Combinatorial time-loops: Probabilistic inference on time-series based on recurrence analysis with symbolic methods

Recurrence quantification analysis (RQA) is inspired by Poincaré’s early studies and describes non-linear characteristics of dynamical systems. It achieves this by identifying similarities between states, pairing each observation with every other. The identification of recurrences maps trajectories to the realm of binary states, which is a fertile ground for combinatorial methods. In this work, we utilize symbolic methods from analytic combinatorics to perform inference on the dynamics of systems represented as time-series data. Our case studies include simulated data from Gaussian noise, a periodic signal, and autoregressive processes. We demonstrate the detection of significant motifs: specific sequences of consecutive states that are repeated either within a series or between two series. The framework successfully identifies patterns in systems such as random walks and noisy periodic signals. With combinatorial constructions tailored for special cases, our method provides exact probabilities for inference within the recurrence analysis of dynamical systems. The methods have been implemented and are available in open-source software:AnalyticComb.jl and SymbolicInference.jl.