Isabelle BOURNAUD & Jean-Gabriel GANASCIA. LAFORIA-IBP. Université Paris 6. 4, place Jussieu - Boîte 169,. 75252 Paris Cedex 05. France. Fax: (33) 1 44 ...
Conceptual Clustering of Complex Objects: A Generalization Space based Approach
Isabelle BOURNAUD & Jean-Gabriel GANASCIA LAFORIA-IBP Université Paris 6 4, place Jussieu - Boîte 169, 75252 Paris Cedex 05 France Fax: (33) 1 44 27 70 00 Tel: (33) 1 44 27 71 19 e-mail: {bournaud, ganascia}@laforia.ibp.fr
Conceptual Clustering of Complex Objects: A Generalization Space based Approach Isabelle BOURNAUD & Jean-Gabriel GANASCIA LAFORIA-IBP, Université Paris 6 4, place Jussieu - Boîte 169, 75252 Paris Cedex 05, France {bournaud, ganascia}@laforia.ibp.fr
Abstract A key issue in learning from observations is to build a classification of given objects or situations. Conceptual clustering methods address this problem of recognizing regularities among as set of objects that have not been pre-classified, so as to organize them into a hierarchy of concepts. Early approaches have been limited to unstructured domains, in which objects are described by fixed sets of attribute-value pairs. Recent approaches in structured domains use a first order logic based representation to represent complex objects. The problem addressed in this paper is to provide a basis for the analysis of complex objects clustering represented using conceptual graphs formalism. We propose a new clustering method that extracts a hierarchical categorization of the provided objects from an explicit space of concepts hierarchies, called Generalization Space. We give a general algorithm and expose several complexity factors.
Keywords Conceptual clustering of complex objects, Conceptual graphs, Generalization Space, Hierarchical categorization.
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Conceptual Clustering of Complex Objects: A Generalization Space based Approach
1.
Introduction
A key issue in learning from observations is to build a classification of given objects or situations. This kind of inductive learning, where the objects have not been assigned classes by a teacher, is often called unsupervised learning. It consists in searching for regularities in the training objects, so as to organize them into categories (Dietterich, 1990 ; Fisher and Pazzani, 1991; Michalski and Stepp, 1983). Conceptual clustering and traditional techniques developed in cluster analysis and numerical taxonomy differ in two major aspects. First, in the fact that the conceptual clustering problem is not only to build a partition of the set of objects into separated clusters, but also to associate a characterization to each cluster acquired in terms used to describe objects, called intentional description of clusters (Thompson and Langley, 1991). Secondly, the clustering quality in the conceptual clustering paradigm is not solely a function of individual objects, but is dependent on concepts that describe the clusters, i.e. the intentional descriptions of the clusters (Decaestecker, 1993). More precisely, conceptual clustering has been defined by Michalski and Stepp (Michalski & Stepp, 1983) as: - Given: a set of objects and their associate descriptions; - Find: . clusterings that group these objects into concepts; . an intentional definition for each concept, . a hierarchical organization for these concepts. We call categorization the result of a conceptual clustering process. The task of concept formation is very similar to conceptual clustering, with the added constraint that learning be incremental (Gennari, et al., 1989). Early systems in unsupervised concept learning have been limited to unstructured domains, in which objects are described by fixed sets of attribute-value pairs; like in CLUSTER /2 (Michalski and Stepp, 1983), COBWEB (Fisher, 1987; Lebowitz, 1987). However, objects frequently have some natural structure, i.e. have components and relations among these components. Attribute-value based representations are not well suited to represent such complex objects (Thompson and Langley, 1991). Recent approaches in structured domains use a first order logic (FOL) based representation to represent complex objects. As for example, we may refer
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to systems such as C L U S T E R / S (Stepp and Michalski, 1986) or LABYRINTH (Thompson and Langley, 1991). The major problem with such representations is that of the number of matching, i.e. the number of different ways two descriptions structurally match. In FOL based language, there are potentially an exponential number of matchings (Neri and Saitta, 1994). One way to limit the matching complexity is to restrict the expressiveness of the generalization language (Levesque and Brachman, 1985). Such restrictions are easier to express and to formalize in the conceptual graphs formalism than in a FOL based representation (Zucker and Ganascia, 1994). The problem addressed in this paper is to provide a basis for the analysis of complex objects clustering represented using conceptual graphs formalism (Sowa, 1984). Conceptual graphs are formally defined with a model theoretic semantics. They have all the expressive power of logic but are more intuitive and readable, with a smooth mapping to natural language (Ellis, 1993b). Moreover, conceptual graphs are often used in data knowledge representation. The main differences between our approaches and related works are first that the complex objects are represented using the conceptual graphs formalism, and secondly, the conceptual clustering process consists in searching through an explicit space of concepts hierarchies, called Generalization Space. A Generalization Space is an inheritance network, whose construction is based on a restricted graphs' generalization. We propose to extract from the GS one categorization corresponding to a hierarchical organization of the acquired concepts. In section 2, we briefly review the classical clustering methods and present our approach based on the notion of Generalization Space. In the following section, we describe a method to build a Generalization Space, as developed by Mineau in the MSG (Mineau, 1990). In section 4, we describe the conceptual clustering method proposed on an example. We finally give a general algorithm and expose several complexity factors. 2.
