On nearness measures in fuzzy relational data models

AbstractIt has been widely recognized that the imprecision and incompleteness inherent in real-world data suggest a fuzzy extension for information management systems. Various attempts to enhance these systems by fuzzy extensions can be found in the literature. Varying approaches concerning the fuzzification of the concept of a relation are possible, two of which are referred to in this article as the generalized fuzzy approach and the fuzzy-set relation approach. In these enhanced models, items can no longer be retrieved by merely using equality-check operations between constants; instead, operations based on some kind of nearness measures have to be developed. In fact, these models require such a nearness measure to be established for each domain for the evaluation of queries made upon them. An investigation of proposed nearness measures, often fuzzy equivalences, is conducted. The unnaturalness and impracticality of these measures leads to the development of a new measure: the resemblance relation, which is defined to be a fuzzified version of a tolerance relation. Various aspects of this relation are analyzed and discussed. It is also shown how the resemblance relation can be used to reduce redundancy in fuzzy relational database systems

Information Flow Model for Commercial Security

Information flow in Discretionary Access Control (DAC) is a well-known difficult problem. This paper formalizes the fundamental concepts and establishes a theory of information flow security. A DAC system is information flow secure (IFS), if any data never flows into the hands of owner’s enemies (explicitly denial access list.

Heuristic algorithm for interpretation of multi-valued attributes in similarity-based fuzzy relational databases

AbstractIn this work, we are presenting implementation details and extended scalability tests of the heuristic algorithm, which we had used in the past [1,2] to discover knowledge from multi-valued data entries stored in similarity-based fuzzy relational databases. The multi-valued symbolic descriptors, characterizing individual attributes of database records, are commonly used in similarity-based fuzzy databases to reflect uncertainty about the recorded observation. In this paper, we present an algorithm, which we developed to precisely interpret such non-atomic values and to transfer the fuzzy database tuples to the forms acceptable for many regular (i.e. atomic values based) data mining algorithms

Application of Conditional Probability in Constructing Fuzzy Functional Dependency

