Symbolic objects: order structure and pyramidal ... - Springer Link

r.~nnals of Operations Research 55( 1995)277-297

277

Symbolic objects: order structure and pyramidal clustering P. Brito

ENST, Dept. lnlormatique, 46 rue Barrault, F-75634 Paris Cedex 13, France and INRIA, Projet CLOREC, Rocquencourt. F-78153 Le Chesnay Cedex, France

We recall a formalism based on the notion of symbohc object IDiday [15], Brito and Diday [811, which allows to generalize the classical tabular model of Data Analysis. We study assertion objects, a particular class of symbolic objects which is endowed with a partial order and a quasi-order. Operations are then defined on symbohc objects. We study the property of completeness, already considered in Bnto and Diday [8], which expresses the duality extension intension. We formalize this notion in the framework of the theory of Galois connections and stud) the order structure of complete assertion objects. We introduce the notion of c-connection, as being a pair of mappings (f, g) between two partially ordered sets which should fulfil given conditions. A complete assertion object is then defined as a fixed point of the composed/-o g: this mapping is called a "'completeness operator" for it "'completes" a given assertion object. The set of complete assertion objects forms a lamce and we state how suprema and infima are obtained. The lattice structure being too complex to allow a clustering study of a data set. we have proposed a pyramidal clustering approach [8]. The symbolic pyramidal clustering method builds a pyramid bottomup, each cluster being de~ribed by a complete assertion object whose extension is the cluster itself. We thus obtain an inheritance structure on the data set. The inheritance structure then leads to the generation of rules.

!.

Introduction

T h e f o r m a l i s m o f s y m b o l i c objects [8, 15] has been i n t r o d u c e d to extend data analysis to d a t a d e s c r i b e d by intension, in the f o r m o f c o n j u n c t i o n s o f properties. This f o r m a l i s m generalizes the t a b u l a r m o d e l o f D a t a Analysis, allowing to represent m u l t i - v a l u e d d a t a as well as c o m p l e x a n d s t r u c t u r e d data. M o r e o v e r , it allows to take into a c c o u n t the b a c k g r o u n d k n o w l e d g e which expresses d o m a i n semantics. F o c u s is m a d e o n the n o t i o n o f extension. Several a u t h o r s have addressed the p r o b l e m o f k n o w l e d g e r e p r e s e n t a t i o n in this context, see for instance, Michalski [30, 31, 33], Sidi [41], Stoffel [43]. T h e analysis o f s y m b o l i c d a t a led to a new a p p r o a c h o f D a t a Analysis, which has been called S y m b o l i c D a t a Analysis: new objects are to be processed, which can no longer be r e p r e s e n t e d by vectors o f Re; new criteria are considered, mainly based

' J.C. Baltzer AG. Science Publishers

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P. Brito/Symbolic objects

on the duality intension/extension of a symbolic object and on generalization, specialization techniques. On the other hand, output results should be expressed in the same formalism used to represent input data. Several works have been developed in Symbolic Data Analysis, see for instance: Diday [18], Brito and Diday [8], De Carvalho [9], Jacq [28], Sebag and Schoenauer [40], Lebbe [29] and Vignes [44] for applications in biology and medicine. In clustering, the approach that aims at obtaining interpreted clusters is known as conceptual clustering. The problem of looking simultaneously for the clusters and their representation has been set out in Diday [11]: several kinds of representation have been used (see Diday and Simon [21]): in Diday [12] a logical representation of clusters is proposed. In the eighties, several methods of con. ceptual clustering have been developed, we may cite Michalski [32], Diday et al, [19], Sidi [41], Michalski and Stepp [35--37], Michalski et al. [34], Stepp and Michalski [42], Fisher [25, 26], and Diday and Roy [20]. More recently, the work of Decaestecker [10] and Bisson [6] is also following in this direction. We start by recalling the formalism of knowledge representation, based on thc notion of symbolic object and we focus on assertion objects, which may be delined as conjunctions of disjunctions on the values taken by the variables. The notion of extension is defined, as the set of objects verifying a given symbolic object. We define a partial order relation (generalization/:specialization order) and a quasi-order on the set of assertion objects. Operations of union and intersection, which will be needed in the sequel, are also defined. We then focus on the property of completeness, which expresses the duality intension,cxtcnsion, this is the criterium that will be considered in the presented conceptual clustering method, An assertion object is said to be complete if it describes exhaustively its extension, and if it is minimal, for the symbolic order, to fultil this condition. This notion is formalized by introducing two m a p p i n g s / and g, which should fulfil certain conditions, f being the mapping that, given an assertion object, determines its extension, and g a mapping that, given a set of obscrvcd objects, determines an assertion object that generalizes it (that is, whose extension contains the initial set). We come close to the theory of Galois connections (see Birkhoff [5], Barbut and Monjardet [2]). The set of complete assertion objects presents a lattice structure, and we state how suprema and infima are obtained. This approach is related to the work of Wille [45], for the binary case, already proposed in Barbut and Monjardet [2]. The lattice structure being too complex to allow for a clustering analysis of a data set, we have developed an approach based on the pyramidal model (Brito and Diday [8]), Generalizing the hierarchical model, for it allows the presence of nondisjoint clusters, pyramidal clustering yields a structure which is much simpler than lattices, since each cluster is an interval of an order 0 (as a consequence, pyramids present no crossing on their graphical representation). We present an algorithm of symbolic pyramidal clustering. It allows to cluster elementary objects, representing the lines of a classical data array, but also assertion objects.

