TRENDS IN MATHEMATICAL PSYCHOLOGY ppt

TRENDS IN MATHEMATICAL PSYCHOLOGY... TRENDS IN MATHEMATICAL PSYCHOLOGY 1984 NORTH-HOLLAND... 1984 All rights reserved.. Library of Comgmm Catmlogl~g 11 PmbUeathm Datm MBin.entry under

TRENDS IN MATHEMATICAL PSYCHOLOGY

TRENDS IN MATHEMATICAL PSYCHOLOGY

1984

NORTH-HOLLAND

Elsevier Science Publishers B.V 1984

All rights reserved No part of this publication may be reproduced stored in a retrieval system or transmitted in any form o r by any means electronic mechanical photocopying recording o r otherwise without the prior

permissionof thecopyright owner

ISBN: 0 4 4 87512 3

Publishers:

ELSEVIER SCIENCE PUBLISHERS B V

P.O Box 1991 1ocN) B Z Amsterdam

T h e Netherlands

Sole disiribur0r.r for the U.S.A und (briadu

ELSEVIER SCIENCE PUBLISHING C O M P A N Y INC

52 Vanderbilt Avenue

New York N.Y 1OUI7

U S A

Library of Comgmm Catmlogl~g 11 PmbUeathm Datm

MBin.entry under title:

Trends in mathematical psychology

(Advances in psychology ; 2 0 )

Papers presented at the 14th European Mathematical

Ssychology Group Meeting, held in Brussels, Sept 12-14,

1983

Includes indexes

Psychometrics Congresses I Degreef, E., 1953-

TI Buggenhaut, J van, 1937- 111 European

Mathematical Psychology Group

Psychology Group Meeting (14th : 1983 : Brussels,

Belgium)

Netherlands) 4 20

IV European Mathematicd'

V Series: Advances in psychology ( h t e r d a m ,

BF39.T74 1984 150' 28'7 84-6032

ISBN 0-444-87512-3 ( U S )

P R I N T E D IN THE N E T H E R L A N D S

PREFACE

T h i s volume g a t h e r s most o f t h e papers p r e s e n t e d a t t h e 1 4 t h Eurooean Mathe-

m a t i c a l Psychology Grouo Meetinq The m e e t i n g took p l a c e i n B r u s s e l s f r o m Sentember 12 t o September 14, 1983 and was h o s t e d by t h e V r i j e U n i v e r s i -

t h e i d e a t o make a s e l e c t i o n o f t h e c o n t r i b u t i o n s and t o g a t h e r them i n a book

I n o r d e r t o s t r u c t u r e t h e whole, we t o o k t h e l i b e r t y t o groun t h e papers

i n t o t h r e e p a r t s , knowing t h a t t h e c l a s s i f i c a t i o n can be discussed;

o f t h e papers, indeed, can f i n d a p l a c e i n more than one p a r t

ble hope t h a t t h e s t u d i e s c o l l e c t e d here, f a i r l y r e o r e s e n t t h e d i f f e r e n t

p e r s p e c t i v e s and t h a t t h e volume as a whole w i l l be a dynamic r e s o u r c e f o r those who want t o keeD a b r e a s t o f f l a t h e m a t i c a l Psychology i n g e n e r a l and

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Tree r e n r e s e n t a t i o n s o f a s s o c i a t i v e s t r u c t u r e s i n semantic and

G , Oe S o e t e , W.S DCa.tbo, C.Pl FUMLU, J.D CmoU

ifleak and s t r o n g models i n o r d e r t o d e t e c t and measure o o v e r t y

an Poisson model f o r misreadings

model and d i c h o t o m i z a t i o n o f graded resnon-

P.G.P! .TanAeen, E.E Qobkam

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PARTICIPANTS

H A B D I , L a b o r a t o i r e de p s y c h o l o g i e , Ancienne Facul t e , Rue Chabot Charny,

21000 Di j o n , France

H ALGAYER, D o n n d o r f o r s t r 93, D-8580 Bayreuth, Federal R e p u b l i c Germany

