A combinatorial multiverse of time-loops: Probabilistic inference on time-series based on recurrence analysis with symbolic methods

Recurrence quantification analysis (RQA) is inspired by Poincare ‘s early studies and describes non-linear character-istics in time-series by identifying similarity between states pairwisely among all observations. Since every pair iseither close or distant, this procedure maps trajectories to the realm of binary states, a fruitful field for the applicationof combinatorial tools. In this work, we leverage symbolic methods from analytic combinatorics to make inferenceon time-series data. Methods were implemented and made available in open-source software (AnalyticComb.jl ; Sym-bolicInference.jl). Study cases include simulated data, precipitation volumes and dengue cases. We demonstrate the detection of significant motifs: specific sequences of consecutive states that are repeated within a series or between two of them. The framework successfully identifies patterns in systems such as random walks and noisy periodic signals. When applied to empirical data, it also highlights the association of dengue peaks (e.g. infectious outbreaks) and rainfall seasonality (e.g. weather fluctuation). Using combinatorial constructions tailored for special cases, our method provides exact probabilities for inference in the analysis of time-series recurrences