Symbolic Structures of Differences (SSD): A Geometrical Approach to Quantifying Complexity in Time Series

We introduce Symbolic Structures of Differences (SSD), a method for quantifying the complexity of time series data based on the local geometry of second-order differences. Unlike global entropy measures, SSD captures the diversity of local sequential patterns by analyzing the signs of first and second-order differences within overlapping triplets, mapping them to a space of 27 unique symbols. We provide a theoretical analysis of SSD, proving its invariance under affine transformations and establishing its relationship to permutation entropy. The method's statistical properties, including robustness to noise and finite-size effects, are examined through Monte Carlo simulations. We validate SSD on a benchmark of synthetic and real-world physiological time series, comparing its performance against four established complexity measures (permutation entropy, sample entropy, Lempel-Ziv complexity, and spectral entropy) in the context of detecting epileptic seizures from EEG data. The results demonstrate that SSD offers a competitive and computationally efficient framework for characterizing dynamical regimes and identifying phase transitions, with unique sensitivity to local geometrical structures.