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Granular Fuzzy Models: Construction, Analysis, and Design Open Access


Other title
Granular Fuzzy Modeling
Reconstruction Criterion
Differential Evolution
Particle Swarm Optimization
Fuzzy C-Means
Conditional Clustering
Type of item
Degree grantor
University of Alberta
Author or creator
Reyes-Galaviz, Orion Fausto
Supervisor and department
Pedrycz, Witold (Computer Engineering)
Examining committee member and department
Reformat, Marek (Computer Engineering)
Musilek, Petr (Computer Engineering)
Gomide, Fernando (University of Campinas, Sao Paulo, Brazil)
Kuru, Ergun (Civil & Environmental Engineering)
Department of Electrical and Computer Engineering
Software Engineering and Intelligent Systems
Date accepted
Graduation date
Doctor of Philosophy
Degree level
Building abstract concepts is essential to humans when acquiring knowledge, realizing processing (reasoning), and communicating findings. Abstraction comes hand in hand with information granules and information granulation. When analyzing digital images, we form groups (clusters) of pixels, colors, and textures that constitute familiar objects. This paramount ability of forming groups of objects (information granules), manipulating them, and producing sound conclusions, is realized in an almost subconscious manner. Information granules are critical when representing and processing knowledge. They become instrumental when solving everyday problems. The objective of this dissertation is to design, analyze, and optimize granular fuzzy models in which information granules play a pivotal role. In the presented considerations, fuzzy models serve as a vehicle for the construction of granular fuzzy models, offering a convenient and efficient way to describe complex and nonlinear systems. Fuzzy models produce numeric results. In contrast, granular fuzzy models produce results in the form of information granules. In this study, we develop a novel approach to the design and optimization of fuzzy relational structures. Subsequently, we transform them into their granular counterpart by experimenting with different strategies to optimally allocate information granularity. This direct generalization (abstraction) of the fuzzy model comes as a sound alternative to solve a system of relational equations, and to assess the quality of the original model. Granular fuzzy models are also developed by exploiting a concept of information granularity. We propose an innovative model coming as a network of associations among information granules, which form the backbone of the overall construct. Interval information granules positioned in the output space induce information granules in the input space, and we develop different strategies to optimize these intervals. Further optimization of the model is proposed by optimally re-distributing the iii induced information granules. The performance of granular models is assessed by considering criteria of coverage and information specificity (information granularity). The associations are further exploited to construct a granular model that is fundamentally structured around information granules regarded as hyperboxes. We develop two novel methods to construct a family of hyperboxes in the input space; one by realizing some constrictions, and the other one by engaging an optimization mechanism. We also propose a new approach to optimally eliminate or reduce possible overlap among hyperboxes. The resulting information granules are compared and assessed in terms of their coverage, and the data captured by those is evaluated by proposing a three-step verification process. Furthermore, in this research we propose a novel approach to refine information granules produced by the Fuzzy C-Means algorithm, and optimize their representation and classification abilities. We develop different strategies to adjust a location of the prototypes so that a certain performance index becomes optimized. The granular fuzzy models are optimized by population-based algorithms, such as particle swarm optimization and differential evolution. The experimental studies involve synthetic data and benchmark datasets from publicly-available repositories.
This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for the purpose of private, scholarly or scientific research. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.
Citation for previous publication
Reyes-Galaviz, Orion F. Pedrycz, Witold (2013). Fuzzy relational structures: Learning alternatives for fuzzy modeling. Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS), Edmonton AB, Canada, pp. 374-379.Reyes-Galaviz, Orion F. Pedrycz, Witold (2015). Granular Fuzzy Models: Analysis, Design, and Evaluation. International Journal for Approximate Reasoning, vol. 64, pp. 1-19.Reyes-Galaviz, Orion F. Pedrycz, Witold (2015). Granular Fuzzy Modeling with Evolving Hyperboxes in Multi-Dimensional Space of Numerical Data. Neurocomputing, vol. 168, Issue 30, pp. 240-253.

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