Fuzzy Similarities in Stars/Galaxies Classification Salvatore Sessa 1 Roberto Tagliaferri 2 Giuseppe Longo Angelo Ciaramella 2 Antonino Staiano 2
3
1
Dipartimento di Costruzioni e Metodi Matematici in Architettura, Universit`a di Napoli, via Monteoliveto, 3 - 80134 Napoli, Italy E-mail:
[email protected] 2 Dipartimento di Matematica e Informatica, Universit`a di Salerno, Via Allende, 84081 Baronissi, Salerno, Italy, and INFM Unit`a di Salerno . E-mail: (robtag, ciaram, staiano)@unisa.it. 3 Dipartimento di Scienze Fisiche, Universit`a di Napoli, Via Cintia, 80126 Napoli, Italy E-mail:
[email protected]
Abstract— By basing on the concept of fuzzy similarity with respect to a continuous triangular norm built via the well known method of ordinal sums, we propose a modification of an our previous algorithm which improves the performance of the Stars/Galaxies classification in astronomical data mining. This algorithm is implemented in Matlab 6 and we make also use of Fuzzy c-Means clustering algorithm for constructing two prototypes with respect to which the fuzzy similarity is calculated. Keywords— Fuzzy Similarity, Neural Network
I. Introduction The popularity of fuzzy logic comes mainly from many successful applications, where linguistic variables are suitably transformed in fuzzy sets, combined via the conjunction and disjunction operations by using a continuous triangular norm t : [0, 1]2 −→ [0, 1] and the related dual conorm, respectively. As implication, the correspondent residuum −→t is also used. For simplicity, in the sequel we write t-norm instead of triangular norm. It is well known that [1] a t-norm t is an associative, commutative and monotone (in both variables) ope5ration such that t(x, 0) = 0 and t(x, 1) = x for any x ∈ [0, 1]. Throughout this paper, we use the standard notation xty instead of t(x, y), where x, y ∈ [0, 1] and we assume t to be continuous in classical sense. The related residuum operator −→t is defined as (x −→t y) = sup{z ∈ [0, 1] : xtz ≤ y}. The most used t-norm in fuzzy logic and the correspondent residua are the following [2]: G¨odel t-norm (minimum): x∧y
=
x →M y
=
min{x, y} ½ 1 if x ≤ y y otherwise
Goguen t-norm (product): xP y x →P y
= ½ xy 1 = y x
if x ≤ y otherwise
Lukasiewicz t-norm: xLy x →L y
= max{0, x + y − 1} = min{1, 1 − x + y}
These t-norms are evidently continuous. Later we shall also use the concept of bi-residuum (with respect the t-norm t) defined as (x ←→t y) =
(x −→t y) ∧ (y −→t x)
For the above t-norms we have respectively: ½ 1 if x = y (x ←→M y) = min{x, y} otherwise ½ (x ←→P y) = (x ←→L y)
1 if x = y min{ xy , xy } otherwise
= 1 − max{x, y} + min{x, y} = = 1 − |x − y|
In order to build an algebraic structure in which one considers a logic more general than the three logics (Lukasiewicz, Product and G¨odel logic) commonly employed in fuzzy set theory (in narrow sense), Hajek [2] has invented the BL-algebras. A continuous t-norm induces a structure of BL-algebra over [0, 1], henceforth we assume such a structure just for formalizing the results of Section 2. In the context of BL-algebras [2], the concept of similarity is a direct generalization of the fuzzified equality. Indeed, if X is a referential finite set and
S : X × X −→ [0, 1] is a fuzzy relation, S is a fuzzy similarity (with respect to the t-norm used) if S(a, a) = 1 (reflexivity), S(a, b) = S(b, a) (symmetry) and S(a, b)tS(b, c) ≤ S(a, c) (transitivity) for all a, b, c ∈ X. A fuzzy similarity is also called fuzzy equivalence relation, or fuzzy equality or indistinguishability operator [3]. It is well known that a fuzzy set µ : X −→ [0, 1] generates a fuzzy similarity S defined as [3]. We shall also use the concept of ordinal sum [1]. Strictly speaking, let {]am , bm [} be a family of non empty, pairwise disjoint open subintervals of [0, 1] and tm be a family of continuous t-norms. Then the function am + (bm − am ) xty =
min(x, y)
´ ³ y−am m , tm bx−a m −am bm −am if x, y ∈ [am , bm ] otherwise
is a continuous t-norm. II. Some results We have the following results. Theorem 1: Let S : X × X → [0, 1] be a fuzzy similarity and let k ∈]0, 1[. Then the fuzzy relation S 0 : X × X → [0, 1] defined by S 0 (a, b) = max{S(a, b), k} is a fuzzy similarity for all a, b ∈ X. Proof: The reflexivity and the symmetry are immediate. Further, we have for all a, b, c ∈ X: S 0 (a, b) t S 0 (b, c) ≤ 1 t k = k ≤ max{S(a, c), k} = S 0 (a, c) if S 0 (a, b) = k ≥ S(a, b) or S 0 (b, c) = k ≥ S(b, c); otherwise, i.e. if S 0 (a, b) = S(a, b) ≥ k and S 0 (b, c) = S(b, c) ≥ k, we have obviously: S 0 (a, b) t S 0 (b, c) = S(a, b) t S(b, c) ≤ S(a, c) ≤ S 0 (a, c) Any k ∈]0, 1[ divides [0, 1] in the subintervals [0, k] and [k, 1]. Considering any continuous t-norm t over [0, k] and the Lukasiewicz t-norm over [k, 1], we build the following t-norm using the above definition of ordinal sum: if x, y ∈ [0, k] k( xk tL ky ) max{x + y − 1, k} if x, y ∈ [k, 1] tL(x, y) = min{x, y} otherwise
