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On clustering with chernoff-typefacesEugene F. Tidmore a & Danny W. Turner aa Baylor University , Waco, TexasPublished online: 27 Jun 2007.

To cite this article: Eugene F. Tidmore & Danny W. Turner (1983) On clustering withchernoff-type faces, Communications in Statistics - Theory and Methods, 12:4, 381-396, DOI:10.1080/03610928308828466

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C O M N . STATIST.-THEOR. METH., 1 2 ( 4 ) , 381-396 (1983)

ON CLUSTERING WITH CHERNOFF-TYPE FACES

F . Eugene Tidmore and Danny W. Turner

Bay lo r U n i v e r s i t y , Waco, Texas

Key W o r d s a n d P h r a s e s : l i n e p r i n t e r f a c e s ; c l u s t e r i n g mult.1- v a r i a t e d a t a ; e v a l u a t i o n o f c l u s t e r i n g w i t h . l i n e p r i n t : t ? r f a c e s ; Rand s t a t i s t i c ; s ing1 .e l i n k a g e ; c o m p l e t e l i n k a g e ; a v e r a g e l i n k a g e ; W a r d ' s m e t h o d .

ABSTRACT

Chernof f (1973) in t roduced a new procedure f o r rep

mu l t id imens iona l da ta by u s i n g car toon-1 i k e faces drawn

pen p l o t t e r , w h i l e Turner and Tidmore (1977) in t roduced

Cherno f f - t ype faces which can be generated on a l i n e p r

The use o f such faces f o r c l u s t e r i n g m u l t i v a r i a t e data

e s e r ~ t ing

by ;I

asyr imetr ic

nter-.

S 21

we1 l known technique. However, t h e r e have been few a t tempts

t o eva lua te t h i s g r a p h i c a l procedure i n a sys temat i c f a s h i o n .

Th is paper r e p o r t s r e s u l t s ob ta ined i n a comparison of t h e

1 i n e p r i n t e r faces c l u s t e r i n g method w i t h severa l nongraph ica l

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 , i n c l u d i n g s i n g l e , comp;ete,

and average l i n k a g e and Ward's minimum v a r i a n c e method.

1 . INTRODUCTION

The concept o f u s i n g two-dimensional c a r t o o n - l i k e faces t o

represen t m u l t i v a r i a t e d a t a p o i n t s was o r i g i n a t e d by C h e r n l ~ f f

(1973). Each o r i g i n a l Chernof f f a c e c o u l d accomodate up t o

18 v a r i a b l e s per case, a l though c e r t a i n n o r m a l i z a t i o n s v i r t u a l l y

reduced t h i s t o 16.

Copyright O 1983 by Marcel Dekker, Inc.

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382 TIDMORE AND TURNER

The Chernof f faces were drawn by a Calcomp p l o t t e r . Turner

and Tidmore (1977) in t roduced asymmet r i c Cherno f f - t ype faces

t h a t cou ld be generated on a l i n e p r i n t e r . Our goal was n o t

t o d u p l i c a t e the o r i g i n a l Chernof f face on a l i n e p r i n t e r , b u t

t o cap tu re t h e essence o f Chernof f faces u s i n g an inexpensive

procedure which would be r e a d i l y a v a i l a b l e t o d a t a a n a l y s t s . A

t y p i c a l l i n e p r i n t e r f a c e appears i n F i g u r e 1 . Our FACES program

( w r i t t e n i n FORTRAN) generates 8 such faces per page o f o u t p u t .

L i n e p r i n t e r faces can handle up t o 12 v a r i a b l e s per case.

The 12 f e a t u r e s t h a t can be c o n t r o l l e d a r e t h e 4 corners, 2 eye-

brows, 2 eye frames, 2 p u p i l s , nose, and mouth. These f e a t u r e s

operate independent ly , w i t h each one being c o n t r o l l e d by a s i n g l e

v a r i a b l e . (Some f e a t u r e s a r e dormant i f t h e r e a re l e s s than

12 v a r i a b l e s . ) The manner i n which a v a r i a b l e X . c o n t r o l s a

f e a t u r e F. o f t h e p r i n t e r face i s b a s i c a l l y as f o l l o w s . The

range o f X . i s broken i n t o 10 equal l e n g t h i n t e r v a l s . Feature

