A statistical analysis of causal decomposition methods applied to Earth system time series

Causal Decomposition (CD) constitutes a novel and widely accepted method for discovering and quantifying the internal causal relationships inherent to complex systems. This new causality bases not on time nor state. Instead, it relies on instantaneous phase coherence between the corresponding Intrinsic Mode Functions (IMFs) of the signals, obtained via Empirical Mode Decomposition (EMD). In this paper we compare the results obtained with two noise-assisted methods: Ensemble EMD (EEMD) and Noise Assisted Multivariate EMD (NA-MEMD). Given the inherent stochastic nature of noise-assisted methods, CD was treated as a randomized experiment. Hence, we repeated procedures to perform a robust statistical analysis. Confidence intervals, adequate normality conditions and differential causality hypothesis testing were then evaluated. The methodology was adjusted through the paradigmatic case of a forced mechanical oscillator, since the causal implications are known in advance. CD was then applied to a couple of series from the Earth system: insolation and oxygen isotope rate. Differential causality was established in favor of insolation at the frequency corresponding to the obliquity cycle. The apparent causality detected with EEMD on other time scales was discarded and attributed to a mode mixing problem. Therefore, NA-MEMD outperformed EEMD with less mode mixing and sharper mode alignment.