Considering the BL-algebra over [0, 1] induced by the above t-norm tL, we can draw the following theorem: Theorem 2: Let Si : X × X −→ [0, 1], i = 1, ...n, be n fuzzy similarities (with respect to tL). Then 0 the fuzzy relation PnSi :0 ×X −→ [0, 1] defined 1 0 by S (x, y) = n i=1 Si (x, y) is a fuzzy similarity (with respect to tL) for all x, y ∈ X, where Si0 (x, y) = max{Si (x, y), k} Proof: It suffices to prove the transitivity. Indeed we have for all x, y, z ∈ X:
= = = ≤ ≤ = =
0 S tL S 0 (y, ¢z) =¡ P ¡ 1(x, ¢ Py) n n 1 0 0 SP i (x, y) tL n i=1 Si (y, z) n ª ©i=1 n max © n1 [( i=1 Si0 (x, y) + Si0 (y, z)) − 1] , k max n1 [(S10 (x, y) + S10 (y, z)) + ... 0 0 ... + (S ©£n1(x, y) + 0Sn (y, z)) −0 n, k} ª max n max{S1 (x, y) + S1 (y, z) − 1, k + ... 0 0 ... + max{S z) − 1, k}, k} © 1 0 n (x, y) + Sn (y, 0 (x, z) + ... + S (x, z)], k} max [S n 1 Pn n 0 1 S (x, z) i i=1 n S 0 (x, z)
For an analogous result, see [7]. III. Applications to stars/galaxies catalogues In the previous paper [4] we used the total fuzzy similarity (with respect to the t-norm L) concept given in [7] for classifying stars and galaxies from an astronomical labeled catalogue of 231000 objects. There we obtained a correct classification of 7738 objects (2867 galaxies and 4871 stars) over 10000 objects randomly chosen from that catalogue, hence a classification rate of 77.38%. Here we follow the method adopted in [4], which was implemented in Matlab 6. For the construction of the two prototypes we use the Fuzzy C-Means clustering algorithm [5] instead of SOM (Self Organizing Maps)[6] used in [4]. It is clear that our referential set X formed by the above 10000 objects. After a preprocessing for selecting the most meaningful features for each object (our test deals with 7 features) we define 7 fuzzy sets µi , i = 1, ..., 7 normalizing the f eaturei for the xobjects via the formula:
The related bi-residuum is given by µx
x ←→tL
if x, y ∈ [0, k] x ←→t y 1 − |x − y| if x, y ∈ [k, 1] y= min{x, y} otherwise
=
f eaturei −minx∈X (f eaturei (x)) maxx∈X (f eaturei (x))−minx∈X (f eaturei (x))
The second step consists in the determination, based on the Fuzzy C-means, of two prototypes:
one for the stars and the other one for the galaxies. In the next step, by putting k=0.5 and using he product t-norm in the interval [0, 0.5], we calculate seven fuzzy similarity Si between the objects of the catalogue and the two prototypes, using the same features of [4]. k=0.5 is interpreted like a threshold under which the membership values of the fuzzy similarities cannot go down. This threshold is strongly dependent by the coordinate values of the prototypes and in some experiments k was chosen as the mean of the values of each prototype. In any case the choice of the threshold k can be derived from some test performed during the preprocessing phase. Finally, we assign the label “Star” or “Galaxy” to each of the 10000 objects on the basis of the maximum similarity S 0 with respect to two prototypes, using Theorem 2. The results are visualized in figure 1, in which are visualized the fuzzy similarity values for each object of the catalog with respect to the stars prototype. The first grouping of similarity values (between 0.58 and 0.66) is formed by stars, while the second one (between 0.54 and 0.56) are galaxies. Clearly, in these two groupings, there are misclassified objects, that is, objects whose features are very similar. The correct classification obtained consists of 7896 objects, that is 5810 stars and 2086 galaxies, so deducing a classification rate of 78.96% as classification global rate. This improves slightly the contents of our previous paper. [4]. 0.66
0.64
0.62
0.6
0.58
0.56
0.54
0.52
0.5
0
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7000
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9000
10000
Fig. 1. Fuzzy Similarity values with respect to the stars prototype
References [1] E. P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer Academic Publishers, Dordrecht, 2001. [2] P. Hajek, Metamathematics of fuzzy logic, Kluwer Academic Publishers, Dordrecht, 1998. [3] L. Valverde, On the Structure of F-indistinguishability operators, Fuzzy Sets and Systems 17, pp. 313–328, 1985.
[4] G. Longo, R. Tagliaferri, S. Sessa, A. Staiano et al., Advanced Data Mining Tools for Exploring Large Astronomical Databases, SPIE’s 46th Annual Meeting International Symposium on Optical Science and Technology, San Diego, CA, USA, 2001. [5] J. C. Bezdek, Pattern Recognition with Fuzzy Objective Function Algorithms, Plenum Press, New York, 1981. [6] T. Kohonen, Self-Organizing Maps, Springer, Berlin, Heidelberg, 1995. [7] E. Turunen, Mathematics Behind Fuzzy Logic, Advances in Soft Computing, Physica-Verlag, Heidelberg, 1999.