Conceptual clustering
2.1. Related approaches Different clustering methods have been developed in the area of research on unsupervised learning. On the one hand, the classic agglomerative clustering methods build a binary tree in a "bottom-up" manner. They begin with as many categories as the number of objects. They compute the similarity between all pairs of categories, and merge the two most similar categories into a single category. The merging process is repeated until all objects have been placed in a single category representing the root of the tree. Examples of systems based on this method are KBG (Bisson, 1993), MK10 (Wolff, 1980).
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On the other hand, the result of divisive methods is a tree built in a "topdown" manner. These methods begin with a single category containing all the objects, and repeatedly subdivide this category until all categories contain only a single object. The partitioning algorithm ("Nuées dynamiques") developed by Diday (Diday, et al., 1982), and which has been used as a basis for developing the CLUSTER's family of systems, uses a top-down approach. These two methods have been both used in Numerical Taxonomy and in Machine Learning. Although they differ in the building process, they both can be viewed as a search through a non explicitly built space of concepts hierarchies (Fisher and Langley, 1985; Gennari, et al., 1989). This is summarized on figure 1. Heuristics
Set of objects
Categorization
Fig.1: Classical approaches principle 2.2.
A new approach
We propose a new approach to conceptual clustering, called COING, which consists in searching a categorization of a set of objects through a space of concepts hierarchies of the objects explicitly built: the Generalization Space. This approach is based on a representation of complex objects using conceptual graphs (Sowa, 1984). 2.2.1 Knowledge representation Our work being related to the MSG developed by Mineau (Mineau, 1990), we have considered the same restrictions on the conceptual graphs: existential graphs, dyadic relations and disjunction of the concepts of a graph. It is not the purpose of this article to detail them here. We will only introduce some terminology and an example used along this article. The term concept will be used within two contexts: - an acquired concept in the conceptual clustering paradigm, designs the entity "category + category description", - a domain concept corresponds to a concept of the conceptual graphs formalism. We call conceptual arc or arc a triplet [domainConcept] -> (Relation) -> [domainConcept] of a conceptual graph (see figure 3). A conceptual arc appearing in an object description is called elementary arc, and one obtained in generalizing an elementary arc is called a generalized arc. Throughout this paper, we will use as an example a set of objects to be clustered (see figure 2). This example, taken from (Gennari, et al., 1989), provides a good example of an application of our approach in spite of the
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fact that these objects do not require the full expression power of the conceptual graphs formalism.
O4
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Fig 2. An example of set of objects In this example, we represent each object using conceptual graphs; figure 3 gives an example of the representation of an object. surface color cell
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number of nuclei number of tails
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Fig. 3: Representation of the O3 domain object using three conceptual arcs 2.2.2 Generalization Space Structure A Generalization Space (GS) is an inheritance network. In this network, a node ni is a pair (cni, dni) where cni, the coverage of ni, is the set of objects covered by ni; and dn i, the description of ni, corresponds to the common characteristics of c n i (i.e. the set of the common conceptual arcs); dn i is a maximally specific description. A node fn is the father of a node n iff: dn is more specific than dfn (arcs of d n are arcs that are not more general than arcs of d fn ) and cn is strictly included in cfn. We call elementary node a node covering exactly one of the provided objects, and generalizing node the other nodes. The Generalization Space is different from the "generalization hierarchy" introduces by Sowa (Sowa, 1984 ). Indeed, in the GS, nodes' descriptions have not to be a connected graph but are only a set of arcs (Mineau, et al., 1990). Figure 4 presents the generalization space obtained with the set of objects presented in figure 2. n1: c = {O} (surface color) [cell]
[concept] [concept]
(number of nuclei) (number of tails)
[concept]
n2: c = (surface color) [cell]
n3: c = (number of tails) [cell]
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n5: c = [cell] (surface color)
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n6: c = [cell] (number of nuclei)
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Fig. 4: The Generalization Space for the set of objects of fig.1: GS1
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2.2.3 The problematic of clustering objects using a GS Any set of nodes of the GS, that cover all the provided objects, may be considered as a categorization of the set of objects. In that way, the GS may be viewed as a space of categorizations. In fact, the GS contains a sub-set of all the potential categorizations of the training objects, according to the restriction used in the building of GS (see § 3.1.). COING 's approach consists in extracting one categorization from the GS. This approach is summarized in the figure 5. Extraction process
Generalization
Set of objects
Generalization Space
Categorization
Fig.5: COING's principle To build a GS from a set of objects represented using conceptual graphs, we use the method developed by Mineau in (Mineau, 1990). We briefly present this method in the next section. To the best of our knowledge, no work has yet dealt with the problem of extracting a categorization from an explicitly built space of concept hierarchies. Section 4 presents more precisely COING 's approach to this problem. 3.
