Recurrence Triangle Features for Unsupervised Time Series Clustering: Application to Human Gait.

Unsupervised classification or clustering of complex time series generated from nonlinear dynamical systems remains challenging, particularly when signals exhibit noise and subtle structural differences. We develop a recurrence-based framework that captures local geometric patterns in recurrence plots by extracting triangular motifs, termed recurrence triangles (RTs), and mapping their relative frequencies to compact feature vectors for clustering. The approach is evaluated on four synthetic systems: the continuous-time Rössler and Lorenz systems and the discrete-time Logistic map and AR(2) model. RT-based features generally achieved higher clustering accuracy than classical recurrence quantification analysis (RQA), although performance depended on the dynamical system and signal length. RQA performed better than RT for AR(2) at longer signal lengths, while a statistical baseline, namely StatACF, also outperformed RT under several conditions, indicating that no single representation was uniformly optimal. We further applied the methods to gait recordings from younger and older adults. RT features achieved the highest observed clustering accuracy among the three representations (73.7%), followed by StatACF (68.4%) and RQA (52.6%). However, the differences among the methods were not statistically significant in the small cohort (n = 19). RT motif distributions also differed between age groups in a marker-dependent manner. These findings suggest that RT distributions capture fine-scale recurrence structure complementary to global recurrence statistics and warrant further validation in larger, independent cohorts.