Pawlak, Z.: Rough sets, International Journal of Computer and Information Science, 11, 341--356, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....value that denotes the confidence that the point belongs to a particular region. It is generally the case that the closer the instance is to a particular centroid, the more confident one can be that it belongs to that region. Work in areas such as fuzzy set theory [Zad65] and rough sets [Paw82] have similarities to the idea of using non binary region memberships. A first guess at a region membership measure is normalised Euclidean distance from the centroid. By normalised, we mean that the distance along each axis is divided by its standard deviation. Doing so ensures that no one ....

Z. Pawlak. Rough sets. International Journal of Computer and Information Sciences, 11:341--356, 1982.

....quality of new notions for most general and simple concepts will be more dicult to assess and both empirically and theoretically developped heuristics have to be incorporated into the learning process. 3 Data Mining with Rough Sets Rough set theory was created by Pawlak in the early eighties [Paw82] and has since then become very popular. One of its applications is data mining. We will present the basic ideas in a very compact way following Bazan [Baz98] We break down the description of rough set data mining techniques into two parts, the description of rough set approximation and of rough ....

Z. Pawlak. Rough sets. International Journal of Computer and Information Sciences, 11(5):341-356, 1982.

....phrases. Artificial intelligence, machine learning, decision tree, data mining, entropy, rough set. c Yang s Scitific Reserach Institute, LLC. All rights reserved. 25 and used in AI, information processing and especially in data mining or Knowledge Discovery in Databases, i.e. KDD, nowadays [2][3] 4] 7] 8] It is another useful and powerful theory after fuzzy set theory for dealing with information with uncertainties. We utilize the basic idea in rough set, and set up an approach to the selection of nodes of decision tree. By designing an experiment, in which all of the values of each ....

....upper approximation of X is the least composed set in S that contains X, the best lower approximation of X is the greatest composed set in S that is contained in X. The original definition of rough set and some further discussions on the features of rough set can be found in the paper of Z Pawlak[2]. In classification applications, an information set may be given as in Table 1. In Table 1, A, B, C, D are called condition attributes, d is called decision attribute. Each line in Table 1 is called a tuple. All tuples in the table constitute U . An equivalence relation corresponds to each ....

Z. Pawlak. Rough Sets. International Journal of Computer and Information Science. 1982, vol.11, No.5, pp. 341-356.

....However, we only have: IVa) apr(X (IVb) apr(X apr(Y ) It is impossible to obtain the lower (upper) approximation of the union (intersection) of some sets from their lower (upper) approximations. Additional properties of rough set approximations can be found in Pawlak [4], and Yao and Lin [17] Equivalence classes of the partition U E are called the elementary granules. They represent the available information. All knowledge we have about the universe are about these elementary granules, instead of about individual elements. With this interpretation, we also have ....

Pawlak, Z. Rough sets, International Journal of Computer and Information Sciences, 11, 341356, 1982.

....and note that this setting is suitable also for other operators based on binary relations. Properties of the ordered sets of the upper and the lower approximations of the elements of an atomic Boolean lattice are studied. 1 Introduction The basic ideas of rough set theory introduced by Pawlak [8] deal with situations in which the objects of a certain universe can be identified only within the limits determined by the knowledge represented by a given indiscernibility relation. The indiscernibility relation enables us to divide objects of the universe U into three disjoint sets with respect ....

Z. Pawlak, Rough Sets, International Journal of Computer and Information

....exclusiveness garanties coveredness even for the case that more information about elements in the ABox become available. Given the Open World Assumption which usually underlies description logic based knowledge representation, this seems to be a crucial requirement. Based on Rough Set theory [10] we also show in [12] that exclusiveness provides a theoretical notion of safe data which ensures good properties w.r.t. noisy data. A learned concept LD should also be supported by the fact that there is at least one example in the ABox, which is an instance of LD and we de ne a witness for LD ....

Z. Pawlak. Rough sets. International Journal of Computer and Information Sciences, 11(5):341-356, 1982.

....even the existence of objects is unknown. Another approach to describe vague values is the use of fuzzy sets. A prerequisite for this approach is probabilistic information, such as the distribution of the concerned attributes. Some relevant papers in this respect are [3, 7, 15] The articles [1, 14, 19, 20, 16] are concerned with vague sets resulting from vague attribute values sometimes called rough sets . To this end, the result is approximated by a lower bound (containing all objects surely contained in the desired crisp result) and an upper bound (containing all objects potentially contained in ....

Z. Pawlak. Rough Sets. International Journal of Computer and Information Sciences, 11(5):341--356, 1982.

....the form of d 1 , can be viewed as the greatest lower bound of all open worlds emanating from d 1 . To conclude this brief survey we mention a recent approach taken by Keen and Rajasekar [KEEN93] who formalise the partitions arising from data dependencies such as FDs in the context of rough sets [PAWL82]. 10 Concluding Remarks We have investigated the additivity problem for NFDs and NINDs. The largest class of sets of NFDs for which additivity holds with respect to DB(R) is the class of monodependent sets of NFDs. In addition, checking whether a set of NFDs is monodependent can be done in time ....

Z. Pwalak, Rough sets. International Journal of Computer and Information Sciences 11, (1982), 341-356.

....Norwegian University of Science and Technology, 7034 Trondheim, Norway 2 Institute of Mathematics, Warsaw University, 02 097 Warsaw, Banacha 2, Poland 3 Polish Japanese Inst. of Computer Techniques, 02 018 Warsaw, Koszykowa 86, Poland 1 Introduction Rough sets and Pawlak information systems [18, 19] have recently gained rather substantial scientific interest. We believe that, especially in this field, successful research requires good co operation between theoreticians and practitioners. This can be enhanced by providing a sophisticated user environment supporting all aspects of the ....