Generalization Space
3.1. Building a GS Building the GS requires to group into clusters all objects that have similarities. One way to "evaluate" similarity between objects is to generalize the objects by searching the common features subsuming the objects. Generalizing objects, in the representation used, requires to match graphs, which is a problem known to be NP complete (Garey and Johnson, 1979). Among the several approaches developed to find a partial solution to this problem, one with a low complexity is the one developed by Mineau in the MSG (Mineau, 1990). This approach consists in searching the largest subsets of common arcs and not the largest common sub graphs. In doing so, it ignores the fact that any two arcs may be connected. In other words, a graph is considered as a set of arcs; matching graphs is viewed as matching arcs. It consists in restricting the generalization language to a one-to-one matching (Hayes-Roth and McDermott, 1978). As a result of this restriction, the complexity of the graph matching process is polynomial. Nevertheless, as previously explained in § 2.2.3., the GS does not contain all the existing generalizations of objects (only the ones that may be expressed in this restricted language).
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3.2. Size of the GS Let n be the number of objects and k the average number of elementary arc in objects' descriptions. The GS nodes are obtained in grouping in the same node all the conceptual arcs covering the same objects. The GS does not contain as many nodes as the number of all possible sub-sets of objects (2n), but only nodes for sub-sets of objects having common features. The number of arcs in the GS, elementary and generalized arcs, is less or equal than 23 * n * k (Mineau, 1990). Since the GS nodes contain at least one arc, the number of nodes of the GS (GS size, noted | GS| ) is linear with the number of objects: | GS| ≤ 8* n * k. In this article, we do not consider the impact of introducing background knowledge on the complexity of the GS's construction. An analysis of such complexity may be founded in (Gey, 1994). 4.
COING's approach
In (Fisher and Langley, 1985), the authors explain that the conceptual clustering process is "composed of three distinct sub processes: the process of deriving a hierarchical classification scheme, the process of aggregating objects into individual classes; and the process of assigning conceptual descriptions to object classes". In our approach, the aggregation and characterization processes are made during the building of the GS. The remaining problem is to derive a classification scheme from the GS. 4.1. Conceptual Clustering based on GS The main categorization structure types that one meets in the literature are partition (mutually exclusive categories), clumping (overlapping categories), hierarchy with overlap, or hierarchy without overlap (Decaestecker, 1993; Fisher and Langley, 1985). As in major conceptual clustering methods, we are interested in finding the fourth kind of structure, i.e. categorizations (Gennari, et al., 1989; Michalski and Stepp, 1981). Indeed, hierarchical organization is very natural for many domains (Langley, 1987). Moreover, hierarchical structures have proven to be useful for organizing and managing large databases because of their searching efficiency (Ellis, 1993a; Levinson, 1984). The goal of COING's approach is to extract a categorization, corresponding to a hierarchical organization of the acquired concepts, from the GS. We briefly present the various alternative hierarchies extractable from the GS, the two stages process of extraction and give a simple extraction algorithm. We then expose the criteria used to choose one hierarchy over another.
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4.2.
