The Clustering Occupation Kernel (CLOCK) Method for Time Series Arising from Dynamical Systems
Authors: Michael L. Wells, Kamel Lahouel, Victor Rielly, Bruno Jedynak
Organizations: Fariborz Maseeh Department of Mathematics and Statistics, Portland State University, Portland, OR 97201 · Translational Genomics Research Institute, Phoenix, AZ 85004
We introduce the Clustering Occupation Kernel (CLOCK) method, a novel representation-based framework for clustering time series arising from dynamical systems. CLOCK embeds observed sequences into a reproducing kernel Hilbert space (RKHS) via a time-averaged occupation kernel, enabling trajectory comparison through their RKHS norm. For systems governed by differential equations, this formulation rigorously approximates the kernel mean embedding of the invariant Sinai-Ruelle-Bowen (SRB) measure, meaning our dissimilarity metric approximates the maximum mean discrepancy (MMD) between attractors. To accommodate general time series, we introduce time lags via Takens' Embedding Theorem to reconstruct the phase space. Crucially, CLOCK natively handles trajectories of unequal lengths and irregularly-spaced observations. We validate the proposed metric using spectral clustering, demonstrating superior performance against benchmark methods on both synthetic datasets and a real-world intracranial large-scale electroencephalogram (EEG) dataset. Finally, ablation studies on the Lorenz-96 system are consistent with the theoretical convergence rates of the occupation kernels and highlight the method's strong robustness to zero-mean Gaussian noise.
Figures & tables
Figure 1: Shown are the synthetic datasets in the (x1,x2) -plane with coloring given by the true cluster labels. We present the true trajectories along with the noisy samples. For higher-dimensional datapoints, we plot only the first two coordinates.
Figure 2: In (a), the log-mean-squared error between the time-averaged occupation kernels and the empirical KME of the SRB measure of the Lorenz-96 attractor is plotted as a function of the log of the trajectory length T , along with a 95% confidence interval (CI). The line with slope −0.5 and intercept −0.96 is plotted for reference. In (b), the ARI of clusterings with the CLOCK method on the Lorenz-96 system is plotted with a 95% CI as a function of T .
Table 1: Results of Numerical Experiments. The CLOCK method is listed along with its accompanying kernel. Information on the kernels used can be found in Appendix C . The Gaussian Multiscale kernel (GM) with one bandwidth per dimension was the default choice. The Gaussian Uniscale kernel (GU) with a single bandwidth for all dimensions was chosen for high-dimensional data. The quadratic kernel (quad) and Laplace kernel were also used. Best results for each metric are in bold.
Appendix figures & tables1 asset
Supplementary material from the paper’s appendix.
Appendix
Figure 3: Plots of evaluation metrics in the noise robustness test of Appendix F . The horizontal axis represents the standard deviation of the Gaussian noise added to each observation. The vertical axis represents one of the evaluation metrics: ARI, NMI or purity. In Figure 3(a) , we have ARI results, in Figure 3(b) , we have results for NMI, and in Figure 3(c) , we have purity results.
Multivariate Time Series (MTS) clustering is an important tool in temporal data mining, aiming to discover latent group structures from complex observations without supervision. Although existing deep clustering methods can learn discriminative temporal representations, the resulting latent clusters are often difficult to relate back to waveform characteristics that practitioners can directly inspect and compare, limiting their ability to assess whether the discovered patterns reflect meaningful temporal behaviors. This paper, therefore, proposes WAVE (Waveform Aligned Visual-temporal Embedding), which treats time series and their deterministically rendered waveform plots as complementary views of the same observations. To produce discriminative representations whose cluster structures can be traced to observable waveform characteristics, WAVE aligns and integrates fine-grained temporal variations with holistic visual patterns, while associating each discovered cluster with its centroid-nearest authentic sample. Accordingly, interpretability in this work specifically refers to waveform-level traceability rather than a general explanation of model decisions. Extensive evaluations across 10 real-world public datasets show that WAVE achieves the highest macro-averaged clustering performance and the best average rank among the compared methods, while qualitative case studies illustrate how the discovered clusters can be inspected through authentic waveform records. The source code is available at https://github.com/Zheng-Zhu1/WAVE.
Zheng Zhu, Zexi Tan, Yuming Deng +1
Huizhou No. 1 High School (High School Student) · Guangdong University of Technology · South China University of Technology
Time-series clustering remains challenging due to the inherent trade-off between clustering effectiveness and computational efficiency. Similarity-based methods often suffer from quadratic complexity caused by pairwise distance computations, while deep learning-based approaches typically rely on costly iterative training and a large number of trainable parameters. In this paper, we propose MSRGC-Net, an efficient time-series clustering framework that integrates multiscale reservoir computing, granular-ball-based anchoring graph construction, and consensus learning. MSRGC-Net adopts a training-free reservoir computing paradigm to extract multiscale temporal representations from raw time series without backpropagation, significantly reducing computational overhead. To capture the intrinsic structure of the resulting representations, granular-ball computing is employed to adaptively model data distributions via density-consistent regions, yielding compact and robust anchor graph representations. Furthermore, a consensus-based anchoring graph optimization strategy is introduced to effectively align multiscale reservoir representations and integrate complementary information across temporal scales. Extensive experiments on widely used univariate and multivariate benchmark datasets demonstrate that MSRGC-Net consistently outperforms state-of-the-art methods in clustering performance while maintaining superior computational efficiency.
Yifan Wang, Lifeng Shen, Shuyin Xia +1
Chongqing Key Laboratory of Computational Intelligence, Key Laboratory of Cyberspace Big Data Intelligent Security, Ministry of Education, Sichuan-Chongqing Co-construction Key Laboratory of Digital Economy Intelligence and Key Laboratory of Big Data Intelligent Computing, College of Computer Science and Technology, Chongqing University of Posts and Telecommunications · Chongqing Ant Consumer Finance Co,. Ltd, Ant Group.
We develop a closure modeling framework for identifying missing components of dynamical systems using Kernel Ridge Regression (KRR). The framework addresses two classes of closure problems: difference-equation closures arising in ODE and PDE settings, and algebraic closures arising from moment closure in kinetic equations. For the first class, we derive an error bound in an ODE setting that quantifies contributions from time integration, approximation of unresolved scales, and interpolation required to couple unresolved-scale effects to the resolved solver. Numerical experiments on the Lorenz-63 system and the Kuramoto-Sivashinsky equation demonstrate accurate long-horizon predictions and substantial improvements over an LSTM-based closure model. For the second class, we consider moment closure for a one-dimensional kinetic equation by modeling discrepancies between kinetic and macroscopic fluxes as a function of the resolved macroscopic variables. We compare global KRR models based on PCA coordinates with spatially local models. While the global model performs well for unimodal initial conditions, its accuracy deteriorates for bimodal initial conditions. Spatially local models with appropriate modeling inputs improve robustness and achieve higher predictive accuracy.
Evan Habbershaw, John Harlim, Senwei Liang
Department of Mathematics, The Pennsylvania State University · Department of Mathematics and Institute for Computational and Data Sciences, The Pennsylvania State University · Department of Mathematics and Statistics, Texas Tech University