Time series clustering promises to uncover hidden structural patterns in data with applications across healthcare, finance, industrial systems, and other critical domains. However, without validated ground truth information, researchers cannot objectively assess clustering quality or determine whether poor results stem from absent structures in the data, algorithmic limitations, or inappropriate validation methods, raising the question whether clustering is "more art than science" (Guyon et al., 2009). To address these challenges, we introduce CSTS (Correlation Structures in Time Series), a synthetic benchmark for evaluating the discovery of correlation structures in multivariate time series data. CSTS provides a clean benchmark that enables researchers to isolate and identify specific causes of clustering failures by differentiating between correlation structure deterioration and limitations of clustering algorithms and validation methods. Our contributions are: (1) a comprehensive benchmark for correlation structure discovery with distinct correlation structures, systematically varied data conditions, established performance thresholds, and recommended evaluation protocols; (2) empirical validation of correlation structure preservation showing moderate distortion from downsampling and minimal effects from distribution shifts and sparsification; and (3) an extensible data generation framework enabling structure-first clustering evaluation. A case study demonstrates CSTS's practical utility by identifying an algorithm's previously undocumented sensitivity to non-normal distributions, illustrating how the benchmark enables precise diagnosis of methodological limitations. CSTS advances rigorous evaluation standards for correlation-based time series clustering.
Time series clustering poses a significant challenge with diverse applications across domains. A prominent drawback of existing solutions lies in their limited interpretability, often confined to presenting users with centroids. In addressing this gap, our work presents $k$-Graph, an unsupervised method explicitly crafted to augment interpretability in time series clustering. Leveraging a graph representation of time series subsequences, $k$-Graph constructs multiple graph representations based on different subsequence lengths. This feature accommodates variable-length time series without requiring users to predetermine subsequence lengths. Our experimental results reveal that $k$-Graph outperforms current state-of-the-art time series clustering algorithms in accuracy, while providing users with meaningful explanations and interpretations of the clustering outcomes.
In this paper, we propose a novel method of model-based time series clustering with mixtures of general state space models (MSSMs). Each component of MSSMs is associated with each cluster. An advantage of the proposed method is that it enables the use of time series models appropriate to the specific time series. This not only improves clustering and prediction accuracy but also enhances the interpretability of the estimated parameters. The parameters of the MSSMs are estimated using stochastic variational inference, a subtype of variational inference. The proposed method estimates the latent variables of an arbitrary state space model by using neural networks with a normalizing flow as a variational estimator. The number of clusters can be estimated using the Bayesian information criterion. In addition, to prevent MSSMs from converging to the local optimum, we propose several optimization tricks, including an additional penalty term called entropy annealing. To our best knowledge, the proposed method is the first computationally feasible one for time series clustering based on general (possibly nonlinear, non-Gaussian) state space models. Experiments on simulated datasets show that the proposed method is effective for clustering, parameter estimation, and estimating the number of clusters.
The detection of abnormal or critical system states is essential in condition monitoring. While much attention is given to promptly identifying anomalies, a retrospective analysis of these anomalies can significantly enhance our comprehension of the underlying causes of observed undesired behavior. This aspect becomes particularly critical when the monitored system is deployed in a vital environment. In this study, we delve into anomalies within the domain of Bio-Regenerative Life Support Systems (BLSS) for space exploration and analyze anomalies found in telemetry data stemming from the EDEN ISS space greenhouse in Antarctica. We employ time series clustering on anomaly detection results to categorize various types of anomalies in both uni- and multivariate settings. We then assess the effectiveness of these methods in identifying systematic anomalous behavior. Additionally, we illustrate that the anomaly detection methods MDI and DAMP produce complementary results, as previously indicated by research.
Time series data can be found in almost every domain, ranging from the medical field to manufacturing and wireless communication. Generating realistic and useful exemplars and prototypes is a fundamental data analysis task. In this paper, we investigate a novel approach to generating realistic and useful exemplars and prototypes for time series data. Our approach uses a new form of time series average, the ShapeDTW Barycentric Average. We therefore turn our attention to accurately generating time series prototypes with a novel approach. The existing time series prototyping approaches rely on the Dynamic Time Warping (DTW) similarity measure such as DTW Barycentering Average (DBA) and SoftDBA. These last approaches suffer from a common problem of generating out-of-distribution artifacts in their prototypes. This is mostly caused by the DTW variant used and its incapability of detecting neighborhood similarities, instead it detects absolute similarities. Our proposed method, ShapeDBA, uses the ShapeDTW variant of DTW, that overcomes this issue. We chose time series clustering, a popular form of time series analysis to evaluate the outcome of ShapeDBA compared to the other prototyping approaches. Coupled with the k-means clustering algorithm, and evaluated on a total of 123 datasets from the UCR archive, our proposed averaging approach is able to achieve new state-of-the-art results in terms of Adjusted Rand Index.
