W. Ziarko, "Variable precision rough set model," Journal of Computer and System Sciencs, 46, 39-59, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....both #X i (Y j ) and #X i (Y j ) can be reduced to inclusion degree. 9. Inclusion Degree and the Variable Precision Rough Set Model Let X and Y be non empty subsets of a finite universe U . The measure c(X, Y ) of the relative degree of misclassification of the set X with respect to set Y (see [16]) is defined as (12) c(X, Y ) if 0, 0, if = 0. It can be easily shown that c(X, Y ) 1 0 (Y X) D 0 ( U Y ) X) This means that c(X, Y ) can be reduced to inclusion degree. # 0.5. Then c(X, Y ) # if and only if D 0 (Y X) #. Thus, the variable precision ....

....c(X, Y ) if 0, 0, if = 0. It can be easily shown that c(X, Y ) 1 0 (Y X) D 0 ( U Y ) X) This means that c(X, Y ) can be reduced to inclusion degree. # 0.5. Then c(X, Y ) # if and only if D 0 (Y X) #. Thus, the variable precision rough set model (see [16]) can be expressed by inclusion degree as follows. Let X U and R be an equivalence relation on U . The # lower approximation # , and the # upper approximation of the set X is defined as U IND(R) D 0 (X E) # . Consequently, the # boundary region of X is given by BNR # X = ....

W.Ziarko, Variable precision rough set model, Journal of Computer and System Sciencs 46: 39-59, 1993.

....how much we loosen our new definitions. Obviously, fi must be restricted to a range of 0 , where the new definitions converge to the original definitions, up to 50 since above this limit one can not reasonably speak of a majority. This extended version of Rough Set Theory was proposed by Ziarko [Ziar93] and is called Variable Precision Rough Set Model. This theory maps to the setting of machine learning as follows: The concepts to be learned correspond to arbitrary sets which are to be described by means of the indiscernibility relation R. R, as a family of equivalence relations, corresponds to ....

....of equivalence relations, corresponds to the classes of attribute values that are known about the examples. The task is to abstract from the full description of the examples to a few simple rules. There are different ways of exploiting Rough Set Theory for the purposes of machine learning. See [Ziar93, References 3, 7, 12, 15, 17] for algorithms. One of them, Wong al86] uses a measure from Rough Set Theory, the discriminant index , to replace the information gain measure of ID3 [Qui86] 4 For the experiments described in this paper, the commercially available software DataLogic R, the underlying theory of which is ....

Ziarko, W. (1991). Variable Precision Rough Set Model. Journal of Computer and System Sciences, 46, 39--59. 18

....to multi attribute filtering can be found in [79, 80] 9 Extensions and variations of RSDA In this section we will briefly describe other directions into which RSDA has branched, some of which are only beginning to be investigated. The variable precision rough set model (VPRS) introduced in [83], is a generalisation of the original RSDA in the direction of relaxing the strict boundaries of equivalence classes; it assumes that rules are only valid within a certain part of the population, and it is able to cope with measurement errors: Hypotheses derived from sample data should not : ....

.... # 77 Sepal width: 35, 37, 39 44 # 35 23 16 20, 24 # 24 Petal length: 10 19 # 14 30 44,46,47 # 46 43 8 50, 52, 54 69 # 50 Petal width: 1 6 # 2 10 13 # 11 22 8 17, 20 25 # 17 information if the majority of available data to which such a rule applies can be correctly classified [83]. The first step in the VPRS analysis is a numerical form of set inclusion in analogy to the rough membership function of (5.4) If X;Y # U , then the relative classification error of X with respect to Y is the value c#X; Y## 8 : # # #### # ## # ; if X ## #; #; otherwise. 9.1) We now ....

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Ziarko, W. (1993). Variable precision rough set model. Journal of Computer and System Sciences, 46.

....to a de conf 100 conf= 100 1 2 3 4 5 6 7 8 9 pattern length Fig. 4. The distribution of the lengths of the classifying patterns. fault class. Only recently, some generalized models of RS have been introduced, in which 100 implications are replaced by approximate dependencies, see e.g. [20]. As mentioned in the previous section, we noticed in our experiments that over 90 of the records was classified by a pattern P with conf(P ) 100 . This phenomenon makes RS and PatMat similar. The problem of generating minimal value reducts is known to be NP hard. Any algorithm generating a ....

