An information-based tool for inferring the nature of deterministic sources in real data

The scope of the paper is to find signatures of the forces controlling complex systems modeled by Langevin equations, by recourse to information-theory quantifiers. We evaluate in detail the permutation entropy (PE) and the permutation statistical complexity (PSC) measures for two similarity classes of stochastic models, characterized by either drifting or reversion properties, and employ them as a reference basis for the inspection of real series. New relevant model parameters arise as compared to standard entropy measures. We determine the normalized PE and PSC curves according to them over a range of permutation orders n and infer the limiting measures for arbitrary large order. We found that the PSC measure is strongly scale-dependent, with systems of the drifting class showing crossovers as n increases. This result gives warning signs about the proper interpretation of finite-scale analysis of complexity in general processes. Conversely, a key n-invariant outcome arises, that is, the normalized PE values for both classes of models keep complementary for any n. We argue that both PE and PSC measures enable one to unravel the nature (drifting or restoring) of the deterministic sources underlying complexity. We conclude by investigating the presence of local trends in stock price series.