A CLASSIFICATION METHOD BASED ON FUZZY CONTEXTS

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1 ] Bu!. ~tiinl. Uillv. Baia Mare, Ser. B, Matematidi-Informaticii, Vol. XVIII(2002), Nr. 2, A CLASSIFICATION METHOD BASED ON FUZZY CONTEXTS U Sandor I RADELECZKI 2 Abstract. The main idea of different fuzzy methods used -for the classification of the elements of a finite set A is to define a fuzzy similarity relation among the elements of the set A. In this paper we present a new method for the construction of this similarity relation using some fundamental notions of Fuzzy Concept Analysis. <:.c MSC: 04A72, 06B23 Keywords: similarity relation, partition tree, concept lattice, fuzzy context. 1. Preliminaries I L. The purpose of this paper is to classify a-fmite set of objects A = {xi' x 2,, x,,} on the basis of their properties ~,~,..., P,,,' (n, men). In fact, a classification of the elements means a partition 11 = {Ai 11::;i ::;k } of the set A (k::; n ), where the blocks Ai of 11 are constituted from objects with "similar" properties. A) Elements of the theory of fuzzy relations, A binary fuzzy relation p defined between the elements of the sets X and Y is a triple p=(x,y,.up), where.up:xxy~[o;i] is a function. The value.up(x,y) express the "strength" of the relation p between the elements x E X and y E Y. The fuzzy relation p = (X, Y,.up) is said t~ be sm~ller tha~ the fuzzy relation R = (X, Y,.u R) if.up(x,y) ::;.ur(x,y) holds for all (x,y) E X x Y. IfX=Y, then p is called homogenou~. Afuzzy tolerance (see e.g. [1] or [3]) is a homogenous fuzzy relation p = (X, X,.uJ satisfying the properties: and.up(x,x)=i,forall XEX. (1).up(X,y)=.up(Y, x), for all x,y E X 319 o (2)

2 If P satisfies in addition the inequality,up{x,z)~ min{,up{x,y),,up{y,z)}, for all x,y,z E X (3) then it is called ajuzzy similarity relation (i.e. ajuzzy equivalence - see e. g. [1] or [8]). Let a E [0; 1]. An a -cut of a fuzzy relation p = (X, Y,,u p) is a crisp (or traditional) binary relation Pa ~ X X Y defmed as P a ={{x,y)exxyi,u)x,y)~a}. (4) If P = (X,X,,uJ is a fuzzy similarity relation, then P a is an equivalence on the set X. Let na stand for the partition induced by Pa on X. It is easy to see that for any at E [0; 1] with at ~ a, na' is a refinement of na. Therefore to any sequence 0::; a l < a 2 <...< a k ::; 1 we can attach a nested sequence of partitions na, ' n a',",..., na. and this may be represented in the form of a partition tree, as shown in Figure (the exa~ple is from [8]). '1 The transitive closure of a homogenous fuzzy relation ~ = (X, X,,up) is the smallest fuzzy relation p = (x,x,,u i» satisfying the inequality (3) and P s P. If P is a fuzzy tolerance, then p always exists and it is a fuzzy equivalence. The composition poe of two fuzzy relations p = (X, y,,uj and e = (Y, Z,,uo) is defmed a~ a fuzzy relation poe = (X,z,,up"o), where:,up"o{x,z) = sup {min {u)x, y);,u.{y, Z)}I y E Y}, for each (x, z) E X x Z. (5) The m-th power of a fuzzy relation p = (X, X,,u p) is defm~4 as p'" = po p"'-i, m > 1 and, '320

3 p' = p. New let X be a finite se! with J X I= n. It is easy to see that there exists a number 1$ k $ n such that pk = P. In this case we also obtain pk = pk+'" for all men. B) The prin~ipal steps of the {uzzy methods The main steps of the several fuzzy classification methods (see e. g: [7]) can be summarised as follows: l.let A = {x" X 2,, x,,} be ~ fmi.te set of objects. The properties p, (1$ i $ m) of the elements of A are defined as fuzzy sets on the universe A characterised by the membership fimctions f.l,:: A ~ [0; 11 1$ i $ m. The value /-1,: (x j ) express "how much" the property p, is valid for the object Xj EA. Now to any object Xj E A is assocjated a point Q j E R'" defmed as Q j = {f.l,~(xj;...;f.lp"(xj) G (6) 2. Introducing a metric d: R'" x R'" ~ {O;1] (this is possible in several ways) a fuzzy tolerance p = (A, A,f.LJ is defined as follows: f.l p {x" xj= 1-" d{x"xj (7) 3. Computing the consecutive powers p2, p\..., pk tk < n) until pk = pk+' by using formula (5), the traf.lsitive closme p of p is obtained as p = pk. 4. By a -cuts of this p, we produce a sequence of nested partitions n,...,n, i.e. a OJ a, partition tree corresponding to a previously established sequence 0 < a, < a 2 <...< a" $ Notions of Formal Concept Analysis A) Crisp contexts and concept lattic.es Given a set G of obj ects and a set M of attributes (or properties) a binary relation J <;;;; G x M is defmed as follows: (g, m) E J if and only if the object g E G has the attribute in EM. (8) The triple (G, M, J) is called a formal context in mathematicalliteratme (see e.g. [6] or [2]). By defming A' = {m E Mj(g,m)E J for all g E A} B' = {g E Gj(g,m)E J for all me B} for all subsets A <;;;; G and B <;;;; M, we establish a Galois conneotion between G an M. The pairs (A, B) with A' =B and B' = A are called thefo.rm..(lilconc~pts of the context (G, M, J). The formal concepts of( G, M, J) together with the partial order defmed by (A" B,)$ (A2' B2) ~ A, <;;;; A2 (or equivalently B2 <;;;; B,) (9) 321

