Hooman Tahayori

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.

Induction of Shadowed Sets Based on the Gradual Grade of Fuzziness

The existing methods of determining an α-cut of a fuzzy set to construct its underlying shadowed set do not fully comply with the concept of shadowed sets, namely, a retention of the total amount of fuzziness and its localized redistribution throughout a universe of discourse. Moreover, no closed formula to calculate the corresponding α-cut is available. This paper proposes analytical formulas to calculate threshold values required in the construction of shadowed sets. We introduce a new algorithm to design a shadowed set from a given fuzzy set. The proposed algorithm, which adheres to the main premise of shadowed sets of capturing the essence of fuzzy sets, helps localize fuzziness present in a given fuzzy set. We represent the fuzziness of a fuzzy set as a gradual number. Through defuzzification of the gradual number of fuzziness, we determine the required threshold (i.e., some α-cut) used in the formation of the shadowed set. We show that the shadowed set obtained in this way comes with a measure of fuzziness that is equal to the one characterizing the original fuzzy set.

Interval type-2 fuzzy sets to model linguistic label perception in online services satisfaction

In this paper, we propose a novel two-phase methodology based on interval type-2 fuzzy sets (T2FSs) to model the human perceptions of the linguistic terms used to describe the online services satisfaction. In the first phase, a type-1 fuzzy set (T1FS) model of an individual’s perception of the terms used in rating user satisfaction is derived through a decomposition-based procedure. The analysis is carried out by using well-established metrics and results from the Social Sciences context. In the second phase, interval T2FS models of online user satisfaction are calculated using a similarity-based data mining procedure. The procedure selects an essential and informative subset of the initial T1FSs that is used to discard the outliers automatically. Resulting interval T2FSs, which are synthesized based on the selected subset of T1FSs only, exhibit reasonable shapes and interpretability.

Concave type-2 fuzzy sets: properties and operations

In this article, concave type-2 fuzzy sets are investigated. The calculation of union and intersection of concave type-2 fuzzy sets using the min t-norm and the max t-conorm are explored and it is proved that the set of concave type-2 fuzzy sets is closed under those operations. It is also shown that the set of LR-normal concave type-2 fuzzy sets forms a commutative semiring under join and meet.

On the calculation of extended max and min operations between convex fuzzy sets of the real line

In this paper we will identify the sets of so-called sub- and pseudo-highest intersection points of convex fuzzy sets of the real line and will explore their properties. Based on the properties of these sets, an algorithm for calculating extended max and min operations between two or more convex fuzzy sets of the real line with general membership functions, not necessarily continuous, is proposed.

Distinguishability of interval type-2 fuzzy sets data by analyzing upper and lower membership functions

In this paper, we deal with the problem of classification of interval type-2 fuzzy sets through evaluating their distinguishability. To this end, we exploit a general matching algorithm to compute their similarity measure. The algorithm is based on the aggregation of two core similarity measures applied independently on the upper and lower membership functions of the given pair of interval type-2 fuzzy sets that are to be compared. Based on the proposed matching procedure, we develop an experimental methodology for evaluating the distinguishability of collections of interval type-2 fuzzy sets. Experimental results on evaluating the proposed methodology are carried out in the context of classification by considering interval type-2 fuzzy sets as patterns of suitable classification problem instances. We show that considering only the upper and lower membership functions of interval type-2 fuzzy sets is sufficient to (i) accurately discriminate between them and (ii) judge and quantify their distinguishability.

Shadowed Fuzzy Sets: A Framework with More Freedom Degrees for Handling Uncertainties Than Interval Type-2 Fuzzy Sets and Lower Computational Complexity Than General Type-2 Fuzzy Sets

In this chapter, the concept of Shadowed Fuzzy Set is introduced and some of its related operations are studied. Shadowed Fuzzy Set enables localization of the underlying uncertainty of fuzzy grades in type-2 fuzzy sets through exploitation of shadowed sets. It provides a capable framework that despite preserving the uncertainties of fuzzy grades in type-2 fuzzy sets, adheres the simplicity of the concept and operations of interval type-2 fuzzy sets.

Matching general type-2 fuzzy sets by comparing the vertical slices

In this paper, we propose a procedure for computing the dissimilarity measure of finite general type-2 fuzzy sets, represented as sequences of vertical slices. Through representing general type-2 fuzzy sets as a sequence of objects, we compute their overall dissimilarity value using suited matching algorithms for generalized sequences. The evaluation of the proposed matching algorithm is performed in the setting of classification, by defining datasets of general type-2 fuzzy sets conceived as labeled patterns. Experimental results show that the matching methodology is robust, accurate, and computationally acceptable.

Operations on Concavoconvex Type-2 Fuzzy Sets

about to represent. Once the membership function has been established (estimated or defined), the concept is described very precisely as the membership values are exact numerical quantities. This seems to raise a certain dilemma of excessive precision in describing imprecise phenomena. In fact, this concern has already sparked a lot of debates starting from the very inception of fuzzy sets.” Concavoconvex fuzzy set is the result of the combination of the concepts of convex and concave fuzzy sets. This paper investigates concavoconvex type-2 fuzzy sets. Basic operations, union, intersection and complement on concavoconvex type-2 fuzzy sets using min and product t-norm and max t-conorm are studied and some of their algebraic properties are explored. In reality, there are situations in which the grade of

Artificial immune system based on interval type-2 fuzzy set paradigm

Abstract: This paper discusses the design and engineering of a biologically-inspired intrusion detection system, based on interval type-2 fuzzy set paradigm, for protecting computer networks. To this end, we have proposed a performance-based Artificial Immune System (AIS) that mimics the workings of an adaptive immune system and consists of a number of running artificial white blood cells, which search, recognize, store and deny anomalous behaviors on individual hosts. The proposed AIS monitors the system through analyzing the set of parameters to provide general information on its state. For the analysis, we have suggested a dynamic technique based on interval type-2 fuzzy set paradigm that enable identifying the system status - i.e. Non-Attack, Suspicious-Non-Attack, Non-Decidable, Suspicious-Attack, Attack. In conclusion, for proving the effectiveness of the suggested model, an exhaustive testing is conducted and results are reported.