J ANDRES, R a t h a u s s t r 53, 53 Bonn 3, Federal R e p u b l i c Germany

J.-P BARTHELEMY, ENST, Deoartement d ' I n f o r m a t i q u e , 46 Rw B a r r a u l t

75634 P a r i s Cedex 13, France

A BOHRER CRS, S e c t i e voor P s y c h o l o g i s c h Onderzoek, Kazerne K l e i n K a s t e e l -

t j e , 1000 B r u s s e l , Belgium

H F J M BUFFART, P s y c h o l o g i sch L a b o r a t o r i urn , Kathol i e k e U n i v e r s i t e i t

Nijmegen, Postbus 9104, 6500HE Nijmegen, The N e t h e r l a n d s

M.A CROON, K a t h o l i e k e Hopeschool T i l b u r g , Hogeschoollaan 225, T i l b u r g , The Nether1 ands

E DEGREEF, CSOO, V r i j e U n i v e r s i t e i t B r u s s e l , P l e i n l a a n 2, 1050 B r u s s e l , Belgium

L DELBEKE, P s y c h o l o g i s c h I n s t i t u u t , K a t h o l i e k e U n i v e r s i t e i t Leuven,

T i e n s e s t r a a t 102, 3000 Leuven, Belgium

G DE MEUR, U n i v e r s i t e L i b r e de B r u x e l l e s , CP 135, 1050 B r u x e l l e s , Belgium

G DE SOETE, D i e n s t v o o r Psychologie, R i j k s u n i v e r s i t e i t Gent, H e n r i

Dunantlaan 2, 9000 Gent, Belgium

M DESPONTIN, CSOO, V r i j e U n i v e r s i t e i t B r u s s e l , P l e i n l a a n 2, 1050 B r u s s e l , Be1 gium

P DICKES, U n i v e r s i t e de Nancy 11, Bd A l b e r t I , BP 3397, 54015 Nancy-Cedex, France

S GROSSBERG, Center f o r A d a p t i v e Systems, Boston U n i v e r s i t y , 111 C u m i n g t o n

S t r e e t , Boston, Massachusetts 02215, USA

P.K.G GUNTHER, S i e b e n g e b i r g s t r 11, D-5330 K o n i g s w i n t e r 41, F e d e r a l R e p u b l i c Germany

M HAHN, M o l l w i t z s t r 5, D-1000 B e r l i n 19, Federal R e p u b l i c Germany

K HERBST, I n s t i t u t filr Psychologie, U n i v e r s i t a t Regensburg, U n i v e r s i t a t s t r

31, D-8400 Regensburg, Federal R e p u b l i c Germany

D HEYER, I n s t i t u t f i r P s y c h o l o g i e , U n i v e r s i t a t K i e l , Ohlshausenstr 40/60, D-2300 K i e l , Federal R e p u b l i c Germany

M.G.H JANSEN, I n s t i t u u t voor Onderwi jskunde, R i j k s u n i v e r s i t e i t Groningen, Westerhaven 16, 9718 AW Groningen , The N e t h e r l a n d s

V.Y KRYLOV, Department o f Mathematical Psychology, I n s t i t u t e o f Psychology, USSR, Academy o f Sciences, Moscow 129366, 13 Yaroslavskaya, USSR

N LAKENBRINK, A u f dem Kampf 11, D-2000 Hamburg 63, Federal R e p u b l i c Germany

X LUONG, L a b o r a t o i r e de Mathematiques, U n i v e r s i t e de Besancon, Besancon, France

R.R MAC DONALD, Department o f Psychology, U n i v e r s i t y o f S t i r l i n g , S t i r l i n g FK94LA S c o t l a n d

C MULLER, L e h r s t u h l fur P s y c h o l o g i e 111, U n i v e r s i t a t Regensburg, Postfach

K R A I N I O , U n i v e r s i t y o f H e l s i n k i , P e t a k s e n t i e 44, 00630 H e l s i n k i , F i n l a n d F.S ROBERTS Qutqers, S t a t e U n i v e r s i t y c f f'ew Jersey Deep o f 'lath a t New Brunswick, H i l l Center f o r t h e '4athematical Sciences, Rush Camous, New Rrunswick, New J e r s e y 08903, 1J.S.A

