# Transformation-Based Fuzzy Rule Interpolation Using Interval Type-2 Fuzzy Sets
## 1 Introduction  
The compositional rule of inference [1] offers an effective mechanism to deal with fuzzy inference for dense rule bases. Briefly, a dense rule base is a rule base where the input universe of discourse is covered completely. Given such a rule base and an observation that is at least partially covered by the rule base, the conclusion can be inferred from certain rules that intersect with the observation. However, for cases where a fuzzy rule base contains \"gaps\" (i.e., the so-called sparse rule base as termed in [2]), if a given observation has no overlap with the antecedent values of any rule, conventional fuzzy inference methods will fail. Fortunately, using fuzzy rule interpolation (FRI) [3; 4], certain conclusions may still be obtained. Moreover, with the help of FRI techniques, the complexity of a rule base can be reduced by omitting fuzzy rules, which may be approximated from their neighbouring ones. Nevertheless, despite these advantages, the application of traditional FRI methods may lead to abnormal and invalid fuzzy conclusions. One particular issue is that the convexity of the derived fuzzy values is not guaranteed [5], but convexity is often a crucial requirement for fuzzy inference in order to attain improved interpretability of the results. In order to overcome such drawbacks, a number of significant extensions to the original FRI methods has been proposed in the literature, including [6; 7; 8; 9]. In particular, the scale and move transformation-based approach (T-FRI) [10; 11] and its generalisation [12] can handle interpolation and extrapolation involving multiple fuzzy rules, with each rule consisting of multiple antecedents. Such work also guarantees the uniqueness, as well as the normality and convexity of the interpolated conclusion. This approach has recently been further enhanced with a backward interpolation technique to enable chained rule interpolation [13]; a dynamic learning method to support efficient use of previously interpolated results for subsequent inference [14]; an adaptive mechanism to facilitate automatic error corrections in interpolative inference [15; 16]; an extended adaptive fuzzy rule interpolation approach for polygonal fuzzy sets [17; 18]; and a system that works through similarity-based reasoning with interpolative conditions [19]. There is little work in the area of FRI that can handle uncertainty in fuzziness itself however. This is because all of the aforementioned approaches are implemented based on conventional fuzzy set representations, i.e., type-1 fuzzy sets [20]. Whilst membership functions play an important role in defining fuzzy sets, it is sometimes extremely difficult, if not impossible, to give precise definitions for such membership functions. More generally, there may be different types of uncertainty in fuzzy rule-based systems that need to be captured or modelled [21]: (1) the variables that are used in the antecedents and the consequences of rules may be indiscernible; (2) the meanings of the words may be vague because the same word can mean different things to different people; (3) an object can belong to a given set or conceptual class to a partial degree, but that degree may itself be uncertain; (4) the obtained rules may be inconsistent when individual views are obtained from a group of experts; (5) observations attainable by inexact knowledge may be noisy and therefore randomly distributed. Most of these types of uncertainty can be difficult to deal with in order to determine the crisp membership functions for the fuzzy sets used. For instance, certain weather conditions are considered as cold by all people, but others may be considered as cold by only certain individuals. That is, different people may indicate different temperatures as their understanding of the concept cold. The membership functions for different people may therefore be very different, depending on their perception, preference, experience, etc.