Granular, Soft and Fuzzy Approaches for Intelligent Systems

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Publisher : Springer
ISBN 13 : 3319403141
Total Pages : 248 pages
Book Rating : 4.3/5 (194 download)

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Book Synopsis Granular, Soft and Fuzzy Approaches for Intelligent Systems by : Janusz Kacprzyk

Download or read book Granular, Soft and Fuzzy Approaches for Intelligent Systems written by Janusz Kacprzyk and published by Springer. This book was released on 2016-11-14 with total page 248 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers a comprehensive report on the state-of-the art in the broadly-intended field of “intelligent systems”. After introducing key theoretical issues, it describes a number of promising models for data and system analysis, decision making, and control. It discusses important theories, including possibility theory, the Dempster-Shafer theory, the theory of approximate reasoning, as well as computing with words, together with novel applications in various areas, such as information aggregation and fusion, linguistic data summarization, participatory learning, systems modeling, and many others. By presenting the methods in their application contexts, the book shows how granular computing, soft computing and fuzzy logic techniques can provide novel, efficient solutions to real-world problems. It is dedicated to Professor Ronald R. Yager for his great scientific and scholarly achievements, and for his long-lasting service to the fuzzy logic, and the artificial and computational intelligence communities. It has been motivated by the authors’ appreciation of his original thinking and groundbreaking ideas, with a special thought to his valuable research on the computerized implementation of various aspects of human cognition for decision-making and problem-solving.

Granular Computing

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Publisher : CRC Press
ISBN 13 : 135183262X
Total Pages : 313 pages
Book Rating : 4.3/5 (518 download)

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Book Synopsis Granular Computing by : Witold Pedrycz

Download or read book Granular Computing written by Witold Pedrycz and published by CRC Press. This book was released on 2018-09-03 with total page 313 pages. Available in PDF, EPUB and Kindle. Book excerpt: Information granules, as encountered in natural language, are implicit in nature. To make them fully operational so they can be effectively used to analyze and design intelligent systems, information granules need to be made explicit. An emerging discipline, granular computing focuses on formalizing information granules and unifying them to create a coherent methodological and developmental environment for intelligent system design and analysis. Granular Computing: Analysis and Design of Intelligent Systems presents the unified principles of granular computing along with its comprehensive algorithmic framework and design practices. Introduces the concepts of information granules, information granularity, and granular computing Presents the key formalisms of information granules Builds on the concepts of information granules with discussion of higher-order and higher-type information granules Discusses the operational concept of information granulation and degranulation by highlighting the essence of this tandem and its quantification in terms of the associated reconstruction error Examines the principle of justifiable granularity Stresses the need to look at information granularity as an important design asset that helps construct more realistic models of real-world systems or facilitate collaborative pursuits of system modeling Highlights the concepts, architectures, and design algorithms of granular models Explores application domains where granular computing and granular models play a visible role, including pattern recognition, time series, and decision making Written by an internationally renowned authority in the field, this innovative book introduces readers to granular computing as a new paradigm for the analysis and synthesis of intelligent systems. It is a valuable resource for those engaged in research and practical developments in computer, electrical, industrial, manufacturing, and biomedical engineering. Building from fundamentals, the book is also suitable for readers from nontechnical disciplines where information granules assume a visible position.

Granular Computing and Intelligent Systems

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Publisher : Springer Science & Business Media
ISBN 13 : 3642198201
Total Pages : 308 pages
Book Rating : 4.6/5 (421 download)

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Book Synopsis Granular Computing and Intelligent Systems by : Witold Pedrycz