F. o f face j (drawn f o r case j ) i s determined by the i n t e r v a l

t h a t X . f a l l s i n f o r case j. Thus, t h e number o f d i s t i n c t faces 12

t h a t can be produced i s 10 . Attempts a t a n a l y t i c a l l y e v a l u a t i n g t h e faces technique

seem t o be few. Chernof f and R i z v i (1975) address the problem

o f a s s i g n i n g v a r i a b l e s t o f e a t u r e s and t h e e f f e c t t h i s has on

c l a s s i f i c a t i o n e r r o r us ing p l o t t e r faces. Our e a r l y work on

e v a l u a t i n g t h e p r i n t e r faces procedure f o r c l u s t e r i n g data

f o l l o w e d t h e example o f Chernof f (1973) and i n v o l v e d t h e use o f

da ta w i t h no predetermined " r i g h t 1 ' answers o r c l u s t e r s . Moreover,

t h e r e s u l t s (Turner and Tidmore [19771) p rov ided no comparisons

w i t h o t h e r c l u s t e r i n g a l g o r i t h m s . I n t h i s paper we take a more

sys temat i c approach t o e v a l u a t i n g t h e e f f e c t i v e n e s s o f t h e faces

c l u s t e r i n q procedure by a p p l y i n g i t t o da ta s e t s f o r which i t i s

known t h a t c l u s t e r s o f a p a r t i c u l a r t ype do e x i s t . Several

nongraphica l c l u s t e r i n g a l g o r i t h m s a r e a l s o a p p l i e d t o these same

da ta se ts t o p r o v i d e a b a s i s f o r comparison. These a l g o r i t h m s

inc lude s i n g l e , complete, and average l i n k a g e and Ward's

minimum var iance method.

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CLUSTERING WITH CHERNOFF-TYPE FACES

FIG. 1 . T y p i c a l L i n e P r i n t e r Face

2. PRELIMINARY DEFINITIONS

Th is sec t i o n in t roduces two parameters, 0 and y, t h a t rwa-

sure geometr ic p r o p e r t i e s o f a c o l l e c t i o n o f subsets ( c l u s t e r s )

o f a g i v e n s e t . Each i n v o l v e s c r i t i c a l r a t i o s o f c e r t a i n

d is tances computed f rom a g iven c l u s t e r i n g ( p a r t i t i o n ) o f a s e t

and i s in tended t o p r o v i d e a measure o f d i f f i c u l t y f o r t h e t < i s k

o f recover ing the g i v e n c l u s t e r i n g f rom the unc lus te red d a t a .

A c l u s t e r w i l l be viewed as a se t o f p o i n t s s a t i s f y i n g

some s i m i l a r i t y c o n d i t i o n w i t h i n a l a r g e r s e t o f p o i n t s f rom

some f i n i t e dimensional space. Given t h a t a data se t con ta ins

c l u s t e r s o f a p a r t i c u l a r t ype , i t i s t h e t a s k o f a c l u s t e r i n g

procedure t o l o c a t e these c l u s t e r s . I n t h e case o f c l u s t e r i n g

based on geometr ic d is tance , t h e r e c o g n i t i o n o f such c l u s t e r s

can have v a r y i n g l e v e l s o f d i f f i c u l t y , depending upon how

c lose s i m i l a r p o i n t s a r e t o each o t h e r compared w i t h d i s t a n c e s

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384 TIDMORE AND TURNER

between c l u s t e r s . For t h i s reason, a parameter B w i l l be

de f ined below t h a t g i v e s t h e c r i t i c a l r a t i o o f d i s s i m i l a r p a i r

d i s tances t o s i m i l a r p a i r d i s tances . Roughly, i f B = 2 , t h e

d is tance between two d a t a p o i n t s n o t considered c l o s e t o each

o t h e r would be a t l e a s t tw ice as g r e a t as t h e d i s t a n c e between

two data p o i n t s which a r e judged as c l o s e o r s i m i l a r t o each

o ther .

L e t S be a s e t and l e t x and y be p o i n t s i n S. A f i n i t e

ordered subset C = { z 0 , Z , , . . . , z i s c a l l e d a cha in connec t ing

x and y i n 2 i f z o = x , z = y and z . E S , 12 i I n - 1. I f ---- there i s a d i s t a n c e f u n c t i o n d d e f i n e d on S , then

i s c a l l e d the norm o f the cha in C. I f S i s f i n i t e , then f o r any

p a i r x and y i n S the re e x i s t s a cha in connec t ing x and y i n S

w i t h minimum norm. Such a cha in w i l l be c a l l e d a minimal

connect ing cha in f o r x and y i n S .

