# Transformation-Based Fuzzy Rule Interpolation Using Interval Type-2 Fuzzy Sets
## 6 Conclusions  
This paper has proposed an extension to the existing transformation-based fuzzy rule interpolation approach through the use of interval type-2 fuzzy sets. This approach offers a more concise way to deal with the uncertainty in fuzzy rule interpolation. Four demonstrative case studies have been presented in order to illustrate the operation of this novel approach. The conceptual and practical comparisons between type-2 fuzzy sets and rough-fuzzy sets have also been provided, with their implications to the extensions to T-FRI pointed out. The investigations have indicated that the proposed approach is of a natural appeal for FRI while dealing with the uncertainty that the conventional T-FRI technique may otherwise be difficult to handle. The work is an effective generalisation of the original T-FRI since the results of applying this extension degenerate to those obtainable by the existing type-1 FRI if the higher-order uncertainty embedded in the type-2 fuzzy sets vanishes.  
There are a number of aspects that could be improved with further research. In particular, only triangle-shaped membership functions are considered for implementation in this work. It would be beneficial to investigate how the approach may be extended (e.g., to enable the use of trapezoidal and other piecewise linear membership functions and to compare the effects of employing such different types of membership function). Furthermore, it would be interesting to evaluate the interpolated results attainable with this technique in comparison with those achievable using alternative FRI methods (e.g., [36; 37; 38]). Furthermore, currently, for simplicity (both mathematically and computationally), the current work is based on interval type-2 fuzzy sets. It would be useful to examine how the proposed work could be further generalised to support the involvement of general type-2 fuzzy sets while performing FRI. Finally, scaled-up real-world applications would help to further evaluate the full efficacy and potential of this research.  
This research was partly supported by Aberystwyth University, UK, and partly by the Scientific and Technological Research Program of Chongqing Municipal Education Commission, China (Grant No. KJ1601316), and the Research Foundation of Chongqing University of Science and Technology, China (Grant No. CK2016B04).  
[PERSON] and [PERSON] conceived of and designed the experiments. [PERSON] performed the experiments. [PERSON] and [PERSON] analysed the data. [PERSON] and [PERSON] wrote the paper.  
The authors declare no conflict of interest. The founding sponsors had no role in the design of the study; in the collection, analyses or interpretation of data; in the writing of the manuscript; nor in the decision to publish the results.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} FOU & Footprint of uncertainty \\ FRI & Fuzzy rule interpolation \\ LA & Lower approximation \\ LMF & Lower membership function \\ T-FRI & Scale and move transformation-based fuzzy rule interpolation approach \\ UA & Upper approximation \\ UMF & Upper membership function \\ \end{tabular}