A two stage process
4.2.1 Skeleton selection As previously said, the GS has an inheritance network structure. It implies that each node of the GS, excluding the most general (and its immediate sons), may have more than one father, i.e. may be generalized by more than one node of the GS. In a hierarchy, each node has only one father. Extracting such a hierarchy structure from the GS may be viewed as the problem of choosing a unique father for each node having more than one father. We call hierarchy skeleton, or skeleton, the structure obtained after a single father has been chosen for each node of the GS. The selection of a skeleton consists in two stages: - choosing, for each node n having more than one father, one father fnc from the set Fn = {fn1, fn2,.., fnm } of node n's fathers. For example, the two potential skeletons for the GS1 of figure 4 are presented in figure 6. The skeleton sk1 is the result of the choice of the node n3 as a father for the node O3, and the skeleton sk2 the choice of the node n2 as a father for O3. GS1
sk1 n2 n4 O1
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Fig.6: Two potential skeletons, sk1 and sk2, for GS1 of fig.4. - updating the coverage of each father fn i ∈ Fn, i ≠ c, in the skeleton where father fn c has been selected. If the resulting coverage of the node fni is the same as the coverage of one of its sons fnis (for example, n2 and n4 of sk1, and n3 and n5 of sk2), then fn i and fni s are grouped in the same node, and the description of the resulting node is updated: arcs of fn i ' s description that generalize an arc of fn i s' description are removed, other arcs of fn i's description are added to the resulting node. Figure 7 presents the updated skeleton sk'1 of sk1 (figure 6): nodes n2 and n4 have been grouped in a "new" node n'2. The updated content of the node n'2 is presented on figure 8.
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sk1
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O5 O6 Fig.7: The resulting skeleton for sk1 of fig.6. n'2: c = (surface color) [cell]
n5
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n6 O5
O6
[light]
(number of nuclei) (number of tails)
[2] [1]
Fig. 8: Content of the node n'2 of fig.7 (different of node n2 of fig.4). Let SK be the set of potential skeletons for a GS. The size of SK is given by the following formula:
Π
(number of father (node i)) | SK| = node i ∈ GS For example, if each node of the GS has 2 fathers, the size of SK is equal to 2| G S |. In practice, this number may be small, e.g. the number of skeletons for the generalization space GS1 of the figure 4 is 2. 4.2.2 Skeleton reduction Given one skeleton ski, different hierarchies may be extracted. These hierarchies differ in the number of generalizing nodes they contain. In order to generate all the different hierarchies, we define an absorption operation over nodes. Absorbing a node n i into its father node fni consists in propagating the description of n i "down into" its sons' descriptions (except into the sons corresponding to elementary nodes). For example, if the node n5 is absorbed by the node n3, as in h1 on figure 10, n5 description is added to its sons' description n6. The resulting node n"6 is presented on figure 9. As a result of the absorbing operation, the resulting hierarchy does not contain as much information as the initial skeleton; when a node is absorbed into its father, the common features between its sons are lost; consequently the resulting hierarchy does not contain a node for each sub-set of objects having common features. n"6 : c = [cell] (number of nuclei)
[3]
[dark]
(surface color)
Fig.9: Content of the node n"6.
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The process of restricting a skeleton to a hierarchy consists in choosing which nodes of the skeleton to absorb. Figure 10 presents two hierarchies, h1 and h2, extracted from the skeleton sk1 of the figure 7. h1 n1 n1 h2 n'2 O1 O2
n3
n3 n5
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6 n"
O1 2O2 n'
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Fig.10: Two instances of the skeleton sk1. Let GNski be the set of generalizing nodes of the skeleton ski, and Iski the set of potential hierarchies of sk i . The size of Isk i | is given by the following formula: | Iski| ≤ 2 |GNski| 4.3. Algorithm Let us first introduce the notion of a GS's (or skeleton's) layer. A layer corresponds to the set of objects or acquired concepts that have the same depth; the depth of a node corresponding to the number of ancestors of this node. For example, the object O3 and the nodes n4 and n5 of GS1 belong to the GS1's layer of depth 3, noted L(3). Figure 11 presents COING 's extraction algorithm based on this notion of layer. Let n be the given node, nfi be a father of n and nsj a son of n; let c be the coverage of a node and d its description. Let depthmax be the maximum depth of the GS (or the skeleton). Stage 1: Skeleton selection For each depth, from 1 to depthmax, analyse the GS's layer L(depth): . For each node n of layer L(depth) that has more than one father {fn1,fn2,...,fnm} belonging to L(depth - 1): - build the pair (n,{fn1,fn2,...,fnm}) - choose➀ a unique father fnk Œ {fn1,fn2,...,fnm} for n - for each node fni, fni π fnk, remove cn from cfni, if fni has a unique son, agglomerate dfni with its sons description. The structure obtained is a skeleton. Stage 2: Skeleton reduction For each depth, from 2 to depthmax - 1, analyze the skeleton's layer L(depth) . For each non elementary node n of layer L(depth), decide (choice ②) whether or not to absorb it with its father fn of layer L(depth - 1) . If so, add dn to all its sons' descriptions of layer L(depth + 1) The structure obtained is one hierarchy of the acquired concepts.