L e t S be a f i n i t e se t , w i t h d i s t a n c e f u n c t i o n d , which has

been p a r t i t i o n e d i n t o c l u s t e r s . T h i s p a r t i t i o n i s s a i d t o s a t i s f y

the T p r o p e r t y i f f o r each c l u s t e r A and each p a i r x and y i n A ,

t h e r e e x i s t s a cha in C connect ing x and y i n A such t h a t

/ I c / / < d(A, S-A), where - denotes se t d i f f e r e n c e .

A p a r t i t i o n P i s n o n t r i v i a l i f P con ta ins a t l e a s t two

c l u s t e r s , a t l e a s t one o f which has two o r more p o i n t s .

For any f i n i t e s e t S w i t h d i s t a n c e d and any n o n t r i v i a l

p a r t i t i o n P o f S t h e r e w i l l e x i s t a un ique l a r g e s t r e a l number B

such t h a t p 1 ~ c I 1 5 d(A, S-A) f o r a l l c l u s t e r s A i n P and a1 1

minimal connect ing chains C f o r p a i r s x and y i n A.

Le t x and y be p o i n t s o f A, a subset o f S which has some

d is tance f u n c t i o n d. The l i nkage d i s t a n c e between x and y ,

re1 a t

f o r x

Clear

ve t o A, w i l l be the norm of a minimal connect ing cha in

and y i n A, and w i l l be denoted by L (x, y ) . L e t A

MA = max i t A ( x , y) : x , y E A).

Y , MA # 0 i f f A c o n t a i n s more than one p o i n t .

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CLUSTERING WITH CHERNOFF-TYPE FACES

Theorem 1 g i ves a computat ional procedure f o r o b t a i n i n g

@ f o r n o n t r i v i a l p a r t i t i o n s .

Theorem 1.

L e t S be a f i n i t e se t w i t h d i s t a n c e d and l e t P be a non-

t r i v i a l p a r t i t i o n o f S. Then B = m, where m = min {d(A, S-A)/MA:

Proo f :

c l u s t e r

For any

connect

Since /

A i s a c l u s t e r i n P hav ing a t l e a s t two p o i n t s ) .

The f i n i t e n e s s o f S i m p l i e s t h a t t h e r e e x i s t s some

A' hav ing a t l e a s t two p o i n t s such t h a t m = d (A1 ,S

c l u s t e r A w i t h a t l e a s t two p o i n t s and any min imal

ing cha in C i n A, m 2 d ( ~ , S - A ) / M ~ , hence mM < d(A A -

C I / < MA, ml / c / I 5 mMA ( d(A, S-A). A lso, MA, = [ I - f o r some min imal connec t ing cha in C ' i n A ' ; t h e r e f o r e , i f r > m

then r / I c ' I I = r M A , > mMA, = d ( A ' , S - A ' ) . For s i n g l e t o n c l u s t e r s

A, I I c / / = 0 f o r any cha in C i n A. Then c l e a r l y m l / c l I 2 d ( ~ , , S-A)

i n t h i s case. I t f o l l o w s t h a t m = B.

The f o l l o w i n g e a s i l y proved theorem r e l a t e 5 t h e T p r o p e r t y

and the q u a n t i t y 6.

Theorem 2.

A n o n t r i v i a l p a r t i t i o n P o f a f i n i t e se t S s a t i s f i e s the

T p r o p e r t y i f f B > 1 .

The use o f a v i s u a l c l u s t e r i n g procedure a l s o

another r a t i o might be r e l a t e d t o successfu l i d e n t

e x i s t i n g c l u s t e r s . Let S be a f i n i t e se t w i t h d i s

d which has been p a r t i t i o n e d i n t o c l u s t e r s C 1 ' C 2 '

suggested chat

i f i c a t i o n o f

tance funct: i o n

..., C k , i . > I .

Let b(S) = max { d ( x , y ) : x , y E S ) and l e t p = min {d(C S.-c.) : ' J 1 5 j 2 k } . Then the parameter y = 6(S) /p i s a r a t i o which

r e l a t e s the d iameter o f t h e se t S t o d i s t a n c e s between p o i n t s

i n d i f f e r e n t c l u s t e r s . For l a r g e va lues o f y the d i s t i n c t i o n

between c e r t a i n p a i r s o f c l u s t e r s may be d i f f i c u l t t o d e t e c t .

