Fuzzy Sets and Systems 159 (2008) 270 – 288 www.elsevier.com/locate/fss Approximation techniques for the transformation of fuzzy sets into random sets Mihai Cristian Florea a, b, ∗ , Anne-Laure Jousselme c , Dominic Grenier b , Éloi Bossé c a Informatique WGZ. inc, Québec, Que., Canada b Université Laval, Québec, Que., Canada c DRDCValcartier, Québec, Que., Canada Received 31 March 2007; received in revised form 11 September 2007; accepted 12 September 2007 Available online 1 October 2007 Abstract With the recent rising of numerous theories for dealing with uncertain pieces of information, the problem of connection between different frames has become an issue. In particular, questions such as how to combine fuzzy sets with belief functions or probability measures often emerge. The alternative is either to define transformations between theories, or to use a general or unified framework in which all these theories can be framed. Random set theory has been proposed as such a unified framework in which at least probability theory, evidence theory, possibility theory and fuzzy set theory can be represented. Whereas the transformations of belief functions or probability distributions into random sets are trivial, the transformations of fuzzy sets or possibility distributions into random sets lead to some issues. This paper is concerned with the transformation of fuzzy membership functions into random sets. In practice, this transformation involves the creation of a large number of focal elements (subsets with non-null probability) based on the -cuts of the fuzzy membership functions. In order to keep a computationally tractable fusion process, the large number of focal elements needs to be reduced by approximation techniques. In this paper, we propose three approximation techniques and compare them to classical approximations techniques used in evidence theory. The quality of the approximations is quantified using a distance between two random sets. © 2007 Published by Elsevier B.V. Keywords: Random sets; Fuzzy sets; Information fusion; Approximation techniques 1. Introduction Probability theory is the first mathematical modelling tool of uncertainty, which was developed in the seventeenth century to represent random events. Randomness in card or die games was one of the first aspects of uncertainty to be recognized. In the recent decades, other aspects of uncertainty were characterized and several taxonomies were proposed by Klir [12] and Smets [23]. Theories such as fuzzy set theory [27], evidence theory [2,22], possibility theory [26], rough sets theory [21] were developed to cope with different aspects of uncertainty, for which proba- bility theory was not quite appropriate. In information fusion applications, the combination of information coming from multiple and various sources is considered. When observing a given situation, more the information is various ∗ Corresponding author. Informatique WGZ. inc, Québec, Que., Canada. E-mail addresses: Mihai-Cristian.Florea@wgz.ca (M.C. Florea), Anne-Laure.Jousselme@drdc-rddc.gc.ca (A.-L. Jousselme). 0165-0114/$ - see front matter © 2007 Published by Elsevier B.V. doi:10.1016/j.fss.2007.09.019