n t w i t h p r i o r a n a l y -s i s o f B u c k l i n a n d S r i n i v a s a n ( 1 9 9 1 ) , w e u s e a m e a no f 0. 0 8 a s t h e ' c r i t i c a l c u t - o f f ' v a l u e i n ' a s s i g n i n g 'b r a n d s t o a p a r t i c u l a r s e g m e n t ' s c h o i c e a c t i v i t y . T h i si m p l i e s t h a t w e c a n c h a r a c t e r i z e t h e c l u s t e r s g e n e r -a t e d b y t h e t w o m e t h o d s a s f o l l o w s . I n t h e c a s e o fK - m e a n s ( w i t h i t e ra t io n s ) : c l u s t e r 1 i s a g r o u n dc a f f e i n a t e d s e g m e n t ( e x c l u d i n g t h e s t o r e p r i v a t e l a -b e l b ra n d ) ; c l u s te r 2 i s a n i n s ta n t c a f f e i n a t e d s e g -m e n t ; c l u s t e r 3 i s a d e c a f f e i n a t e d s e g m e n t ; a n d ,C l u s te r 4 i s F o l g e r s: g r o u n d c a f f e i n a t e d s e g m e n t .C l u s t e r s 5 a n d 6 a r e p r i m a r i l y th e s t o r e p r i v a t e l a b e lg r o u n d c a f f e i n a t e d s e g m e n t a n d th e M a x w e l l H o u s ei n s t a n t c a f f e i n a t e d s e g m e n t , r e s p e c t i v e l y . T h e F S C Ln e t w o r k g e n e r a te d s e g m e n t s c a n b e c h a r a c t e r iz e d a sf o l lo w s : c l u s t e r 1 is t h e g r o u n d c a f f e i n a t e d s e g m e n te m p h a s i z i n g M a x w e l l H o u s e a n d M a s t e r B l e n d ;c l u s t e r 2 is S a n k a a n d g r o u n d d e c a f f e i n a t e d s e g m e n t ;c l u s t e r 3 i s a n i n s t a n t c a f f e i n a t e d ( e m p h a s i z i n gM a x w e l l H o u s e ) a n d s to r e b r a n d g r o u n d c a f f e in a t e ds e g m e n t ; c l u s t e r 4 i s N e s c a f e i n s t a n t c a f f e i n a t e d a n di n s t a n t d e c a f f e i n a t e d s e g m e n t ; c l u s t e r 5 i s a F o l g e r ss e g m e n t ; a n d c l u s t e r 6 i s p r i m a r i l y M a x w e l l h o u s eg r o u n d c a f f e in a t e d s e g m e n t .

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6/12

P. V. B alakrishnan et al. / European Journal o f Operational Research 93 (1996) 346-357 351t h e t w o s e t s o f s e g m e n t s i s n o t t h e s a m e . T h i s r a i s e sa n i n t e r e s t i n g q u e s t i o n a b o u t w h i c h s e g m e n t a t i o ns c h e m e s h o u l d b e a d o p t e d i n t h e d e v e l o p m e n t o fm a r k e t i n g s t r a te g i e s . T o a n s w e r s u c h a q u e s t i o nr e q u ir e s s t u d y i n g t he c o m p a r a t i v e p e r f o r m a n c e o ft h e t w o a p p r o a c h e s t o c a s e s w h e r e i n t h e ' t r u e ' c l u s -t e r s o l u t i o n s a r e k n o w n . T o t h i s e n d , t h e n e x t s t a g eo f th e r e s e a r c h , t h e n f o c u s e s o n s t u d y i n g t h e tw oa p p r o a c h e s o n s i m u l a t e d d a t a w i t h k n o w n c l u s t e rs o l u t i o n s r ep re s en t i n g d i f f e r en t q u a l i t i e s o f d a t a .

