Interval Type-2 Sets Research Articles

A Novel Approach to Type-Reduction and Design of Interval Type-2 Fuzzy Logic Systems

Abstract Fuzzy logic systems, unlike black-box models, are known as transparent artificial intelligence systems that have explainable rules of reasoning. Type 2 fuzzy systems extend the field of application to tasks that require the introduction of uncertainty in the rules, e.g. for handling corrupted data. Most practical implementations use interval type-2 sets and process interval membership grades. The key role in the design of type-2 interval fuzzy logic systems is played by the type-2 inference defuzzification method. In type-2 systems this generally takes place in two steps: type-reduction first, then standard defuzzification. The only precise type-reduction method is the iterative method known as Karnik-Mendel (KM) algorithm with its enhancement modifications. The known non-iterative methods deliver only an approximation of the boundaries of a type-reduced set and, in special cases, they diminish the profits that result from the use of type-2 fuzzy logic systems. In this paper, we propose a novel type-reduction method based on a smooth approximation of maximum/minimum, and we call this method a smooth type-reduction. Replacing the iterative KM algorithm by the smooth type-reduction, we obtain a structure of an adaptive interval type-2 fuzzy logic which is non-iterative and as close to an approximation of the KM algorithm as we like.

Properties of interval type-2 fuzzy sets in decision support systems

Definitions and methods of designing interval type-2 fuzzy sets in fuzzy inference systems for control problems of complex technical objects in conditions of uncertainty are considered. The main types of uncertainties, that arise when designing fuzzy inference systems and depend on the number of expert assessments, are described. Methods for assessing intra-uncertainty and inter-uncertainty are proposed, taking into account the different number of expert assessments at the stage of determining the types and number of membership functions. Factors influencing the parameters and properties of interval type-2 fuzzy during experimental studies are determined. Such factors include the number of experiments performed, external factors, technical parameters of the control object, and the reliability of the components of the computer system decision support system. The properties of the lower and upper membership functions of interval type-2 fuzzy sets are investigated on the example of the Gaussian membership function, which is one of the most used in the problems of fuzzy inference systems design. The main features and differences in the methods of determining the lower and upper membership functions of interval type-2 fuzzy sets for different types of uncertainties are taken into account. Methods for determining the footprint of uncertainty, as well as the dependence of its size on the number of expert assessments, are considered. The footprint of uncertainty is characterized by the lower and upper membership functions, and its size directly affects the accuracy of the obtained solutions. Methods for determining interval type-2 fuzzy sets using regulation factors of membership function parameters for intra-uncertainty and weighting factors of membership functions for inter-uncertainties have been developed. The regulation factor of the function parameters can be used to describe the lower and upper membership functions while determining the size of the footprint of uncertainty. Complex interval type-2 sets are determined to take into account inter-uncertainties in the problems of fuzzy inference systems design.

A discriminative subject-specific spatio-spectral filter selection approach for EEG based motor-imagery task classification

Motor-imagery tasks generate event related synchronization and de-synchronization in certain subject-specific frequency ranges of the subject’s ElectroEncephaloGraphy (EEG) signals. The selection of frequency ranges for each subject is important for obtaining better classification accuracy of motor-imagery based Brain Computer Interface (BCI). Further, the spatial filters extracted corresponding to the selected spectral ranges also influence the classification accuracy. In this paper, a subject-specific spatio-spectral filter selection approach using a cognitive fuzzy inference system for classification of the motor-imagery tasks in a two step approach is presented. The cognitive fuzzy inference system (CFIS) employs an evolving interval type-2 system to classify the non-stationary features. The classifier employs a meta-cognitive sequential algorithm to determine both the structure and parameters of the CFIS. In the first step, the CFIS classifier is used to find the desired spectral filters by eliminating those frequency bands that do not affect the classification performance. In the second step, CFIS is used to eliminate those spatial filters which do not affect the performance. The performance of CFIS based spatio-spectral scheme has been evaluated using two publicly available BCI competition data sets and compared with other existing algorithms like FBCSP, DCSP and BSSFO. The results indicate that the proposed approach outperforms the CSP method by approximately 15–18% and other algorithms like FBCSP, DCSP by 8–10%. Compared to a recently proposed algorithm BSSFO, it achieves an improvement of 2%, but is simpler in comparison to BSSFO. The main impact of the work is its ability to handle non-stationarity using interval type-2 sets and provide good classification performance. In general, the proposed CFIS algorithm can be applied in the field of expert and intelligent systems where it is necessary to deal with non-stationary signals.