Download or read book Granular Computing and Intelligent Systems written by Witold Pedrycz and published by Springer Science & Business Media. This book was released on 2011-04-28 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt: Information granules are fundamental conceptual entities facilitating perception of complex phenomena and contributing to the enhancement of human centricity in intelligent systems. The formal frameworks of information granules and information granulation comprise fuzzy sets, interval analysis, probability, rough sets, and shadowed sets, to name only a few representatives. Among current developments of Granular Computing, interesting options concern information granules of higher order and of higher type. The higher order information granularity is concerned with an effective formation of information granules over the space being originally constructed by information granules of lower order. This construct is directly associated with the concept of hierarchy of systems composed of successive processing layers characterized by the increasing levels of abstraction. This idea of layered, hierarchical realization of models of complex systems has gained a significant level of visibility in fuzzy modeling with the well-established concept of hierarchical fuzzy models where one strives to achieve a sound tradeoff between accuracy and a level of detail captured by the model and its level of interpretability. Higher type information granules emerge when the information granules themselves cannot be fully characterized in a purely numerical fashion but instead it becomes convenient to exploit their realization in the form of other types of information granules such as type-2 fuzzy sets, interval-valued fuzzy sets, or probabilistic fuzzy sets. Higher order and higher type of information granules constitute the focus of the studies on Granular Computing presented in this study. The book elaborates on sound methodologies of Granular Computing, algorithmic pursuits and an array of diverse applications and case studies in environmental studies, option price forecasting, and power engineering.

Hierarchical Modular Granular Neural Networks with Fuzzy Aggregation

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Publisher : Springer
ISBN 13 : 3319288628
Total Pages : 101 pages
Book Rating : 4.3/5 (192 download)

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Book Synopsis Hierarchical Modular Granular Neural Networks with Fuzzy Aggregation by : Daniela Sanchez

Download or read book Hierarchical Modular Granular Neural Networks with Fuzzy Aggregation written by Daniela Sanchez and published by Springer. This book was released on 2016-02-23 with total page 101 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this book, a new method for hybrid intelligent systems is proposed. The proposed method is based on a granular computing approach applied in two levels. The techniques used and combined in the proposed method are modular neural networks (MNNs) with a Granular Computing (GrC) approach, thus resulting in a new concept of MNNs; modular granular neural networks (MGNNs). In addition fuzzy logic (FL) and hierarchical genetic algorithms (HGAs) are techniques used in this research work to improve results. These techniques are chosen because in other works have demonstrated to be a good option, and in the case of MNNs and HGAs, these techniques allow to improve the results obtained than with their conventional versions; respectively artificial neural networks and genetic algorithms.

Type-2 Fuzzy Granular Models

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Publisher : Springer
ISBN 13 : 3319412884
Total Pages : 93 pages
Book Rating : 4.3/5 (194 download)

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Book Synopsis Type-2 Fuzzy Granular Models by : Mauricio A. Sanchez

Download or read book Type-2 Fuzzy Granular Models written by Mauricio A. Sanchez and published by Springer. This book was released on 2016-08-26 with total page 93 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this book, a series of granular algorithms are proposed. A nature inspired granular algorithm based on Newtonian gravitational forces is proposed. A series of methods for the formation of higher-type information granules represented by Interval Type-2 Fuzzy Sets are also shown, via multiple approaches, such as Coefficient of Variation, principle of justifiable granularity, uncertainty-based information concept, and numerical evidence based. And a fuzzy granular application comparison is given as to demonstrate the differences in how uncertainty affects the performance of fuzzy information granules.

Handbook of Granular Computing

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Publisher : John Wiley & Sons
ISBN 13 : 0470724153
Total Pages : 1148 pages
Book Rating : 4.4/5 (77 download)

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Book Synopsis Handbook of Granular Computing by : Witold Pedrycz

Download or read book Handbook of Granular Computing written by Witold Pedrycz and published by John Wiley & Sons. This book was released on 2008-07-31 with total page 1148 pages. Available in PDF, EPUB and Kindle. Book excerpt: Although the notion is a relatively recent one, the notions and principles of Granular Computing (GrC) have appeared in a different guise in many related fields including granularity in Artificial Intelligence, interval computing, cluster analysis, quotient space theory and many others. Recent years have witnessed a renewed and expanding interest in the topic as it begins to play a key role in bioinformatics, e-commerce, machine learning, security, data mining and wireless mobile computing when it comes to the issues of effectiveness, robustness and uncertainty. The Handbook of Granular Computing offers a comprehensive reference source for the granular computing community, edited by and with contributions from leading experts in the field. Includes chapters covering the foundations of granular computing, interval analysis and fuzzy set theory; hybrid methods and models of granular computing; and applications and case studies. Divided into 5 sections: Preliminaries, Fundamentals, Methodology and Algorithms, Development of Hybrid Models and Applications and Case Studies. Presents the flow of ideas in a systematic, well-organized manner, starting with the concepts and motivation and proceeding to detailed design that materializes in specific algorithms, applications and case studies. Provides the reader with a self-contained reference that includes all pre-requisite knowledge, augmented with step-by-step explanations of more advanced concepts. The Handbook of Granular Computing represents a significant and valuable contribution to the literature and will appeal to a broad audience including researchers, students and practitioners in the fields of Computational Intelligence, pattern recognition, fuzzy sets and neural networks, system modelling, operations research and bioinformatics.