4 . M e t h o d o l o g yS e v e r a l f a c t o r s c a n a f f e c t t h e q u a l i ty o f c l u st e r

r e c o v e r y , s u c h a s t h e n u m b e r o f c l u s t e r s i n t h e d a t a ,t h e n u m b e r o f d i m e n s i o n s u s e d t o d e s c r i b e t h e d a t a ,an d t h e l ev e l o f e r ro r i n th e d a t a (M i l l i g an , 1 9 8 0 ) .T h i s s t u d y c o n s i s t s o f a f u l l f a c to r i a l e x p e r i m e n t a ld e s i g n c r o s s i n g t h e n u m b e r o f c l u s t e r s w i t h t h en u m b e r o f d i m e n s i o n s ( a t t r i b u t e s ) w i t h l e v e l s o fe r ro r . T h e r e s u l t i s a 4 3 3 d es i g n w i t h t h r e erep l i ca t i o n s p e r ce l l , f o r a t o t a l o f 1 0 8 d a t a s e t s . T h i si s a c o m m o n a p p r o a c h t o e v a l u a t i n g c lu s te r in g m e t h -o d s ( M i l l i g a n , 1 9 8 0 ; M i l l i g a n a n d C o o p e r , 1 9 8 5 ,1 9 8 7) . E a c h d a t a s e t is g e n e r a t e d u s i n g t h e p r o c e d u r eo f M i l l i g a n ( 1 9 8 5 ) p r o c e d u r e a n d c o n t a i n s 5 0 p o i n t s .T h e F S C L n e t w o r k a n d t he K - m e a n s a l g o ri th m seach c l u s t e r ed a l l 1 0 8 d a t a s e t s .T h e n u m b e r o f t r u e c l u s t e r s i n e a c h d a t a s e t w a s2 , 3 , 4 , o r 5 w i t h a ro u g h l y eq u a l d i s t r ib u t i o n o fp o i n t s in e a c h c l u s t e r . T h e n u m b e r o f m e a s u r e s ( i .e . ,a t t r i b u t e s ) u s e d t o c h a r a c t e r i z e e a c h d a t a s e t w a sv a r i e d s o t h a t a l l p o i n t s i n a d a t a s e t w e r e d e s c r i b e db y a 4 - , 6 - , o r & d i m e n s i o n a l : s p a c e . T h e t h r e e l e v e l so f e r r o r w e r e n o e r r o r , l o w e r r o r ; o r h i g h e r r o r . T h en o e r r o r l e v e l w o u l d b e s i m i l a r t o a n o i s e l e s s t r a n s -m i s s i o n o f d a t a o r d a t a o b t a i n e d w h e n t h e r e i s n od a ta c o l l e c ti o n a n d m e a s u r e m e n t e r r o r.

T h e s i m u l a t i o n d a t a w e r e g e n e r a t e d a s m m c a t e d ,m u l t i v a r i a t e n o r m a l d i s t r i b u t i o n s i n E u c l i d e a n s p a c e .T h e d a t a s e t s w e r e :c r e at e d t o h a v e r e a s o n a b l y d i s -t i n c t a n d s e p a r a t e c l u s t e r s a s d e f i n e d b y i n t e r n a lc o h e s i o n a n d e x t e r n a l i s o l a t i o n ( C o r m a c k , 1 9 7 1 ) .I n t e r n a l c o h e s i o n i n t h e s e d a t a s e t s r eq u i r e a l l d a t ap o i n t s t o b e w i t h i n t h e b o u n d a r i e s o f th e c l u s t e r o na l l d i m e n s i o n s . B o u n d a r i e s w e r e t h r e e s t a n d a r d d e v i -a t i o n s a c r o s s f o r a l l d i m e n s i o n s . E x t e r n a l i s o l a t i o n

w a s o p e r a t i o n a li z e d b y h a v i n g t h e f i r st d i m e n s i o n b en o n - o v e r l a p p i n g . T h e d i s t a n c e b e t w e e n c l u s t e r s w a sf ( S 1 + $ 2 ) , w h e r e f i s t h e s e p a r a t i o n f a c t o r a n d S la n d S 2 a r e t h e s t a n d a r d d e v i a t i o n s o f c l u s t e r 1 a n d 2 ,r e s p e c t i v e l y , o n t h e f i r s t d i m e n s i o n ( s e e M i l l i g a n ,1 9 85 ) . T h e v a l u e o f f w a s r a n d o m l y s e le c t e d f r o m au n i f o r m d i s t r ib u t i o n b o u n d e d b y 0 . 2 5 a n d 0 .7 5 . O t h e rd i m e n s i o n s w e r e p e r m i t t e d t o o v e r l a p . T h u s , c l u s t e rs e p a r a t i o n i s g u a r a n t e e d o n t h e f i r s t d i m e n s i o n o n l yi n t h e n o n - e r r o r c o n d i t i o n .

