Bayesian Nonparametric Dynamical Clustering of Time Series

TL;DR

This work tackles clustering of time series with evolving dynamics by introducing a Bayesian nonparametric framework that couples an HDP prior over switching linear dynamical systems with Gaussian process priors for amplitude variation and a GP-based monotone warping model for temporal alignment. The model jointly discovers an unbounded number of morphologies and tracks their evolution, while aligning misaligned observations within a principled probabilistic setting. Efficient variational inference is developed in both off-line and on-line forms, enabling scalable learning and streaming applications demonstrated on ECG heartbeat clustering and breathing estimation. The approach provides interpretable hyperparameters for controlling cluster growth and plasticity, and outperforms fixed-cluster baselines by avoiding proliferation while capturing dynamic morphologies with principled uncertainty quantification.

Abstract

We present a method that models the evolution of an unbounded number of time series clusters by switching among an unknown number of regimes with linear dynamics. We develop a Bayesian non-parametric approach using a hierarchical Dirichlet process as a prior on the parameters of a Switching Linear Dynamical System and a Gaussian process prior to model the statistical variations in amplitude and temporal alignment within each cluster. By modeling the evolution of time series patterns, the method avoids unnecessary proliferation of clusters in a principled manner. We perform inference by formulating a variational lower bound for off-line and on-line scenarios, enabling efficient learning through optimization. We illustrate the versatility and effectiveness of the approach through several case studies of electrocardiogram analysis using publicly available databases.

Figures (9)

• Figure 1: Several heartbeat examples from record e1302 of European ST-T Database taddei_1992. All of them were manually annotated as Normal except the fifth one which was annotated as Ventricular goldberger_2000.
• Figure 2: Graphical model. A number of linear dynamics are generated to represent the different clusters that evolve across time. Conditioned on a specific setting of the switch variable $s_n=m$, an observation $\mathbf{y}_n$ is generated by LDS $m$ through a linear projection given by equations (\ref{['eq:latent_trans']}) and (\ref{['eq:emission_trans']}) from the last observation generated by this same LDS.
• Figure 3: Model enhancing plasticity. A total of 50 heartbeats are represented, manually annotated as Normal, from record 114 of the MIT-BIH Arrhythmia Database [11:30.000,12:20.000]. The resulting GP $f(t)$ (a) and $x(t)$ (b), after evolving along with these sequences, was sampled over a dense time axis. The mean of the posterior is shown as a black line. A 95% confidence region for the posterior is shown in cyan blue. Kernel hyperparameters for $k_{\theta}(t,t')$ are $\sigma^{2}_{f} = 16.0^{2}$, $l = 2.5$, $\sigma^{2}_{n} = 5.0^{2}$. LDS priors: $S_{\omega} = 10.0^{2}I, S_{\epsilon} = 5.0^{2}I$, where $I$ is the identity matrix. For clarity purposes, no warping function has been used. This model embraces the totality of the heartbeats in a single cluster.
• Figure 4: Model enhancing stability. The same 50 heartbeats as of Figure \ref{['fig:plasticity_model']} are represented, from record 114 of the MIT-BIH Arrhythmia Database. The resulting GP $f(t)$ (a) and $x(t)$ (b), after evolving along with these sequences, was sampled over a dense time axis. Kernel hyperparameters for $k_{\theta}(t,t')$ are $\sigma^{2}_{f} = 16.0^{2}$, $l = 2.7$, $\sigma^{2}_{n} = 5.0^{2}$. LDS priors: $S_{\omega} = 4.0^{2}I, S_{\epsilon} = 15.0^{2}I$. For clarity purposes no warping function has been used. Again, the model embraces the totality of the heartbeats in a single cluster.
• Figure 5: The GP model for each one of the beats depicted in Figure \ref{['fig:ecg_e1302']} is shown. Our method correctly includes all the beats in the same evolving cluster, except for the fifth one that gives rise to a new cluster.