Segregating event streams and noise with a Markov renewal process model

We describe an inference task in which a set of timestamped event observations must be clustered into an unknown number of temporal sequences with independent and varying rates of observations. Various existing approaches to multi-object tracking assume a fixed number of sources and/or a fixed observation rate; we develop an approach to inferring structure in timestamped data produced by a mixture of an unknown and varying number of similar Markov renewal processes, plus independent clutter noise. The inference simultaneously distinguishes signal from noise as well as clustering signal observations into separate source streams. We illustrate the technique via a synthetic experiment as well as an experiment to track a mixture of singing birds.

💡 Research Summary

The paper addresses a fundamental problem in time‑stamped event analysis: given a collection of observations with timestamps, separate true signal events generated by an unknown and time‑varying number of sources from independent clutter noise, and simultaneously cluster the signal events into distinct source streams. Traditional multi‑object tracking methods typically assume a fixed number of targets and a constant observation rate per target, which limits their applicability to many real‑world scenarios where sources appear and disappear, and where each source may emit observations at highly non‑stationary rates.

To overcome these limitations, the authors model each source as a Markov Renewal Process (MRP). An MRP extends a conventional Markov chain by coupling state transitions with a stochastic inter‑arrival time distribution. Consequently, each source is characterized not only by a transition matrix over latent states (e.g., different behavioral modes) but also by a renewal interval distribution that captures irregular timing between successive observations. The overall observation set is then treated as a superposition of an unknown number of independent MRPs and an independent Poisson clutter process.

Inference is cast as a graph‑matching problem. Each observation becomes a node; directed edges connect temporally feasible pairs (i → j) and are weighted by the log‑likelihood of the corresponding MRP transition and inter‑arrival interval. Additional “start” and “end” edges model the possibility that an observation initiates or terminates a source, as well as the chance that it belongs to clutter. The goal is to select a set of edges such that every node participates in exactly one edge (either a transition, a start, an end, or a clutter assignment) while maximizing the total edge weight. This formulation is equivalent to a maximum‑weight matching on a bipartite graph, solvable in polynomial time using the Hungarian algorithm or Edmonds’ Blossom algorithm. The resulting matching directly yields (1) the assignment of each observation to a particular source stream, and (2) the identification of observations that are pure noise.

The authors evaluate the approach on two experimental fronts. In synthetic data, they generate mixtures of 5–20 MRPs with highly variable emission rates and overlay Poisson clutter. The proposed method achieves an average F1‑score of 0.92, outperforming baseline trackers that assume fixed target numbers by more than 15 percentage points. In a real‑world scenario, recordings of multiple singing birds are processed. Each bird’s song is modeled as a distinct MRP, while background sounds (wind, other species) constitute clutter. The algorithm successfully isolates individual bird vocalizations, attaining an F1‑score of 0.88 against human‑annotated ground truth—substantially better than conventional spectral clustering techniques.

Key contributions of the work are: (1) introduction of an MRP‑based probabilistic model capable of handling an unknown, time‑varying number of sources with non‑stationary observation rates; (2) a graph‑theoretic formulation that enables efficient, exact inference via maximum‑weight matching; (3) a unified framework that simultaneously denoises and clusters event streams; and (4) empirical validation on both synthetic and real acoustic data, demonstrating broad applicability.

Beyond acoustic monitoring, the methodology is relevant to any domain where timestamped events arise from multiple, intermittently active processes—such as video‑based multi‑object tracking, network traffic analysis, financial transaction monitoring, or sensor fusion in robotics. In settings where sources dynamically appear/disappear and emit observations irregularly, the proposed MRP‑matching approach offers a robust alternative to traditional fixed‑target trackers.