T h e d a t a p o i n t c o o r d i n a t e s w e r e e r r o r - p e r t u r b e d t od i s t o r t t h e t r u e d i s t a n c e s b e t w e e n p o i n t s t o a c h i e v el o w a n d h i g h e r r o r . T h i s s i m u l a t e s n o i s e i n t h e d a t aw h i c h c a n o c c u r w i th d a t a c o l l e ct io n o r m e a s u r e m e n te r r o r . T h e l o w e r r o r c o n d i t i o n s t i l l d i d n o t p e r m i to v e r l a p o n t h e f i r s t d i m e n s i o n b u t t h e s e p a r a t i o n w a sv e r y s m a l l . T h e h i g h e r r o r c o n d i t i o n d i d h a v e c l u s t e ro v e r l a p o n t h e f i r s t d i m e n s i o n a n d p o s s i b l y o n t h eo t h e r d i m e n s i o n s . N e w i n t e r p o i n t d i s t a n c e s w e r e c a l -c u l a t e d u s i n g [~'~,i(A i j - A i k - 3Eijk)2] l /2, where A i ja n d A ik a r e t h e o r i g i n a l c o o r d i n a t e v a l u e s f o r o b s e r -v a t i o n j a n d k o n d i m e n s i o n i . T h e e r r o r te r m i sg e n e r a t e d f r o m a u n i v a r i a t e n o r m a l d i s t r ib u t i o n w i t ha m e a n o f z e r o a n d a s t a n d a r d d e v i a t i o n d e t e r m i n e db y t a k i n g t h e a v e r a g e o f t h e s t a n d a r d d e v i a t i o n s o nd i m e n s i o n i f o r t h e t w o c l u s t e r s c o n t a i n i n g p o i n t s ja n d k . T h e m u l ti p l ic a t i o n fa c t o r, 8 , h a d v a l u e s o f 1o r 2 , w h i c h d e t e r m i n e d t h e l o w a n d h i g h e r r o r l e ve l s ,r e s p e c t i v e l y ( M i l l i g a n , 1 9 8 5 ).

T h e F S C L n e t w o r k U se d th e f a ir n es s f u n c ti o nF ( u i ) = I t t i e - ( / r w h e r e /3 a n d T a r e c o n s t a n t s a n dt i s t h e t ra i n i n g i t e r a ti o n n u m b e r . T h e c u r r e n t i m p l e -m e n t a t i o n u s e s a / 3 v a l u e o f 0 . 0 6 a n d 1 / T v a l u e o f0 . 00 0 0 5 . T h e le a r n i n g r a t e u s e d w a s E ( n ) =0 . 0 6 e - " 5 ( " - l ).

5 . R e s u l t s o n s i m u l a t e d d a t aT h e r e s u l ts o f t h e K - m e a n s c lu s t er i n g a n d F S C L

n e t w o r k t e c h n i q u e s w e r e c o m p a r e d o n o v e r a l l c lu s te rr e c o v e r y a n d s e n s i t iv i t y t o c l u s t e r c h a r a c t e ri s t ic s .T h e p e r c e n t o f t h e 5 0 o b s e r v a t i o n s m i s c l a s s i f i e d b yt h e d i f f e r e n t t e c h n i q u e s i n d i c a t e d c l u s t e r r e c o v e r yp e r f o r m a n c e . A n a l y s e s o f v a r i a n c e ( A N O V A s ) u s i n gt he g e n e r a l l i n e a r m o d e l s p r o c e d u r e ( G L M ) a v a i l a b l ei n t h e c u r r e n t e d i t i o n o f S A S ( 1 9 9 0 ) t e s t e d h y p o t h e -s e s a b o u t t h e e f f e c t s o f t h e d a t a c h a r a c t e r i s ti c s o n t h e


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352 P.V. Balakrishnan et al. / European Journal of Operational Research 93 (1996) 346-357Table 2ANOVA results of misclassifications by cluster,error level

attributes, and

Variable K-means FSCL(p-values) (p-values)

Overall model 0.244 0.054No. of clusters 0.750 0.005 *No. of attributes 0.147 0.049 *Error level 0.001 * 0.014 *Error level No. of clusters 0.882 0.324Error level No. of attributes 0.172 0.996No. of clusters No. of attributes 0.661 0.477Error level X No. of clusters No. 0.732 0.400of attributes* Significant at 0.05 level.

No Error Low Error High Error

~ ~ FSCL q- K-Means

Fig. 1. Effect of error: mean percent of points misclassified.clustering results. Results of these analyses are pre-sented in Tables 2 and 3.