Industry 4.0 Interoperability, Analytics, Security, and Case Studies

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Publisher : CRC Press
ISBN 13 : 1000338045
Total Pages : 202 pages
Book Rating : 4.0/5 (3 download)

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Book Synopsis Industry 4.0 Interoperability, Analytics, Security, and Case Studies by : G. Rajesh

Download or read book Industry 4.0 Interoperability, Analytics, Security, and Case Studies written by G. Rajesh and published by CRC Press. This book was released on 2021-01-31 with total page 202 pages. Available in PDF, EPUB and Kindle. Book excerpt: All over the world, vast research is in progress on the domain of Industry 4.0 and related techniques. Industry 4.0 is expected to have a very high impact on labor markets, global value chains, education, health, environment, and many social economic aspects. Industry 4.0 Interoperability, Analytics, Security, and Case Studies provides a deeper understanding of the drivers and enablers of Industry 4.0. It includes real case studies of various applications related to different fields, such as cyber physical systems (CPS), Internet of Things (IoT), cloud computing, machine learning, virtualization, decentralization, blockchain, fog computing, and many other related areas. Also discussed are interoperability, design, and implementation challenges. Researchers, academicians, and those working in industry around the globe will find this book of interest. FEATURES Provides an understanding of the drivers and enablers of Industry 4.0 Includes real case studies of various applications for different fields Discusses technologies such as cyber physical systems (CPS), Internet of Things (IoT), cloud computing, machine learning, virtualization, decentralization, blockchain, fog computing, and many other related areas Covers design, implementation challenges, and interoperability Offers detailed knowledge on Industry 4.0 and its underlying technologies, research challenges, solutions, and case studies

Uncertainty Modeling

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Publisher : Springer
ISBN 13 : 3319510525
Total Pages : 292 pages
Book Rating : 4.3/5 (195 download)

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Book Synopsis Uncertainty Modeling by : Vladik Kreinovich

Download or read book Uncertainty Modeling written by Vladik Kreinovich and published by Springer. This book was released on 2017-01-31 with total page 292 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book commemorates the 65th birthday of Dr. Boris Kovalerchuk, and reflects many of the research areas covered by his work. It focuses on data processing under uncertainty, especially fuzzy data processing, when uncertainty comes from the imprecision of expert opinions. The book includes 17 authoritative contributions by leading experts.

A Guided Tour of Artificial Intelligence Research

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Publisher : Springer Nature
ISBN 13 : 3030061647
Total Pages : 808 pages
Book Rating : 4.0/5 (3 download)

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Book Synopsis A Guided Tour of Artificial Intelligence Research by : Pierre Marquis

Download or read book A Guided Tour of Artificial Intelligence Research written by Pierre Marquis and published by Springer Nature. This book was released on 2020-05-08 with total page 808 pages. Available in PDF, EPUB and Kindle. Book excerpt: The purpose of this book is to provide an overview of AI research, ranging from basic work to interfaces and applications, with as much emphasis on results as on current issues. It is aimed at an audience of master students and Ph.D. students, and can be of interest as well for researchers and engineers who want to know more about AI. The book is split into three volumes: - the first volume brings together twenty-three chapters dealing with the foundations of knowledge representation and the formalization of reasoning and learning (Volume 1. Knowledge representation, reasoning and learning) - the second volume offers a view of AI, in fourteen chapters, from the side of the algorithms (Volume 2. AI Algorithms) - the third volume, composed of sixteen chapters, describes the main interfaces and applications of AI (Volume 3. Interfaces and applications of AI). Implementing reasoning or decision making processes requires an appropriate representation of the pieces of information to be exploited. This first volume starts with a historical chapter sketching the slow emergence of building blocks of AI along centuries. Then the volume provides an organized overview of different logical, numerical, or graphical representation formalisms able to handle incomplete information, rules having exceptions, probabilistic and possibilistic uncertainty (and beyond), as well as taxonomies, time, space, preferences, norms, causality, and even trust and emotions among agents. Different types of reasoning, beyond classical deduction, are surveyed including nonmonotonic reasoning, belief revision, updating, information fusion, reasoning based on similarity (case-based, interpolative, or analogical), as well as reasoning about actions, reasoning about ontologies (description logics), argumentation, and negotiation or persuasion between agents. Three chapters deal with decision making, be it multiple criteria, collective, or under uncertainty. Two chapters cover statistical computational learning and reinforcement learning (other machine learning topics are covered in Volume 2). Chapters on diagnosis and supervision, validation and explanation, and knowledge base acquisition complete the volume.