W. Ziarko, Variable Precision Rough Set Model, J. of Computer and System Sciences, Vol. 46, pp. 39--59, 1993.

....In many applications only partial knowledge about approximated concepts is given. Hence quite often first a parametrized family of concept approximations is built and next by tuning of the parameters the best, in a sense, approximation is chosen (see e.g. variable precision rough set model [40]) in approximation spaces. In this paper we follow this approach in generalized approximation spaces. We discuss rough set model based on approximation spaces with uncertainty functions and rough inclusions. Both elements of approximation space are parametrized and for the proper application of ....

....measurements or mistakes made during collecting the data. In such situations the notions of full inclusion and non empty intersection used in approximations definition are too restrictive. Some extensions in this direction have been proposed by Ziarko in the variable precision rough set model [40]. The indiscernibility relation can be employed in order to define not only approximations of sets but also approximations of relations. Investigations on relation approximation are well motivated both from theoretical and practical points of view. Let us mention two examples. The equality ....

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Ziarko W.: Variable Precision Rough Sets Model, Journal of Computer and Systems Sciences, Vol. 46, No. 1, 1993, pp. 39-59.

....conditioned to the decision classes) expanded with all possible values (possibly conditioned to the decision classes) or treated as a special value in its own right. ffl Computation of partitions and rough set approximations, either in the standard sense or within the variable precision model [25]. ffl Sampling of subtables for validation purposes. ffl Discretization of numerical attributes with various discretization algorithms. Since rough set methods are well suited for applications that take on a coarsegrained view of the world, it is typically desirable to pre process the data so ....

Ziarko, W.: Variable precision rough set model. Journal of Computer and System Sciences 46 39--59 This article was processed using the L A T E X macro package with LMAMULT style

....Features currently offered by computational kernel include among others: Completion of decision tables with missing values according to various completion strategies. Computation of partitions of rough set approximations, either in the standard sense or within the variable precision model [6]. Sampling of subtables for validation purposes. Discretization of numerical attributes with various discretization algorithms. Computation of reducts (both in standard sense as well as object related ones) Various approximation algorithms (e.g. genetic algorithms [5] are offered as ....

W. Ziarko (1993). Variable precision rough set model . J. of Computer and System Sciences, 46, pp. 39--59.

....respect to attributes B. The value B X (x) measures the closeness of fy 2 U : InfB (x) InfB (y)g and X. The formulae for the lower and upper set approximations can be generalized to some arbitrary level of precision 2 [ 1 2 ; 1] by means of the variable precision rough membership function [39] (see below) Possible ties in the case of = 0:5 can be resolved by assigning the objects in question to the interior of the set. Note that the lower and upper approximations as originally formulated are obtained as a special case with = 1:0. B X = fx j B X (x) g B X = fx j B X (x) ....

....of objects known from the training sample. It is also desirable to approximate subsets of all objects (including also new unseen objects) The best known technique for such applications is the so called called boundary region thinning. It is related to the variable precision rough set approach [39]. Another technique is used in tuning of decision rules. For instance, better quality of new objects classification may be achieved by introducing some degree of inconsistency of the rules on the training objects. This technique is an analogue of the well known techniques for decision tree ....

W. Ziarko (1993), Variable Precision Rough Set Model , J. of Computer and System Sciences, 46, pp. 39--59.

.... ; FQ 3 ) 0 = 0 Similarity R1 (FD 3 ; FQ 3 ) 0 = 1 Gamma 1=4 = 0:75 Similarity R1 (FD 3 ; FQ 3 ) 0:4 = 1 5 Variable Precision Rough Sets Another interesting extension to the rough set model, for information retrieval purposes, is the variable precision rough set model [Wong Ziarko, 1987, Ziarko, 1993] In Pawlak s standard model, an element belongs to the lower approximation of a set S if all its related elements belong to S. For the upper approximation at least 1 of its related elements should be in S. In graded rough sets [Yao Wong, 1992] the degree of overlap is considered. For some n ....

