Grey relational analysis between hesitant fuzzy sets with applications to pattern recognition

Hesitant Fuzzy Sets (HFSs) is an important tool to deal with uncertain and vague information. There have been lots of fuzzy measures for it from different views. However, these fuzzy measures are more or less inappropriate in the applications. The distance and similarity measures only consider the closeness of the HFSs, while the correlation coefficients only consider the linear fashion. They are merely one side of the HFSs intrinsic fuzzy measures. Therefore, in this paper, we apply the grey relational analysis to the HFSs for the first time and define the HFSs grey relational degree to express the closeness. Furthermore, we creatively propose the difference and slope concept of the HFSs. Based on the difference and slope we define the HFSs slope grey relational degree to represent the linear fashion. Sequentially, combining the HFSs grey relational degree and HFSs slope grey relational degree together, we construct the HFSs synthetic grey relational degree, which takes both the closeness and the linear fashion into consideration. With the help of the proposed HFSs synthetic grey relational degree we propose the hesitant fuzzy grey relational recognition methodology. Finally, we apply the HFSs synthetic grey relational degree to deal with the pattern recognition problems. Compared with some examples, the performance of the proposed HFSs synthetic grey relational degree outperforms the existing HFSs fuzzy measures in the accuracy and integrity.