3. DESIGN OF EXPERIMENT I

Our f i r s t experiment invo lved 48 da ta s e t s w i t h the number

o f p o i n t s i n an i n d i v i d u a l da ta s e t rang ing (randomly) f rom

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386 TIDMORE AND TURNER

28 t o 40. A f a c t o r i a l d e s i g n i n v o l v i n g f o u r f a c t o r s was imple-

mented as f o l l o w s : F a c t o r 1 : a t f o u r l e v e l s ( a i n t h e f o l l o w -

i n g i n t e r v a l s - 11.1, 1.31, 11.4, 1.61, K1.8, 2 .21, L2.4 , 2 .81 ) ;

F a c t o r 2: y a t t h r e e l e v e l s (y = 2, 6 , 1 0 ) ; F a c t o r 3 : d imens ion

o f space a t two l e v e l s (6, 12 ) ; F a c t o r 4 : c o n s t r u c t e d number

o f c l u s t e r s a t two l e v e l s ( 2 , 3 ) . T h i s s t r u c t u r e accoun ts f o r

t h e 48 s e t s o f d a t a t h a t were c o n s t r u c t e d u s i n g an i n t e r a c t i v e

FORTRAN program (SIMDAT) we des igned f o r t h i s purpose. The

"shape" o f an i n d i v i d u a l c l u s t e r i n t h i s expe r imen t i s b e s t

d e s c r i b e d as r e c t a n g u l a r p a r a l l e l a p i p e d and moreover , c l u s t e r s

w i t h i n a s i n g l e d a t a s e t do n o t o v e r l a p ( i . e . , t h e r e e x i s t s sepa r -

a t i n g hype rp lanes ) .

F i v e c l u s t e r i n g p rocedures were a p p l i e d t o each o f t h e 48

d a t a s e t s d e s c r i b e d above. The f i v e a r e s i n g l e l i n k a g e , comp le te

1 inkage, average l inkage (g roup average method) , Ward1 s minimum

v a r i a n c e method, and l i n e p r i n t e r faces. S i n g l e , comple te and

average l i n k a g e were implemented u s i n g BMDP program PIM w i t h

Euc l idean d i s t a n c e ( d i s t a n c e m a t r i x i n p u t , see D i x o n and Brown

(1979) ) .

Ward's method was implemented u s i n g program CLUSTAR (see

Romesburg and M a r s h a l l (1980)) and l i n e p r i n t e r f a c e s were , genera ted u s i n g o u r program FACES (see T u r n e r and T idmore (1980)) .

S ince t h e f o u r n o n g r a p h i c a l methods a r e h i e r a r c h i c a l , t h e i r

o u t p u t does n o t a u t o m a t i c a l l y i n c l u d e a c h o i c e f o r t h e number

o f c l u s t e r s t o use i n t h e f i n a l p a r t i t i o n o f a d a t a s e t . I n

t h i s expe r imen t , f o r t h e n o n g r a p h i c a l h i e r a r c h i c a l methods,

we a lways chose t h e a l g o r i t h m ' s s o l u t i o n wh ich cor responded t o

t h e des igned c o r r e c t number o f groups. However, when faces

was a p p l i e d t o a s e t o f d a t a , t h e o n l y knowledge conce rn ing

c o r r e c t number o f groups was t h a t i t was e i t h e r two o r t h r e e .

The dependent v a r i a b l e i n t h i s expe r imen t i s a measure

o f agreement between two p a r t i t i o n s o f a s e t c a l l e d t h e Rand

s t a t i s t i c (see Rand (1971) ) . I t i s t h e p r o p o r t i o n o f a l l p a i r s

o f p o i n t s t h a t t h e two p a r t i t i o n s ag ree on where agreement

means t h e p a r t i t i o n s b o t h have t h e p a i r t o g e t h e r i n some c l u s t e r

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CLUSTERING WITH CHEWOFF-TYPE FACES 387

o r they bo th have t h e p a i r i n separate c l u s t e r s . C l e a r l y , the

va lue o f t h e Rand s t a t i s t i c ranges f rom 0 t o 1 w i t h 0 be ing

t o t a l disagreement and 1 be ing t o t a l agreement. Our da ta

c o n s i s t s of t h e 240 va lues o f Rand ob ta ined by a p p l y i n g each

o f our 5 c l u s t e r i n g methods t o each o f t h e 48 c o n s t r u c t e d da ta

se ts and computing t h e v a l u e o f Rand between the method's

p a r t i t i o n and the c o r r e c t ( cons t ruc ted) p a r t i t ion. (The Rand

va lue used f o r faces f o r a d a t a set was the average o f two va lues ,

one f o r each a u t h o r ' s face p a r t i t i o n o f the data se t . The

a u t h o r s ' face p a r t i t i o n s were g e n e r a l l y q u i t e s i m i l a r . )

