Attribute-Oriented Knowledge Discovery in Rough Relational Databases

The rough relational database model was developed for the management of uncertainty in relational databases. A particular type of knowledge discovery, attribute oriented induction of rules from generalized data is described in this paper. Rough Relational Databases In rough sets (Pawlak 1984) an approximation space is defined on some universe U by defining some equivalence relation that partitions the universe into equivalence classes called elementary sets, based on some definition of ‘equivalence’ as it relates to the application domain. Any finite union of these elementary sets is called a definable set. A rough set X U, however, can be defined in terms of the definable sets in terms of its lower (RX) and upper ( R X) approximation regions: RX = {x U |[x]R X} and R X = {x U | [x]R X }. We may refer to RX as the positive region, UR X as the negative region, and R X – RX as the boundary or borderline region of the rough set X. The lower and upper approximation regions, then, allow the distinction between certain and possible inclusion in a rough set. The rough relational database model captures all the essential features of the theory of rough sets including the notion of indiscernibility of elements through the use of equivalence classes and the idea of denoting an indefinable set by its lower and upper approximation regions The attribute domains in this model are partitioned by equivalence relations designated by the database designer or user. Within each domain, a group of values that are considered indiscernible form an equivalence class. The query mechanism uses class equivalence rather than value equality in retrievals. A user may not know the particular attribute value, but might be able to think of a value that is equivalent to the value required. For example, if the query requests "COLOR = 'BROWN'", the result will contain all colors that are defined as equivalent to BROWN, such as TAN, SORREL, or CHESTNUT. Therefore, the exact wording of a query is less critical. The rough relational database (Beaubouef and Petry 2000) retains significant features of the ordinary relational database. Both models represent data as a set of relations containing tuples. The relations themselves are also sets. The tuples of a relation are its elements, and like the elements of Copyright © 2007, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. sets in general, are unordered and nonduplicated. A tuple ti takes the form (di1, di2, ..., dim), where dij is a domain value of a particular domain set Dj. Let P(Di) denote the powerset(Di) . Definition. A rough relation R is a subset of the set cross product P(D1) P(D2) P(Dm). A rough tuple t is any member of R, which implies that it is also a member of P(D1) P(D2) P(Dm). If ti is some arbitrary tuple, then ti = (di1, di2, ..., dim) where dij Dj. Definition. Tuples ti = (di1, di2, ..., dim) and tk = (dk1, dk2, ..., dkm) are redundant if [dij] = [dkj] for all j = 1,..., m, and where [dij] denotes the equivalence class to which dij belongs. Attribute Oriented Generalization Generalization of data is typically performed with utilization of concept hierarchies on an attribute-byattribute basis, applying a separate concept hierarchy for each of the generalized attributes included in the relation of task-relevant data. The basic steps / guidelines for attribute-oriented generalization in an object-oriented database are summarized below (Han and Kamber 2006): 1. An initial query to the database provides the starting generalization relation R which contains the set of data that is relevant to the user’s generalization interest. 2. If there is a large set of distinct values for an attribute but there is no higher level concept provided for the attribute, the attribute should be removed in the generalization process. 3. If there exists a higher-level concept in the concept tree for an attribute value of a tuple, the substitution of the value by its higher-level concept generalizes the tuple. 4. Two generalized tuples may become similar enough to be merged, so we include an added attribute, Count, to keep track of how many objects have been merged to form the current generalized relation. The value of the count of a tuple should be carried to its generalized tuple and the counts should be accumulated when merging identical tuples in generalization. 5. The generalization is controlled by providing levels that specify how far the process should proceed. If the number of distinct values of an attribute in the given relation is larger than the generalization threshold value, further generalization on this attribute should be performed. If the number of tuples in a generalized relation is larger than their generalization threshold value, the generalization should proceed further. We can then extract characteristic rules from generalized data.