  Inference of causality in time series has been principally based on the
prediction paradigm. Nonetheless, the predictive causality approach may
overlook the simultaneous and reciprocal nature of causal interactions observed
in real world phenomena. Here, we present a causal decomposition approach that
is not based on prediction, but based on the instantaneous phase dependency
between the intrinsic components of a decomposed time series. The method
involves two assumptions: (1) any cause effect relationship can be quantified
with instantaneous phase dependency between the source and target decomposed as
intrinsic components at specific time scale, and (2) the phase dynamics in the
target originating from the source are separable from the target itself. Using
empirical mode decomposition, we show that the causal interaction is encoded in
instantaneous phase dependency at a specific time scale, and this phase
dependency is diminished when the causal-related intrinsic component is removed
from the effect. Furthermore, we demonstrate the generic applicability of our
method to both stochastic and deterministic systems, and show the consistency
of the causal decomposition method compared to existing methods, and finally
uncover the key mode of causal interactions in both the modelled and actual
predator prey system. We anticipate that this novel approach will assist with
revealing causal interactions in complex networks not accounted for by current
methods.