Before r e p o r t i n g the r e s u l t s o f experiment I , we i l l u s t - - a t e

two o f t h e d a t a s e t s used by showing t h e p a r t i t i o n s ob ta ined

us ing faces. F i g u r e 2 shows a p a r t i t i o n ob ta ined u s i n g t h e

faces procedure on a da ta set c o n s t r u c t e d f o r the p a i r f3 = 1 . 1 ,

y = 2. Cases 602, 108, and 930 were i n c o r r e c t l y p laced i n

c l u s t e r 1. F i g u r e 3 shows t h e p a r t i t i o n ob ta ined u s i n g t h e

faces procedure f o r another da ta se t . The two cons t ruc ted c l u s -

t e r s , w i t h B = 2 .2 , y = 10, can be ob ta ined by combining c l u s t e r s

I and I I i n t o a s i n g l e c l u s t e r , w i t h c l u s t e r I l l as the second

c l u s t e r . A p a r t i t i o n w i t h t h e T p r o p e r t y , B = 1 .7 , can be

ob ta ined by d e l e t i n g 704 f rom the p a r t i t i o n represented i n

F i g u r e 3. The reader should n o t e c a r e f u l l y i n F igures 2 and 3

how the faces a r e arranged. I n p a r t i c u l a r , a c l u s t e r may

c o n t a i n two faces t h a t do n o t look a l i k e , bu t t h e r e i s a cha in

o f faces connect ing the two. Thus, we see t h a t us ing a

g raph ica l method, l i k e faces, i s n o t n e c e s s a r i l y r e s t r i c t e d

t o j u s t grouping o b j e c t s t h a t a r e ( p a i r w i s e ) s i m i l a r t o each

o ther .

4. RESULTS OF EXPERIMENT I

Most o f the r e s u l t s r e p o r t e d below a r e based on ou tpu t

generated by runn ing our da ta through BMDP a n a l y s i s o f vari<ance

program P2V. Since the va lues o f t h e dependent v a r i a b l e (Fhnd

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TIDMORE AND TURNER

FIG. 2. Faces fo r Data Set 122

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F I G . 3. Faces f o r D a t a Set 235

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390 TIDMORE AND TURNER

s t a t i s t i c ) tended t o be i n t h e upper end o f t h e i n t e r v a l L O , 13,

an a r c s i n e t r a n s f o r m a t i o n was used.

The B f a c t o r had a s i g n i f i c a n t e f f e c t (P -va lue = .08) w i t h

l a r g e r va lues o f Rand assoc ia ted w i t h l a r g e r va lues of B, and

the main e f f e c t o c c u r r i n g between l e v e l 1 and l e v e l 2 o f B.

The y f a c t o r was h i g h l y s i g n i f i c a n t (P-value - .0007) w i t h t h e

t r e n d be ing l a r g e r va lues o f Rand assoc ia ted w i t h sma l le r va lues

o f y, and the main e f f e c t o c c u r r i n g between l e v e l s 1 and 2.

The dimension o f space and c o r r e c t number o f groups f a c t o r s

were bo th i n f l u e n t i a l ( r e s p e c t i v e P-values o f . O 7 and .01)

w i t h h igher dimension o r h i g h e r number o f groups hav ing a

d i m i n i s h i n g e f f e c t on Rand.

C l u s t e r i n g method was found t o have a h i g h l y s i g n i f i c a n t -4

e f f e c t (P-value < 10 ) . The s i n g l e l i n k a g e a l g o r i t h m was

s i g n i f i c a n t l y b e t t e r than t h e o t h e r c l u s t e r i n g procedures, hav ing

an average Rand va lue o f .98. The o t h e r methods, ranked by

average Rand va lue (which i s shown i n parentheses) were

faces (.89), average l i n k a g e ( .88) , Ward's method ( .87) , and

complete 1 inkage (. 84) .

There were a l s o i n t e r a c t i o n terms t h a t were s i g n i f i c a n t .

Gamma by t r u e number o f groups had P-value = .02 w i t h Rand

having aber ran t h i g h va lues when y = 10 and c o r r e c t number

of groups equals 3. A lso , c l u s t e r i n g method by t r u e number o f

groups was impor tan t (P-value 2 .01) w i t h faces buck ing t h e t r e n d

by hav ing an "ou t -o f -1 ine" h i g h mean Rand v a l u e when t h e t r u e

number o f groups was 3. The l a s t s i g n i f i c a n t two-way i n t e r a c t i o n

was between c l u s t e r i n g method and gamma, w i t h P-value a .0003.