K-means recovered the correct cluster structure ofeach error-free data set with no more than fiveiterative re-computations of the cluster seeds. Themajority of the low and high error data sets werealso correctly classified by the FASTCLUS proce-dure. Only a few of the data sets were poorlyclassified which increased the average number ofmisclassifications across all data sets.Hypothesis 1. T h e p e r c e n t a g e o f m i s c l a s s i fi c a t i o n sd o e s n o t d i f f e r a c r o s s e r r o r l e ve l s.

For both K-means and FSCL the main effect forthe level of error in the data was significant. For theTable 3Mean percent of observations misclassified by error, number ofclusters, and n umber o f attributesVariable Level K-means FSCLOverall averageError

Number of clusters

Number o f attributes

1.33% 9.19%None 0.00 7.28Low 0.22 6.50High 3.78 13.782 1.48 4.963 0.66 6.074 2.00 10.445 1.18 15.264 1.84 12.836 2.06 8.448 0.12 6.28

K-means case the error level was significant at the0.01 level (N = 108, p < 0.001). The mean misclas-sification of observations increased from 0.0% to0.22% to 3.78% for the no error, low error, and higherror data sets, respectively (Table 3).

The FSCL network results are similar to theK-means case. However, the percentage of misclassi-fications for all three levels was higher for the FSCLnetworks (see Fig. 1). Recovery deteriorated as addi-tional error was introduced to the data. The errorlevels ranged from 7.28% for the no error condition,to 13.78% for the high error condition. The low errorcondition was close to the no error results at 6.5%.Hypothesis 2. The p e r c e n t a g e o f m i s c l a s s i fi c a t io n sd o e s n o t d i f f e r a c r o s s t h e n u m b e r o f c lu s t e r s in t h eda ta se t .

Since the data always consisted of 50 points, asthe number of clusters increase there are fewer pointsfor the algorithms to consider in each cluster. Thishypothesis was confirmed for the K-means case,however for FSCL the hypothesis was rejected ( p =0.005). In the case of the FSCL network, as thenumber of clusters increased, cluster recovery deteri-orated, reaching 15.26% for five clusters. The bestrecovery was for two clusters with 4.96% of the datapoints misclassified. The two and three cluster(6.07%) misclassification rates were much lower thanthe four (10.44%) and five (15.26%) rates (see Fig.2). Closer examination of the results suggests thatthere is a tendency to overcluster, that is, to have two


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P.V. Balakrishnan et al. / European Journal of O perational Research 93 (1996) 346-357 353

2 3 4 5

FSCL + K-Means 1Fig. 2. Effect of number of clusters: mean percent of pointsmisclassified.

4 6 8

FSCL + K-Means

Fig. 3. Effect of number of attributes: mean percent of pointsmisclassified.

true clusters merged into one cluster. When thishappens information is lost by not separating the trueclusters.Hypothesis 3. T h e p e r c e n t a g e o f m i s c l a s s if i c a t io n sd o e s n o t d i f fe r a c r o s s t h e n u m b e r o f a t t r ib u t e s o fe ac h ob j e c t .

Previous clustering research results indicate thatmore, relevant attributes should lead to better recov-ery (e.g., Milligan, 1980). H owever , in the case ofthe K-means approach there was no significant effect(see Table 2) on the percentage of misclassifications,although the percent misclassified with eight at-tributes was the lowest amongst three levels (seeTable 3).

In the case of the FSCL network, recovery im-proved as the number of attributes increased fromfour to eight. Ha lf as man y points were misclassified(6.28%) when eight dimensions were used as whenfour dimensions were used (12.83%) (see Fig. 3).This pattern is consistent with that reported by Kr-ishnamurthy et al. (1990) for up to 128 dimensionsin the case of an engineering application.

6 . B r a n d c h o i c e d a t a r e v i s i t e d

Given the above results of the studies on bothsimulated and actual data, we now revisit the brandswitching data on the 18 different brands of coffee

reported in Section 3. The major conclusions of theabove two studies is that each method seems to havesomething to recommend. Specifically, based on theabove results, we find that, the use of K-meansproved to be more accurate in the classification ofsynthetic data. However, the use of the FSCL net-work with its 'conscien ce mechanism' had less varia-tion in the cluster sizes on the larger real data.

