Scope: Statistical Confidence of Oscillatory Processes with EMD

scope is the Python-based package for detecting oscillatory signals in observational or experimental time series with the EMD technique and assessing their statistical significance vs. power-law distributed background noise.

Oscillatory processes in real datasets of various origins are often contaminated by a combination of white and coloured noise with a power-law spectral dependence, so that the EMD-revealed intrinsic mode functions need to be rigorously tested against the periodic components generated by noise.

To do so, we compute the EMD energy spectrum containing the total energy and dominant period of each EMD-revealed intrinsic mode and the noise confidence limits for modal energy. This allows us to identify the significant mode(s) with the energy beyond the confidence limits, which is expected to be of a non-noise origin and associated with a quasi-periodic oscillatory process of interest.

The developed package does not assume the physical origin of the input dataset, making it readily applicable for analysing oscillatory processes across various fields of science and industry.

Main Features

The project consists of the following main parts:

• Perform EMD analysis of the original time series and reveal the set of intrinsic modes using Empirical Mode Decomposition in Python (Quinn et al. 2021). See function scope.emd.emd_modes().

• Estimation of the power-law index and energy of superimposed background noise from the Fourier power spectrum as described in Vaughan (2005). See function scope.fourier.fit_fourier().

• Estimation of the dominant period of each EMD-revealed intrinsic mode from the global wavelet spectrum produced with Torrence & Compo Wavelet Analysis Software (Torrence & Compo 1998). See function scope.emd.emd_period_energy().

• Calculate EMD confidence limits using the method proposed by Kolotkov et al. (2016). See function emd_noise_conf().

• Visualise results with plot_signal(), plot_modes(), plot_fft_spectrum(), and plot_emd_spectrum().

Contents:

• API Reference
  • scope.fourier module
  • scope.emd module
  • scope.utils package
• Tutorial
  • Applying EMD to Extract Intrinsic Mode Functions (IMFs)
  • Estimating the Empirical Trend
  • Fourier Analysis of the Detrended Signal
  • Computing the EMD Energy Spectrum
  • Computing Confidence Limits for the EMD Energy Spectrum
  • Identifying Statistically Significant EMD Modes
• Theory
  • emd_period_energy
  • fit_fourier
  • emd_noise_conf