Naïve Bayesian Rough Sets Under Fuzziness

Transcription

1 IJMSA: Vol. 6, No. 1-2, January-June 2012, pp Serials Publiations ISSN: Naïve ayesian Rough Sets Under Fuzziness G. GANSAN 1,. KRISHNAVNI 2 T. HYMAVATHI 3 1,2,3 Department of Mathematis, Adikavi Nannaya University, Rajahmundry, Andhra adesh, India 1 mail: prof.ganesan@yahoo.om, 2 krishnavenibadam@yahoo.o.in Abstrat: In 2010, Yiyu Yao and ing Zhou have disussed the Naïve ayesian Rough Set Model and in priniple it is also viewed as the ayesian Deision Theoreti Rough Set Model. This approah deals with the three ways probabilisti approximations on a given onept and the orresponding ayesian approah using parameterized rough set model. In 2005, G. Ganesan and. Raghavendra Rao analyzed the signifiane of piking the thresholds in a given fuzzy input under Pawlak s onventional Rough Sets approah. In this paper, we extended the onept of thresholds disussed earlier for the parameterized rough set model and using it the theory is extended for Naïve ayesian Rough Sets Model under Fuzziness. Keywords: Naïve ayesian Rough Sets, ayesian Deision Theoreti Rough Sets, Parameterized Rough Sets 1. INTRODUTION In reent days, Rough omputing has beome one of the emerging areas of researh due to its enormous appliations. This theory was initiated by Z. Pawlak in 1982, using onventional approahes in Mathematis. onsidering its appliations in information systems, various researhers have ontributed several tools on rough sets. From the ineption of this theory, the impliations of probability to derive the degree of unertainty were studied by Pawlak, Skrowran, Greo, Yao, Slezak, Ziarko, ing Zhou et. Reently, Yiyu Yao and ing Zhou disussed the Naïve ayesian Rough Set Model in [8] and earlier to this, the initial approah in this regard was disussed in [7] by Slezak. Parallel to these studies, sine fuzziness involves in various tehnial issues, the researhers suh as Dubois, ade, Nakamura, iswas et. have been working on hybridized models on roughness and fuzziness. In 2005, G.Ganesan and. Raghvendra Rao [1] disussed the importane of defining the thresholds in rough fuzzy omputing. In this paper, we introdued the thresholds on fuzzy onepts and implemented Naïve ayesian Rough Set Model on the fuzzy onepts. This paper has been organized into five setions. Setion Two deals with the onepts of Rough Sets whih inludes Deision Theoreti Rough Sets, obabilisti Rough Sets and Naïve ayesian Rough Sets Model; Setion Three deals with the analysis on introduing a threshold on a fuzzy onept in rough omputing. In setion Four, we disuss the Naïve ayesian Rough Set Model on Fuzzy onepts with a threshold. 2. DISION THORTI AND PROAILISTI ROUGH STS In 1982, Pawlak introdued the theory of rough sets [3, 5] whih projeted various diretions in tehnial aspets suh as knowledge disovery, data mining, information retrieval et. This theory

2 20 G. Ganesan,. Krishnaveni and T. Hymavathi gives two way approximations namely lower and upper approximations. For given finite universe of disourse U and an equivalene relation, we define the equivalene lass of any x U to be [x] = {y U/xy}. The family of equivalene lasses U x U is a partition of the universe U. For a given onept, Pawlak defined the lower approximation as apr ( ) { x U / } and upper approximation as apr ( ) { x U / }. Aording to Pawlak, for a given onept, three disjoint regions an be defined namely positive, negative and boundary regions whih are defined as follows: Positive Region: POS ( ) { x U / } oundary Region: ND ( ) { x U / ^ } Negative region : NG ( ) { x U / } Understanding the limitations of Pawlak s restritive model, several researhers foused on generalizing this approah towards parameterized rough set model, probabilisti rough set model and generalized rough set model. Many probability measures were introdued in rough sets. In eighties, Pawlak expressed the measures of lower and upper approximations respetively as q( ) ( ) and U q( ) ( ) and they are referred to be the rough probabilities [4] by Pawlak. arlier to it, U Pawlak referred the auray of the rough approximation on by ( ) whih was viewed as and is interpreted as the probability that an element belongs to the lower approximation given that the element belongs to the upper approximation. However, all these probabilities do not be amiable for the implementation of aye s Theorem. In 1994, Pawlak and Skowron [6] defined rough membership funtion by onsidering degrees of overlap between equivalene lasses and a onept to be approximated and is viewed as the onditional probability of an objet belongs to given that the objet is in [x] (for simpliity, we [ ] denote [x] with [x]) whih is given as x [ x ] Using the definition quoted above, in [8], the positive, boundary and negative regions are defined as follows: POS( ) x U / 1 ND( ) x U / 0 1 NG( ) x U / 0

