Definition of the Subject
This article describes the dominance‐based rough set approach (DRSA) to granular computing and data mining. DRSA was first introduced asa generalization of the rough set approach for dealing with multicriteria decision analysis, where preference order is important. The ordering isalso important, however, in many other problems of data analysis. Even when the ordering seems absent, the presence or the absence of a property canbe represented in ordinal terms, because if two properties are related, the presence, rather than the absence, of one property should make more (or less)probable the presence of the other property. This is even more apparent when the presence or the absence of a property is graded or fuzzy, because inthis case, the more credible the presence of a property, the more (or less) probable the presence of the other property. Since the presence ofproperties, possibly fuzzy, is the basis of any granulation, DRSA can be seen as a general basis for...
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Abbreviations
- Case-based reasoning:
-
Case-based reasoning is a paradigm in machine learning whose idea is that a new problem can be solved by noticing its similarity to a set of problems previously solved. Case-based reasoning regards the inference of some proper conclusions related to a new situation by the analysis of similar cases from a memory of previous cases. Very often similarity between two objects is expressed on a graded scale and this justifies application of fuzzy sets in this context. Fuzzy case-based reasoning is a popular approach in this domain.
- Decision rule:
-
Decision rule is a logical statement of the type “if…, then…”, where the premise (condition part) specifies values assumed by one or more condition attributes and the conclusion (decision part) specifies an overall judgment.
- Dominance‐based rough set approach (DRSA):
-
DRSA permits approximation of a set in universe U based on available ordinal information about objects of U. Also the decision rules induced within DRSA are based on ordinal properties of the elementary conditions in the premise and in the conclusion, such as “if property \( { f_{i1} } \) is present in degree at least \( { \alpha_{i1} } \) and … property f ip is present in degree at least α ip , then property f iq is present in degree at least α iq ”.
- Fuzzy sets:
-
Differently from ordinary sets in which an object belongs or does not belong to a given set, in a fuzzy set an object belongs to a set in some degree. Formally, in universe U a fuzzy set X is characterized by its membership function \( { \mu_X\colon U \rightarrow [0,1] } \), such that for any \( { y \in U } \), y certainly does not belong to set X if \( { \mu_X(y)=0 } \), y certainly belongs to X if \( { \mu_X(y)=1 } \), and y belongs to X with a given degree of certainty represented by the value of \( { \mu_X(y } \)) in all other cases.
- Granular computing:
-
Granular computing is a general computation theory for using granules such as subsets, classes, objects, clusters, and elements of a universe to build an efficient computational model for complex applications with huge amounts of data, information, and knowledge. Granulation of an object a leads to a collection of granules, with a granule being a clump of points (objects) drawn together by indiscernibility, similarity, proximity, or functionality. In human reasoning and concept formulation, the granules and the values of their attributes are fuzzy rather than crisp. In this perspective, fuzzy information granulation may be viewed as a mode of generalization, which can be applied to any concept, method, or theory.
- Ordinal properties and monotonicity:
-
Ordinal properties in description of objects are related to graduality of the presence or absence of a property. In this context, it is meaningful to say that a property is more present in one object than in another object. It is important that the ordinal descriptions are handled properly, which means, without introducing any operation, such as sum, averages, or fuzzy operators, like t-norm or t‑conorm of Łukasiewicz, taking into account cardinal properties of data not present in the considered descriptions, which would, therefore, give not meaningful results. Monotonicity is strongly related to ordinal properties. It regards relationships between degrees of presence or absence of properties in the objects, like “the more present is property f i , the more present is property f j ”, or “the more present is property f i , the more absent is property f j ”. The graded presence or absence of a property can be meaningfully represented using fuzzy sets. More precisely, the degree of presence of property f i in object \( { y \in U } \) is the value given to y by the membership function of the set of objects having property f i .
- Rough set:
-
A rough set in universe U is an approximation of a set based on available information about objects of U. The rough approximation is composed of two ordinary sets called lower and upper approximation. Lower approximation is a maximal subset of objects which, according to the available information, certainly belong to the approximated set, and upper approximation is a minimal subset of objects which, according to the available information, possibly belong to the approximated set. The difference between upper and lower approximation is called boundary.
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Greco, S., Matarazzo, B., Słowiński, R. (2009). Granular Computing and Data Mining for Ordered Data: The Dominance-Based Rough Set Approach. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_251
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