Centroid of triangular and Gaussian type-2 fuzzy sets

In this paper, we present exact computation procedures for the centroid of type-2 fuzzy sets with either triangular or Gaussian secondary membership functions. We assume the standard formulation of the centroid based on the sup-min extension principle. The centroid of a triangular type-2 fuzzy set can be considered as a subsequent analytical extension of the KM type-reduction procedure dedicated to interval type-2 sets. The centroid of a Gaussian type-2 fuzzy set is recursively expressed by an independent procedure.

Fuzzy multiple attributes group decision-making based on the interval type-2 TOPSIS method

Type-2 fuzzy sets involve more uncertainties than type-1 fuzzy sets. They provide us with additional degrees of freedom to represent the uncertainty and the fuzziness of the real world. In this paper, we present an interval type-2 fuzzy TOPSIS method to handle fuzzy multiple attributes group decision-making problems based on interval type-2 fuzzy sets. We also use some examples to illustrate the fuzzy multiple attributes group decision-making process of the proposed method. The proposed method provides us with a useful way to handle fuzzy multiple attributes group decision-making problems in a more flexible and more intelligent manner due to the fact that it uses interval type-2 sets rather than traditional type-1 fuzzy sets to represent the evaluating values and the weights of the attributes.

Fuzzy Subsethood for Fuzzy Sets of Type-2 and Generalized Type-<formula formulatype="inline"><tex Notation="TeX"> ${n}$</tex></formula>

In this paper, we use Zadeh's extension principle to extend Kosko's definition of the fuzzy subsethood measure S(G, H) to type-2 fuzzy sets defined on any set X equipped with a measure. Subsethood is itself a fuzzy set that is a crisp interval when G and H are interval type-2 sets. We show how to compute this interval and then use the result to compute subsethood for general type-2 fuzzy sets. A definition of subsethood for arbitrary fuzzy sets of type-n > 2 is then developed. This subsethood is a type-(n - 1) fuzzy set, and we provide a procedure to compute subsethood of interval type-3 fuzzy sets.

Detection of hard cuts and gradual transitions from video using fuzzy logic

In this paper, we introduce a novel approach for the detection and classification of shot boundaries based on interval type-2 fuzzy logic. We consider fuzzy low level features (Fuzzy Histogram (FH) and Fuzzy Co-occurrence Matrix (FCM)) and evaluate a fuzzy rule base modelled by interval type-2 sets to detect cut boundaries. After detection of cuts, a recovery mechanism is used to reduce false positives caused by illumination variation (e.g., camera flash light). The fuzzy rules used for gradual transition form a unified approach to detect various types gradual transitions like dissolves, fades and wipes, by a single method. Test results indicate that very high precision and recall could be achieved for the test videos, particularly the recall value of gradual transitions.

Centroid of a type-2 fuzzy set

Abstract In this paper, we introduce the centroid and generalized centroid of a type-2 fuzzy set (both of which are essential for implementing a type-2 fuzzy logic system), and explain how to compute them. For practical use, we show how to compute the centroid of interval and Gaussian type-2 fuzzy sets. An exact computation procedure is provided for an interval type-2 set, whereas an approximation is provided for both interval and Gaussian type-2 sets. Examples are given that compare the exact computational results with the approximate results.