# Innovative Approaches for Fuzzy Rule-based Interpolative Reasoning

• Mou Zhou

Student thesis: Doctoral ThesisDoctor of Philosophy

### Abstract

Fuzzy reasoning, which is part of approximate reasoning, is a significant research field in fuzzy set theory. It involves the process of deriving fuzzy conclusions from fuzzy premises by applying IF-THEN rules and inference rules. In real-world applications, traditional fuzzy reasoning may not always produce satisfactory reasoning results and may even fail to produce any inferred results. This is mainly because the rules in a rule base may not be able to cover the entire problem space. Linear fuzzy interpolative reasoning methods, commonly referred to as fuzzy rule interpolation (FRI), have been developed to address such issues and strongly support approximate inference. One of the main FRI methods enables the derivation of a fuzzy conclusion by selecting appropriate rules close to the observation from the given (sparse) rule base and subsequently constructing a new intermediate (interpolated or extrapolated) fuzzy rule to perform the interpolation. Traditional ways of implementing the critical rule selection process are typically based on the exploitation of Euclidean distances between the observation and rules. It is conceptually straightforward for implementation but applying this distance metric may systematically lead to inferior results because it fails to reflect the variations of the relevance or significance levels amongst different domain features. To address this important issue, a novel transformation-based FRI approach is presented, on the basis of utilising the Mahalanobis distance metric. The new FRI method works by transforming a given sparse rule base into a coordinates system where the distance between instances of the same category becomes closer while that between different categories becomes further apart. In so doing, when an observation is present that matches no rules, the most relevant neighbouring rules to implement the required interpolation are more likely to be selected. Following this, the scale and move factors within the classical transformation-based FRI procedure are also modified by Choquet integral. Several metric learning methods are discussed, all of which can be utilised to learn the necessary Mahalanobis matrix, showcasing the versatility of this approach. A systematic experimental investigation over a range of benchmark classification problems demonstrates that the proposed approach remarkably outperforms the existing state-of-the-art FRI methods with respect to accuracy and efficiency. The rule selection process has so far been generally conducted on the basis of distance metrics. Although this is intuitively interpretable and has enabled the development of various FRI techniques, it is heavily reliant on massive distance calculations, causing inefficient inference processes, especially when confronted with rapid response requirements and big data tasks. The thesis proposes a novel rule-ranking-based FRI mechanism (RT-FRI) that circumvents the inherent limitations of the decades-old distance-based approach to FRI. It utilises ranking scores of the given rules and unmatched observation generated by fusing the antecedent attributes through aggregation functions to accomplish the rule selection process. Nevertheless, practical applications may find the monotonicity constraint of aggregation functions overly strict. The constraint requires that the output is larger or equal to another input only if all the antecedent values of an input are not smaller than the corresponding values of the other input. This means that if any attribute of the former input is not greater than its counterpart in the latter input, the output will not be monotonic. Invalid entire monotonicity leads to the inexplicable rationale of RT-FRI. To address the issue of strict monotonicity constraints imposed by aggregation functions, a rule-ranking-based FRI with directional monotonicity (DMRT-FRI) is further introduced. Experimental results demonstrate that RT-FRI is a highly efficient technique with DMRT-FRI being capable of achieving high levels of both accuracy and efficiency. A fuzzy rule base system uses natural language terms to describe objects, helping bridge the gap between automated methods and user expectations. An FRI-based diagnostic system was developed for mammographic mass identification, but it focused on the interpretability of the fuzzy interpolative reasoning system with geometric mass features. The thesis goes beyond describing the basic property of FRI and focuses on breast cancer detection using the more effective FRI methods proposed above. It presents a detailed account of feature extraction and selection processes from real mammographies, where 63 numerical features are extracted from the original images to preserve and represent their properties. The most important subset of features, including shape, margin, and texture features, is then selected using a classical feature selection algorithm to reduce the dimensionality of the feature space. A comparative study is conducted to systematically evaluate the effectiveness and efficiency of the novel proposed FRI approaches as the core of a diagnosis system, demonstrating their ability to handle mammogram datasets.
Date of Award 2023 English Aberystwyth University Changjing Shang (Supervisor) & Qiang Shen (Supervisor)

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