title: Analysis for Data with Uncertainty based on Fuzzy Theory
Creators: name: Yang, Youpeng
Publisher: name: University of Liverpool Repository
Year: 2029

description: Since the introduction by Zadeh in 1965, Fuzzy Sets have been used to analyze data with uncertainty as a major framework for mathematically quantifying and handling uncertainty. Together with Fuzzy Sets, fuzzy theory has been widely used to construct the approximate quantitative relations between input and output. It is based on Fuzzy Theory, and I have investigated how to systematically and efficiently analyze data with uncertainty, whose results are presented in this thesis.
First, the Standard Fuzzy Sets (SFSs) has been extented as a basis for methods of the analysis of data with uncertainty. In SFSs, a membership degree uniquely characterizes a fuzzy set because a non-membership degree is just its complement. To more flexibly handle uncertainty, Atanassov extended FSs to Intuitionistic Fuzzy Sets (IFSs) by introducing hesitation. Due to the constraint of total information, however, the ordered pairs of membership and non-membership degrees cannot cover the areas around the point of (1, 1) in the Cartesian product of [0,1]×[0,1] under the IFSs. To address this issue, Yager further extended IFSs to Pythagorean Fuzzy Sets (PFSs) and Q-rung Orthopair Fuzzy Sets (QOFSs). Though these extended Fuzzy Sets loosen the relationship between membership and non-membership degrees, there are still areas that cannot be covered by their ordered pairs. Therefore, a novel form of Negative Hesitation Fuzzy Sets (NHFSs) has been proposed. NHFSs further loosen the relationship between membership and non-membership degrees by separating the hesitation into positive and negative values according to their different physical meanings so that the ordered pairs of membership and non-membership degrees can cover the whole Cartesian product of [0,1]×[0,1]. I also propose a similarity measure to utilize NHFSs in pattern recognition problems. With the help of the soft set theory, NHFSSs are applied to decision-making problems.
Second, the study of NHFSs has been applied to decision-making problems. Decision-making is one of the most critical topics in the application of fuzzy theory. I have applied NHFSSs to decision-making problems and develop a novel score function for IFSs based on the Poincaré metric. Numerical examples and real case studies show the advantage of the novel score function.
Third, the granular computing (GrC) has been considered and investigated as efficient granulation methods. GrC was introduced based on fuzzy theory, which is considered an alternative to Fuzzy Sets in data analysis. In conventional GrC, only the magnitude and the change of magnitude of data are used for granulation. In this regard, The permutation entropy (PeEn) has been introduced as a complexity measure for granulation, where the monotonicity of time series can be used to construct information granules. I applied the proposed granulation method to pattern recognition and electroencephalogram (EEG) classification problems and demonstrated that the proposed granulation method combined with the support vector machine (SVM) algorithm outperforms conventional neural networks.