Knowledge Discovery by Relation Approximation: A Rough Set Approach

In recent years, rough set theory [1] has attracted attention of many researchers and practitioners all over the world, who have contributed essentially to its development and applications. With many practical and interesting applications rough set approach seems to be of fundamental importance to AI and cognitive sciences, especially in the areas of machine learning, knowledge acquisition, decision analysis, knowledge discovery from databases, expert systems, inductive reasoning and pattern recognition [2]. The common issue of the above mentioned domains is the concept approximation problem which is based on searching for description – in a predefined language L – of concepts definable in other language L∗. Not every concept in L∗ can be exactly described in L, therefore the problem is to find an approximate description rather than exact description of unknown concepts, and the approximation is required to be as exact as possible. Usually, concepts are interpretable as subsets of objects from a universe, and the accuracy of approximation is measured by the closeness of the corresponding subsets. Rough set theory has been introduced as a tool for concept approximation from incomplete information or imperfect data. The essential idea of rough set approach is to search for two descriptive sets called the lower approximation containing those objects that certainly belong to the concept and the ”upper approximation” containing those objects that possibly belong to the concept. Most concept approximation methods realize the inductive learning approach, which assumes that a partial information about the concept is given by a finite sample, so called the training sample or training set, consisting of positive and negative cases (i.e., objects belonging or not belonging to the concept). The information from training tables makes the search for patterns describing the given concept possible. In practice, we assume that all objects from the universe U are perceived by means of information vectors being vectors of attribute values (information signature). In this case, the language L consists of boolean formulas defined over conditional (effectively measurable) attributes. The task of concept approximation is possible when some information about the concept is available. Except the partial information above the membership function given by training data set, the domain knowledge is also very useful