Graph Theory with Applications to Engineering and Computer ... This outstanding introductory treatment of graph theory and its applications has had a long life in the instruction of advanced undergraduates and graduate students in all areas that require knowledge of this subject. The first nine chapters constitute an excellent overall introduction, requiring only some knowledge of set theory and matrix algebra.

Graph theory with applications to engineering and computer science, by Narsingh Deo. Pp xvii, 478. 1974. SBN 0 13 363473 6 (Prentice-Hall)

Basic rough set model introduced by Pawlak in 1982 has been extended in many directions to enhance their modeling power. One such attempt is the notion of rough sets on fuzzy approximation spaces by De et al in 1999. This basic model uses equivalence relation for its definition, which decompose the universal set into disjoint equivalence classes. These equivalence classes are called granules of knowledge. From the granular computing point of view the basic rough set model is unigranular in character. So, in order to handle more than one granular structure simultaneously, two types of multigranular rough sets, called the optimistic and pessimistic multigranular rough sets were introduced by Qian et al in 2006 and 2010 respectively. In this paper, we introduce two types of multigranular rough sets on fuzzy approximation spaces (optimistic and pessimistic), study several of their properties and illustrate how this notion can be used for prediction of rainfall. The introduced notions are explained through several examples.

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Properties of Multigranular Rough Sets on Fuzzy Approximation Spaces and their Application to Rainfall Prediction

Time-varying classifiers, namely, evolving classifiers, play an important role in a scenario in which information is available as a never-ending online data stream We present a new unsupervised learning method for numerical data called evolving Internal-eXternal Fuzzy clustering method (Fuzzy eIX) We develop the notion of double-boundary fuzzy granules and elaborate on its implications Type 1 and type 2 fuzzy inference systems can be obtained from the projection of Fuzzy eIX granules We perform the principle of the balanced information granularity within Fuzzy eIX classifiers to achieve a higher level of model understandability Internal and external granules are updated from a numerical data stream at the same time that the global granular structure of the classifier is autonomously evolved A synthetic nonstationary problem called Rotation of Twin Gaussians shows the behavior of the classifier The Fuzzy eIX classifier could keep up with its accuracy in a scenario in which offline-trained classifiers would clearly have their accuracy drastically dropped

Unsupervised Fuzzy eIX: Evolving Internal-eXternal Fuzzy Clustering

Since its inception rough set theory has proved itself to be one of the most important models to capture impreciseness in data. However, it was based upon the notion of equivalence relations, which are relatively rare as far as applicability is concerned. So, the basic rough set model has been extended in many direct ions. One of these extensions is the covering based rough set notion, where a cover is an extension of the concept of partition; a notion which is equivalent to equivalence relation. From the granular computing point of view, all these rough sets are unigranular in character; i.e . they consider only a singular granular structure on the universe. So, there arose the necessity to define multig ranular rough sets and as a consequence two types of mult igranular rough sets, called the optimistic mult igranular rough sets and pessimistic rough sets have been introduced. Four types of covering based optimistic multig ranular rough sets have been introduced and their properties are studied. The notion of equality of sets, which is too stringent for real life applications, was extended by Novotny and Pawlak to define rough equalities. Th is notion was further extended by Tripathy to define three more types of approximate equalities. The covering based optimistic versions of two of these four approximate equalities have been studied by Nagaraju et al recently. In th is article, we study the other two cases and provide a comparative analysis.

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Covering Based Optimistic Multigranular Approximate Rough Equalities and their Properties

Time-varying classifiers, namely, evolving classifiers, play an important role in a scenario in which information is available as a never-ending online data stream. We present a new unsupervised learning method for numerical data called evolving Internal-eXternal Fuzzy clustering method (Fuzzy eIX). We develop the notion of double-boundary fuzzy granules and elaborate on its implications. Type 1 and type 2 fuzzy inference systems can be obtained from the projection of Fuzzy eIX granules. We perform the principle of the balanced information granularity within Fuzzy eIX classifiers to achieve a higher level of model understandability. Internal and external granules are updated from a numerical data stream at the same time that the global granular structure of the classifier is autonomously evolved. A synthetic nonstationary problem called Rotation of Twin Gaussians shows the behavior of the classifier. The Fuzzy eIX classifier could keep up with its accuracy in a scenario in which offline-trained classifiers would clearly have their accuracy drastically dropped.

