Recurrences reveal shared causal drivers of complex time series

TL;DR

SHREC reframes driver reconstruction as a dynamics-inspired problem: an unobserved driver shapes multiple responses whose recurrences encode the driver’s states. It builds a consensus recurrence graph from adaptive, fuzzy proximity embeddings and traverses it via diffusion or Leiden-based community detection to recover a time-resolved driver. The method achieves strong, data-efficient performance across diverse domains, reveals higher-order interactions, and links reconstruction accuracy to percolation on the recurrence graph and to the dominant unstable periodic orbits of the driver. These results establish a physics-based, unsupervised approach for extracting shared causal forces from noisy observational data with broad applicability in biology, ecology, and engineering.

Abstract

Unmeasured causal forces influence diverse experimental time series, such as the transcription factors that regulate genes, or the descending neurons that steer motor circuits. Combining the theory of skew-product dynamical systems with topological data analysis, we show that simultaneous recurrence events across multiple time series reveal the structure of their shared unobserved driving signal. We introduce a physics-based unsupervised learning algorithm that reconstructs causal drivers by iteratively building a recurrence graph with glass-like structure. As the amount of data increases, a percolation transition on this graph leads to weak ergodicity breaking for random walks -- revealing the shared driver's dynamics, even from strongly-corrupted measurements. We relate reconstruction accuracy to the rate of information transfer from a chaotic driver to the response systems, and we find that effective reconstruction proceeds through gradual approximation of the driver's dynamical attractor. Through extensive benchmarks against classical signal processing and machine learning techniques, we demonstrate our method's ability to extract causal drivers from diverse experimental datasets spanning ecology, genomics, fluid dynamics, and physiology.

Figures (12)

• Figure 1: The shared dynamics reconstruction method. An unobserved driver influences an ensemble of response systems, each containing unique internal dynamics and random measurement filters. Here, we use the Rössler dynamical system to drive an ensemble of Lorenz systems with random parameters that have been filtered with random Gaussian response functions. Timepoint-wise weighted recurrence networks are separately calculated for each response using using topological data analysis, and then aggregated to produce a consensus graph. This graph is traversed with either community detection (discrete time) or a diffusive flow (continuous time) in order to reconstruct the driver dynamics.
• Figure 2: Applying the shared recurrences (SHREC) algorithm to diverse datasets. (A) Example driver signals (blue) and their reconstructions (black) from response variables for four different experimental time series. (B) Runtimes versus Spearman correlation scores for different reconstruction methods. Error bars denote bootstrapped one-sided standard errors around random model initializations trained on random subsets of the original time series. The different methods sorted by overall score are annotated above each panel.
• Figure 3: Shared reconstruction identifies higher-order interactions. (A) Dynamics driven by pairwise interactions versus a single multiplicative driver. (B) Accuracy of network inference methods in identifying two multiplicative driving nodes among a network of $N$ interacting genes. Error ranges are standard errors across $100$ random networks with one multiplicative driver pair among $N=12$ genes.
• Figure 4: Driver reconstruction across a synchronization transition. The Kuramoto order parameter $\langle R(t) \rangle_t$ (black) and the accuracy of driver reconstruction (blue) as the driver forcing strength increases. In this figure, $N=10^3$, $K = \Delta = 1$ in Eq. \ref{['kuramoto']}, and error bars are standard deviations over $50$ randomly-initialized replicates.
• Figure 5: Correlation between chaoticity and accuracy across thousands of dynamical systems. (A) Accuracy of driver reconstruction across thousands of random skew-product dynamical systems. Each system is produced by sampling pairs from a set of $135$ named chaotic systems (e.g. Lorenz, Rössler, etc) and using one system as a driver for replicates of the other system. Linear fits with 95% bootstrapped confidence intervals highlight significant Spearman correlations between the reconstruction accuracy and the Lyapunov exponents of the driver (red, $\rho=0.16 \pm 0.03$) and response (blue, $\rho=-0.20 \pm 0.03$). (B) For each individual dynamical system, the average accuracy of a reconstruction in which it appears as a response variable, versus one in which it appears as a driver.