Relationships among time series can be exploited as inductive biases in learning effective forecasting models. In hierarchical time series, relationships among subsets of sequences induce hard constraints (hierarchical inductive biases) on the predicted values. In this paper, we propose a graph-based methodology to unify relational and hierarchical inductive biases in the context of deep learning for time series forecasting. In particular, we model both types of relationships as dependencies in a pyramidal graph structure, with each pyramidal layer corresponding to a level of the hierarchy. By exploiting modern - trainable - graph pooling operators we show that the hierarchical structure, if not available as a prior, can be learned directly from data, thus obtaining cluster assignments aligned with the forecasting objective. A differentiable reconciliation stage is incorporated into the processing architecture, allowing hierarchical constraints to act both as an architectural bias as well as a regularization element for predictions. Simulation results on representative datasets show that the proposed method compares favorably against the state of the art.
PyPOTS is an open-source Python library dedicated to data mining and analysis on multivariate partially-observed time series, i.e. incomplete time series with missing values, A.K.A. irregularlysampled time series. Particularly, it provides easy access to diverse algorithms categorized into four tasks: imputation, classification, clustering, and forecasting. The included models contain probabilistic approaches as well as neural-network methods, with a well-designed and fully-documented programming interface for both academic researchers and industrial professionals to use. With robustness and scalability in its design philosophy, best practices of software construction, for example, unit testing, continuous integration (CI) and continuous delivery (CD), code coverage, maintainability evaluation, interactive tutorials, and parallelization, are carried out as principles during the development of PyPOTS. The toolkit is available on both Python Package Index (PyPI) and Anaconda. PyPOTS is open-source and publicly available on GitHub https://github.com/WenjieDu/PyPOTS.
Time series data, spanning applications ranging from climatology to finance to healthcare, presents significant challenges in data mining due to its size and complexity. One open issue lies in time series clustering, which is crucial for processing large volumes of unlabeled time series data and unlocking valuable insights. Traditional and modern analysis methods, however, often struggle with these complexities. To address these limitations, we introduce R-Clustering, a novel method that utilizes convolutional architectures with randomly selected parameters. Through extensive evaluations, R-Clustering demonstrates superior performance over existing methods in terms of clustering accuracy, computational efficiency and scalability. Empirical results obtained using the UCR archive demonstrate the effectiveness of our approach across diverse time series datasets. The findings highlight the significance of R-Clustering in various domains and applications, contributing to the advancement of time series data mining.
The Cauchy-Schwarz (CS) divergence was developed by Pr\'{i}ncipe et al. in 2000. In this paper, we extend the classic CS divergence to quantify the closeness between two conditional distributions and show that the developed conditional CS divergence can be simply estimated by a kernel density estimator from given samples. We illustrate the advantages (e.g., rigorous faithfulness guarantee, lower computational complexity, higher statistical power, and much more flexibility in a wide range of applications) of our conditional CS divergence over previous proposals, such as the conditional KL divergence and the conditional maximum mean discrepancy. We also demonstrate the compelling performance of conditional CS divergence in two machine learning tasks related to time series data and sequential inference, namely time series clustering and uncertainty-guided exploration for sequential decision making. The code of conditional CS divergence is available at https://github.com/SJYuCNEL/conditional_CS_divergence.
In this work, we address the problem of unsupervised moving object segmentation (MOS) in 4D LiDAR data recorded from a stationary sensor, where no ground truth annotations are involved. Deep learning-based state-of-the-art methods for LiDAR MOS strongly depend on annotated ground truth data, which is expensive to obtain and scarce in existence. To close this gap in the stationary setting, we propose a novel 4D LiDAR representation based on multivariate time series that relaxes the problem of unsupervised MOS to a time series clustering problem. More specifically, we propose modeling the change in occupancy of a voxel by a multivariate occupancy time series (MOTS), which captures spatio-temporal occupancy changes on the voxel level and its surrounding neighborhood. To perform unsupervised MOS, we train a neural network in a self-supervised manner to encode MOTS into voxel-level feature representations, which can be partitioned by a clustering algorithm into moving or stationary. Experiments on stationary scenes from the Raw KITTI dataset show that our fully unsupervised approach achieves performance that is comparable to that of supervised state-of-the-art approaches.