....(S) fxjjS R(x)j ng Thus x belongs to apr n R (S) if at most n members of R(x) are not in S and it belongs to apr nR (S) if more than n members of R(x) are in S. This gives us a family of graded approximation operators by simply varying n. Variable precision rough sets [Wong Ziarko, 1987, Ziarko, 1993] offer a probabilistic approach to rough sets by extending the idea of graded rough sets. In essence it also considers the size of R(x) Thus we now have, apr Rfl (S) fxj jS R(x)j jR(x)j 1 Gamma flg 14 apr Rfl (S) fxj jS R(x)j jR(x)j flg where fl 2 [0; 1] Variable precision ....

[Ziarko, 1993]Ziarko, W. Variable precision rough set model. Journal of Computer and System Sciences, 46, pp. 39-59, 1993. 23

....criterion, concept approximation. One can find such approach as a basis for approximate reasoning in applications of fuzzy logic, neural networks, clustering etc. In e.g. 5, 8] some methods for calibration of tolerance relation to obtain (sub ) optimal granule description are given and in [32] a possibility of tuning information granules by boundary region thinning is presented. One of the main problem in soft computing is that the language L 0 in which the evidence about objects is represented and the language L 1 used to describe knowledge about the problem to be solved are different ....

Ziarko W.: Variable precision rough sets model, Journal of Computer and Systems Sciences, Vol. 46, No. 1, 1993, pp. 39-59.

....table. This happens, e.g. when the number of examples supporting the decision rule is relatively small. Therefore one can use approximate decision rules instead of consistent decision rules to construct the classification algorithm for a decision table A. Different methods (see [1] 10] 14] [20]) are now widely used to generate approximate decision rules. In our experiments (see Section 4) we used the method of approximate rules generation described in [1] 3 Decision rule synthesis by matching new objects against data tables The method of classification algorithms construction ....

Ziarko, W.: Variable precision rough set model. Journal of Computer and System Sciences 40 (1993) 39--59

....one decision attribute. For each attribute value pair, decision rules in a table are converted into a simple form. For example, a 1 a 2 d 1 d 2 is converted into a 1 a 2 d 1 : 1 In this paper, these ideas can be applied not only to deterministic domains but also to probabilistic domains[10]. For further information about non deterministic reduct networks, reader could refer to [6] a 1 a 2 d 1 d 2 Q Q Q Qs j j j j3 Figure 1: A Small Reduct Network a 1 a 2 d 1 d 2 Q Q Q Qs 6 Figure 2: A Revised Small Reduct Network (2) Calculation of Reducts From the ....

Ziarko, W. (1993). Variable Precision Rough Set Model. Journal of Computer and System Sciences, 46, 39-59.

....comprises of several steps that, in consequence, lead to removing some descriptors from a particular rule. Usually, after shortening, the number of rules decreases as repetitions occur in the set of shortened rules. There are several methods leading to this goal, for details review e.g. 1] 4] [13]. 5 Rules as attributes In the classical approach, once we have decision rules we are at the end of classifier construction. But there is also other way of treating the rules since they describe relations existing in our data. Therefore we may treat them as features of objects. In this view, the ....

Ziarko W., Variable Precision Rough Set Model, Journal of Computer and System Sciences, 40 (1993), pp. 39-59

....by applying boolean reasoning [1] The solutions are represented by tolerance reducts and relative tolerance reducts. 1 Introduction We discuss a generalization of the approximation space definition introduced in [8] Our investigations are motivated by the results of [3] 7] 18] and [20] concerning sets with the boundary regions less crisp than in the case presented in [8] as well as by papers [6] 13] 17] on relation approximation. Investigations on relation approximation are well motivated both from theoretical and practical points of view. The equality approximation [4] ....

....object x belongs to an object set X we have to answer a question whether its tolerance set I(x) is included in X. Hence we take as a primitive notion a vague inclusion function rather than the fuzzy or rough membership function. Our approach allows to unify different cases considered in [3] 8] [20]. One can define a variety of the lower and upper set approximations in the case when the relation y 2 I(x) is a tolerance (and not equivalence) relation. In general, experiments related to a particular approximation space can show which type of approximation is the best e.g. from the point of ....

