Non-uniform state space reconstruction and coupling detection

We investigate the state space reconstruction from multiple time series derived from continuous and discrete systems and propose a method for building embedding vectors progressively using information measure criteria regarding past, current and future states. The embedding scheme can be adapted for different purposes, such as mixed modelling, cross-prediction and Granger causality. In particular we apply this method in order to detect and evaluate information transfer in coupled systems. As a practical application, we investigate in records of scalp epileptic EEG the information flow across brain areas.

💡 Research Summary

The paper addresses a fundamental limitation of conventional uniform‑delay state‑space reconstruction, namely that applying the same embedding delay and dimension to all variables often fails to capture the rich, nonlinear interdependencies present in multivariate continuous or discrete dynamical systems. To overcome this, the authors introduce a non‑uniform embedding scheme that builds the reconstruction vector progressively, guided by information‑theoretic criteria that evaluate the contribution of each candidate past, present, or future observation to the prediction of the target series.

The algorithm works as follows. For each time series (x_i(t)) a pool of candidate components is formed, consisting of past values (x_i(t-\tau)), the current value (x_i(t)), and future values (x_i(t+\tau_f)) over a range of lags. For each candidate the mutual information with the target and the conditional entropy given the already selected components are computed. The candidate that yields the largest reduction in the conditional entropy (i.e., the greatest increase in predictive information) is added to the embedding vector. Redundancy is controlled by discarding candidates whose conditional mutual information with the existing vector exceeds a preset threshold. This greedy, information‑maximising procedure continues until no further significant gain is observed, resulting in an embedding where each variable may have a distinct delay and a distinct number of copies.

Having obtained non‑uniform embeddings for two subsystems A and B, the authors define an information‑transfer measure that quantifies how much knowledge of A’s embedding reduces the uncertainty of B’s present state beyond what B’s own embedding already provides. Formally, the transfer from A to B is
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