The n a t u r e o f t h i s i n t e r a c t i o n was t h a t mean Rand va lues tended

t o drop sharp1 y from l e v e l one o f gamma (y - 2) t o l e v e l two

(y 2 6) and then remain r e l a t i v e l y cons tan t o r r i s e s l i g h t l y

a t l e v e l th ree ( y - 10) except f o r s i n g l e l i n k a g e , which remained

cons tan t f rom l e v e l one t o l e v e l two and then dropped moderate ly

a t l e v e l t h r e e (y - 10) . Faces a l s o e x h i b i t e d a crossover e f f e c t

by hav ing a lower mean than t h e o t h e r methods a t l e v e l one o f

gamma, b u t h i g h e r mean than t h e o t h e r s (except f o r s i n g l e l i n k a g e )

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a t l e v e l s two and t h r e e . F i n a l l y , one three-way i n t e r a c t i o n

i n v o l v i n g c l u s t e r i n g method, be ta , and t r u e number o f groups bas

s i g n i f i c a n t b u t we p r o v i d e no d e s c r i p t i o n o f t h e n a t u r e o f t h i s

e f f e c t i n t h i s r e p o r t .

Up t o t h i s p o i n t we have examined r e l a t i o n s h i p s among

p a r t i t i o n s generated by v a r i o u s c l u s t e r i n g methods and c o r r e s -

ponding c o r r e c t p a r t i t i o n s o f

Another i n t e r e s t i n g ques t ion

ing a l g o r i t h m was faces most

( f o r each da ta s e t ) the va lue

generated us ing faces and t h e

a se t o f rnu l t i r l imensional point. .

s t o ask which nongraphica l c l u s t e r -

i k e ? To answer t h i s , we computed

o f Rand between t h e p a r t i t i o n

p a r t i t i o n generated by each o f

the o t h e r f o u r a lgor i thms. These va lues were averaged across

a l l the da ta se ts w i t h t h e f o l l o w i n g r e s u l t s . Faces p a r t i t i o n s

were m s t s i m i l a r t o s i n g l e l i n k a g e p a r t i t i o n s w i t h the mean

va lue o f Rand being .91. Average l i n k a g e f o l l o w e d w i t h mean

Rand o f .88, so t h e marg in o f v i c t o r y was n o t l a r g e , b u t impc~r-

t a n t , when combined w i t h t h e r e s u l t s r e p o r t e d i n s e c t i o n 6

below.

5 . DESIGN OF EXPERIMENT I I

Our second experiment invo lved 10 se ts o f da ta hav ing t h e

f o l l o w i n g genera l s t r u c t u r e : 60 t o 90 s ix-d imensional p o i n t s

per s e t ; 2 to 5 m u l t i v a r i a t e normal c l u s t e r s (groups) per set.;

covar iance m a t r i c e s n o t n e c e s s a r i l y equal ; c l u s t e r s may o v e r i 3 p .

There was no systemat ic v a r i a t i o n o f model parameters f o r these

data sets . However, each data se t was cons t ruc ted w i t h a d e f i n i t e

geometr ic c o n f i g u r a t i o n i n mind f o r t h e u n d e r l y i n g c l u s t e r

s t r u c t u r e . We inc luded no r e a l l y "easy" c o n f i g u r a t i o n s (e.g.,

w i d e l y separated s p h e r i c a l groups) and, i n f a c t , some would be

considered q u i t e d i f f i c u l t (e.g., i n t e r s e c t i n g h y p e r e l l i p s o i ~ d s

w i t h equal mean v e c t o r s and n o n t r i v i a l covar iance s t r u c t u r e ) .

Since a d e t a i l e d d e s c r i p t i o n o f a l l 10 c o n f i g u r a t i o n s would se

leng thy , we s h a l l d e s c r i b e o n l y one, da ta s e t 858, i n d e t a i l .

