Rolly Intan

Fuzzy functional dependency and its application to approximate data querying

Reviews a new definition of fuzzy functional dependency based on conditional probability and its application to approximate data reduction related to the operation of projection in classical relational databases in order to construct fuzzy integrity constraints. We introduce the concept of partial fuzzy functional dependency, which expresses the fact that a given attribute X does not determine Y completely, but in the partial area of X it might determine Y. Finally, we discuss another application of fuzzy functional dependency in constructing fuzzy query relations for data querying and approximate joins of two or more fuzzy query relations in the framework of an extended query system.

Conditional Probability Relations in Fuzzy Relational Database

In 1982, Buckles and Petry [1] proposed fuzzy relational database for incorporating non-ideal or fuzzy information in a relational database. The fuzzy relational database relies on the spesification of similarity relation [8] in order to distinguish each scalar domain in the fuzzy database. These relations are reflexive, symmetric, and max-min transitive. In 1989, Shenoi and Melton extended the fuzzy relational database model of Buckles and Petry to deal with proximity relation [2] for scalar domain. Since reflexivity and symmetry are the only constraints placed on proximity relations, proximity relation is considered as generalization of similarity relation. In this paper, we propose design of fuzzy relational database to deal with conditional probability relation for scalar domain. These relations are reflexive and not symmetric.We show that naturally relation between fuzzy information is not symmetric. In addition, we define a notion of redundancy which generalizes redundancy in classical relational database. We also discuss partitioning of domains with the objective of developing equivalence class.

Generalization of Rough Sets with alpha-Coverings of the Universe Induced by Conditional Probability Relations

Standard rough sets are defined by a partition induced by an equivalence relation representing discernibility of elements. Equivalence relations may not provide a realistic view of relationships between elements in real-world applications. One may use coverings of, or non-equivalence relations on, the universe. In this paper, the notion of weak fuzzy similarity relations, a generalization of fuzzy similarity relations, is used to provide a more realistic description of relationships between elements. A special type of weak fuzzy similarity relations called conditional probability relation is discussed. Generalized rough set approximations are proposed by using a-coverings of the universe induced by conditional probability relations.

Generalized fuzzy rough sets by conditional probability relations

In 1982, Pawlak proposed the concept of rough sets with a practical purpose of representing indiscernibility of elements or objects in the presence of information systems. Even if it is easy to analyze, the rough set theory built on a partition induced by equivalence relation may not provide a realistic view of relationships between elements in real-world applications. Here, coverings of, or nonequivalence relations on, the universe can be considered to represent a more realistic model instead of a partition in which a generalized model of rough sets was proposed. In this paper, first a weak fuzzy similarity relation is introduced as a more realistic relation in representing the relationship between two elements of data in real-world applications. Fuzzy conditional probability relation is considered as a concrete example of the weak fuzzy similarity relation. Coverings of the universe is provided by fuzzy conditional probability relations. Generalized concepts of rough approximations and rough membership functions are proposed and defined based on coverings of the universe. Such generalization is considered as a kind of fuzzy rough set. A more generalized fuzzy rough set approximation of a given fuzzy set is proposed and discussed as an alternative to provide interval-value fuzzy sets. Their properties are examined.

Degree of Similarity in Fuzzy Partition

In this paper, we discuss preciseness of data in terms of obtaining degree of similarity in which fuzzy set can be used as an alternative to represent imprecise data. Degree of similarity between two imprecise data represented in two fuzzy sets is approximately determined by using fuzzy conditional probability relation. Moreover, degree of similarity relationship between fuzzy sets corresponding to fuzzy classes as results of fuzzy partition on a given finite set of data is examined. Related to a well known fuzzy partition, called fuzzy pseudopartition or fuzzy c-partition where c designates the number of fuzzy classes in the partition, we introduce fuzzy symmetric c-partition regarded as a special case of the fuzzy c-partition. In addition, we also introduce fuzzy covering as a generalization of fuzzy partition. Similarly, two fuzzy coverings, namely fuzzy c-covering and fuzzy symmetric c-covering are proposed corresponding to fuzzy c-partition and fuzzy symmetric c-partition, respectively.

Generalization of Rough Membership Function Based on \alpha -Coverings of the Universe

In 1982, Pawlak proposed the concept of rough sets with practical purpose of representing indiscernibility of elements. Even it is easy to analyze, the rough set theory built on a partition induced by equivalence relation may not provide a realistic view of relationships between elements in the real-world application. Here, coverings of, or non-equivalence relations on, the universe can be considered to represent a more realistic model instead of partition in which a generalized model of rough sets was proposed. In this paper, based on a-coverings of the universe, a generalized concept of rough membership functions is proposed and defined into three values, minimum, maximum and average. Their properties are examined.

Knowledge-based representation of fuzzy sets

A fuzzy set is considered to represent deterministic uncertainty called fuzziness. In deterministic uncertainty of fuzzy sets, one may subjectively determine the membership function of a given element by his knowledge. Different persons with different knowledge may provide different membership functions for elements in a universe with respect to a given fuzzy set. Here, knowledge plays important roles in determining or defining a fuzzy set. By adding the component of knowledge, we generalize the definition of fuzzy set based on probability theory. Some basic operations are re-defined. In addition, by using a fuzzy conditional probability relation, granularity of knowledge is given in two frameworks, crisp granularity and fuzzy granularity. Also, two asymmetric similarity classes or subsets are considered. When fuzzy sets represent problems or situations, a granule of knowledge might describe a class (group) of knowledge (persons) who has similar point of view in dealing the problems. Objectivity and individuality measures are proposed in order to calculate degree of objectivity and individuality, respectively of a given element of knowledge.

Generating Fuzzy Thesaurus by Degree of Similarity in Fuzzy Covering

A notion of fuzzy covering over a set of terms was proposed as a generalization of a well known fuzzy partition. Degree of similarity relationship between fuzzy classes (subsets) on the set of terms (subjects) as results of the fuzzy covering is approximately calculated using a fuzzy conditional probability relation by which degree of similarity relationship between two terms can be determined. An associated function was defined to calculate degree of association between two terms. Special attention will be given to generate a concept of fuzzy thesaurus using the associated function. Its some properties are discussed and examined. A simple example is given to demonstrate the concept.

A Proposal of Probability of Rough Event Based on Probability of Fuzzy Event

Probability and fuzziness are different measures for dealing with different uncertainties in which they be able tob e combined and considered as a complementary tool. In this paper, the relationship between probability and fuzziness are discussed based on the process of perception. In probability, set theory is used to provide a language for modeling and describing random experiments. Here, as a generalization of crisp set, fuzzy set is used to model fuzzy event as defined by Zadeh. Similarly, rough set can be also used to represent rough event in terms of probability measure. Special attention will be given to conditional probability of fuzzy event as well as conditional probability of rough event. Their several combinations of formulation and properties are defined.

Multi-rough Sets and Generalizations of Contexts in Multi-contexts Information System

In previous paper, we introduced a concept of multi-rough sets based on a multi-contexts information system (MCIS). The MCIS was defined by a pair I = (U,A), where U is a universal set of objects and A is a set of contexts of attributes. For every A i ∈A is a set of attributes regarded as a context or background, consequently, if there are n contexts in A, where A = {A i ,...,A n }, it provides n partitions. A given set of object, X ⊆ U, may then be represented into n pairs of lower and upper approximations defined as multi-rough sets of X. In this paper, our primary concern is to discuss three kinds of general contexts, namely AND-general context, OR-general context and OR + -general context.