E.E ROSKAM, Vakgroep Mathematische P s y c h o l o g i e , K a t h o l i e k e U n i v e r s i t e i t Nijmegen Postbus 9104, 6500 HE Nijmegen, The N e t h e r l a n d s

M ROUBENS, F a c u l t e P o l y t e c h n i q u e de Mons, 9 Rue de Houdain, 8-7000 Mons, Belgium

SUCK, U n i v e r s i t a t Osnabruck, P o s t f a c h 4469, 45 Osnabruck, F e d e r a l

Republ i c Germany

TEROUANNE, UER Mathematiques, U n i v e r s i t e Paul V a l e r y , BP 5043, 34032

M o n t p e l l i e r - Cedex, France

VAN ACKER, 1 Chaussee de Wavre, 1050 B r u x e l l e s , B e l g i u m

VAN BUGGENHAUT, CSOO V r i j e U n i v e r s i t e i t B r u s s e l , P l e i n l a a n 2

J.C VAN SNICK F a c u l t 6 des Sciences Economiques e t S o c i a l e s U n i v e r s i t e de

1 ' E t a t a Mons, 17 P l a c e Warocque, 7000 Mons, Belgium

V.F VENDA Department o f E n g i n e e r i n g Psychology, I n s t i t u t e of Psychology, USSR Academy o f Sciences, Moscow 129366, 13 Yaroslavskaya USSR

R VERHAERT, CSOO, V r i j e U n i v e r s i t e i t B r u s s e l , P l e i n l a a n 2 , 1050 B r u s s e l ,

B e l g i u m

Participants

N D VERHELST, Subfacul t e i t der psychologie, vakgroep PSM, R i jksuniversi-

t e i t Utrecht, Sint-Jacobsstraat 14, 3511 BS Utrecht, The Netherlands

Ph VINCKE, I n s t i t u t de Statistique, Universite Libre de Bruxelles,

CP 210, 8-1050 Bruxelles, Belgium

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P A R T I PERCEPTION, LEARNING AND MEMOR Y

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E Degreef and J Van Bu genhaut (editors)

0 Elsevier Science Dublisfers B V (North-Holland), 1984

TREE REPRESENTPTIONS OF ASSOCIATIVE STPUCTUpE5 I N SFFWNTIC AND

E P J SOD1 C VFF1ORY RE SF ARCH Herve Abdi

L a b o r a t o i r e de Ps.vcholoaie, D i j o n Jean-Pierre Barth12l&ny F.PJ.q.T., P a r i s Xuan Luona

L a b o r a t o i r e de Mathematioues, Besanqon lnle exnose some research i n the area o f psycholorry o f

The u t i l i z a t i o n o f c l u s t e r i n q methods f o r a t t e s t i n a t h e o r a a n i r a t i o n o f memo-

r v o r r e v e a l i n g i t s s t r u c t u r e has been s t r o n y l y advocated r e c e n t l y by some authors i n d i f f e r e n t areas o f c o g n i t i v e Fsycholooy (see, among o t h e r s :

M i l l e r (1969) , Henley (1969), F r i e n d l y (19781, Rosenbera e t a1 (1968) , (1972), ( 1 9 8 2 ) ) Most o f t h e used methods amount t o r e n r e s e n t the o r i g i n a l m a t r i x

b y an U l t r a w t r i c Tree Recently, t h e r e has been an a t t e m p t t o b u i l d sow

methods l e a d i n g t o r e n r e s e n t a t i o n s l e s s s t r i n g e n t than the c l a s s i c a l U1 t r a -

m e t r i c Tree, i e the A d d i t i v e Tree (see C a r r o l l & Chang (1973), Cunningham (1974), (1978); S a t t a t h R Tversky (1977)) Ye nronose h e r e a f t e r an (econo-

m i c a l ) h e u r i s t i c g i v i n a an A d d i t i v e Tree from a o r o x i m i t y m a t r i x and i l l u s t r a -

t e i t w i t h some examples b o r r o w i n g f r o m o u r c u r r e n t research o r from c l a s s i -

c a l oaners

T h i s naner i s t h r e e f o l d : we f i r s t d e s c r i b e

2 A RUNCH OF EXAWLES

2.1 RAPTLET': : " ' i l ' f T C THE GHOSTS"

I n 1932, R a r t l e t t asked i: few suhdects t o read an l r w r i c a n I n d i a n f o l k t a l e (named "The '.tar o f the chosts"! and t o r e c a l l the s t o r i t on r e v e r a l occasions ( a w t h o d c a l l e d "rerleatec' r e n r o d u c t i o n " ) ; i n a v a r i a n t o f the method ( i e