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39 2 TIDMORE AND TURNER

The geomet r i c i dea o f s e t 858 was t h r e e m u l t i v a r i a t e normal groups

ar ranged i n a t r i a n g u l a r c o n f i g u r a t i o n w i t h some o v e r l a p a t

t h e " v e r t i c e s " . T h i n k i n g i n t h r e e d imens ions (even though o u r

d a t a i s s i x - d i m e n s i o n a l here) one can v i s u a l i z e an e l l i p s o i d

w i t h m a j o r a x i s a l o n g t h e x a x i s and one end a t t h e o r i g i n , a

s i m i l a r e l l i p s o i d a l o n g t h e y a x i s , and a t h i r d e l l i p s o i d c o n n e c t i n g

t h e ends o f t h e f i r s t

normal p o p u l a t i o n s hav

ance m a t r i c e s , one can

m a n i p u l a t e t h e p o p u l a t

s e t 858, s p e c i f i c a l l y ,

wo. By s t a r t i n g w i t h t h r e e m u l t i v a r i a t e

ng z e r o mean v e c t o r s and d i a g o n a l c o v a r i -

use v a r i o u s l i n e a r t r a n s f o r m a t i o n s t o

ons i n t o t h e d e s i r e d ar rangement . Data

c o n s i s t e d o f t h e f o l l o w i n g :

Group 1 2 3

Sample S i z e 20

Mean V e c t o r (4.6,0,4.6,0,4.6,0) (2.4,0,-4.9,0,2.4.0) (9,0,-.8,0,9,0)

Covar iance M a t r i x 6 0 . 5 0 5 0 2.3 0 -2.7 0 1 .3 0 1 0 . 5 0 0 0

To measure t h e degree o f o v e r l a p between a p a i r o f groups

o f sample p o i n t s , we compute t h e pe rcen tage o f a l l sample p o i n t s

t h a t i n t r u d e t h e o p p o s i t e g r o u p ' s 90% p r o b a b i l i t y e l l i p s o i d .

Fo r d a t a s e t 858, t h e pe rcen tages were 4 .3 f o r g roups 1 and 2 ,

13.5 f o r 1 and 3 , and 0 f o r 2 and 3.

F i g u r e 4 shows t h e c l u s t e r e d f a c e s f o r d a t a s e t 858. As i n

expe r imen t I , each o f t h e f i v e c l u s t e r i n g methods was a p p l i e d

t o each o f t h e 10 s e t s o f d a t a w i t h t h e Rand s t a t i s t i c b e i n g

t h e measure o f agreement between two p a r t i t i o n s .

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FIG. 4. FACES for Data Set 858 as clustered by one author. The

Rand statistic for this partition (relative to the correct

partition) i s .76. Ward's method generated the best partition

with Rand value equal .81.

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6. RESULTS OF EXPERIMENT I I

We a g a i n used BMDP program P2V t o p e r f o r m t h e necessa ry

compu ta t i ons w i t h t h e a r c s i n e t r a n s f o r m a t i o n b e i n g a p p l i e d t o

' t h e dependent v a r i a b l e Rand.

B l o c k i n g on t h e t e n d a t a s e t s was e f f e c t i v e (P -va lue <

w i t h d a t a s e t s hav ing h i g h e r pe rcen tage o v e r l a p b e i n g more d i f f i -

c u l t f o r t h e c l u s t e r i n g methods t o p a r t i t i o n c o r r e c t l y . - 4

Method o f c l u s t e r i n g was a l s o s i g n i f i c a n t (P -va lue < 10 )

w i t h Ward's method b e i n g b e t t e r t han t h e o t h e r methods. The

methods, ranked by mean v a l u e o f Rand, were Ward 's ( . 7 9 ) , faces

( . 7 3 ) , average l inkage (. 7 0 ) , comple te 1 inkage ( .69 ) , and

s i n g l e l i n k a g e ( . 4 7 ) .

Us ing compu ta t i ons l i k e those d e s c r i b e d a t t h e end o f

s e c t i o n 4, i t t u r n e d o u t t h a t faces p a r t i t i o n s were most s i m i l a r

t o Ward's method p a r t i t i o n s (average Rand = .75) w i t h average

l inkage n e x t (average Rand = . T I ) .

7. SUMMARY AND CONCLUDING REMARKS

I n t h i s paper we have r e p o r t e d on two expe r imen ts t h a t were

des igned m a i n l y t o h e l p e v a l u a t e t h e f a c e s method f o r v i s u a l l y

c l u s t e r i n g m u l t i v a r i a t e da ta . Two new measures, B and y ,

o f c e r t a i n geomet r i c p r o p e r t i e s o f a c l u s t e r i n g ( p a r t i t i o n ) o f

a g i v e n s e t were i n t r o d u c e d and found t o be r e l a t e d t o t h e

c a p a b i l i t y o f c l u s t e r i n g a l g o r i t h m s t o r e c o v e r t h e g i v e n p a r t i t i o n .