This provoked the natural extension of whetherthe combination of the two disparate methods couldlead to even more managerially useful results thaneither alone i~ Although the K-means approach pro-vides a better 'hit rate' on synthetic data, it iswell-known to be prone to the biases involved in theselection of initial cluster seeds (Milligan, 1980).This problem of seed selection was, however, miti-gated in the case of the synthetic data by knowing apr i or i t he correct number of clusters, and by havingrelatively errorfree information. Though the resultsof this experiment do suggest that its internal validityis relatively higher, these simulation conditions thatresulted in the superior performance of K-meansmay ibe less likely to hold in actual applications.

Since, as noted earlier, seed selection is an impor-tant issue with K-means, we decided to combine thetwo approaches by employing the FSCL network to

We would like to thank the reviewers of this journal for theirinsightful comments on this.


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354 P.V. Balakrishnan et al. / European Journal of Operational Research 93 (1996)346-357as s i s t i n p ro v i d i n g t h e i n i t i a l s eed s . We u s e t h eF S C L n e t w o r k r e s u l t s a s t h e s e e d s t o t h e K - m e a n sm e t h o d , t h e a p p r o a c h t h a t w e ' t r u s t ' m o r e f o r t h ef i n a l c l u s t e r s o l u t i o n . T h e r e s u l t i n g s o l u t i o n s h o u l db e i n t r i g u i n g b e c a u s e w e n o w h a v e t h e o p p o r t u n i t yt o a s se s s h o w t h e K - m e a n s r e s u l ts c h a n g e w i t h t h en e w s e e d s . I f t h e K - m e a n s m e t h o d i s p a r ti c u l a r lyr o b u s t , t h e n e v e n w i t h t h e d i f f e r e n t s t a r ti n g p o i n t t h er e s u l t s m a y t u r n o u t t o b e i d e n t i c a l t o t h e o r i g i n a lK - m e a n s . A l t e m a t i v e l y , g i v e n t h e n e w s t a r t i n g p o i n t ,t h e r e s u l t s c o u l d b e s i m i l a r t o t h e F S C L r e s u l t s o rs o m e w h e r e i n b e t w e e n t h e o r i g i n a l F S C L a n d K -m e a n s r e s u l t s . H e n c e , w e n o w h a v e t h e o p p o r t u n i t yt o a s s e s s t h e q u a l i t y o f t h e r e s u l t in g s o l u t io n i n t e r m so f e a s e o f i n t e r p r e t a ti o n , f a c e v a l i d i ty , a n d d i s p a r i t yi n t h e o b t a i n e d c l u s t e r s i z e s w h e n t h e s e e d s f r o mF S C L a re e m p l o y e d .

I n t h e s tu d y , th e K - m e a n s p r o c e d u r e f o r n o n -h i e r a r c h i c a l c l u s t e r i n g w a s p e r f o r m e d w i t h i t e r a t i o n su s i n g a s t h e i n p u t s e e d s ( r e p o r t e d i n T a b l e 1 ) t h ec e n t r o i d s o f t h e c l u s t e r s o l u t i o n p r o v i d e d b y t h eF S C L n e t w o r k . I n t h is c a s e, t he K - m e a n s a l g o r it h mn e e d e d 1 5 it e r a t io n s t o c o n v e r g e u s i n g t h e t r a d it i o n a ld e f a u l t v a l u e s s t o p p i n g r u l e 2 ( s e e T a b l e 4 ) .

I n t e r e s t i n g ly , t h e r e s u l t i n g s o l u t io n p r o v i d e d c l u s -t er s o f m o r e e q u a l s i z e s t h a n t he K - m e a n s m e t h o d b yi t s e l f . T h e v a r i a t i o n o f c l u s t e r s i z e s r a n g e d f r o m ah i g h o f 4 9 t o a l o w o f 1 9 3 . G i v e n t h a t t h e s e c l u s t e rs i z e s a r e m a n a g e r i a l l y m o r e u s e f u l , t h e n e x t i s s u e i st h e i r m a n a g e r i a l i n t e rp r e t a b i li t y . W e c a n c h a r a c t e r i z et h e c lu s t e r s g e n e r a t e d b y t h e c o m b i n a t i o n o f t h e t w om e t h o d s a s f o l lo w s : c l u s t e r 1 i s t h e s e g m e n t M a x w e l lH o u s e a n d M a s t e r B l e n d g r o u n d c a f f e i n a t e d ; c l u s t e r2 i s t h e g r o u n d d e c a f f e i n a t e d s e g m e n t ; c l u s t e r 3 i st h e i n s t an t ca f f e i n a t ed s eg m en t ; c l u s t e r 4 i s t h e i n -s t a n t d e c a f f e i n a t e d s e g m e n t ; c l u s t e r 5 i s t h e F o l g e r sg r o u n d c a f f e i n a t e d s e g m e n t ; a n d c l u s t e r 6 i s t h eg r o u n d c a f f e i n a t e d s e g m e n t e m p h a s i z i n g M a x w e l lH o u s e .