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Completeness is the main criterium guiding the algorithm: we build a pyramid whose clusters are described by a complete assertion object, the extension of which should be the cluster itself. The method yields an inheritance pyramid, the clusters being automatically interpreted by conjunctions of properties on the variables. We briefly compare the pyramidal symbolic clustering method to other conceptual clustering methods, namely, to C L U S T E R 2 (Michalski [331), COBWEB l Fisher [25, 26]) and U N I N E M (Lebowitz [46]). The inheritance structure between clusters allows the generation of rules. Two methods of rule generation are presented. Finally, an application is presented. 2.

The symbolic objects

In general terms, a symbolic object is defined as a description that is expressed by means of a conjunction of statements on the values taken by the variables. Let f+ be the set of observed objects, f~ = {+'l . . . . . ~,} C_ II. where I] is the entire population under study. Each object is characterized by variables .v,: II ~ O,, i = ! . . . . . p. Let 1," = (.vl . . . . . y?). We then have I::

H~Ol ,,-' ~

x -..

x 0,,,,

( Yl (,.,-9 .....

yp(~))

The point ()',(~,) . . . . . Yv(")) E ( = Ot x . . . x Op is called description ~ 1 ~ , and (' = Oj x ... x O v the description space. 2 I.

THE E L E M E N T A R Y EVENTS

An elementary event, denoted e = [Y, = V,], where 1 < i < p. I/t c O,, expresses the condition "'variable )'t takes its values in I.',". We then define the mapping associated to e: e: ( --. {true,false},

such that e(x I . . . . . . ~v) = true iff x, E Vi. Let us also consider the composed mapping e" = e o Y!:I, where Ylt denotes the restriction of Y to ~L e" : ~ - - ,

{true,false}.

,; --, e(Y(~,')). e'(~.,) = true iffy,(~) is an element of V,.

P. Britoj£:vmbolic objects

280

I)EFINITION

The extension o r e on ~} is defined as exts~e = {~,.' ¢ ~}/e'{w} = true} = e" I(true).

I)fit-INITION

The virtual extension o t e , on ( . is defined as extce=

{ x 6 C/e(x} = true} = e

I(true).

The following p r o p o s i t i o n results immediately from the definitions o f extension and virtual extension. PR(}t}(}SI'II{}N 1

The image b~ Y o f the extension on ~} of an e l e m e n t a r y event e is contained. but not necessarily equal, to the virtual extension o f e, I~XA.M I'Ltl 1

Let ~} {~'l,wz.w~.,,:4} be a set of o b s e r v e d objects, described by two variables, .vl - cohmr. ()l = {t}Im'. red green, w h i t e , . . . }: v: = size, O, = {hig, medium. .,'mall }:

~'1 ~'. .'~ -'a

{_'onsider the elementary event e =

t" I

V-

him' ~lute hlue ,k're¢'n

medium htg htg ~mall

[size = {hig, ,,wdhml}]. Then

ext{~e = {,,.'
[co~our = {blue, red}].

[colour = {h/iw. red}] A 1-':1,,,'.:3} ~ [colour = {Alue}] A [,,'i..-{, = {/~ig. medium}]. e is not complete, g ( . l { e ) ) = [{'ohmr = {blue}] A [size = {hig. medium}] is complete.