4 form a complete lattice L(G, M, J) which is called the concept lattice of the context K=(G,M,J). Remark: If A = An, then the pair (A, A') is a folmal concept of the context (G, M, J). For any g E G we define the concept y(g) = ( {g}", {g }'). It is easy to see that y( g) is the smallest concept CA. B) with, g EA. B) Fuzzy contexts and concept lattices The general formulation of the notions below can be found in [4]. According to our aim here we present them only in a particular form: A fuzzy context is a triple (G, M, J) where G is a set of objects, M is a set of attributes and I = (G, M, f..l,) is a binary fuzzy relation defined by a membership function f..l, : G x M ~ [0; 1]. The value f..l,(g, m) express "how milch is valid" the attribute me M for the object g E G. For each a E [0; 1] the a-cut Ia = { (g, m)1 f..l,(g, m);?: a} deg:rmines a "traditional" context Ka = ( G, M, Ja) and a "traditional" or c~isp ~oncept lattice La = ( G, M, J a) (corresponding to the context Ka). In our particular case the fuzzy concept lattice L (G, M, J) of the fuzzy context (G, M, J) is defined by identifying it to the set {((G, M,J),a) I a ~O; Ij} corresponding to all concept lattices of the fuzzy context K = (G,M,I) [5]. (For a more detailed formulation see [4].) 3. The principle of our classification method Given a finite set A = {x"x 2,,x,,} of objects and a finite set M = {p',~,...,f,,,} of attributes interpreted as fuzzy sets with universe A and with different membership functions f..lp,: A ~ [0;11 1:::;i:::; m, a fuzzy relation J = (A,M,f..lJ and a fuzzy context K=(A,M,I) is defined as follows: Let y)x;} associate the concept ( {xj, {xj) defined by the crisp context Ka = (G,M,IJ, where a E [0;1]. Further, we consider a fuzzy set M(x,) with universe M to any object x, E A, by defining its membership function f..l x, : M ~ [0; 1] as (10) f..l,,(p. ) = f..l,(x" p. ), for all p. EM, 1:::; k :::;m. ( 11 ) The similarity of two fuzzy sets M(xJ and M{xJ is defined as it is usual in literature (see 322

5 We note that S( A, B)= 1-11 AVB II = 1 iff A=B. Now we define a fuzzy tolerance T = (A, A, fir) as follows fir{x;>xj= S{M(Xi)' M{xJ). sup {a E [0;1]1 yjx') = Yak)} (12) Clearly, the above supremum always exits, and we have J1T (x i,x j ) = J1T (x j,xi) E [0; I] by definition. Since S(M(xJ,M(X,)) = 1 and since Ya(Xi) = Ya(Xi) holds for all a E [0;1], we get fir (Xi' Xi) = 1, for all Xi E A - proving that T is a fuzzy tolerance. In what follows, our construction uses the same steps as the formerly presented fuzzy methods (see Subsection I.B), for instance, we proceed constructing a fuzzy similarity relation S = (A,M,fiJ by computing the powers T 2,T\...,T k of the fuzzy tolerance Tuntil Tk = Tk+l. Concluding remarks: The origin of our method comes from an application of the fuzzy contexts in Group Technology, namely, to classify some technological objects on the basis of their common attributes [5]. The advantage of the method consists in the fact that it does not need the construction of an additional R m metric used by the majority of fuzzy methods. Acknowledgement: The support by Hungarian National Foundation for Scientific Research (GrmH No. T029525, T and T034137) and by Istvan Szechenyi Grant of Hungarian Academy of Science is gratefully acknowledged. The author wishes to express his thanks to professor T. T6th for his advice. REFERENCES [1] Dubois, D., Prade, H.: Fuzzy Sets and Systems, Theory and Applications; Mathematics in Science and Engineering, Vol. 144, Academic Press, New York, [2] Ganter, B., Wille R.: Formal Concept Analysis; Mathematical Foundations, Springer Verlag, Berlin, [3] Negoi!a, C. V. and Ralescu. D. A.: Applications of Fuzzy Sets to System Analysis, Birkhauser, Basel (1975). [4] Pollandt, S.: Fuzzy-BegrifJe, Formale Begriffsanalyse Unscharfer Daten, Springer Verlag, Berlin, [5] Radeleczki, S. and T6th, T.: Concept lattices and jilzzy methods and their application in Group Technology, Research report, Miskolc University, 1999 (in Hungarian). [6] Wille, R.: Restructuring lattice theory: an approach based on hierarchies of concepts, In: I. 323

6 Rival (ed.), Ordered Sets, , Reidel, Dordrecht-Boston, [7] Xu, H. and Wang, H. P.: Part family formation for GT applications based on fuzzy mathematics, Int. J. Prod. Res. Vol. 27. No.9. (1989), [8] Zadeh, L. A.: Similarity relations andfuzzy orderings, Inf. Sci. 3 (1971), Received: University of Miskolc, Institute of Mathematics, 3515 Miskolc-Egyetemvaros, Hungary matradi@gold.uni-rniskolc.hu 324