" y e r i a l r e n r o d u c t i o n " ) a chain o f d i f f e r e n t s u h i e c t s i s used, the f i r s t

heinq shown the o r i c l i n a l t e x t and then r e c a l l i n n f o r the second s u b j e c t who would nass i t t o a t h i r d and s o on These c l a s s i c a l e x n e r i w n t s o f P a r t l e t t

w i l l serve here t o i l l u s t r a t e a s e t o f i n f o r m a t i c nrocedures, the aim of which i s t o b u i l d s o w distances hetween t e x t s

I t must be c l e a r t h a t when we sneak of t h e t e x t oiven by a s u b i e c t , we

c o u l d sqeak as w e l l o f a s e t o f themes o r ideas aiven by a suhdect D r o v i - dino an adequate codinc; o f the raw data

The t e x t s are f i r s t t r a n s f o m d i n a d i s k f i l e , then f o r each t e x t we b u i l d the Lexicon associated w i t h i t This Lexicon c o u l d be e i t h e r a Boolean Lexicon (i e i t rnerelv i n d i c a t e s the Presence o r the Absence o f the i t e m

o f Vocabulary) o r an i n t e o e r Lexicon ( i e i t i n d i c a t e s t h e numher o f Occur- rences o f each i t e m ) From the d i C f e r e n t Lexicons (Boolean or I n t e g e r ) we

b u i l d - by union - a General Lexicon t h a t d e f i n e s the Vocabularv shared hy the d i f f e r e n t t e x t s

R ) Construction o f distances hetween t e x t s

k n e n d i n q on the p o i n t o f view adonted, we c o u l d d e f i n e d i f f e r e n t distances;

as an i l l u s t r a t i o s we examine t h r e e ways:

( i \ the t e x t s as suhsets of the Vocabulary

( i i ) the t e x t s as R i - o a r t i t i n n s o f the Vocahularv

( i i i )a "orobabi 1 i s ti c " q e n e r a 1 i t a t i o n

Dpncte by L i the Lexicon associated w i t h a t e x t T i , the aeneral Lexicon hy

V = 0 L i and by the comlement o f L i i n V

i

Tree representations associative structures

d ( T i , T j ) = 2( l L i n L i l + l E q n l ) ( l L i ~ ~ l + l E n L j l )

= 2( lT"Kil)( I L i A L j I ) ( i i i ) I n o r d e r t o take e x n l i c i t l y account o f t h e I n t e g e r Lexicons we c o u l d

T h i s r e s e a r c h l a y s on t h e b o r d e r between t h e work on t h e o r o a n i z a t i o n o f t h e semantic memory and t h e work unon the " i m p l i c i t Dsycholoqy" The ourpose i s

t o d e s c r i b e t h e s u b j e c t i v e o r s a n i z a t i o n o f t h e q u a l i f i e r s o f t h e c h a r a c t e r

As a m a t t e r o f f a c t , i t has o f t e n been n o t e d t h a t we t e n d t o qroup s u b j e c -

t i v e l y some f e a t u r e s o f c h a r a c t e r as i f we has an " I m o l i c i t Theory o f Per-

s o n a l i t y " ( c f e.a., Rosenherrr e t a1 (1972), 'Veoner and V a l l a c h e r ( 1 9 7 7 ) )