P. Brito, Symbolic objects

287

Let a be a complete assertion object (i.e. h ( a ) = a) and E its extension, E = l ( a ) . Then, g(E) = a and E = h'(E). Conversely, if E = h'(E) and a = g(E) then a is complete and E = f(a). IHEOREM I

The set o f all complete assertion objects constitutes a lattice for the symbolic order defined above, where inf and sup are given by: inf(al, a2) = h(al N a2) = g o l ( a l N a2), s u p ( a l . a 2) = al Ua2.

DEFINITION

Given a set o f observed objects ~L a concept o f ~ is defined as a couple (E, a) such that E C_ ~L a C ,4, a is complete and E = l ( a ) . A concept is hence defined as a couple extension intension, it consists on a class together with its description. Here, we generalize the definition presented by Wille [45] for the binary, case, based on the philosophical notion of concept: "'Traditional philosophy considers a concept to be determined by its extent and its intent: the extent consists o f all objects belonging to the concept, while the intent is the multitude of all attributes (or properties) valid for all those objects" [45]. In Machine Learning, however, the notion of concept generally refers to the intension of the class (Rendell [39]L Another point which should be stressed is the fact that we only consider conjunctive concepts, in other words, a couple (E. al is a concept if E can be described by a (discriminant) conjunction, a.

THEOREM 2

The set o f concepts o f ~L with the order defined by (Ei,al) respectively, the assertion objects representing them. We can then write: S ::~ S I VS-,.

4.3.2, Merging method

Let Pl, P2, PI2 E P be such that P~2 = Pl C~P2 ¢ O, and sl:, s~. s 2, respectively, the assertion objects representing them. Then: S 1 /~S'~ ~

5.

512.

Application

In this example, we cluster a set of 30 cat species, which are described by 14 qualitative variables, the 15th variable, that has not been taken into account in the clustering process, indicates the zoological genera. This data is originally from Dorst and Dandelot [22]. In figure 1 we can see the obtained pyramid. The symbolic pyramidal clustering method in this case produced an incomplete pyramid (that is, the cluster corresponding to the whole set is not formed); five terminal clusters can be observed: cluster cluster cluster cluster cluster

1: cat 2: cat 3: cat 4: cat 5: cat

species species species species species

13, 15, 17, 18, 2t, 22, 24, 28, 29, 30; 13, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29; 9, 12, 20, 25; 5, 7, 9, 10, 11, 12, 14; 1, 2, 3, 4, 5, 6, 7, 8, 9, I0, I1, 14.

The descriptions by assertion objects allow us to characterize the clusters: .

Cluster 1 consists of cat species whose fur is not striped nor mottled, that have non-retractile claws, that present a diurnal or mixed predatory behaviour, have the hyoid bone, whose shoulder height is less than 50cm, weight less than 80 kg and length less than 150 cm, tail is of medium size, have well developed canines, preys are small or mixed and hunt.

•

The cat species that constitute cluster 2 have non-retractile claws, nocturnal or mixed predatory behaviour, hyoid bone, shoulder of height less than 50cm, weight less than 10kg and length less than 150cm, well-developed canines and small preys.

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292

30151813 ~ 21 172B24 2':3 16232527 19~L::K) I2 9 1014 7 11 5 8 3 4 2 l 6

Figure 1. Pyramid obtained on the 30 cat species.

•

Cluster 3 consists of cat species whose fur is not striped nor mottled and have short hair. non-retractile claws, diurnal or mixed predatory behaviour, hyoid bone, shoulder height less than 70cm. weight less than 80 kg, length less than 150cm, short tail, well-developed canines, small preys and do not climb trees.

•

Cluster 4 consists of cat species whose fur is not striped nor mottled, have non-retractile claws, whose shoulder is shorter than 70cm, weight over 80kg, length less than 150cm, have small or mixed preys and do not climb l Fees.

•

Finally, cluster 5 is composed of cat species of length greater than 80cm and weight over 10kg.

