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Rough fuzzy hybridization is a method of hybrid intelligent system or soft computing, where Fuzzy set theory is used for linguistic representation of patterns, leading to a fuzzy granulation of the feature space. Rough set theory is used to obtain dependency rules which model informative regions in the granulated feature space.
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In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies measure properties such as countable additivity. The difference between a probability measure and the more general notion of measure is that a probability measure must assign value 1 to the entire space.
Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely false. By contrast, in Boolean logic, the truth values of variables may only be the integer values 0 or 1.
In mathematics, fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set. At the same time, Salii (1965) defined a more general kind of structure called an "L-relation", which he studied in an abstract algebraic context; fuzzy relations are special cases of L-relations when L is the unit interval [0, 1]. They are now used throughout fuzzy mathematics, having applications in areas such as linguistics, decision-making, and clustering.
Lotfi Aliasker Zadeh was a mathematician, computer scientist, electrical engineer, artificial intelligence researcher, and professor of computer science at the University of California, Berkeley. Zadeh is best known for proposing fuzzy mathematics, consisting of several fuzzy-related concepts: fuzzy sets, fuzzy logic, fuzzy algorithms, fuzzy semantics, fuzzy languages, fuzzy control, fuzzy systems, fuzzy probabilities, fuzzy events, and fuzzy information. Zadeh was a founding member of the Eurasian Academy.
In music, transposition refers to the process or operation of moving a collection of notes up or down in pitch by a constant interval.
The shifting of a melody, a harmonic progression or an entire musical piece to another key, while maintaining the same tone structure, i.e. the same succession of whole tones and semitones and remaining melodic intervals.
Granular computing is an emerging computing paradigm of information processing that concerns the processing of complex information entities called "information granules", which arise in the process of data abstraction and derivation of knowledge from information or data. Generally speaking, information granules are collections of entities that usually originate at the numeric level and are arranged together due to their similarity, functional or physical adjacency, indistinguishability, coherency, or the like.
Multiple-criteria decision-making (MCDM) or multiple-criteria decision analysis (MCDA) is a sub-discipline of operations research that explicitly evaluates multiple conflicting criteria in decision making. It is also known as multiple attribute utility theory, multiple attribute value theory, multiple attribute preference theory, and multi-objective decision analysis.
Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. It uses measures of possibility and necessity between 0 and 1, ranging from impossible to possible and unnecessary to necessary, respectively. Professor Lotfi Zadeh first introduced possibility theory in 1978 as an extension of his theory of fuzzy sets and fuzzy logic. Didier Dubois and Henri Prade further contributed to its development. Earlier, in the 1950s, economist G. L. S. Shackle proposed the min/max algebra to describe degrees of potential surprise.
In computer science, a rough set, first described by Polish computer scientist Zdzisław I. Pawlak, is a formal approximation of a crisp set in terms of a pair of sets which give the lower and the upper approximation of the original set. In the standard version of rough set theory described in Pawlak (1991), the lower- and upper-approximation sets are crisp sets, but in other variations, the approximating sets may be fuzzy sets.
In mathematics, the membership function of a fuzzy set is a generalization of the indicator function for classical sets. In fuzzy logic, it represents the degree of truth as an extension of valuation. Degrees of truth are often confused with probabilities, although they are conceptually distinct, because fuzzy truth represents membership in vaguely defined sets, not likelihood of some event or condition. Membership functions were introduced by Aliasker Zadeh in the first paper on fuzzy sets (1965). Aliasker Zadeh, in his theory of fuzzy sets, proposed using a membership function operating on the domain of all possible values.
In computer science and operations research, Genetic fuzzy systems are fuzzy systems constructed by using genetic algorithms or genetic programming, which mimic the process of natural evolution, to identify its structure and parameter.
A fuzzy concept is an idea of which the boundaries of application can vary considerably according to context or conditions, instead of being fixed once and for all. This means the idea is somewhat vague or imprecise. Yet it is not unclear or meaningless. It has a definite meaning, which can be made more exact only through further elaboration and specification - including a closer definition of the context in which the concept is used. The study of the characteristics of fuzzy concepts and fuzzy language is called fuzzy semantics. The inverse of a "fuzzy concept" is a "crisp concept".
Connectionist expert systems are artificial neural network (ANN) based expert systems where the ANN generates inferencing rules e.g., fuzzy-multi layer perceptron where linguistic and natural form of inputs are used. Apart from that, rough set theory may be used for encoding knowledge in the weights better and also genetic algorithms may be used to optimize the search solutions better. Symbolic reasoning methods may also be incorporated.
Sankar Kumar Pal is a computer scientist and the president of the Indian Statistical Institute (ISI), Kolkata. He is also a National Science Chair, Government of India. Pal is a computer scientist with an international reputation on pattern recognition, image processing, fuzzy neural network, rough fuzzy hybridization, soft computing, granular mining, and machine intelligence. He pioneered the development of fuzzy set theory, and neuro-fuzzy and rough-fuzzy computing for uncertainty modelling with demonstration in pattern recognition, image processing, machine learning, knowledge-based systems and data mining. This has made him widely recognized across the world and made India a leader in these disciplines in international scenario. He founded the Machine Intelligence Unit in 1993, and the Center for Soft Computing Research: A National Facility in 2004, both at the ISI. In the process he has created many renowned scientists.
In set theory a serial relation is a homogeneous relation expressing the connection of an element of a sequence to the following element. The successor function used by Peano to define natural numbers is the prototype for a serial relation.
Fuzzy mathematics is the branch of mathematics including fuzzy set theory and fuzzy logic that deals with partial inclusion of elements in a set on a spectrum, as opposed to simple binary "yes" or "no" inclusion. It started in 1965 after the publication of Lotfi Asker Zadeh's seminal work Fuzzy sets. Linguistics is an example of a field that utilizes fuzzy set theory.
An audio watermark is a unique electronic identifier embedded in an audio signal, typically used to identify ownership of copyright. It is similar to a watermark on a photograph.
Named set theory is a branch of theoretical mathematics that studies the structures of names. The named set is a theoretical concept that generalizes the structure of a name described by Frege. Its generalization bridges the descriptivists theory of a name, and its triad structure, with mathematical structures that define mathematical names using triplets. It deploys the former to view the latter at a higher abstract level that unifies a name and its relationship to a mathematical structure as a constructed reference. This enables all names in science and technology to be treated as named sets or as systems of named sets.
Ismat Beg, FPAS, FIMA, is a Pakistani mathematician and researcher. Beg is a Distinguished National Professor at Lahore School of Economics, Higher Education Commission and an honorary full professor at the Mathematics Division at the Ruggero Santilli Institute for Basic Research, Florida, US. He has an enthusiastic and interactive teaching style and is famous for saying “please come on the board” when posed with a question in class. This helps uplift the students’ confidence.