Multidimensional scaling method for complex time series based on the Wasserstein–Fourier distance in complex systems

In this paper, we propose a multidimensional scaling (MDS) method based on the Wasserstein–Fourier (WF) distance to analyze and classify complex time series from a frequency domain perspective in complex systems. Three properties with rigorous derivation are stated to reveal the basics structure of MDS method based on the WF distance and validate it as an excellent metric for time series classification. Our proposed method solves the problem of non-equal-length sequence distances faced by traditional metrics and the problem that the Kullback–Leibler divergence and Jensen–Shannon divergence cannot measure completely non-overlapping distributions. Its practicability for multiple data sets and robustness to noise are demonstrated in the paper. Compared with the MDS methods based on Euclidean distance, Chebyshev distance, correlation distance, complexity-invariant distance and dynamic time warping, our proposed method is more effective and has minimal losses and errors.