Clusters I, 2 and 3 only contain cat species of genus felts. The cheetah, the only cat of genus acinonyx, appears in cluster 5 but rather isolated from the other members of the cluster. The group lion-tiger-jaguar-leopard (cats 1-2-3-4) is characterized by the presence of short hair. non-retractile claws, diurnal or nocturnal predatory behaviour (but not mixed), round ears, absence of the hyoid bone, shoulder of length over 70cm, weight over 80kg and length over 80cm, short or medium tail and underdeveloped canines. These cats all belong to the genus pantera; the fifth element of

P. Brito/Symbolic objects

293

Table I Data 1. 2. 3. 4. 5. 6, 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

Lion Tiger Jaguar Leopard Ounce Cheetah Puma Neofi..nebulosa Serval Ocelot Lynx Caracal Viverrin cat Jaguarundi Chaus cat Golden cat Merguay cat Marguerit cat Cafer cat Bieti cat Bengal cat Rouilleux cat Malaysian cat Borneo cat Nigripes cat Manul cat Marble cat Tigrin cat Temminck cat Andean cat

121 123 123 123 321 II1 122 123

2 2 2 2 2 1 1 2

1212 2 2 1 3 1 1 1 2 1 3 1 2 2 1 1 2 1 4 3 1

1 2 1 1 1 2 1 1 2 1 1 1 1 2 1 1 2 1 1 1

3

3

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3

2 2 2 2 2 3 3 3 2 3 2 3 2 3 3 2 3 3 3 3 3

1 2 2 1 1 2 1 1 1 1 2

3 3 3 3 2 3 2 2

3 3 3 3 2 2 3 2

3

3

3 3

3 3 2 2 2 2 2 2

2 2 1 2 3 3 3 3

~

1

3 3

~

3

3 I

2 2 1 1 122 223 3

3

1 1 3 3

1 3 3

1 3

I I 3 3 3 3

1 1 I I 1 2 2 2 2 2 2 1 2 2 3 2 3 2 3 2 2 2 3 2 31 32 32 32 32 31 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 2 2

2 I 1 1 1 2 1 1 2 1 1 2

1 I 1 1 1 4 3 2 3 3 3 3 3 3 3 3 3 3

2 1 1 1 1 1 2 1 1 1 1 1

3 3 3 3 3 3 3 3 3 3 3 3

this genus, the ounce (cat number 5) is a little apart. This can be explained by the fact that on several variables on which cat species I-2-3-4 agree cat 5 takes a different value. We can also observe that the cheetah is the only cat in cluster 5 having retractile claws. 6.

Conclusion and perspectives

The formalism of symbolic objects presented in this paper aims at formalizing knowledge representation. By combining a set theoretic approach with a logical representation, it allows to bridge the gap between Data Analysis and Machine Learning, thus opening the way to the combination of techniques from both

P. Brito/Symbolic objects

294

Table 2 Variables ~'l

fur: 1 2 3 4

~',

hair: I short hair 2 long hair

without spots with spots striped mottled

~,~

claws: I retractile 2 non-retractile

r'4

predatory behaviour: I diurnal 2 diurnal and nocturnal 3 nocturnal

~,~

shape of ears: I round 2 pointed

,.~

hyoid bone: 1 present 2 absent

,'-

shoulder height: I height < 50cm 2 50cm ,v height < 70cm 3 height ;, 70cm

,,~

weight: I weight < 10kg "~ 10kg < weight < 80kg 3 weight > 80kg

~',~

length: I length ~ 80cm 2 80cm ,~ length < 150cm 3 length > 150cm

w,l~ )

tail-length (% body length) I short 2 medium 3 long

r'll

canines: I well-developed 2 underdeveloped

~'lz

prey: 1 big "~ big or small

~'la

tree-climbing: 1 climbs trees 2 doesn't climb trees

r,14

hunting: 1 yes "~ no

I'lr,

zoological classification: 1 genus panthera 2 genus neofilis 3 genus fells 4 genus acinonyx

domains. Moreover, it allows to make a step further in the processing of higher level data, since symbolic objects generally represent aggregates of objects which become the units to be subjected to analysis. The pyramidal clustering method we present allows to cluster the defined objects (thus generalizing the usual Data Analysis data) and, by considering settheoretic criteria, to obtain an inheritance structure whose clusters have an

P. Brito/Symbolic objects

295

immediate conceptual interpretation. We have hence a structure allowing to represent and learn knowledge. Perspectives of future development include extending the method to imprecise or uncertain data, on the one hand, and to take advantage of Data Analysis techniques (such as factorial analysis) to simplify the method, on the other hand. References Ill

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I~1 [gl

[t)] t l~1] II I1 I121 [13] [14] [I 51

[16] [17]

[I 8] [It)]

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