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Ziarko W. : Variable Precision Rough Set Model, Journal of Computer and System Sciences, vol. 46 (1), 1993, 39-59. 9

....than RS to discover non deterministic, noisy associations rather than clear cut functional dependencies. Only recently, some generalized models of RS have been introduced, in which exact dependencies are replaced by approximate dependencies, repairing this de ciency in a RS manner, see e.g. [10] The Logical Analysis of Data (LAD) method developed by a group around P. Hammer[2] is very similar to the Rough Set method, but has the further drawback that it works only for two valued (Boolean) attributes. The patterns found by LAD form a subset of the frequent patterns that constitute our ....

W. Ziarko, Variable Precision Rough Set Model, J. of Computer and System Sciences, Vol. 46, pp. 39-59, 1993.

....of a rule. In this case the disjointness property will not in general be true. Chapter 3 Rough Set Theory Rough sets were introduced by Z. Pawlak [26, 27] as a mathematical theory for formalizing approximate reasoning. This theory have been refined and extended by Pawlak himself and others [27, 28, 47, 48]. Rough set theory have been used for data analysis and machine learning. The development of rough sets as a fundament for decision rule generation owes much to A. Skowron and his colleagues [3, 4, 37, 38, 40] The aim of this chapter is to establish the basic notions and ideas necessary for rule ....

....general trends. In the following section, other rough set approaches to induce such rules are reviewed, before the presentation of Mollestad s approach commences in Sect. 5.2. 5.1.2 Rough Set Approaches to Generalized Knowledge Variable precision rough sets were introduced by W. P. Ziarko [47, 48]. The underlying idea is a generalization of the lower approximations of decision classes allowing rules to be generated from objects also in the boundary region. This leads to rules that are not true, that is have probability less than 1 (in the training table) These rules cover a larger number ....

Ziarko, W. P. (ed.) Variable Precision Rough Set Model , J. of Computer & System Science, 46 (1), pp.39-59, 1993

....table, we see that the error of assigning ### ## to # is small (1 observation) and that ### ##### is true up to 2 observations. There are several possibilities to reduce the precision of prediction to cope with measurement error. One possibility is the so called variable precision rough set model [17], which assumes that rules are only valid within a certain part of the 14 Gnther Gediga and Ivo Dntsch population. The advantage of this approach is that it uses only two parameters (the precision parameter and #) to describe the quality of a rule system; the disadvantages are that precision and ....

....unsupervised learning and non parametric distribution estimation. Whereas significance testing uses the same theoretical assumptions as the classical RSDA, the rough entropy based method SORES and the probabilistic granule analysis (as well as other approaches such as the variable precision model [17]) allow some error within prediction rules. Although all of these are RSDA based, from a strictly modelling point of view, these methods are partially incompatible competitors; their particular strengths and weaknesses need to be determined by further investigation. ....

Ziarko, W. (1993). Variable precision rough set model. Journal of Computer and System Sciences, 46.

....A f the family of functions S f is defined in the following way: S f : U Theta P(AT ) Gamma U Theta P(AT ) S f (X; AX ) fx 2 X : x O f 6= g; AX ) for all X and for all A f such that X r(AX ; O) A f AX 3. 3 Lower operator Remark: For a further explanation of Rough Sets theory see [6, 7, 11, 14]. A family of functions l C , is defined where C represents the concept which lower approximation is going to be defined with respect to the descriptor AX : l C : U Theta P(AT ) Gamma U Theta P(AT ) l C (X; AX ) fz 2 X : x Cg; AX ) X r(AX ) Property If l C (X; AX ) Y; A Y ) ....

...., is defined where C represents the concept which lower approximation is going to be defined with respect to the descriptor AX : l C : U Theta P(AT ) Gamma U Theta P(AT ) l C (X; AX ) fz 2 X : x Cg; AX ) X r(AX ) Property If l C (X; AX ) Y; A Y ) then Y ae X 3. 4 Upper operator [6, 7, 11, 14] The family of functions uC , is defined as the upper approximation of objects belonging to C with respecto to the descriptor AX : uC : U Theta P(AT ) Gamma U Theta P(AT ) uC (X; AX ) fz 2 X : z C 6= g; AX ) X r(AX ) Let s see now how all the operations that have been studied can ....