The most i n t e r e s t i n g f e a t u r e o f t h e faces method i s t h a t

i t seems t o be f l e x i b l e . Wh i l e f a c e s was n o t t h e o v e r a l l "w inne r1 '

i n e i t h e r expe r imen t w i t h r e s p e c t t o r e c o v e r i n g c o r r e c t

p a r t i t i o n s , i t was second b o t h t imes . Moreover , i n each

expe r imen t , f a c e s genera ted p a r t i t i o n s most s i m i l a r t o t hose

genera ted by t h e w i n n i n g a l g o r i t h m , w h i c h was s i n g l e l i n k a g e i n

expe r imen t I and Ward 's method i n expe r imen t I I . Needless t o

say, t hese two a l g o r i t h m s have q u i t e d i f f e r e n t c l u s t e r i n g

s t r a t e g i e s .

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The r e s u l t s o f these exper iments suggest the f o l l o w i n g

s t r a t e g y f o r u t i l i z i n g t h e g r a p h i c a l procedure faces. Suppose

we have a set o f data t h a t i s t o be p a r t i t i o n e d i n t o a number

o f c l u s t e r s and t h e r e i s , a p r i o r i , no compe l l i ng reason t o

p r e f e r any s p e c i f i c c l u s t e r i n g method. Then c l u s t e r the datz

us ing severa l procedures l i k e the nongraphica l methods

i l l u s t r a t e d h e r e i n and a l s o c l u s t e r u s i n g faces. Now, determ ne

which o f the methods faces i s most s i m i l a r t o and use i t f o r

your a n a l y s i s . The idea i s t h a t faces " p o i n t s t o t h e

winner".

I n t e r e s t i n g examples and more d i s c u s s i o n o f t h e use o f

faces and o t h e r g raph ica l c l u s t e r i n g methods can be found i n

Wang (1978), F ienberg (1979), and Turner (1981).

Bl BLIOGRAPHY

Chernof f , Herman (1973). Usinq Faces t o Represent P o i n t s i n k-dimensional Space ~ r a ~ h i c a l l y , J . Amer. S t a t i s t . A S S O C . - ~ ~ , 361-68.

Chernof f , Herman & R i z v i , M. Haseeb (1975). E f f e c t on C l a s s i - f i c a t i o n E r r o r o f Random Permutat ions o f Features i n Represent ing M u l t i v a r i a t e Data by Faces. J. Amer S t a t i s t . - Assoc. 70, 548-54.

Dixon, W. J. & Brown, M. B., E d i t o r s (1979). BMDP Biomedica'! Computer Programs. U n i v e r s i t y o f C a l i f o r n i a Press.

F ienberg, S. E. (1979) Graph ica l Methods i n S t a t i s t i c s . Arne-. S t a t i s t . 33 (41, 165-178.

Rand, W. M. (1971). O b j e c t i v e C r i t e r i a f o r t h e E v a l u a t i o n o f C l u s t e r i n g Methods. J. Amer. S t a t i s t . Assoc. 66, 846-850.

Romesburg, C. H. 6 M a r s h a l l , K. (1980). CLUSTAR and CLUSTID: Computer Programs f o r H i e r a r c h i c a l C l u s t e r A n a l y s i s . Amel-. S t a t i s t . 34 (3 ) , 186.

Turner . D. W. (1981). Graphica l Methods f o r Represent ing P o i n t s i n n - ~ i r n e n s i o n a l Space. A b s t r a c t s Amer. Math. Soc. 2 (61, 516. (Repr in ts a v a i l a b l e f rom the a u t h o r . )

Turner , D . W. & Tidmore, F. E, (1977). C l u s t e r i n g w i t h Chernof f - t y p e Faces. Proceeding o f t h e American S t a t i s t i c a l A s s o c i a t i o n , S t a t i s t i c a l Computing Sec t ion , 372 - 377.

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396 TIDMORE AND TURNER

Turner , D . W. & Tidmore, F. E . (1980). FACES - A FORTRAN Program f o r Generat ing Cherno f f - t ype Faces on a L i n e P r i n t e r . Arner. S t a t i s t . 34 ( 3 ) , 187.

Wang, Peter C. C. (1978). Graph ica l Represen ta t ion o f M u l t i - v a r i a t e Data. New York: Academic Press.

R e c e i v e d F e b r u a r y , 1980; R e v i s e d J a n u a r y , 1982.

Recommended b y William H . R o g e r s , T h e Rand C o r p . S a n t a M o n i c a , CA

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