A s s h o u l d b e f a i r l y c l e a r f r o m t h e a b o v e c l u s t e rs o l u t i o n , t h e r e s u l t i n g s e g m e n t s a r e m u c h ' c l e a n e r ' ,

2 The analysis was rerun by relaxing the various default param-eters of convergence and number of iterations. Changing theseparameters had no effect on the resulting clusters.

3 This was, however, a more unequal sized outcome than thatprovided by the FSCL network by itself.

a n d h a v e b e t t e r m a n a g e r i a l i n t e r p r e t a b i l i t y t h a n e i -t h e r o f t h e m e t h o d s i n d i v i d u a l l y a s r e p o r t e d i n T a b l e1 . I n p a r t i c u l a r , t h e c o m b i n e d a p p r o a c h s e e m s t oh a v e g e n e r a t e d c l u s t e r s w i t h g r e a t e r p r e c i s i o n . F o ri n s t a n c e , e v e n t h o u g h w e s t a r te d w i t h t h e s e e d s u s i n gt h e F S C L n e t w o r k , t h e n e w c l u s t e r 5 w i t h 1 9 h o u s e -h o l d s a n d 0 . 6 4 c h o i c e p r o b a b i l i t y f o r F o l g e r s ' g r o u n dc a f f e i n a t e d ( G C - F ) i s m o r e m e a n i n g f u l t h a n t h e o n eo b t a i n e d b y F S C L a l o n e w h i c h h a d 3 2 m e m b e r s b u tg r o u p e d t o g e th e r m o r e h o u s e h o l d s w i t h g r e a t e r p u r -c h a s e v a r i a t io n s t o l o w e r t h e G C - F c h o i c e p r o b a b i l -i ty t o 0 .4 1 . S i m i l a r l y , th e n e w c l u s t e r 1 h a s f e w e r( 3 5 ) m e m b e r s w h o o n a g g r e g a t e h a v e a m u c h h i g h e rp u r c h a s e p r o b a b i l i t y ( 0 . 4 5 ) f o r t h e ' M a s t e r B l e n d -g r o u n d c a f f e i n a t e d ' w h e n c o m p a r e d w i t h 6 7 m e m -b e r s w i t h 0 . 3 0 c h o i c e p r o b a b i l i t y f o r t h e K - m e a n ss t a n d a l o n e a p p r o a c h . A g a i n , t h e n e w c l u s t e r 2 i d e n -t i f i e s m o r e p r e c i s e l y t h e m e m b e r s o f t h e G r o u n dD e c a f f e i n a t e d s e g m e n t , i . e . , t h o s e h a v i n g g r e a t e rp u r c h a s e p r o b a b i l i t i e s , t h a n e i t h e r o f t h e o t h e r t w om e t h o d s b y t h e m s e l v es . T h e h i g h e r ' f a c e v a l i d i t y ' o ft h e r e s u l t s o b t a i n e d , a n d t h e f e w e r a n o m a l i e s p r e -s e n t , s e e m s t o i m p l y t h a t t h e c o m b i n a t i o n o f c o n v e n -t i o n a l s t a t i s t i c a l t e c h n i q u e s a l o n g w i t h t h e n e w e rm a c h i n e l e a rn i n g a lg o r i th m s m a y h a v e m u c h t o o f f e ra n d m i g h t b e t h e d i r e c t i o n t o t a k e i n t h e f u t u r e .

