An Improved Causal Decomposition Python Algorithm with Statistical Corroboration

Abstract

Causal Decomposition based on Empirical Mode Decomposition (EMD) has proved to be a powerful tool for identifying causal relationships between time series. This method is based on the phase coherence of the respective oscillatory modes of the signals, known as Intrinsic Mode Functions (IMFs). Hence, a correct alignment of the respective modes of the signals is crucial. Unlike other methods, Causal Decomposition makes no assumption of linearity in the studied signals. Therefore, it is widely applicable to time series emerging from complex systems for which linearity hypothesis generally fail to hold. The decomposition in oscillatory modes is achieved with noise-assisted versions of EMD, which are known to improve the performance of the decomposition, reducing the mode mixing. However, adding noise introduces a stochastic element in the result, that is henceforth treated as a random variable. In the present work we introduce our Python version of the Causal Decomposition algorithm, which incorporates refinements for the selection of the decomposition based on energy considerations. These improvements aim to reduce the outlier results attributable to an incorrect mode alignment. The algorithm was tested on synthetic time series generated using a model of a mechanical oscillator with two masses and two modulated nonlinear forcing terms. A subsequent statistical analysis over multiple realizations showed less dispersion and fewer outliers compared to the previous version of the algorithm.