Intelligent Decision Support Systems

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Publisher : Springer Nature
ISBN 13 : 3030963187
Total Pages : 446 pages
Book Rating : 4.0/5 (39 download)

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Book Synopsis Intelligent Decision Support Systems by : Salvatore Greco

Download or read book Intelligent Decision Support Systems written by Salvatore Greco and published by Springer Nature. This book was released on 2022-05-31 with total page 446 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a collection of essays written by leading researchers to honor Roman Słowiński’s major scholarly interests and contributions. He is well-known for conducting extensive research on methodologies and techniques for intelligent decision support, where he combines operational research and artificial intelligence. The book reconstructs his main contributions, presents cutting-edge research and provides an outlook on the most promising and advanced domains of computer science and multiple criteria decision aiding. The respective chapters cover a wide range of related research areas, including decision sciences, ordinal data mining, preference learning and multiple criteria decision aiding, modeling of uncertainty and imprecision in decision problems, rough set theory, fuzzy set theory, multi-objective optimization, project scheduling and decision support applications. As such, the book will appeal to researchers and scholars in related fields.

Information Processing and Management of Uncertainty in Knowledge-Based Systems

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Publisher : Springer Nature
ISBN 13 : 3030501469
Total Pages : 779 pages
Book Rating : 4.0/5 (35 download)

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Book Synopsis Information Processing and Management of Uncertainty in Knowledge-Based Systems by : Marie-Jeanne Lesot

Download or read book Information Processing and Management of Uncertainty in Knowledge-Based Systems written by Marie-Jeanne Lesot and published by Springer Nature. This book was released on 2020-06-05 with total page 779 pages. Available in PDF, EPUB and Kindle. Book excerpt: This three volume set (CCIS 1237-1239) constitutes the proceedings of the 18th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, IPMU 2020, in June 2020. The conference was scheduled to take place in Lisbon, Portugal, at University of Lisbon, but due to COVID-19 pandemic it was held virtually. The 173 papers were carefully reviewed and selected from 213 submissions. The papers are organized in topical sections: homage to Enrique Ruspini; invited talks; foundations and mathematics; decision making, preferences and votes; optimization and uncertainty; games; real world applications; knowledge processing and creation; machine learning I; machine learning II; XAI; image processing; temporal data processing; text analysis and processing; fuzzy interval analysis; theoretical and applied aspects of imprecise probabilities; similarities in artificial intelligence; belief function theory and its applications; aggregation: theory and practice; aggregation: pre-aggregation functions and other generalizations of monotonicity; aggregation: aggregation of different data structures; fuzzy methods in data mining and knowledge discovery; computational intelligence for logistics and transportation problems; fuzzy implication functions; soft methods in statistics and data analysis; image understanding and explainable AI; fuzzy and generalized quantifier theory; mathematical methods towards dealing with uncertainty in applied sciences; statistical image processing and analysis, with applications in neuroimaging; interval uncertainty; discrete models and computational intelligence; current techniques to model, process and describe time series; mathematical fuzzy logic and graded reasoning models; formal concept analysis, rough sets, general operators and related topics; computational intelligence methods in information modelling, representation and processing.

