Handling uncertainty: from type-1 to interval type-2 fuzzy sets and systems - Aladi, Jabran (2016)
Fuzzy logic systems (FLSs) are widely accepted for their ability to model and handle uncertainty. Type-2 fuzzy sets (T2 FSs) were introduced as an extension of type-1 fuzzy sets (T1 FSs). They are characterised by membership functions (MFs) that are themselves fuzzy for which the membership degrees are expressed as FSs on [0,1], and have been widely accepted as capable of modelling uncertainty with more detail than T1 FSs. Interval type-2 fuzzy sets (IT2 FSs) are a special type of (general) T2 FSs and currently the most widely used, due to their reduction in computational cost. The study of T2 FLSs is a rapidly growing research area with a wide range of application domains. Capturing the uncertainty arising from system noise has been a core feature of FLSs for many years. Since the concept of T2 FSs was introduced, a recurring question in research considering the application of T2 FSs is, ‘How much uncertainty in a given context warrants the use of T2 FSs and systems over their T1 counterparts?’ In other words, while a main issue in the application of FLSs is the estimation of parameters such as the type of fuzzy sets (FSs) and their parameters, as well as the number of rules, an even more fundamental question is whether T1 or T2 FSs should be used. More specifically, ‘How should T2 FSs be shaped in order for them to capture the uncertainties in a given application?’ Although there is experimental evidence showing improvements in terms of the uncertainty handling of interval type-2 fuzzy logic systems (IT2 FLSs) over their T1 counterparts, no systematic way of determining the potential advantages of employing T2 FLSs over T1 has yet been developed. In an effort to relate the size of the footprint of uncertainty (FOU) of employed IT2 FSs to uncertainty levels and vice versa in a given application, this thesis shows the relationship between the size of the FOU of IT2 FSs and the uncertainty levels in a given application and explains how this knowledge can be exploited to inform the design of FLSs. To provide insight into this challenging aim, a detailed investigation of the ability of both T1 and IT2 FLSs to model different levels of uncertainty/noise is conducted. Design methodologies that systematically vary (blur) the size of the FOU of the IT2 FSs are introduced, enabling the comparison of FLSs that are equivalent in all but the size of the FOUs of the employed FSs. We describe an application-driven investigation into the relationship between the FOU size of the FSs and the level of uncertainty in applications by using time series prediction (TSP) as a well-defined and well-controlled sample application. Thus, TSP is used as a platform to comprehensively compare different FLSs with various FOUs. Through contrasting the performance of these resulting FLSs in the face of inputs with varying uncertainty levels in a rich set of TSP experiments, a distinct pattern of performance arising from the different levels-of-uncertainty and FOU-size combinations is explored and captured, showing a direct relationship between FOU size and uncertainty levels. For example, as the noise level increases, the FOU size that gives the best performance increases. Based on this, we provide guidelines for the selection of appropriate FOU sizes for given levels of uncertainty in a given application and propose an approach to quantifying the commonly used linguistic labels, ‘low’, ‘medium’ and ‘high’ through FS models. Finally, going beyond the question of selecting the most appropriate FOU at design time, we conduct some initial work on the appropriate adjustment of FOUs at run time, i.e., when uncertainty levels vary. Specifically, we explore the application of optimisation methods to refine FOU sizes in IT2 FSs.