Equivalence partition based morphological similarity clustering for large-scale time series

Data clustering belongs to the category of unsupervised learning and plays an important role in the dynamic systems and big data. The clustering problem of sampled time-series data is undoubtedly much more challenging than that of repeatable sampling data. Most of the existing time-series clustering methods stay at the level of algorithm design, lacking rigorous theoretical foundation and being inefficient in dealing with large-scale time series. To address this issue, in this paper, we establish the mathematical theory for the large-scale time series clustering of dynamic system. The main contributions of this paper include proposing the concept of time series morphological isomorphism, proving that translation isomorphism and stretching isomorphism are equivalent relations, developing the calculation method of morphological similarity measure, and establishing a new time series clustering method based on equivalent partition and morphological similarity. These contributions provide a new theoretical foundation and practical method for the clustering of large-scale time series. Simulation results in typical applications verify the validity and practicability of the aforementioned clustering methods.