I n t h i s e x p e r i m e n t we s e l e c t f i f t y e i q h t q u a l i f i e r s o f t h e c h a r a c t e r ( u s i n a some Thesauruses and a b i t o f l i t e r a t u r e , ) These q u a l i f i e r s a r e then

o r i n t e d on s e o a r a t e cards and a i v e n i n d i v i d u a l l y t o t w e n t y - e i o h t s u b j e c t s

w i t h t h e r e q u e s t t h a t he o r she s o r t t h e cards i n t o D i l e s w i t h t h e c o n s t r a i n t

t h a t " t h e cards i n a same w i l e g i v e t h e f e e l i n g t o no t o o e t h e r " ; s u b j e c t s were f r e e t o choose t h e numher o f D i l e s f o r s o r t i n n ( f o r a r e v i e w o f t h e

p r o and c o n t r a o f t h e s o r t i n a method, see Rosenbern ( 1 9 8 2 ) )

N o t i c e t h a t one c o u l d " fuzzy " t h e "boolean" d i s t a n c e i n (i) and ( i i ) by

t a k i n g t h e f u z z y e q u i v a l e n t o f t h e u n i o n and i n t e r s e c t i o n , i .e Min and Max

*

6

Hence, each subject exoresses his oninion by a nartition on the s e t of the

n u a l i f i e r s , a n d f o r usino as a 0-rlethodoloay ( c f Kerlinoer (1973)) the afore evoked distances between nartitions could e a s i l y be used The p a r t i -

t i o n given by a subject i s associated with a matrix whose rolls and columns ren-esert the Q u a l i f i e r s , a n d where we p u t a 1 a t the intersection of a row

and a column i f the nualifiers are not sorted i n the same n i l e by the sub-

i e c t Obviously t h i s i s a distance matrix ( c f W l l e r ( 1 9 6 9 ) \ , and so will

he the matrix definetiby the sum o f the matrices of tbe d i f f e r e n t subjects

I n this matrix we simnly count the number of suhjects who do n o t n u t tooe- ther the q u a l i f i e r s

by the sum o f the so-called incidence matrix (where a 1 means t + a t the q u a -

l i f i e r s are i n the same n i l e ) , f n r commodity reasons t h i s i s the m a t r i x we qive l a t e r ( c f Table 3 )

I t must be noted, in nassinq, t h a t the Data obtained a n d consenuently the distance matrix denend uoon the w t h o d s desinned f o r o b t a i n i n n such Data

I n n a r t i c u l a r , other wthods ( e 0 word associations, o r distances between words in free-recall, e t c ) lead to other results (see Pbdi (1383))

hierarchical clusterinn usinq Johnson's (1967) connectedness a n d diarceter mthod These results are recalled i n 4 ( c f Table 4 and Fioure 5 )

The second i s extracted from a study conducted b v Henlev (1969)

tained from twenty one subjects a n estimation of the distance between twelve

animal terms ( f o r each subject she simoly counts the number of terms sepa-

r a t i n g the terms of i n t e r e s t i n a l i s t given by the subiect nrovided w i t h the instruction t o " l i s t a l l the animals thev could"); the matrices are then standardized and the mean f o r each cell (e.4 across subjects) give the entrv o f a dissimilarity matrix

Finallv, Friendly (1979), i n a naoer akind t o the two orevious ones,

Pually vie could have & f i r e d a matrix o f co-occurences

Fiftv college students, each sorted fortv-eiaht words ac-

She oh-

Tree representations of associative structures

s t r o n c r l y advocate t h e use o f h i e r a r c h i c a l c l u s t e r i n ? t o diaaram t h e memory

o r p a n i z a t i o n

e x c e p t t h a t t h e o r i o i n a l d a t a came from f r e e r e c a l l exDeriments The data were "a n r i o r i " o r q a n i z e d i n t h r e e subsets: animal terms, n a r t s o f t h e huran body, vegetables The a i r o f t h e s t u d y was t h e r e c o v e r y o f t h i s a D r i o r i

t h e Sranh, i e b e i n q a t r e e bfe d e t a i l t h i s p o i n t i n a moment

3.1 TRFFS

A q e n e r a l d e f i n i t i o n o f a Tree can be a c y c l e f r e e , connected, u n d i r e c t e d aranh; f o r convenience assume a v a l u a t i o n on t h e edqes, and d e f i n e t h e d i s - tance between two v e r t i c e s as t h e l e n a t h o f t h e n a t h f r o m one v e r t e x t o t h e

o t h e r

The U l t r a m e t r i c Tree c o u l d he c h a r a c t e r i z e d - besides o t h e r c o n d i t i o n s - by

t h e c l a s s i c a l " u l t r a m e t r i c i n e q u a l i t y " Three v e r t i c e s - say x,y,z - on

