High order $\mathit{\alpha }$-planes integration: A new approach to computational cost reduction of General Type-2 Fuzzy Systems

Abstract

Nowadays, there are different representations of Generalized Type-2 Fuzzy Sets that consider a non-uniform distribution of the uncertainty, for example, the Geometric approach, the Z-Slices method and the $\mathit{\alpha }$-planes approximation. Each representation has advantages and disadvantages, however, the present work is focused on the $\mathit{\alpha }$-planes representation, and this representation consists on realizing a horizontal discretization, the solution of each horizontal slice, and then the integration of these planes. Each horizontal slice results in an Interval Type-2 Fuzzy System, so, the computational cost is proportional to the discretization level, which means, is proportional to the number of $\mathit{\alpha }$-planes used for modeling the Generalized Type-2 FS. The aim of this work is reducing the computational cost of Generalized Type-2 FS by a new approach of $\alpha$-planes representation. In this paper the Newton–Cotes quadrature for the $\alpha$-planes integration is proposed, achieving in this way a high-level discrete integration compared with the conventional $\alpha$-planes integration. The proposed approach aims at reducing the number of $\alpha$-planes necessary to obtain a good approximation of Generalized Type-2 FS. In order to validate the proposed approach, a set of experiments was realized with a randomly generated Generalized Type-2 Fuzzy Sets, and they are realized with a different number of $\alpha$-planes in order to compare the performance of the proposed approach with respect to the conventional approach. On the other hand, the proposed approach was also applied to a control problem, as an example of applications of the proposed approach to real-world problems.

Introduction

Type-2 Fuzzy Sets were originally proposed by Zadeh and later studied in more detail by Mendel and John (2002) and the conceptual difference of Type-2 Fuzzy Sets with respect to Type-1 Fuzzy Sets is the inclusion of a new secondary domain related with handling uncertainty. The emergence of Generalized Type-2 FS provides a new ability for handling uncertainty in modeling additionally to the vagueness modeling provided by the Type-1 FS. These properties of the FSs allow their application in many kinds of problems, for example, control systems Schouten et al. (2002), Castillo and Melin (2014), Caraveo et al. (2016), Cervantes and Castillo (2015), image processing Melin et al. (2010), Gonzalez et al. (2015), diagnosis systems Luo et al. (2010), Nilashi et al. (2017), classification, industrial problems Hannan et al. (2015), Wati (2016), etc.

It has been shown in previous works that the performance of an IT2 FS improves on the results obtained with T1 FS, and some researchers that have reported this are in Castillo et al. (2016), however, the uncertainty modeling provided by an IT2 FS is incomplete, because the uncertainty is not uniform as assumed in this model. On the other hand, there exist representations that allow the modeling of non-uniform uncertainty, that correspond to approximations of GT2 FS, and these approaches have the potential to overcome the performance obtained by IT2 FS and they are, for example, the Geometric approach (Coupland and John, 2008), the Z-Slices method Wagner and Hagras (2011), Wagner and Hagras (2010), and the $\mathit{\alpha }$-planes approximation (Mendel et al., 2009b).

The different representations have advantages and disadvantages and have been implemented successfully in many applications, however, the main problem of the implementation of the mentioned representations of GT2 FS is the computational cost, so, the aim of the present work has been focused on reducing the computational cost of the $\mathit{\alpha }$-planes representation of GT2 FS.

Then, considering that the $\mathit{\alpha }$-planes representation is based on a discretization of the GT2 FS, it is expected that the approximation accuracy would be proportional to the discretization level. However, the computational cost is also proportional to the discretization, considering that each slice of GT2 FS for the discretization represents an IT2 FS (also called $\mathit{\alpha }$-plane), and this represents a problem for applications that require a good approximation of the theoretical model of GT2 FS.

The proposed approach, then, aims at reducing the number of $\mathit{\alpha }$-planes that approximate the GT2 FS, and this reduction can be possible with the implementation of High-order $\mathit{\alpha }$-planes integration. This is because the conventional method for $\mathit{\alpha }$-planes integration can be expressed as a numerical integrator of order one, then, the proposal is that with the implementation of High-order integration, the number of planes required for a good approximation can be significantly reduced.

To test the improvement with the proposed approach a set of experiments with random GT2 FS composed of different kinds of GT2 MFs were performed. In this case, 30 experiments and a statistical test were performed to demonstrate that the proposed approach exhibits an improvement with respect to the conventional method.

The structure of the present paper is as follows: in Section 2 the basics of Type 2 FS are presented, Section 3 outlines the GT2 MFs used in the present work, Section 4 explains the Closed Newton–Cotes rules, in Section 5 the proposed approach of $\mathit{\alpha }$-planes integrations is presented, Section 6 contains the experimentation and the obtained results, and finally Section 7 offers the conclusion and the future work.