Quaternion model of Pythagorean fuzzy sets and its distance measure

Pythagorean fuzzy set (PFS), one of several non-standard fuzzy sets, introduces a number pair (a, b) that satisfies the condition a2+b2 <= 1 to express membership grade. The above expression suggests that PFS provides a more extensive description space for describing fuzzy information, thereby attracting much attention in scientific research and engineering practice. Many studies have attempted to further exploit the potential of PFS in a reasonable manner. Consequently, this paper proposes a quaternion model of Pythagorean fuzzy set (QPFS). In QPFS, membership, non-membership, and hesitation function are expressed in the form of quaternion. A major advantage of QPFS over PFS is that its description space of fuzzy information stretches from the real plane to the hypercomplex plane. This is useful for capturing the composite features of fuzzy information as well as describing multi-dimensional fuzzy information. We then define the basic logical operations of QPFS, including complement, union, and intersection, and also derive the properties of these operations. As an additional contribution, this paper presents a distance measure (QPFSD) in the quaternion model of Pythagorean fuzzy sets (QPFSs). QPFSD is a strict distance measure that satisfies the axioms of distance measure, i.e., nonnegativity, symmetry, and triangle inequality. When QPFSs collapse into Pythagorean fuzzy sets (PFSs), QPFSD becomes the distance measure of real space. It has been demonstrated through some numerical examples that QPFSD is an effective method for measuring difference between QPFSs. Furthermore, we apply QPFSD to domains such as expert evaluation and data-driven environments in the QPFS framework. Experimental results related to expert evaluation suggest that QPFSD and the models involving it can provide an intuitive decision-making. Experimental results based on Iris data set demonstrate that with the increase of the number of training sets, the recognition rate of methods associated with QPFSD also increases for unknown targets. With the help of iris data, based on QPFSD, we investigate the performance of QPFS, CPFS, PFS and FS, respectively. The average recognition rate associated with QPFS is the highest among these. Through the use of iris dataset experiments and numerical example, this paper compares QPFSD and other distance measures in the CPFS and PFS frameworks, respectively. As a result of the dataset experiment, the recognition of methods pertaining to QPFSD is highest within CPFS and PFS frameworks. The numerical experiments indicate that QPFSD is highly sensitive to the differences between fuzzy information compared with other distance measures.