As the notion of equality in mathematics is too stringent and less applicable in real life situations, Novotny and Pawlak introduced approximate equalities through rough sets. Three more types of such equalities were introduced by Tripathy et al. as further generalisations of these equalities. As rough set introduced by Pawlak is unigranular from the granular computing point of view, two types of multigranulations rough sets called the optimistic and the pessimistic multigranular rough sets have been introduced. Three of the above approximate equalities were extended to the multigranular context by Tripathy et al. recently. In this paper, we extend the last but the most general of these approximate equalities to the multigranular context. We establish several direct and replacement properties of this type of approximate equalities. Also, we illustrate the properties as well as provide counter examples by taking a real life example.

On Multigranular Approximate Rough Equivalence of Sets and Approximate Reasoning

Graph theory has found its applications in various fields of computation involved in day to day life. A problem solving approach that incorporates graph theory has an added advantage of being simple and visually more comprehensible [9]. Data mining, image processing, astrology and astronomy, theoretical computer science, artificial intelligence and compiler optimization et al., every evolving field utilizes efficient algorithms involving graphs and their properties. Graph coloring has major applications in the field of compiler optimization. This paper proposes a heuristic approach of graph coloring for code optimization and hence, improvement in compiler performance and accuracy. Color-based merging of colored graphs reduces the use of temporary variables, increasing efficiency in memory utilization. A comparative analysis has been carried out in order to present the advantages of the proposed algorithm [5] [6].

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An Optimal Code Heuristic Approach for Compiler Optimization using Graph Coloring Technique

There have been three approaches to find the measures of fuzzy sets, which extend the count function for crisp sets The first one is due to Deluca and Termini; second one a measure theoretic approach due to Zadeh and a bag theoretic approach due to Tripathy et al Intuitionistic fuzzy sets are generalizations of fuzzy sets So, efforts have been made recently to extend the above three approaches to find measures for cardinalities of intuitionistic fuzzy sets The extension of the approach due to Deluca and Termini to the context of intuitionistic fuzzy sets by Tripathy et al [9] In this paper, we define measures of intuitionistic fuzzy sets to define the count of an intuitionistic fuzzy set in the direction of Zadeh Also, we establish several properties of these measures

On intuitionistic fuzzy measures and the count of intuitionistic fuzzy sets

Data mining processes were initially invented and developed for finding out meaningful information and deriving rules from more recent data sets. With growth in the size of power systems due to the increased requirements of industries and cities for uninterrupted power supply, data mining in electrical systems has emerged as a very efficient tool for continuous assessment of power systems. In this paper we propose Rough Set based knowledge discovery for predicting faults and probability of failure in high voltage equipment present in a real electrical systems. The approach has several different steps viz. analysis of real time data, creation and population of databases, pre-processing of data and use of rough sets based data mining algorithm to finally determine the set of rules for knowledge discovery. The methodology has been validated by presenting a case study and application of the algorithm on real time data.

http://ieeexplore.ieee.org/iel7/7005020/7033740/07034152.pdf

A Monte Carlo-rough based mixed-integer probablistic non-linear programming model for predicting and minimizing unavailability of power system components

Prateek Saraf

Papers

On Multigranular Approximate Rough Equivalence of Sets and Approximate Reasoning

An Optimal Code Heuristic Approach for Compiler Optimization using Graph Coloring Technique

On intuitionistic fuzzy measures and the count of intuitionistic fuzzy sets

A Monte Carlo-rough based mixed-integer probablistic non-linear programming model for predicting and minimizing unavailability of power system components