3 Na Ï ve ayesian Rough Sets Under Fuzziness 21 In 2009, Greo et. al [2] disussed the parameterized roughest model by generalizing the above said definitions. In this model, two thresholds namely and are used to define the probabilisti regions and the positive, boundary and negative regions are modified as follows: POS ( ) / (, ) x U ND(, ) ( ) x U / NG ( ) / (, ) x U These obabilisti regions will lead three way deisions namely aeptane, deferment and rejetion respetively for any objet x in U. ut, however, in several ases, it is easy to ompute the [ ] [ ] probability of the existene of a ategory [x] for a given onept using x x Hene, by aye s Theorem, [ x ] an be obtained by ([ ]) [ ] ( ) [ ] x x x ( ) ( ) [ x ] () where Now, [ x ] [ x ] 1 [ x ] 1 On applying Logarithm, we get ( ) log log log ( ) 1 and similarly, ( ) log log log ( ) 1 Thus, we obtain

4 22 G. Ganesan,. Krishnaveni and T. Hymavathi and ( ) log where log log ( ) 1 ( ) log where log log ( ) 1 Thus the Positive, oundary and Negative regions with respet to aye s Theorem are defined as follows: (/ ) POS(, ) ( ) x U / log ([ ]/ x ) (/ ) ND(, )( ) x U / log ([ ]/ x ) (/ ) NG(, ) ( ) x U / log ([ ]/ x ) Now, we shall disuss the onventional approah on dealing the fuzzy sets to approximate under rough omputing, whih was disussed in [1] 3. ANALYSIS OF FUZZY ST USING A THRSHOLD onsider a set D, alled R-domain [1], satisfying the following properties: (a) D (0, 1) (b) If a fuzzy onept is under omputation, eliminate the values µ (x) and ( ) x U from the domain D, if they exist. () After the omputation using, the values removed in (b) may be inluded in D provided A must not involve in further omputation onsider the universe of disourse U = {x 1, x 2,, x n }. Let,, 1 2, be the thresholds assume one of the values from the domain D, where D is onstruted using the fuzzy onepts A an d. For a given threshold an d a fu zzy set A, th e Stron g -ut is given by A[ ] { x U / A( x ) }. The union and intersetion of fuzzy sets are by the maximum and minimum of orresponding membership values respetively. Using these definitions, the following properties were derived in [1]. (a) A[ 1 ] A[ 2 ] = A[ ] where = min( 1, 2 ) (b) A[ 1 ] A[ 2 ] = A[ ] where = max ( 1, 2 ) () (A )[ ] = A[ ] [ ] (d) (A )[ ] = A[ ] [ ]