Pawlak, Z.: Rough sets. International Journal of Computer and Information Sciences 11 (1982) 341--356

....location (Burrough and Frank 1996; Bittner 1999) A second component of uncertainty is due to error in observing the coverage values. Among different error components, at least noise is inevitable. There is a lot of research in modeling error, uncertainty, or imprecision, proposing rough sets (Pawlak 1982), fuzzy sets (Zadeh 1965) or stochastic methods (Koch 1999) None of these methods are considered in interoperability standards up to now. However, observing two features independently may disturb their topological relation under these circumstances. Therefore, features taken or derived from ....

Pawlak, Z., 1982: Rough Sets. International Journal of Computer and Information Sciences, 11(5): 341-356.

....provide algorithmic ways to find this best approximation. We also state a number of miscellaneous results and discuss some open problems. 1. Introduction In this paper we look at fundamental methodological issues underlying the concept of a rough set. Rough sets have been introduced by Pawlak [14] to serve as approximate descriptions of sets that are unknown, incompletely specified, or whose exact specification is complex. The approach pioneered by Pawlak allows us to reason about such sets given only their representations as rough This is an extended version of the first part of the ....

.... the structures hP(U) Theta P(U) kn ; in i and hD I Theta D I ; kn ; in i (this latter one under any of the finiteness conditions) form complete bilattices [6, 5] We will now discuss connections between the concept of a rough set as defined above and the original one introduced by Pawlak [14]. Pawlak observed that when an information system 8 V.W. Marek and M. Truszczy nski Contributions to the Theory of Rough Sets I = hU ; Ai satisfies any of the finiteness conditions then, for every set X U , there exists a greatest definable set X 0 such that X 0 X and, similarly, there ....

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Pawlak, Z. Rough Sets, International Journal of Computer and Information Sciences 11 (1982), pp. 341-356.

....corresponding relation S standard transitive on U , semi partial standard ordering relation on U and standardequivalence relation on U , respectively. 2 Classical Crisp Rough Sets and their Crisp Generalizations Assume that R j U U is an equivalence relation on U and X jU . Following PAWLAK [24 26] we define the R lower approximation RX and the R upper approximation RX of X as follows Definition 3 1. RX = def mY Y U RY j Xr 2. RX = def kY Y U RY X = #p. Furthermore, following PAWLAK we define Definition 4 1. X is said to be R exact = def RX = RX 2. X is said to be ....

Z. PAWLAK. Rough sets. International Journal of Computer and Information Sciences 11, 341--356, 1982.

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Z. Pawlak, Rough sets, International Journal of Computer and Information Sciences 11 (1982), 341--356.

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Z. Pawlak, Rough sets, International Journal of Computer and Information Sciences 11 (5) (1982) 341 -- 356.

....restricted by a tolerance relation R. We study R approximations, R definable sets, R equalities, and investigate briefly the structure of R rough sets. 1 Introduction In rough set theory it is usually assumed that the knowledge about objects is restricted by some indiscernibility relation (see [10], for example) Indiscernibility relations are equivalences which are interpreted so that two objects are equivalent if we cannot distinguish them by using our information. This means that the objects of the given universe U can be classified by the knowledge represented by an equivalence ....

Z. PAWLAK, Rough Sets, International Journal of Computer and Information Sciences 5 (1982), 341--356.

....show how we can by using a matrix represention of preimage relations determine families of sets, which are dense in dependence spaces defined by preimage relations. In rough set theory it is usually assumed that the knowledge about objects 1 is restricted by some indiscernibility relation (see [43, 45], for example) Indiscernibility relations are equivalences which are interpreted so that two objects are equivalent if we cannot distinguish them by using our information. In an information system an indiscernibility relation arises naturally when one considers a given set of attributes: two ....

....C( A) is said to be dependent on B( A) denoted by B C, if CD (C) CD (B) We present a method based on dense families which for a given dependency B C finds all 5 minimal subsets D of B such that D B. 1. 4 Rough Sets We generalize the lower and upper approximations defined by Pawlak [43]. For any tolerance R on U the lower R approximation of a set X( U) is XR = fx 2 U j x=R Xg and its upper R approximation is X R = fx 2 U j x=R X 6= g: Here x=R is the R neighborhood fy 2 U j xRyg of x. The set BR (X) X R Gamma XR is referred to as the R boundary of X . The idea ....

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Z. PAWLAK, Rough Sets, International Journal of Computer and Information Sciences 5 (1982), 341--356.

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Pawlak, Z.: Rough sets, International Journal of Computer and Information Science, 11, 341--356, 1981.

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Z. Pawlak, Rough sets, International Journal of Computer and Information Science 11 (1981) 341--356. 15

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Z. Pawlak: Rough sets, International Journal of Computer and Information Science, 11, 341--356, 1981.

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Pawlak, Z.: Rough Sets, International Journal of Computer and Information Sciences, 11, 1982, 341--356.

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Pawlak, Z.: Rough Sets. International Journal of Computer and Information Sciences, 11, 1982, 341-356.

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Pawlak, Z.: Rough sets. International Journal of Computer and Information Sciences 11 (1982) 341-356.

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Pawlak,Z.,(1982), "Rough sets", International Journal of Computer and Information Sciences, Issue. 11, pp. 413-433.

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Pawlak, Z., (1982), "Rough Sets", International Journal of Computer and Information Sciences, Issue. 11, pp. 413-433.

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Z. Pawlak. "Rough sets". International Journal of Computer and Information Science, 11:341--356, 1982.

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