W. Ziarko, Variable Precision Rough Sets Model, Journal of Computer and System Sciences, vol. 46. 1993, 39-59.

....to include the postscript figure in Figure 1. setlength epsfxsize 3.0in begin figure [htb] begin center epsfbox roughfig.eps end center vskip 0.1in caption A Distribution label FigMixture end figure 3. 3 Recommended Bibliography Style Please see the tail of this document[1, 2, 3]. You should use the command: begin thebibliography 9 bibitem Pawlak Pawlak, Z. 1991) it Rough Sets. Kluwer Academic Publishers, Dordrecht. bibitem TsumotoTsumoto, S. and Tanaka, H. 1995) Algebraic Formulation of Empirical Learning Methods based on Rough Sets and Matroid Theory. ....

Ziarko, W. (1993). Variable Precision Rough Set Model. Journal of Computer and System Sciences, 46, 39-59. Bulletin of International Rough Set Society Volume 1, Number 2 roughfig.eps

....or RAS (U; R) and X U , the rough membership function associated with X is defined by X : U [0; 1] X (u) jX R(u)j jR(u)j : This provides a numeric characterization of rough sets. Based on the definition of rough membership function, a variable precision rough set model is proposed in [30, 10]. For 0 ff fi 1, the ff and fi approximation of X is defined by R ff X = fu 2 U j X (u) 1 Gamma ffg R fi X = fu 2 U j X (u) 1 Gamma fig: Though the rough membership function and the accuracy of approximation are well defined for finite universe U , it uses the cardinality which may be ....

W. Ziarko. "Variable precision rough set model". Journal of Computer and System Science, 46:39--59, 1993. This article was processed using the L a T E X macro package with LLNCS style

....jRXj : Furthermore, the rough membership function associated with X is defined by X : U [0; 1] X (u) jX R(u)j jR(u)j : This provides a numeric characterization of rough sets. Based on the definition of rough membership function, a variable precision rough set model is proposed in [ Ziarko, 1993; Katzberg and Ziarko, 1996 ] For 0 ff fi 1, the ff and fi approximation of X is defined by R ff X = fu 2 U j X (u) 1 Gamma ffg R fi X = fu 2 U j X (u) 1 Gamma fig: Though the rough membership function and the accuracy of approximation are well defined for finite universe U , it ....

W. Ziarko. "Variable precision rough set model". Journal of Computer and System Science, 46:39--59, 1993.

....conditioned to the decision classes) expanded with all possible values (possibly conditioned to the decision classes) or treated as a special value in its own right. ffl Computation of partitions and rough set approximations, either in the standard sense or within the variable precision model [15]. ffl Sampling of subtables for validation purposes. ffl Discretization of numerical attributes with various discretization algorithms. Since rough set methods are well suited for applications that take on a course grained view of the world, it is typically desirable to preprocess the data so ....

W. Ziarko (1993), Variable Precision Rough Set Model, Journal of Computer and System Sciences, 46, pp. 39--59.

....include a element(e.g. 7) which does not belong to the class c. However, we have common elements between [x] R2 and C, so we say that R 2 classfies c partially. We can describe this relation in terms of variable precision rough set theory, which is the extension of originial rough set theory [11], as follows: R 2 fi c iff [x] R2 C 6= OE and fi = 1 0 card [x] R2 C card [x] R2 ; where fi denotes the misclassfication rate of R 2 . So, this means that if a case satisfies R 2 , then this case belong to c with the accuracy 1 0 fi. In this way, we can develop rule ....

Ziarko,W. Variable Precision Rough Set Model, Journal of Computer and System Sciences, 46, 39-59, 1993.

....generalized version of the rough set model is proposed. The generalized rough set model is introduced to overcome these shortcomings by incorporating the available statistical information. The generalized rough sets model is an extension of the concept of the variable precision rough sets model [Zia93a]. Our new approach will deal with the situations where uncertain objects may exist, different objects may have different importance degrees, and different classes may have different noise ratios. The standard rough set model and the VP model of rough sets [Zia93b] become a special case of the ....

....degree d is assigned and the set of importance degree is d(obj i ) f4; 3; 4; 4; 3; 4g (i = 1; 2; 6) 4.2. 2 Noise Tolerance in Uncertain Information Systems To manage noise in uncertain information systems, we adopt the concept of relative classification error which was introduced by Ziarko [Zia93a]. The main idea is to draw some boundary region between positive region and negative region, according to some classification factors. The goal is to generate some strong rules which are almost always correct. In the real world, each class (positive class and negative class) in the information ....