5 Na Ï ve ayesian Rough Sets Under Fuzziness 23 (e) A [ ] = A[1 ] (f) (A ) [ ] = A [ ] [ ] (g) (A ) [ ] = A [ ] [ ] Using the mathematial tool derived as above, in [1], rough set approah on fuzzy sets using a threshold is introdued as disussed below. 3.1 Rough Approximations on Fuzzy Sets Using Let be any partition of U, say { 1, 2,, t }. For the given fuzzy onept, the lower and upper approximations with respet to an be defined as ( [ ]) and ( [ ]) respetively opositions Here, by using the properties of rough sets, the following propositions [1] an be obtained. (a) ( A ) A (b) ( A ) A () ( A ) A (d) ( A ) A (e) ( A ) ( 1 A) (f) 1 ( A ) ( A) Now, we shall hybridize the onepts dealt in the above two setions whih gives the approah of dealing a fuzzy onepts under Naïve ayesian obabilisti Rough Sets. 4. NAÏV AYSIAN PROAILISTI ROUGH STS MODL FOR A FUZZY ONPT Sine, in the above both setions, the same threshold has been used, for different purposes, to make the homogeneity, in this paper, we replae the threshold to obtain a Strong ut on fuzzy sets with. Hene, for a given fuzzy onept F with the threshold, the probabilisti positive, boundary and negative regions are respetively defined on the approximation spae U/ as POS ( F) x U / 1 ND ( F) x U / 0 1 NG ( F) x U / 0

6 24 G. Ganesan,. Krishnaveni and T. Hymavathi For given parameters and, the regions of the parameterized rough sets model are given by POS [ ] (,, ) ( F) x U / F ND ( F) x U / (,, ) NG ( F) x U / (,, ) and the Regions of Naïve ayesian Rough Sets Model are given by (/ ) POS(,, ) ( F) x U / log ([ ]/( [ ]) x F ) (/ ) ND(,, )( F) x U / log ([ ]/( [ ]) x F ) (/ ) NG(,, ) ( F) x U / log ([ ]/( [ ]) x F ) where ' and are given by ( ) ( ) log log and log log ( ) 1 ( ) 1 Using the operty disussed in Setion 3, these definitions an further be modified as ( ) / log / POS(, ', ) F x U [ ]/ x F [1 ] ( ) / log / ND(, ', ) F x U [ ]/ x F [1 ] ( ) / log / NG(, ', ) F x U [ ]/ x F [1 ] Thus, we obtain the three way approximations of a fuzzy set under Naïve ayesian Rough Set with the threshold. 5. ONLUSION In this paper, we extended the work on Naïve ayesian Rough Sets Model for the fuzzy onepts using the threshold to be hosen from R Domain. Sine, the present work yields three way risp approximations, it is planned to extend this work towards obtaining fuzzy approximations.

7 Na Ï ve ayesian Rough Sets Under Fuzziness 25 Referenes [1] G. Ganesan,. Raghavend Rao, Rough Set: Analysis of Fuzzy Sets Using Thresholds, UG Sponsored National onferene on Reent Tends in omputational Mathematis, Marh 2004, Gandhigram Rural Institute, Tamilnadu, by Narosa Publishers, pp , [2] Greo, S., Matarazzo,. and S lowi_nski, R., Parameterized Rough Set Model Using Rough Membership and ayesian on_rmation Measures, International Journal of Approximate Reasoning, 49, , [3] Pawlak Z., Rough Sets, International Journal of omputer and Information Sienes, 11, , [4] Pawlak Z., Wong S.K.M. and Ziarko W. Rough Sets: obabilisti Versus Deterministi Approah, International Journal of Man-Mahine Studies, 29, 81-95, [5] Pawlak Z., Rough Sets, Theoretial Aspets of Reasoning About Data, Dordreht: Kluwer Aademi Publishers, [6] Pawlak, Z. and Skowron, A., Rough Membership Funtions, in: Yager, R.R., Fedrizzi, M. and Kaprzyk, J., ds., Advanes in the Dempster-Shafer Theory of videne, John Wiley and Sons, New York, , [7] Slezak, D. and Ziarko, W., The Investigation of the ayesian Rough Set Model, International Journal of Approximate Reasoning, 40, 81-91, [8] Yiyu Yao and ing Zhou, Naive ayesian Rough Sets, oeedings of RSKT 2010, LNAI 6401, pp , 2010.