Fig. 11: COING's algorithm
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As presented in this algorithm, extracting one categorization with a hierarchical structure requires to make two different choices noted choice➀ and choice② on figure 11. We explain, in the next section, which criteria may be used in order to make these choices. 4.4. Extraction criteria Let HC be the set of potential c a t e g o r i z a t i o n s (i.e. hierarchical organization of the acquired concepts) that can be extracted from a GS. The size of HC, function of the number of generalizing nodes of the GS, is given by the following formula:
Σ
| HC| = | Iski| ski ∈ {SK} Consequently, choosing one categorization from HC requires to define criteria. For preferring one categorization over another one, a general criterion may be express as such: Construct categories that maximize similarity within categories and that concurrently minimize similarity between categories (Gluck and Corter, 1985; Hanson and Bauer, 1989). This criterion is a tradeoff between within-category similarity and intercategory dissimilarity of objects. In our framework, this criterion may be interpreted in terms of choice➀ (skeleton selection) and of choice➁ (skeleton reduction). This framework allows to provide an intuitive interpretation of these complex choices w.r.t. the GS. Choice➀ As previously explained, selecting a skeleton can be viewed as choosing a unique father for each node of the GS. Choosing a father for a node corresponds to decide which father is the most important one, according to the clustering task. Three types of criterion may be used for choosing one father: - A local criterion: Select, among the fathers of a node, the one that maximizes an entropic measure; - A global criterion: select the father such that the resulting categorization maximizes the category cohesion; - an preference bias: Select a father based on a user defined knowledge. This kind of background knowledge may help to determine which features are relevant in a given situation (Hanson and Bauer, 1989). For example, the relation number of tails is more important than number of nuclei. Choice② Different hierarchies for a given skeleton differ in the number of nodes they contain. In other words, the reduction process depends on whether one wishes to have a concept hierarchy with a lot of generalized nodes covering few objects, i.e. a rather specific hierarchy, or if one wishes that generalized nodes cover a lot of objects, i.e. the hierarchy is not as deep as the skeleton, many nodes are absorbed. As per now, the criterion used is
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only based on a maximal number of nodes per hierarchy. This criterion guides the reduction process to reach a hierarchy that satisfies this simple criterion. 5.
Conclusion
Conceptual graphs are well suited to represent complex objects but, to our knowledge, conceptual clustering based on a Generalization Space had not yet been analyzed. In this paper, we propose a basis for describing clustering task using a GS. We present an algorithm to extract a categorization from the GS. We show that classical conceptual clustering methods approaches criteria may be expressed in terms of choices during the extraction algorithm. Because the GS, under certain restrictions, has a tractable size, it may well be used as a basis for clustering objects described using conceptual graphs. However, further empirical evaluation of the approach would clarify its interest in terms of performance. One of the most interesting perspective to our work lies in its potential use to revise categorizations. Indeed, we are not so interested in enhancing existing conceptual clustering methods than to develop an approach to propose alternative hierarchies of a set of objects. References Bisson, G. (1993). KBG: Induction de Bases de Connaissances en Logique des Prédicats. PhD thesis, Paris XI - Orsay. Decaestecker, C. (1993). Apprentissage et outils statistiques en classification incrémentale. Revue d'intelligence artificielle 7 - n°1: pp. 33-71. Diday, E., Lemaire, J., Pouget, J. and Testu, F. (1982).Elements d'Analyse des données. Dunod ed. Ellis, G. (1993a). Efficient Retrieval from Hierarchies of Objects Using Lattice Operations. Proc. First International Conference on Conceptual Structures, Lecture Notes in AI, n°699, pp. 274-293. Ellis, G. (1993b). Managing Complex Objects. University of Quennsland, Australia, Report. Fisher, D. and Langley, P. (1985). Approaches to conceptual clustering. Proc. Ninth International Joint Conference on Artificial Intelligence, pp. 691-697. Fisher, D. H. (1987). Knowledge Acquisition Via Incremental Conceptual Clustering. Machine Learning 2: pp. 139-172. Garey, M. and Johnson, D. (1979).Computers and intractability: A guide to the theory of NP-completeness. ed. San Fransisco, CA: W. H. Freeman.
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