Information Processing and Management of Uncertainty in Knowledge-Based Systems. Theory and Foundations

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Publisher : Springer
ISBN 13 : 3319914766
Total Pages : 773 pages
Book Rating : 4.3/5 (199 download)

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Book Synopsis Information Processing and Management of Uncertainty in Knowledge-Based Systems. Theory and Foundations by : Jesús Medina

Download or read book Information Processing and Management of Uncertainty in Knowledge-Based Systems. Theory and Foundations written by Jesús Medina and published by Springer. This book was released on 2018-05-30 with total page 773 pages. Available in PDF, EPUB and Kindle. Book excerpt: This three volume set (CCIS 853-855) constitutes the proceedings of the 17th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, IPMU 2017, held in Cádiz, Spain, in June 2018. The 193 revised full papers were carefully reviewed and selected from 383 submissions. The papers are organized in topical sections on advances on explainable artificial intelligence; aggregation operators, fuzzy metrics and applications; belief function theory and its applications; current techniques to model, process and describe time series; discrete models and computational intelligence; formal concept analysis and uncertainty; fuzzy implication functions; fuzzy logic and artificial intelligence problems; fuzzy mathematical analysis and applications; fuzzy methods in data mining and knowledge discovery; fuzzy transforms: theory and applications to data analysis and image processing; imprecise probabilities: foundations and applications; mathematical fuzzy logic, mathematical morphology; measures of comparison and entropies for fuzzy sets and their extensions; new trends in data aggregation; pre-aggregation functions and generalized forms of monotonicity; rough and fuzzy similarity modelling tools; soft computing for decision making in uncertainty; soft computing in information retrieval and sentiment analysis; tri-partitions and uncertainty; decision making modeling and applications; logical methods in mining knowledge from big data; metaheuristics and machine learning; optimization models for modern analytics; uncertainty in medicine; uncertainty in Video/Image Processing (UVIP).

Emerging Approaches in Design and New Connections With Nature

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Publisher : IGI Global
ISBN 13 : 1799867269
Total Pages : 348 pages
Book Rating : 4.7/5 (998 download)

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Book Synopsis Emerging Approaches in Design and New Connections With Nature by : Özdamar, Esen Gökçe

Download or read book Emerging Approaches in Design and New Connections With Nature written by Özdamar, Esen Gökçe and published by IGI Global. This book was released on 2021-12-03 with total page 348 pages. Available in PDF, EPUB and Kindle. Book excerpt: In today’s changing and transforming socio-economic, political, cultural, and technological paradigms, we encounter many methodologies, approaches, proposals, and practices in reconsidering the disappearing or emerging relations in the human/nonhuman-environment-nature interaction. These approaches, proposals, and practices range from new methods of urban gardening to biophilic design and augmented/immersive environments. However, these human-centric approaches, which only aim to meet their needs or emerge as technology-oriented replicas and representations of nature, lead to a departure from a holistic approach to the natural and artificial environment. Therefore, how can new and emerging approaches or methodologies draw a holistic framework for environmental health, sustainability, wellness, and co-existence between environments for all living beings? Emerging Approaches in Design and New Connections With Nature covers a variety of topics related to the intersection between nature, environment, and ways of living and provides a comprehensive guide to biophilic design and the idea of design and nature, including benefits, theories, and effects. Covering topics such as biophilic design and sustainability, soundscapes and landscapes, and urban environments and design, it is ideal for architects, designers, urban planners, landscape designers, policymakers, engineers, interior designers, practitioners, students, academicians, and researchers.

Nature-Inspired Design of Hybrid Intelligent Systems

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Publisher : Springer
ISBN 13 : 331947054X
Total Pages : 838 pages
Book Rating : 4.3/5 (194 download)

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Book Synopsis Nature-Inspired Design of Hybrid Intelligent Systems by : Patricia Melin

Download or read book Nature-Inspired Design of Hybrid Intelligent Systems written by Patricia Melin and published by Springer. This book was released on 2016-12-08 with total page 838 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book highlights recent advances in the design of hybrid intelligent systems based on nature-inspired optimization and their application in areas such as intelligent control and robotics, pattern recognition, time series prediction, and optimization of complex problems. The book is divided into seven main parts, the first of which addresses theoretical aspects of and new concepts and algorithms based on type-2 and intuitionistic fuzzy logic systems. The second part focuses on neural network theory, and explores the applications of neural networks in diverse areas, such as time series prediction and pattern recognition. The book’s third part presents enhancements to meta-heuristics based on fuzzy logic techniques and describes new nature-inspired optimization algorithms that employ fuzzy dynamic adaptation of parameters, while the fourth part presents diverse applications of nature-inspired optimization algorithms. In turn, the fifth part investigates applications of fuzzy logic in diverse areas, such as time series prediction and pattern recognition. The sixth part examines new optimization algorithms and their applications. Lastly, the seventh part is dedicated to the design and application of different hybrid intelligent systems.

