A new spectral distance based on adaptive selection algorithm for non-stationary time series

We propose a distance metric to quantify the dissimilarity between time series from the perspective of the frequency domain, especially non-stationary time series, called the Wasserstein Hilbert marginal spectral distance (WHMS). It calculates the Wasserstein distance between the Hilbert marginal spectral (HMS) of the time series, where HMS is the integral of Hilbert spectrum. Hilbert spectrum is obtained by the Hilbert transform performed on the intrinsic mode functions (IMFs) generated after the Empirical Mode Decomposition (EMD) operation. On this basis, we propose an IMF adaptive selection algorithm to improve the feature extraction accuracy of the HMS, free from a large number of prior experiments to set the threshold. We demonstrate that WHMS distance can be used as a general measure of time series from both theoretical and practical aspects. Combined with multidimensional scaling (MDS), we comprehensively evaluate the proposed method through simulation experiments and empirical data. The results confirm the good applicability of the WHMS distance combined with the IMF adaptive selection algorithm, which can distinguish different types of complex systems, is robust to noise, and is superior to Wasserstein–Fourier (WF) distance, Wasserstein–Fourier distance with short time Fourier transform (STFT), Euclidean distance and Chebyshev distance.