https://doi.org/10.1140/epjs/s11734-026-02334-7
Regular Article
Combinatorial time-loops: probabilistic inference on time-series based on recurrence analysis with symbolic methods
1
Instituto de Psiquiatria, Universidade de São Paulo, Dr. Ovídio Pires de Campos, 785-Cerqueira César, 05403-903, São Paulo, SP, Brazil
2
Instituto de Física, Universidade Federal da Bahia, Campus Universitário de Ondina-Ondina, 40210-340, Salvador, BA, Brazil
3
Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, P.O. Box 601203, 14412, Potsdam, Germany
4
Instituto de Matemática e Estatística, Universidade Federal da Bahia, Campus Universitário de Ondina-Ondina, 40170-110, Salvador, BA, Brazil
5
Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Straße 32, 14476, Potsdam, Germany
a
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Received:
11
November
2025
Accepted:
20
April
2026
Published online:
11
May
2026
Abstract
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. With combinatorial constructions tailored for special cases, our method provides exact probabilities for inference on small datasets. 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 noisy periodic signals, auto-regressive processes and non-linear systems with chaotic behavior. The methods were implemented and made available in open-source software: AnalyticComb.jl and SymbolicInference.jl.
Supplementary Information The online version contains supplementary material available at https://doi.org/10.1140/epjs/s11734-026-02334-7.
© The Author(s) 2026
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