Dynamic Time Warping (DTW) is used for matching pairs of sequences and celebrated in applications such as forecasting the evolution of time series, clustering time series or even matching sequence pairs in few-shot action recognition. The transportation plan of DTW contains a set of paths; each path matches frames between two sequences under a varying degree of time warping, to account for varying temporal intra-class dynamics of actions. However, as DTW is the smallest distance among all paths, it may be affected by the feature uncertainty which varies across time steps/frames. Thus, in this paper, we propose to model the so-called aleatoric uncertainty of a differentiable (soft) version of DTW. To this end, we model the heteroscedastic aleatoric uncertainty of each path by the product of likelihoods from Normal distributions, each capturing variance of pair of frames. (The path distance is the sum of base distances between features of pairs of frames of the path.) The Maximum Likelihood Estimation (MLE) applied to a path yields two terms: (i) a sum of Euclidean distances weighted by the variance inverse, and (ii) a sum of log-variance regularization terms. Thus, our uncertainty-DTW is the smallest weighted path distance among all paths, and the regularization term (penalty for the high uncertainty) is the aggregate of log-variances along the path. The distance and the regularization term can be used in various objectives. We showcase forecasting the evolution of time series, estimating the Fr\'echet mean of time series, and supervised/unsupervised few-shot action recognition of the articulated human 3D body joints.
In this paper, we consider the task of clustering a set of individual time series while modeling each cluster, that is, model-based time series clustering. The task requires a parametric model with sufficient flexibility to describe the dynamics in various time series. To address this problem, we propose a novel model-based time series clustering method with mixtures of linear Gaussian state space models, which have high flexibility. The proposed method uses a new expectation-maximization algorithm for the mixture model to estimate the model parameters, and determines the number of clusters using the Bayesian information criterion. Experiments on a simulated dataset demonstrate the effectiveness of the method in clustering, parameter estimation, and model selection. The method is applied to real datasets commonly used to evaluate time series clustering methods. Results showed that the proposed method produces clustering results that are as accurate or more accurate than those obtained using previous methods.
Temporal Neural Networks (TNNs), inspired from the mammalian neocortex, exhibit energy-efficient online sensory processing capabilities. Recent works have proposed a microarchitecture framework for implementing TNNs and demonstrated competitive performance on vision and time-series applications. Building on these previous works, this work proposes TNN7, a suite of nine highly optimized custom macros developed using a predictive 7nm Process Design Kit (PDK), to enhance the efficiency, modularity and flexibility of the TNN design framework. TNN prototypes for two applications are used for evaluation of TNN7. An unsupervised time-series clustering TNN delivering competitive performance can be implemented within 40 uW power and 0.05 mm^2 area, while a 4-layer TNN that achieves an MNIST error rate of 1% consumes only 18 mW and 24.63 mm^2. On average, the proposed macros reduce power, delay, area, and energy-delay product by 14%, 16%, 28%, and 45%, respectively. Furthermore, employing TNN7 significantly reduces the synthesis runtime of TNN designs (by more than 3x), allowing for highly-scaled TNN implementations to be realized.
Being able to capture the characteristics of a time series with a feature vector is a very important task with a multitude of applications, such as classification, clustering or forecasting. Usually, the features are obtained from linear and nonlinear time series measures, that may present several data related drawbacks. In this work we introduce NetF as an alternative set of features, incorporating several representative topological measures of different complex networks mappings of the time series. Our approach does not require data preprocessing and is applicable regardless of any data characteristics. Exploring our novel feature vector, we are able to connect mapped network features to properties inherent in diversified time series models, showing that NetF can be useful to characterize time data. Furthermore, we also demonstrate the applicability of our methodology in clustering synthetic and benchmark time series sets, comparing its performance with more conventional features, showcasing how NetF can achieve high-accuracy clusters. Our results are very promising, with network features from different mapping methods capturing different properties of the time series, adding a different and rich feature set to the literature.
In the past two decades, the field of applied finance has tremendously benefited from graph theory. As a result, novel methods ranging from asset network estimation to hierarchical asset selection and portfolio allocation are now part of practitioners' toolboxes. In this paper, we investigate the fundamental problem of learning undirected graphical models under Laplacian structural constraints from the point of view of financial market times series data. In particular, we present natural justifications, supported by empirical evidence, for the usage of the Laplacian matrix as a model for the precision matrix of financial assets, while also establishing a direct link that reveals how Laplacian constraints are coupled to meaningful physical interpretations related to the market index factor and to conditional correlations between stocks. Those interpretations lead to a set of guidelines that practitioners should be aware of when estimating graphs in financial markets. In addition, we design numerical algorithms based on the alternating direction method of multipliers to learn undirected, weighted graphs that take into account stylized facts that are intrinsic to financial data such as heavy tails and modularity. We illustrate how to leverage the learned graphs into practical scenarios such as stock time series clustering and foreign exchange network estimation. The proposed graph learning algorithms outperform the state-of-the-art methods in an extensive set of practical experiments. Furthermore, we obtain theoretical and empirical convergence results for the proposed algorithms. Along with the developed methodologies for graph learning in financial markets, we release an R package, called fingraph, accommodating the code and data to obtain all the experimental results.