In real-world application, information is mostly imprecise or ambiguous. Therefore, the motivation of extending classical (crisp) relational database [Codd, 1970] to fuzzy relational database by Buckles and Petry [1982] stems from the need to be able to process and represent vague, imprecise and partially known (incomplete) information. The concept of fuzzy relational database proposed by Buckles and Petry [1982] are necessary to be extended to a more generalized concept of fuzzy relational database, since the data value in domain attributes of the fuzzy relational model is still considered as a subset of atomic data. In this case, each data value stored in the more generalized concept of fuzzy relational database is considered as a fuzzy subset. An important feature of a relational database is to express constraints in sense of relation of data, known as integrity constraints (ICs). For instance, if a relational database contains information about student ID-number, course, unit, term and grade, some constrains such as: “A given ID-number, course, and term give a unique grade”, “number of courses are no more than 6 courses for a given ID-number and term” and “total units are no more than 16 for a given ID-number and term” might be hold. Many types of integrity constraints have been provided since 1970s along with the Codd’s relational database, such as multi-valued dependency proposed by Fagin [1977], join dependency [Nicolas, 1978] [Rissanen, 1978], etc. Among them, functional depen¬dencies (FDs) [Berstein, Swenson, & Tsichritzis, 1975] are one of the most important and widely used in database design. As we extend the classical relational database to fuzzy relational database, it would be necessary to consider integrity constraints that may involve fuzzy value. In fact, fuzzy integrity constraints, such as: “The higher an education someone has, the higher salary he should get”, “almost equally qualified employees should get more or less equal salary” will arise naturally and usefully in real-world application. Therefore, the objective of extending FDs to fuzzy functional dependencies (FFDs) is in necessary to apply FDs in fuzzy relational database [Intan, Mukaidono, 2000a, 2003, 2004]. Various definitions and the notion of a fuzzy functional dependency have been devised since 1988. Among them, Raju and Majumdar [1988] defined FFD based on the membership function of the fuzzy relation; Tripathy, [1990] proposed definition of the FFD in terms of fuzzy Hamming weight; Kiss, [1991] constructed FFD using weighted tuples; Chen [1995], Cubero [1994] and W. Liu [1992,1993] introduced definition of the FFD based on the equality of two possibility distributions, and they used a certain type of implication and expression of cut off; Liao [1997] gave design of the FFD by introducing semantic proximity. In this book, some properties of conditional probability and its relation with fuzzy sets are studied and discussed as an alternative concept to measure similarity of fuzzy labels. Even it could be understood that interpretation of numerical value between fuzzy sets and probability measures are philosophically distinct, basic operations, such as, intersection and union of two fuzzy values can be interpreted as maximum intersection and minimum union of two events. Considering this reason, it is necessary to define three approximate conditional probabilities of two fuzzy events based on minimum, independent and maximum probability intersection between two (fuzzy) events. Moreover, conditional probability of two fuzzy events can be interpreted as probabilistic matching of two fuzzy sets [Baldwin, Martin, Pilsworth, 1995], [Baldwin, Martin, 1996] and as basis of getting similarity of two fuzzy sets and constructing equivalence classes inside their domain attribute. By using this property and Cartesian product operation of fuzzy sets, a concept of fuzzy functional dependency (FFD) is proposed and defined to express integrity constraints that may involve fuzzy value, called fuzzy integrity constraints. It can be proved that the concept of FFD satisfies classical/ crisp relational database by example. Also, inference rules which are similar to Armstrong’s Axioms [Armstrong, 1974] for the FFDs are both sound and complete. Next, a concept of partial FFD is introduced to express the fact as usually found in data that a given attribute domain X do not determine Y completely, but in the partial area of X, it might determine Y. For instance, in the relation between two domains student’s name and student’s ID, student’s ID determines student’s name. It means a given student’s ID certainly gives a unique student’s name. On the other hand, a given student’s name may give more than one student’s ID because it is possible to have more than one student who has the same name. However, in a partial area of student’s name where some students have unique names, student’s name can be considered to determine student’s ID. In addition, approximate data reduction and projection of relations are investigated in order to get relation among the partitions of data values. Here, data values might be considered as crisp as well as fuzzy data. Finally, this book discusses the application of FFDs in constructing fuzzy query relation for query data and approximate natural join of two or more fuzzy query relations in the framework of extended query system [Intan, Mukaidono, 2001, 2002]. The structure of the book is following. In Chapter 2, some basic definitions and notations, such as conditional probability, classical relational database, functional dependency, fuzzy sets, transformation fuzzy set and probability, and fuzzy relational database are recalled. Chapter 3 firstly introduces conditional probability of two fuzzy sets based on the possibility theory [Baldwin, Martin, Pilsworth, 1995]. The next, it provides three approximate interpretations in constructing conditional probability of two fuzzy events (sets) based on minimum, independent and maxi¬mum probability intersection between two (fuzzy) events [Intan, Mukaidono, 2004]. Chapter 4 is devoted to the construction of FFDs based on the concept of conditional probability relations. It is proved that inference rules (Reflexivity, Augmentation and Transitivity) which are similar to Armstrong’s Axioms for FFDs are both sound and complete. A special attention will be given to partial FFD in order to find relation between two partial areas of two attribute domains [Intan, Mukaidono, 2004]. In Chapter 5, the application of FFDs in approximating data reduction and query data are presented [Intan, Mukaidono, 2001, 2002]. This chapter also discussed two other operations called projection and join operations in the relation to approximate data reduction and extended query system respectively [Intan, Mukaidono, 2004]. This book will be closed by summary including suggestion for future work in Chapter 6

Relational visual cluster validity

The assessment of cluster validity plays a very important role in cluster analysis. Most commonly used cluster validity methods are based on statistical hypothesis testing or finding the best clustering scheme by computing a number of different cluster validity indices. A number of visual methods of cluster validity have been produced to display directly the validity of clusters by mapping data into two- or three-dimensional space. However, these methods may lose too much information to correctly estimate the results of clustering algorithms. Although the visual cluster validity (VCV) method of Hathaway and Bezdek can successfully solve this problem, it can only be applied for object data, i.e. feature measurements. There are very few validity methods that can be used to analyze the validity of data where only a similarity or dissimilarity relation exists – relational data. To tackle this problem, this paper presents a relational visual cluster validity (RVCV) method to assess the validity of clustering relational data. This is done by combining the results of the non-Euclidean relational fuzzy c-means (NERFCM) algorithm with a modification of the VCV method to produce a visual representation of cluster validity. RVCV can cluster complete and incomplete relational data and adds to the visual cluster validity theory. Numeric examples using synthetic and real data are presente