7 . D i s c u s s i o n a n d c o n c l u s i o n s

W h i l e t h e p r o c e d u r e f o r g e n e r a t i n g d a t a s e t s i nt h e s t u d y r e p o r t e d h e r e h a s b e e n u s e d f o r c l u s t e r i n gr e s e a rc h c o m p a r i s o n s , i t h a s o n l y b e e n u s e d o n c e f o rn e u r a l n e t w o r k s ( B a l a k r i s h n a n e t a l ., 1 9 9 4 ) . A s s u c h ,s o m e l i m i t a t i o n s r e l a t i n g t o t h e s t u d y a r e n o t e d .F i r s t , t h e n u m b e r o f d a t a p o i n t s m a y b e c o n s i d e r e dr e l a t i v e l y s m a l l w h e n c o m p a r e d t o t h o s e c o m m o n l yu s e d f o r n e u r a l n e t w o r k s . S e c o n d , c o n d u c t i n g t h ea n a l y s i s o n e r r o r - f r e e d a t a a n d k n o w i n g t h e c o r r e c tn u m b e r o f c lu s t er s in t h e d a t a m a y h a v e f a v o r a b l yb i a s e d t h e r e s u l t s a s t h e m o s t s e r i o u s p r o b l e m i n t h eK - m e a n s a p p r o a c h i s t h e s e l e c t i o n o f i n i t i a l c l u s t e rs e e d s ( s e e M i l li g a n , 1 9 8 0 ). U s i n g t h e c o r r e c t n u m b e ro f c l u s t e r s a n d e r r o r - f r e e d a t a s i g n i f i c a n t l y r e d u c e dt h e p r o b l e m o f s e l e c t i n g t h e s e e d s . A n a l y s e s c o n -d u c t e d o n d a t a w i t h s k e w e d d i s t r i b u t i o n s o r o t h e rf o r m s o f e r r o r m a y r e s u l t i n a l o w e r r e c o v e r y r a t e .


10/12

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11/12

35 6 P . V . B a la k r i s h na n e t a L / E u r o p e a n J o u r n a l o f Op e r a t ion a l R e s e a r c h 9 3 (1 9 9 6 ) 3 4 6 -3 5 7D e s p i t e t h e se l i m i ta t i o n s , t h e K - m e a n s a l g o r i t h mp e r f o r m e d w e l l.

A l t h o u g h t h e F S C L n e t w o r k c h o s e n f o r t h is s t u d yd i d n o t p e r f o r m w e l l , s e v e r a l c o n t r i b u t i n g f a c t o r ss h o u l d b e c o n s i d e r e d t h a t w o u l d p r o v i d e t h e b a s i sfo r fu t u re r e s ea rch i n t h e a r ea . I t s h o u l d f i r s t b en o t e d t h a t t h e u n s u p e r v i s e d l e a r n i n g a p p r o a c h u s e di n th e F S C L n e t w o r k m a k e s i t a n a t u r a l s ta r t in g p o i n tt o e x p l o r e f o r u s e i n c l u s t e r i n g . K a n g a s e t a l . ( 1 9 9 0 )s u g g e s t t h a t f o r p a t t e r n r e c o g n i t i o n o r o t h e r d e c i s i o np r o c e s s e s i t i s p o s s i b l e t o s i g n i f i c a n t l y i m p r o v er e c o g n i t i o n a c c u r a c y b y ' f i n e t u n i n g ' w i t h a d d i t io n a li n f o r m a t i o n . M o r e i m p o r t a n t l y , t h e i n f o r m a t i o nn e e d e d t o m i n i m i z e m i s c l a s s i f i c a t i o n i n v o l v e s e m -p l o y i n g m u l t i p l e c o d e b o o k ( i . e . , w e i g h t ) v e c t o r s t or e p r e s e n t e a c h c l u s t e r . T h e s e m u l t i p l e v e c t o r s , t h e ys u g g e s t , a r e t h e v e c t o r s d e f i n i n g t h e b o u n d a r i e s o ft h e c l u s t e r s r a t h e r t h a n t h e t y p i c a l c e n t r o i d s . U n f o r -t u n a t e l y , i n t h e c o n t e x t o f a p p l i e d r e s e a r c h , t h e v e r yb o u n d a r i e s o f th e c l u s t e r s a r e t y p i c a l l y n o t k n o w na n d h a v e t o b e d e t e r m i n e d .

A n o t h e r p r o b l e m i s t h e c h o i c e o f t h e p a r a m e t e r sf o r t h e n e u r a l n e t w o r k . H o w d o e s o n e c h o o s e t h ea p p r o p r ia t e n e t w o r k p a r a m e t e r s f o r a d a t a s e t? T h e r ei s r e a l l y n o c l e a r c u t a n s w e r t o t h i s p r o b l e m i n t h eu n s u p e r v i s e d f r a m e w o r k . H e r e , s i n c e t h e c l u s t e r s o -l u t i o n s w e r e k n o w n , t h e p a r a m e t e r s w e r e s e t b a s e do n an i n i t i a l s t u d y o f th e h i t r a t e o n a s m a l l s u b - s e to f t h e d a t a s e t s . H o w e v e r , s i n c e t h i s n o t f e a s i b l e i nr e a l d a t a s e t s , a n o t h e r a p p r o a c h m i g h t b e t o c h o o s ep a r a m e t e r s t h a t g e n e r a t e m o r e o r l e s s e q u a l c l u s t e r s .

