Join and Meet Operations for Type-2 Fuzzy Sets With Nonconvex Secondary Memberships

In this paper, we will present two theorems for the join and meet operations for general type-2 fuzzy sets with arbitrary secondary memberships, which can be nonconvex and/or nonnormal type-1 fuzzy sets. These results will be used to derive the join and meet operations of the more general descriptions of interval type-2 fuzzy sets presented in a paper by Bustince Sola et al. ('Interval type-2 fuzzy sets are generalization of interval-valued fuzzy sets: Towards a wider view on their relationship,' IEEE Trans. Fuzzy Syst., vol. 23, pp. 1876-1882, 2015), where the secondary grades can be nonconvex. Hence, this study will help to explore the potential of type-2 fuzzy logic systems which use the general forms of interval type-2 fuzzy sets which are not equivalent to interval-valued fuzzy sets. Several examples for both general type-2 and the more general forms of interval type-2 fuzzy sets are presented

Sistema de inferência fuzzy geral do tipo-2 aplicado à classificação

This work proposes the development of a new tool based on general type-2 fuzzy sets to be applied to digital classification of data. The classification problem considered here regards the identification of areas of forest in satellite images. The goal is to assist users in tasks related to monitoring forest. The developed digital classifier employs an inference mechanism called "general type-2 scaled inference" to classify pixels in images according to their vegetation cover. Such classifier is innovative because, besides using general type-2 fuzzy sets, it can use specific and generic rules base (both in a hierarchical way) to reclassify pixels that remain unclassified. Such hierarquical reclassification leads to a compact rule base (with few rules). The reason why one should use type-2 fuzzy inference is that they present better performance than their type-1 counterparts, in spite of their bigger computational cost. The carried out tests showed, for sure, that the proposed system is better than the conventional fuzzy classifier usually employed in similar applications and its performance is comparable to the statistical likelihood classifier, proving to be an alternative choice to this last one.Propõe-se, nesta tese, o desenvolvimento de uma nova ferramenta baseada em conjuntos fuzzy gerais do tipo-2 para aplicação em processos de classificação digital de dados. O problema de classificação a ser considerado está relacionado à identificação de regiões de floresta em imagens de satélite com o objetivo de auxiliar em tarefas de monitoramento florestal. O classificador digital desenvolvido utiliza um mecanismo de inferência denominado de "inferência escalonada fuzzy geral do tipo-2" para classificar os pixels das imagens de satélite de acordo com sua cobertura vegetal. Tal classificador é inovador pois, além de utilizar conjuntos fuzzy tipo-2 gerais, pode utilizar tanto uma base de regras específica quanto uma base genérica (ambas de forma hierárquica) para reclassificar pontos que, do contrário, permaneceriam sem classificação. Isto permite a obtenção de uma base de regras compacta (composta de poucas regras). A justificativa para o uso de sistemas de inferência do tipo-2 é que estes, apesar do custo computacional maior, apresentam desempenho superior aos sistemas do tipo-1 equivalentes. Os testes realizados mostram que, de fato, o sistema proposto é melhor do que o classificador fuzzy convencional usualmente empregado em aplicações semelhantes e possui desempenho comparável ao classificador estatístico da máxima verossimilhança, sendo uma alternativa viável ao último

IMPROVING UNDERSTANDABILITY AND UNCERTAINTY MODELING OF DATA USING FUZZY LOGIC SYSTEMS

The need for automation, optimality and efficiency has made modern day control and monitoring systems extremely complex and data abundant. However, the complexity of the systems and the abundance of raw data has reduced the understandability and interpretability of data which results in a reduced state awareness of the system. Furthermore, different levels of uncertainty introduced by sensors and actuators make interpreting and accurately manipulating systems difficult. Classical mathematical methods lack the capability to capture human knowledge and increase understandability while modeling such uncertainty. Fuzzy Logic has been shown to alleviate both these problems by introducing logic based on vague terms that rely on human understandable terms. The use of linguistic terms and simple consequential rules increase the understandability of system behavior as well as data. Use of vague terms and modeling data from non-discrete prototypes enables modeling of uncertainty. However, due to recent trends, the primary research of fuzzy logic have been diverged from the basic concept of understandability. Furthermore, high computational costs to achieve robust uncertainty modeling have led to restricted use of such fuzzy systems in real-world applications. Thus, the goal of this dissertation is to present algorithms and techniques that improve understandability and uncertainty modeling using Fuzzy Logic Systems. In order to achieve this goal, this dissertation presents the following major contributions: 1) a novel methodology for generating Fuzzy Membership Functions based on understandability, 2) Linguistic Summarization of data using if-then type consequential rules, and 3) novel Shadowed Type-2 Fuzzy Logic Systems for uncertainty modeling. Finally, these presented techniques are applied to real world systems and data to exemplify their relevance and usage

Type-2 Fuzzy Alpha-cuts

Systems that utilise type-2 fuzzy sets to handle uncertainty have not been implemented in real world applications unlike the astonishing number of applications involving standard fuzzy sets. The main reason behind this is the complex mathematical nature of type-2 fuzzy sets which is the source of two major problems. On one hand, it is difficult to mathematically manipulate type-2 fuzzy sets, and on the other, the computational cost of processing and performing operations using these sets is very high. Most of the current research carried out on type-2 fuzzy logic concentrates on finding mathematical means to overcome these obstacles. One way of accomplishing the first task is to develop a meaningful mathematical representation of type-2 fuzzy sets that allows functions and operations to be extended from well known mathematical forms to type-2 fuzzy sets. To this end, this thesis presents a novel alpha-cut representation theorem to be this meaningful mathematical representation. It is the decomposition of a type-2 fuzzy set in to a number of classical sets. The alpha-cut representation theorem is the main contribution of this thesis. This dissertation also presents a methodology to allow functions and operations to be extended directly from classical sets to type-2 fuzzy sets. A novel alpha-cut extension principle is presented in this thesis and used to define uncertainty measures and arithmetic operations for type-2 fuzzy sets. Throughout this investigation, a plethora of concepts and definitions have been developed for the first time in order to make the manipulation of type-2 fuzzy sets a simple and straight forward task. Worked examples are used to demonstrate the usefulness of these theorems and methods. Finally, the crisp alpha-cuts of this fundamental decomposition theorem are by definition independent of each other. This dissertation shows that operations on type-2 fuzzy sets using the alpha-cut extension principle can be processed in parallel. This feature is found to be extremely powerful, especially if performing computation on the massively parallel graphical processing units. This thesis explores this capability and shows through different experiments the achievement of significant reduction in processing time.The National Training Directorate, Republic of Suda

Approximated Type-2 fuzzy set operations

Type-2 fuzzy sets, an elaboration over type-1 fuzzy sets, are an interesting method for handling uncertainty in rules and parameters in fuzzy systems. However, their adoption has not been as wide as one could have expected. In this paper we provide a simple introduction to type-2 fuzzy sets; then we propose a novel method for calculating operations on type-2 fuzzy sets with normal type-1 membership values, for which we redefine set ordering. Finally, based on the max ordering of fuzzy set and highest degree of separation, we propose an approximation for performing the operations, which ensures that the calculation is accurate for the most important parts of the membership values