Towards multicriteria clustering: An extension of the k-means algorithm

Abstract

The research within the multicriteria classification field is mainly focused on the assignment of actions to pre-defined classes. Nevertheless the building of multicriteria categories remains a theoretical question still not studied in detail. To tackle this problem, we propose an extension of the well-known k-means algorithm to the multicriteria framework. This extension relies on the definition of a multicriteria distance based on the preference structure defined by the decision maker. Thus, two alternatives will be similar if they are preferred, indifferent and incomparable to more or less the same actions. Armed with this multicriteria distance, we will be able to partition the set of alternatives into classes that are meaningful from a multicriteria perspective. Finally, the examples of the country risk problem and the diagnosis of firms will be treated to illustrate the applicability of this method.

Introduction

In many situations, decision makers have to group alternatives (objects or actions) into homogeneous classes. Several examples can be cited:

• in finance, especially in business failure prediction, credit risk assessment or country risk assessment;

• in marketing, for the analysis of the characteristics of customers to design the market penetration strategies;

• in medical diagnosis, for the classification of patients into diseases groups, on the basis of a set of symptoms.

The two mostly used techniques for grouping objects with similar properties are: classification and clustering. Both are often confused, but some important differences exist between them. Classification techniques use supervised learning; what means that the objects are assigned to pre-defined classes. On the contrary, clustering is an unsupervised technique that finds potential groups in data such that, the objects within a same cluster are more similar to each other than to objects in other clusters. Several approaches such as expert systems, neural networks, mathematical programming, multicriteria decision aid (MCDA) [19], [20], … , have been explored to deal with this problem. This paper is focused on the MCDA approach.

In the MCDA methodology, grouping alternatives into homogeneous classes consists in assigning a set of alternatives A={a₁,a₂,…,a_n} evaluated on m criteria {g₁,g₂,…,g_m} to one of the categories while examining their intrinsic value. The categories are pre-defined by norms called profiles, which separate them or play the role of central reference objects in these categories. The assignment of an alternative to a specific class results from a comparison of its evaluation on all criteria with the profiles defining the categories.

Among the MCDA methods developed to solve these kinds of problems we can mention: Tricotomic Segmentation [14], [17], [18], ELECTRE TRI [15], [21], [25], PROAFTN [3], Filtering methods based on concordance and non discordance [16], the UTADIS method [7], [11] based on the criteria aggregation model and the Rough set approach [22]. These procedures can be considered as supervised learning methods. As far as we know, in MCDA literature, there are few unsupervised learning techniques [6]. In this paper we are focused on this problematic: regrouping alternatives into a restricted number of categories which will remain as homogeneous as possible. Recently, Zopounidis and Doumpos [26] made a literature review on MCDA classification and sorting methods. We refer the interested reader to this study.

The k-means algorithm [12] is one of the most widely used unsupervised technique. This method allows to group the alternatives into categories in such a way that the distances between the alternatives, within a same category are the shortest, while the distances between the centers of different categories are the largest. Following the multicriteria approach, we are interested in defining a notion of distance between alternatives that takes into account the multicriteria nature of the problem.

In this paper, we present an extension of the k-means algorithm to the multicriteria framework. The intuition behind the method is the following: all actions within the same cluster are preferred, indifferent and incomparable to more or less the same actions following the decision maker preferences. Let us note that this resembles, in the spirit at least, to the sociometric idea of structural equivalence [24]. To quantify this similarity we introduce a multicriteria distance based on the preference structure (P,I,J) defined by the decision maker. The originality of this approach is the application of this new measure within the well-known k-means framework.

The outline of this paper is as follows. In Section 2, we will introduce some important concepts to develop our MCDA clustering approach. The algorithm itself will be presented in Section 3. Section 4 will be dedicated to the study of two applications: the country risk problem and the diagnosis of firms. Finally we will conclude with some general remarks about the method and directions for future research.