Significance test for components of the multivariate empirical mode decomposition with application to solar activity indices

Although the significance of univariate nonstationary oscillation components obtained by empirical mode decomposition (EMD) can already be well tested, there are few studies paying attention to the multivariate case. More complicated than the univariate series, testing the significance of multivariate components has two key issues to be addressed: I) all variates have to be decomposed at the aligned scales, and II) the reference based on white noise has to be constructed for testing. We propose a method based on noise-assisted multivariate empirical mode decomposition (NA-MEMD) and with Bonferroni correction to handle the multiple testing issue. In methodology, NA-MEMD can simultaneously decompose all variates of multivariate series into several multivariate components with their members aligned at the same scales. By adding one white noise channel, the corresponding confidence intervals can be constructed from the perspective of energy as a reference for identifying significant information-bearing components. The effectiveness of our proposed method can be verified by wavelet power spectrum and two simulation experiments, in which it shows better performance than two benchmark methods. Our proposed method can well control the family-wise error rate and is relatively robust to key parameters, as justified by the simulation studies. The applicability of our method for general multivariate series is then demonstrated by a trivariate series comprising three solar activity indices of sunspot number, solar flux and cosmic rays. Our proposed method provides an effective tool to extract the components with information significantly more than white noise for general multivariate series.