Intelligent Systems: Concepts, Methodologies, Tools, and Applications

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Publisher : IGI Global
ISBN 13 : 1522556443
Total Pages : 2351 pages
Book Rating : 4.5/5 (225 download)

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Book Synopsis Intelligent Systems: Concepts, Methodologies, Tools, and Applications by : Management Association, Information Resources

Download or read book Intelligent Systems: Concepts, Methodologies, Tools, and Applications written by Management Association, Information Resources and published by IGI Global. This book was released on 2018-06-04 with total page 2351 pages. Available in PDF, EPUB and Kindle. Book excerpt: Ongoing advancements in modern technology have led to significant developments in intelligent systems. With the numerous applications available, it becomes imperative to conduct research and make further progress in this field. Intelligent Systems: Concepts, Methodologies, Tools, and Applications contains a compendium of the latest academic material on the latest breakthroughs and recent progress in intelligent systems. Including innovative studies on information retrieval, artificial intelligence, and software engineering, this multi-volume book is an ideal source for researchers, professionals, academics, upper-level students, and practitioners interested in emerging perspectives in the field of intelligent systems.

Design of Intelligent Systems Based on Fuzzy Logic, Neural Networks and Nature-Inspired Optimization

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Publisher : Springer
ISBN 13 : 3319177478
Total Pages : 637 pages
Book Rating : 4.3/5 (191 download)

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Book Synopsis Design of Intelligent Systems Based on Fuzzy Logic, Neural Networks and Nature-Inspired Optimization by : Patricia Melin

Download or read book Design of Intelligent Systems Based on Fuzzy Logic, Neural Networks and Nature-Inspired Optimization written by Patricia Melin and published by Springer. This book was released on 2015-06-12 with total page 637 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents recent advances on the design of intelligent systems based on fuzzy logic, neural networks and nature-inspired optimization and their application in areas such as, intelligent control and robotics, pattern recognition, time series prediction and optimization of complex problems. The book is organized in eight main parts, which contain a group of papers around a similar subject. The first part consists of papers with the main theme of theoretical aspects of fuzzy logic, which basically consists of papers that propose new concepts and algorithms based on fuzzy systems. The second part contains papers with the main theme of neural networks theory, which are basically papers dealing with new concepts and algorithms in neural networks. The third part contains papers describing applications of neural networks in diverse areas, such as time series prediction and pattern recognition. The fourth part contains papers describing new nature-inspired optimization algorithms. The fifth part presents diverse applications of nature-inspired optimization algorithms. The sixth part contains papers describing new optimization algorithms. The seventh part contains papers describing applications of fuzzy logic in diverse areas, such as time series prediction and pattern recognition. Finally, the eighth part contains papers that present enhancements to meta-heuristics based on fuzzy logic techniques.

Computing with Words in Information/Intelligent Systems 1

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Author :
Publisher : Physica
ISBN 13 : 3790818739
Total Pages : 523 pages
Book Rating : 4.7/5 (98 download)

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Book Synopsis Computing with Words in Information/Intelligent Systems 1 by : Lotfi A. Zadeh

Download or read book Computing with Words in Information/Intelligent Systems 1 written by Lotfi A. Zadeh and published by Physica. This book was released on 2013-03-09 with total page 523 pages. Available in PDF, EPUB and Kindle. Book excerpt: These two volumes consIstmg of Foundations and Applications provide the current status of theoretical and empirical developments in "computing with words". In philosophy, the twentieth century is said to be the century of language. This is mainly due to Wittgenstein who said: "The meaning of a word is its use in the language game". "The concept game is a concept with blurred edges". In the first phrase, "the language game" implies the everyday human activity with language, and in the latter, "game" simply implies an ordinary word. Thus, Wittgenstein precisely stated that a word is fuzzy in real life. Unfortunately this idea about a word was not accepted in the conventional science. We had to wait for Zadeh's fuzzy sets theory. Remembering Wittgenstein's statement, we should consider, on the one hand, the concept of "computing with words" from a philosophical point of view. It deeply relates to the everyday use of a word in which the meaning of a word is fuzzy in its nature.