Wojciech Ziarko, (1993). Variable Precision Rough Set Model, Journal of Computer & System Science, Vol. 46, No. 1, 39-59

.... rules exist) Although the method essentially applies to the extraction of deterministic rules, it can be extended to derive the maximal generalized nondeterministic rules with decision probabilities by adapting the extended model of rough set called the Variable Precision Rough Set Model (VPRS) [14]. 5 Principles and Algorithm Rough set theory has been successful in many areas, such as knowledge based system in medical diagnosis, machine learning and industry [3] In this section, a new algorithm DBMAXI is presented which is feasible to be used in large database to compute all the maximal ....

Wolciech Ziarko, Variable Precision Rough Set Model, Journal of Computer & System Science, Vol. 46, No. 1, 1993, 39-59

....of necessity and possibility operators. The number of worlds accessible from a world w, and in which a proposition is true, is not taken into consideration. It is therefore not surprising that similar efforts have been attempted in both rough sets and modal logics to incorporate such information [5, 6, 7, 8, 9, 10, 42, 44]. The results of these studies lead to graded and probabilistic interpretations of modal logics and rough sets. 3.1 Graded rough sets Graded modal logics extend modal logic by introducing a family of graded modal operators 2 n and 3 n , where n 2 N and N is the set of natural numbers [5, 7, 8, ....

.... of probabilistic rough set operators in the framework of Bayesian decision theory can be found in Yao and Wong [42] Pawlak and Skowron referred to the conditional probabilities as rough membership functions [29] The variable precision rough set model proposed by Ziarko adopted a similar notions [44]. In fact, the lower and upper approximations defined by Ziarko are exactly the same as those defined in this paper. Compared with modal and graded modal logics, there is a lack of systematic study on probabilistic modal logic. It will be interest to apply the results in probabilistic rough sets ....

Ziarko, W., "Variable precision rough set model," Journal of Computer and System Science, Vol. 46, pp. 39-59, 1993.

....knowledge of the information signature of x with respect to attributes B. B X (x) j[x] B X j j[x] B j 2 [0; 1] The formulae for the lower and upper set approximations can readily be generalized to some arbitrary level of precision 2 [ 1 2 ; 1] by means of the rough membership function [7], as shown below. Possible ties in the case of = 0:5 can be resolved by assigning the objects in question to the interior of the set. Note that the lower and upper approximations as originally formulated are obtained as a special case with = 1:0. B X = fx j B X (x) g B X = fx j B X ....

W. Ziarko (1993), Variable Precision Rough Set Model, Journal of Computer and System Sciences, 46, pp. 39--59.

.... can be interpreted as a frequency based estimate of Pr(x 2 X j x; B) B X (x) j[x] B Xj j[x] B j 2 [0; 1] The formulas for the lower and upper set approximations can readily be generalized to some arbitrary level of precision 2 [ 1 2 ; 1] by means of the rough membership function [7], as shown below. Possible ties in the case of = 0:5 can be resolved by assigning the objects in question to the interior of the set. Note that the lower and upper approximations as originally formulated are obtained as a special case with = 1:0. B X = fx j B X (x) g B X = fx j B ....

W. Ziarko (1993), Variable Precision Rough Set Model, J. Comp. Syst. Sci., 46, pp. 39--59.

....(x) I Y (x) for any x 2 U; i) If X is a family of pair wise disjoint sets of U , then I S X (x) P X2X I X (x) for any x 2 U , provided that I is an equivalence relation. The above ideas can be generalized in the same way as proposed by Ziarko in the variable precision rough model Ziarko (1993). Let fi be a real number such that 0 fi 0:5: Approximations can be defined now as follows: I fi (X) fx 2 U : I X (x) 1 Gamma fig; I fi (X) fx 2 U : x) fig: Note that if fi = 0 , we get the previous case. Remark. We could also assume that 0:5 fi 1 and consequently I fi ....

Ziarko, W. (1993). Variable Precision Rough Set Model. Journal of Computer and System Sciences, vol. 40(1), pp. 39-59.

