A novel Gustafson–Kessel based clustering algorithm using <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si194.svg" display="inline" id="d1e4449"> <mml:mi>n</mml:mi> </mml:math> -Pythagorean fuzzy sets

The Gustafson-Kessel (GK) algorithm, an extension of the fuzzy c-means (FCM) clustering method, effectively handles non-spherical clusters but struggles with uncertainty in membership assignments. To address this limitation, we propose the n -Pythagorean Fuzzy Gustafson-Kessel ( n -PyGK) algorithm, which incorporates an inherent hesitation degree to enhance clustering performance. The proposed algorithm is evaluated on both synthetic and real-world datasets, including the Iris dataset, using nine clustering metrics. We analyze its behavior under varying parameter settings and compare its performance with traditional clustering algorithms. Experimental results demonstrate that n -PyGK offers improved clustering accuracy and greater flexibility in parameter selection, enabling optimal performance for specific clustering indices.