H Abdi, ].-P Bartliildmy X Luong

So t h e n r o b l e m i s t o h u i l d un t h e h e t i e r u i t r a m e t r i c a n n r o x i m a t i o n of a c i - ven s i m i l a r i t y

( c f <neath and 5okal (1973), H a r t i g a n ( 1 9 7 1 ) ) I n s t e a d o f i m n o s i n ? t h e

u l t r a w t r i c i n e q u a l i t y on t h e r e n r e s e n t a t i o n o f a d i s s i m i 1 a r i t v ( o r d i s t a n c e )

m a t r i x , some a u t h o r s have nronosed t o weaken t h e u l t r a m t r i c i n e q u a l i t y i n

o r d e r t o o b t a i n a m r e o e n e r a l and more n a t u r a l r e D r e s e n t a t i o n ( c f C a r r o l and Chano ( 1 9 7 3 ) , Cunnincham (1?174), (1978) ; S a t t a t h and Tverskv (1977) cal1c.C' f o l l o w i n g t h e a u t h n r s we? rlhtcd t r e e , f r e e t r e e , n a t h l e n g t h t r e e o r

u n r o o t e d t r e e ( c f amonn o t h e r s : Cicrr'an (1958) , Fakami and V a l ! (1964) , nuneman (1971), Dobson (1974))

f o l l o w i n g i n e o u a l i t y , h o l d i n q f o r e v e r y f o u r - u n l e t - say x,y,u,v - :

d ( x , y ) t d(u,v) s 'lax [d(x.u) + d ( y , v ) ; d ( x , v ) + d ( y , u ) l

I t can be shown t h a t t h e u l t r a m e t r i c i n e q u a l i t y i s s t r o n ? e r than the a d d i -

t i v e i n e q u a l i t y which, i n t u r n , i m o l i e s t h e t r i a n g l e i n e o u a l i t y

The a l g o r i t h m s f o r f i t t i n r l an a d d i t i v e t r e e t o a n r o x i m i t v m a t r i x a r e l e s s numerous than those desioned f o r t h e s a e c i e s u l t r a m e t r i c t r e e The c o n s t r u c -

t i o n o f t h e a d d i t i v e t r e e can be s e a a r a t e d ( f o r t h e c l a r i t y of t h e exolana-

t i o n ) i n t o two p a r t s :

1) t h e f i n d i n g o f t h e t r e e - s t r u c t u r e (we darc! n o t use t h e t e r m s k e l e t o n )

2 ) t h e e s t i m a t i o n o f t h e v a l u a t i o n o f t h e branches o f t h e t r e e

The c l a s s i c a l apnroach ( a s i l l u s t r a t e d b y Cunninqham (1974), (1978);

S a t t a t h and Tversky ( 1 9 7 7 ) ) makes a d i r e c t use o f t h e a d d i t i v e i n e q u a l i t v

f o r the f i n d i n g o f t h e t r e e - s t r u c t u r e , and then e s t i v a t e s t h e v a l u a t i o n

w i t h a l e a s t - s q u a r e method F o r o u r n a r t , we pronose a n o t h e r anproach t h a t

we d e t a i l i n a moment and c o n t r a s t w i t h t h e c l a s s i c a l one

3.2 CONSTR!KTION Or TIIE :REF STRUCTURE

f r o m A and f o r a l l (u,v} f r o v E ( w i t h x+u, x+v, ypu, y p v ) , t h e f o l l o w i n g

i n e n u a l i t y i s v e r i f i e d

d(x,y) t d(u,v) < f l i n Id(x,u) + d(y,v); d(x,v) + d(.v,u)l (1)

I f E corresponds t o t h e t e r m i n a l v e r t i c e s o f a t r e e , t h e n t h e f o l l o w i n g

i n e q u a l i t v h o l d s f o r a l l f o u r - u n l e t (x,y,u,v):

d(x,y) + d(u,v) i d(x,u) + d(y,v) s d(x.v) + d(v,u)