....the multiset theory and the theory of fuzzy sets. This work was supported by grant No. 8 S503 021 06 from State Committee for Scientific Research, Poland 1 Introduction The author s considerations on the subject of a question of conceptions of rough sets presented by Pawlak [1] 2] and Ziarko [3] and conceptions of Blizad s multisets [4] and Zadeh s fuzzy sets [5] were an inspiration to write this short paper. The generalization of Ziarko s conception of rough sets proposed in this paper is a result of these considerations. 2 A contextual approximates space Definition 1 . The structure ....

....6= X Y = X 1 Y 8k 2 I(X k Y ) k = 1) iii) X i Y i j ) X j Y , iv) X i Y , X i X Y , The expression X i Y we read: X is context of Y in the degree i. From condition (i) immediatly follows Fact 1 . a) x 2 X ) fxg 0 X, b) X 6= X 1 ; EXAMPLE (see Ziarko [3]) Let U be a nonempty finite set, C a partition of U , c a measure of the relative degree of inclusion of two subset of the universe U defined in the following way: for any X; Y U c(X; Y ) 8 : 1 Gamma card(X Y ) card(X) if card(X) 0 0 if card(X) 0; where card denotes set ....

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W. Ziarko, Variable Precision Rough Set Model, Journal and System Sciences, No. 1,1993, p. 39-59.

....a generalization of set approximation in these spaces. Our intention is to give a general tool for the investigation of relation approximations. 2 Generalized approximation spaces In this section we present a generalization of the lower and upper approximations of sets, introduced in [3] and [9]. First we recall the definitions from [3] 9] An approximation space is an ordered pair R= U; IND) where U is a nonempty set and IND U ThetaU is an equivalence relation called the indiscernibility relation. By [x] IND we denote an equivalence class of the relation IND defined by the ....

....spaces. Our intention is to give a general tool for the investigation of relation approximations. 2 Generalized approximation spaces In this section we present a generalization of the lower and upper approximations of sets, introduced in [3] and [9] First we recall the definitions from [3] [9]. An approximation space is an ordered pair R= U; IND) where U is a nonempty set and IND U ThetaU is an equivalence relation called the indiscernibility relation. By [x] IND we denote an equivalence class of the relation IND defined by the object x. The lower and upper approximations of a ....

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Ziarko W. : Variable Precision Rough Set Model, Journal of Computer and System Sciences, vol. 46 (1), 1993, 39-59.

....= paroxysmal] jolt = yes] M1 = yes] However, induction of inclusive rules gives us two problems. First, SI 3 It is notable that this rule is a kind of probabilistic proposition with two statistical measures, which is one kind of an extension of Ziarko s variable precision model(VPRS) [10]. 4 It is also notable that the contrapositive of this proposition is :R :d, which means that if R is not observed, we can delete a disease d from diagnostic candidates. It also means that [x] R is a positive region of :d, that is, x] R will be a lower approximation of :d. procedure ....

Ziarko,W. Variable Precision Rough Set Model, Journal of Computer and System Sciences, 46, 39-59, 1993.

....examples are almost invariably imprecise, so one equivalence class may fall into more than one concepts. In the rough sets, if a particular equivalence class fall into multiple concepts, then usually either all candidate concepts are presented (Pawlak 1991) or the most likely concept is chosen (Ziarko 1993). The most likely concept is chosen based on the conditional probability distributions of positive and negative objects occurring within the equivalence class. Given an attribute value system S = hU; C [ fdg; V; fi and an equivalence relation R(C) on U , Ziarko (1993) defined a fi approximation ....

....most likely concept is chosen (Ziarko 1993) The most likely concept is chosen based on the conditional probability distributions of positive and negative objects occurring within the equivalence class. Given an attribute value system S = hU; C [ fdg; V; fi and an equivalence relation R(C) on U , Ziarko (1993) defined a fi approximation space ASP for the condition attributes C as a quadruple hU; R(C) P; fii, where P is the probability measure described above and fi is a user specified real number in the range [0; 0:5) The equivalence classes induced by the relation R(C) are called elementary sets in ....

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Ziarko, W. 1993. "Variable Precision Rough Set Model," Journal of Computer and System Sciences, 46(1):39-59.