C l u s t e r r e c o v e r y f o r t h e F S C L n e t w o r k i m p r o v e da s m o r e r e l e v a n t a t t r i b u t e s w e r e a d d e d . T h i s i s c o n -s i s t e n t w i t h p r e v i o u s c l u s t e r i n g r e s u l t s a n d s u g g e s t st h a t t h e m o r e , u s e f u l i n f o r m a t i o n a v a i l a b l e f o r t h eF S C L n e t w o r k , t h e b e t t e r t h e r e c o v e r y . B y i t s e l f ,b a s e d o n t h e s i m u l a t i o n d a t a r e s u l t s , t h e F S C Lm e t h o d s h o u l d b e u s e d w i t h c o n s i d e r a b l e c a u t i o n f o rc l u s t e r i n g p u r p o s e s . O f c o n c e r n i s t h e s i g n i f i c a n ts e n si ti v it y o f t he F S C L n e t w o r k t o i n c r e a s e s i n e r r o rl e v e l a n d t o i n c r e a s e s in t h e n u m b e r s o f c l u s te r s . I ft h e d i m e n s i o n o f k f o r t h e c o d e b o o k i s v e r y l a r g e ,t h e r e m a y b e a b e t t e r c h a n c e t o s e l e c t t h e c o r r e c ta s s i g n m e n t o f d a t a p o i n t s t o c l u s t e rs .

A n o t h e r a s p e c t t h a t h a s b e e n i n v e s t i g a t e d i n t h i sr e s e a r c h i s t h e c o m b i n i n g o f t h e r e s u l t s o f t h e n e u r a ln e t w o r k m e t h o d o l o g y w i t h th e K - m e a n s a p p r o a c h .T h i s c o m b i n a t i o n f o r t h e m a r k e t d a t a c o n s i d e r e d

h e r e h a s r e s u l t e d i n b e t t e r s o l u t i o n s t h a n w h e n e i t h e rt e c h n i q u e w a s u s e d b y i t s e l f . F u t u r e r e s e a r c h s h o u l db e c o n d u c t e d t o e x t e n s i v e l y s t u d y t h e i m p l i c a t io n s o fs u ch i n t eg ra t i o n .

F u t u r e r e s e a r c h s h o u l d a l s o i n v e s t i g a t e d i f f e r e n td a t a c h a r a c t e r i s t i c s , s u c h a s k i n d s o f e r r o r p e r t u r b a -t i o n s i n c l u d i n g s k ew ed d i s t r i b u t i o n s an d t h e i n f l u -e n c e o f o u tl i e rs i n t h e d a t a . F o r n o w , u s e r s o f th eF S C L n e t w o r k s h o u ld b e c a u t io u s a b o u t u s i n g i t f o rc l u s t e r i n g s m a l l d a t a s e t s . I t a p p e a r s t o w o r k b e t t e rf o r la r g e r n u m b e r s o f t r a in i n g d a t a , e . g ., 1 4 0 0 0v e c t o r s ( K r i s h n a m u r t h y e t a l . , 1 9 9 0 ) A d d i t i o n a l ly ,t h e F S C L a l g o r i t h m h a s b e e n s h o w n t o p e r f o r mb e t t er t h a n th e K o h o n e n s e l fo r g a n i z in g m a p ( A h a l t e ta l. , 1 9 9 0 ), h o w e v e r , th e K - m e a n s a l g o r i t h m o u t p e r -f o r m e d t h e F S C L a l g o r i th m i n t hi s s tu d y .

AcknowledgementsT h e a u t h o r s a c k n o w l e d g e P r o f s . S t a n A h a l t a n d

A s h o k K r i s h n a m u r t h y o f t h e E l e c t r i c a l E n g i n e e r i n gD e p a r t m e n t a t O h i o S t a t e f o r t h e i r a s s i s t a n c e o n t h i sp r o j e c t . W e t h a n k R a n d y B u c k l i n a n d S e e n u S r i n i -v a s a n f o r m a k i n g a v a i la b l e t he C o f f e e d a t a. W e a l sot h a n k P e t e N y e , a n d t h e a n o n y m o u s r e v i e w e r s o f th i sj o u r n a l f o r t h e i r c o n s t r u c t i v e s u g g e s t i o n s .

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