( o r any o f t h e f i v e i n e q u a l i t i e s t h a t can he deduc?d f r o m t h i s

m u t a t i o n )

Then i f { x , y l i s a l o o s e c l u s t e r on a t r e e , from ( 2 ) and (1) f o

d(x,y) + d(u,v) < d(x,u) + d(y,v) = d(x,v) + d(y,u)

So, i n t h e f o l l o w i n p t r e e w i t h t e r m i n a l v e r t i c e s x,y,u,v

Y

( 2 ) one by p e r -

1 ows :

{x,y} on t h e one hand, and { u , v l on t h e o t h e r a r e l o o s e c l u s t e r s , and ( 2 )

i s then d e f i n e d I n o a r t i c u l a r , x and y, u and v a r e t o adont t h e f o l l o -

10 H Abdi, J.-R BarfkiEmy a d X Luong

L e t ( E,d) be an a d d i t i v e t r e e and k equal t o 2

d i s t a n c e g i v e s t o e v e r y branch o f t h e t r e e t h e l e n g t h 1

the:

The s o - c a l l e d c a n o n i c a l

tlw we can g i v e

Prooosi t i o n : I f t h e c a n o n i c a l d i s t a n c e between trio t e r m i n a l v e r t i c e s

- say a,b - i s two, then k-Sc(a,h) i s maximal and k-Sc(a,u)=k-Sc(b,u)

compute then a s c o r e m a t r i x Me l o o k f o r t h e p a i r w i t h t h e maximal s c o r e

- say a,b -, t h e n we merne a,b t o F i v e c and we Dose:

3.3 ESTIMATIOK OF THE VALUPTION OF THE EDGES

o f t h e branches and t denotes t h e transDosed m a t r i x

Voreover S a t t a t h and Tverskv i n d i c a t e t h e e x i s t e n c e - b u t w i t h o u t g i v i n g

i t - o f a method a v o i d i n n t h e comnutation o f t h e i n v e r s e o f a m a t r i x ( 8 )

o f t h e d i s t a n c e ; i e we embed t h e d i s s i m i l a r i t i e s i n an E u c l i d e a n space, and use t h e g e o m e t r i c a l D r o n e r t i e s o f t h i s sDace i n o r d e r t o o b t a i n an e s t i -

The q e n e r a l i d e a o f o u r nrocedure i s t o make a o e m t r i c a l e s t i m a t i o n

d(z,u) = [d(a,u) + d(b,u) - d(a,b)l / 2

I n t h e more g e n e r a l case where d i s a d i s s i m i l a r i t y , we i n t r o d u c e a " c e n t r a l

f r o m E Now z r e o r e s e n t s { a , b l ; we make t h e h y n o t h e s i s t h a t z must be

b r o u q h t n e a r e r t o 0 According t o t h i s , we can determine by t h e geometry

t h e new d i s s i m i l a r i t y d(z,u) f o r aT1 u f r o m E - I a , b l The nrocessus i s

t h e n r e i t e r a t e d p a r a l l e l l y w i t h t h e s c o r e procedure d e s c r i b e d p r e v i o u s l y

3.4 EVALUATInN OF THE ALGnRITHt’

The nrocedure we oronosed i s c l e e r l v ‘ i c l r i s t i c , and so, i t c o u l d he i n t ? -

r e s t i n o t o have an e v a l u a t i o n o f t h e q u a l i t y o f i t One n o s s i b l e way can be

t n oenerate some random d i s t a n c e m a t r i c e s , t o h u i l d t h e a d d i t i v e t r e e , t h e n

t o r e h u i l d the d i s t a n c e m a t r i x and t o n e a s u m C k c‘ecirec c f f i t between t h e

o r i g i n a l m a t r i x and t h e r e h u i l t m a t r i x

choose as a measure o f f i t t h e c l a s s i c a l o r o d u c t mment c o r r e l a t i o n ! h u t i t

i s we11 known t h a t r i s s t r o n a l y r e l a t e d t o some o t h e r measures o f f i t , e 0