....of which is incomplete and vague. This theory in itself is complementary in relation to Zadeh s set theory (1965) and Pawlak s rough set theory (1982, 1992) The authors considerations on this subject were, however, inspired not only by Pawlak s and Zadeh s works, but also by the conceptions of Ziarko (1993) and Blizard (1989a, 1989b) The generalization of rough sets proposed in this paper is a result of these considerations. The paper consists of four sections. In Section 1 we introduce the notion of an approximation space and give some examples of such spaces. In Section 2 we define the ....

....1, immediately follows Fact 1 In CAS the following conditions are satisfied: a) x 2 X ) fxg 0 X, b) X 6= 9k 2 I(k 1 X k ; c) X 6= X Y = X 1 Y , d) X 6= X Y 6= 9k 2 I(X k Y k 1) Below we give two examples of contextual spaces. Example 1 (see: Ziarko 1993) . Let U be a nonempty finite set, C a partition of U , c a measure of the relative degree of inclusion of two subsets of the universe U defined in the following way: for any X; Y U c(X; Y ) 8 : 1 Gamma card(X Y ) card(X) if card(X) 6= 0 0 if card(X) 0; where card ....

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Ziarko, W. 1993. "Variable Precision Rough Set Model." Journal of Computer and System Sciences, vol. 46, no. 1: 39-59.

....will result in discovering stronger rules having good interpretation characteristics. In this study we induce partly discriminating rules using a modified version of LEM2 algorithm. The above motivation is somehow similar to the concept of so called Variable Precision Rough Set Model introduced by Ziarko (1993). 4 Analysis of the ESWL information system 4.1 Looking for reducts Let us consider the ESWL information system extended to 435 patients described by 33 attributes and classified by two classifications Y 1 and Y 2 . For both classifications Y 1 and Y 2 lower and upper approximations of ....

Ziarko W. 1993: Variable precision rough set model. Journal of Computer and System Sciences, 40, pp. 39-59.

....which are described in Section 2, exact decision rules cannot be derived by standard standard methods[2 4,6] Our objective in this paper is to suggest a method for generating classification rules from incomplete information. The proposed method is based on an extension of the rough set model[12]. Statistical information is used to define the positive and negative regions of a concept. Each classification rule generated by our learning system is characterized by an uncertainty factor which is in fact the probability that an object matching the condition part of the rule belongs to the ....

....i ) 1 fX i 2 R (C)g and R(C) Y ) P (Y jX i ) 0 fX i 2 R (C)g; respectively. The above definitions do not make use of the statistical information in the boundary region R(C) Y ) Gamma R(C) Y ) For this reason, a number of extensions to the original rough set model have been proposed[1,5,8,10,12]. In our approach we attempt to rectify this limitation by introducing a fi approximation space. A fi approximation space AS P is a triple U; R(C) P , where P is a probability measure described in Subsection 3.1 and fi is a real number in the range (0:5; 1] The fi approximation space AS P ....

Ziarko, W. Variable Precision Rough Set Model. Journal of Computer and System Sciences, Vol. 46, No. 1, 1993, pp.39-59.

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W. Ziarko, "Variable precision rough set model," Journal of Computer and System Sciencs, 46, 39-59, 1993.

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Ziarko, W.: Variable precision rough set model. Journal of Computer and Systems Science 46 (1993) 39--59

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W. Ziarko, "Variable precision rough sets model," Journal of Computer and System Sciences, vol. 46, pp. 39--59, 1993.

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W. Ziarko. Variable precision rough sets model. Journal of Computer and Systems Sciences, 46(1):39--59, 1993.

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Ziarko, W.: Variable Precision Rough Set Model, Journal of Computer and Systems Science, 46(1), 1993, 39--59.

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W. Ziarko. "Variable precision rough set model". Journal of Computer and System Science, 46:39--59, 1993.

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Ziarko, W. (1993). Variable Precision Rough Set Model, Journal of Computer and System Sciences, 40(1), 39-59.

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Ziarko, W. (1993). Variable Precision Rough Set Model, Journal of Computer and System Sciences, vol. 40(1), pp. 39-59. This article was processed using the L a T E X macro package with LLNCS style

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