We state theoretical properties for $k$-means clustering of Symmetric Positive Definite (SPD) matrices, in a non-Euclidean space, that provides a natural and favourable representation of these data. We then provide a novel application for this method, to time-series clustering of pixels in a sequence of Synthetic Aperture Radar images, via their finite-lag autocovariance matrices.
Autonomous driving systems validation remains one of the biggest challenges car manufacturers must tackle in order to provide safe driverless cars. The high complexity stems from several factors: the multiplicity of vehicles, embedded systems, use cases, and the very high required level of reliability for the driving system to be at least as safe as a human driver. In order to circumvent these issues, large scale simulations reproducing this huge variety of physical conditions are intensively used to test driverless cars. Therefore, the validation step produces a massive amount of data, including many time-indexed ones, to be processed. In this context, building a structure in the feature space is mandatory to interpret the various scenarios. In this work, we propose a new co-clustering approach adapted to high-dimensional time series analysis, that extends the standard model-based co-clustering. The FunCLBM model extends the recently proposed Functional Latent Block Model and allows to create a dependency structure between row and column clusters. This structured partition acts as a feature selection method, that provides several clustering views of a dataset, while discriminating irrelevant features. In this workflow, times series are projected onto a common interpolated low-dimensional frequency space, which allows to optimize the projection basis. In addition, FunCLBM refines the definition of each latent block by performing block-wise dimension reduction and feature selection. We propose a SEM-Gibbs algorithm to infer this model, as well as a dedicated criterion to select the optimal nested partition. Experiments on both simulated and real-case Renault datasets shows the effectiveness of the proposed tools and the adequacy to our use case.
Due to the wider availability of modern electronic health records, patient care data is often being stored in the form of time-series. Clustering such time-series data is crucial for patient phenotyping, anticipating patients' prognoses by identifying "similar" patients, and designing treatment guidelines that are tailored to homogeneous patient subgroups. In this paper, we develop a deep learning approach for clustering time-series data, where each cluster comprises patients who share similar future outcomes of interest (e.g., adverse events, the onset of comorbidities). To encourage each cluster to have homogeneous future outcomes, the clustering is carried out by learning discrete representations that best describe the future outcome distribution based on novel loss functions. Experiments on two real-world datasets show that our model achieves superior clustering performance over state-of-the-art benchmarks and identifies meaningful clusters that can be translated into actionable information for clinical decision-making.
Clustering is frequently used in the energy domain to identify dominant electricity consumption patterns of households, which can be used to construct customer archetypes for long term energy planning. Selecting a useful set of clusters however requires extensive experimentation and domain knowledge. While internal clustering validation measures are well established in the electricity domain, they are limited for selecting useful clusters. Based on an application case study in South Africa, we present an approach for formalising implicit expert knowledge as external evaluation measures to create customer archetypes that capture variability in residential electricity consumption behaviour. By combining internal and external validation measures in a structured manner, we were able to evaluate clustering structures based on the utility they present for our application. We validate the selected clusters in a use case where we successfully reconstruct customer archetypes previously developed by experts. Our approach shows promise for transparent and repeatable cluster ranking and selection by data scientists, even if they have limited domain knowledge.
We introduce deep Markov spatio-temporal factorization (DMSTF), a generative model for dynamical analysis of spatio-temporal data. Like other factor analysis methods, DMSTF approximates high dimensional data by a product between time dependent weights and spatially dependent factors. These weights and factors are in turn represented in terms of lower dimensional latents inferred using stochastic variational inference. The innovation in DMSTF is that we parameterize weights in terms of a deep Markovian prior extendable with a discrete latent, which is able to characterize nonlinear multimodal temporal dynamics, and perform multidimensional time series forecasting. DMSTF learns a low dimensional spatial latent to generatively parameterize spatial factors or their functional forms in order to accommodate high spatial dimensionality. We parameterize the corresponding variational distribution using a bidirectional recurrent network in the low-level latent representations. This results in a flexible family of hierarchical deep generative factor analysis models that can be extended to perform time series clustering or perform factor analysis in the presence of a control signal. Our experiments, which include simulated and real-world data, demonstrate that DMSTF outperforms related methodologies in terms of predictive performance for unseen data, reveals meaningful clusters in the data, and performs forecasting in a variety of domains with potentially nonlinear temporal transitions.