4 S O T QESULTS

The d i f f e r e n t f i g u r e s and t a b l e s a r e q i v e n a t t h e end o f t h e oaDer

J 1 THE WAR OF THF GHOSTS

!,!e examine h e r e t h e r e s u l t s o b t a i n e d w i t h t h e d i s t a n c e between t e x t s seen

as B i - p a r t i t i o n s ( a s d e f i n e d i n 2 1 R i i ) F i g u r e 1 g i v e s t h e a d d i t i v e

t r e e o b t a i n e d w i t h t h e “ r e p e a t e d r e n r o d u c t i o n ” method have i n c l u d e d

, lree representations of associative structures 13

t h e o r i g i n a l t e x t f o r ease o f comparison F o r convenience, we g i v e t h e

d i s t a n c e m a t r i x (Table 1) an6 t h e "ao r e s u l t i n g o f a F a c t o r i a l A n a l y s i s o f

o t h e r , s o do L 1 and UPIC b u t O R I G and L 120 a r e q u i t e f a r f r o m each o t h e r

So t h e two methods d i f f e r when a g e n e r a l view i s n o t o b v i a u s

The e f f i c i e n c y o f o u r a l g o r i t h m apoears here: we needed seven

P r o b a b l y because t h e i m p l i c i t model h e r e i s one o f c l u s t e r r a t h e r

To be more o r e c i s e , t h e s o r t i n g i n s t r u c -

between s u b j e c t s ( c f t h e d a t a m a t r i x ) l e a d s o h v i o u s l y t o a " c l u s t e r " s o l u -

t i o n N e v e r t h e l e s s , s o r e d i s t i n c t c l u s t e r s o f q u a l i f i e r s can he i d e n t i f i e d and he i n t e r o r e t e d i n t e r m n f " i m a l i c i t p s y c h o l o n y " , h u t t h i s i n t e r p r e t a -

t i o n s (see t e x t ) The s u b j e c t s a r e denoted by t h e l e t t e r ( s ) b e g i n n i n g t h e

L a b e l , t h e f i g u r e s f o l l o w i n o t h e l e t t e r ( s ) i n d i c a t e t h e number o f days be- tween t h e f i r s t p r e s e n t a t i o n o f " t h e war o f t h e Ghosts" and t h e r e c a l l

C R I ~ N A L L I

F i o u r e 1

A d d i t i v e t r e e o b t a i n e d f r o m t h e d i s t a n c e m a t r i x of Table 2 The s u b j e c t s a r e denoted by t h e l e t t e r ( s ) b e q i n n i n o t h e l a b e l ,

t h e f i g u r e f o l l o w i n ? t h e l e t t e r ( s ! - i n d i c a t e t h e number o f days

between t h e f i r s t P r e s e n t a t i o n n f t h e '%r o f t h e Ghosts" and t h e r e c a l l

The s u b j e c t s a r e denoted bv the l e t t e r ( s ) beginnincl the l a h e l ,

the f i g u r e s follorrina the l e t t e r ( s ) i n d i c a t e the numher o f days

between the f i r s t n r e s e n t a t i o n of “ t h e ‘Jar of the Ghosts” and

t h e r e c a l l

Fiaure 3 Additive t r e e obtained from the distance matrix b u i l t f r o p the

d a t a o f R a r t l e t t (1932): "Serial reproduction"

The l a b e l s g i v e the s e r i a l o r d e r

Co-occurence m a t r i x for 48 Enalish nouns

The c e l l s indicate the number o f subjects who o u t the words i n the

same p i l e (number of subjects: 5 0 )

Data from W l l e r ( 1 9 6 9 )

s

5

Tree representations associative structures 23

o m

A I D COUNSEL

vow

HONOR

E A S E REGRET

P r o x i m i t y m a t r i x between animal terms f r o m l!enley (1969)

D r Y LION

MOUSE

F i o u r e 9

E u c l i d e a n r e p r e s e n t a t i o n ( f a c t o r i a l a n a l y s i s o f d i s t a n c e ) o f t h e

d i s t a n c e m a t r i x o f Table 5 ( H e n l e y ' s animal t e r m s )