Epilogue

Introduction

Fuzzy set theory was first introduced by Zadeh in 1965. Fuzzy sets and systems have been evolving for over 50 years and have been chosen as the methodology of tower systems that can release satisfactory performance in situations of scattered uncertainty and inaccuracies. Therefore, fuzzy logic system (FLS) has been successfully implemented in many practical applications including modeling and control, time series prediction (TSP), and data mining.

The type 1 fuzzy logic system (T1 FLS) is the most well-known and widely used type of FLS. Nevertheless, in recent years, there is a significant increase in research into more and more forms of fuzzy logic, such as the interval type 2 fuzzy logic system (IT2 FLS) and more recently undefined type 2 FLS (T2 FLS). This transition motivated me to realize that type 1 fuzzy sets (T1 FS) can only handle a limited level of uncertainty, while real-world applications often face multiple sources and higher levels of uncertainty. In 1975, Zadeh recognized these potential limitations and introduced the concept of a (generic) type 2 fuzzy set (T2 FS), an extension of the T1 FS. As an increasingly pervasive model, the T2 FS is considered to be potentially best suited for modeling uncertainty. The extra complexity arises including footprint of uncertainty (FOU) and 3D, and provides a level of self-domination for T2 FS compared to T1 FS. However, due to the same complexity, FLS with T2 FS is computationally very burdensome compared to FLS with T1 FS.

The computational complexity of using T2 FLS has led to the introduction of the simplified IT2 FLS, the most surprisingly used type of T2 FLS today. The IT2 FLS uses IT2 FS, a special sample of unqualified T2 FS with all secondary membership levels of 1 (Nie et al., 2008). Many researchers prefer IT2 FLS over T1 FLS due to its potential to model and mitigate the effects of uncertainty. From this, it is important to note that many of the research works considering T2 FS in the study do not quantify the "high" method, but they expect diamonds to perform well in the "high level of uncertainty" squatter.

So, the main problem with the use of FLS is the type of fuzzy set and its parameter and interpretation of parameters such as the number of rules, but an increasingly fundamental problem is whether you should use T1 or T2 FS. Although there are experimental traces of improvement in terms of handling uncertainty in IT2 compared to its T1 counterpart, no systematic method has been developed to determine the potential soundness of using T2 FS over T1 FS.

TSP can be used as a well-defined sample application (Aladi et al., 2014) to obtain a usage-led survey of the relationship between the FOU size of FS and the level of uncertainty in use. This can be done by diamond marking the T1 FLS for the TSP and creating a variety of FLSs that continuously increase the FOU size over several steps. At the same time, we inject increasing levels of noise to provide a priori known and well-defined/understood source of uncertainty. Hence, using time series prediction provides a platform to explore the behavior of FLS with different FOU sizes for different levels of noise/uncertainty. The main goal is not to decompress the optimal performance in the predictions, but to illuminate the approaching size of the FOU for a given level of uncertainty/noise. The results are obvious to use, but they provide important insights into the relationship between FOU size and level/uncertainty in a given setup.