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.
Clustering multivariate time series data is a crucial task in many domains, as it enables the identification of meaningful patterns and groups in time-evolving data. Traditional approaches, such as crisp clustering, rely on the assumption that clusters are sufficiently separated with little overlap. However, real-world data often defy this assumption, exhibiting overlapping distributions or overlapping clouds of points and blurred boundaries between clusters. Fuzzy clustering offers a compelling alternative by allowing partial membership in multiple clusters, making it well-suited for these ambiguous scenarios. Despite its advantages, current fuzzy clustering methods primarily focus on univariate time series, and for multivariate cases, even datasets of moderate dimensionality become computationally prohibitive. This challenge is further exacerbated when dealing with time series of varying lengths, leaving a clear gap in addressing the complexities of modern datasets. This work introduces a novel fuzzy clustering approach based on common principal component analysis to address the aforementioned shortcomings. Our method has the advantage of efficiently handling high-dimensional multivariate time series by reducing dimensionality while preserving critical temporal features. Extensive numerical results show that our proposed clustering method outperforms several existing approaches in the literature. An interesting application involving brain signals from different drivers recorded from a simulated driving experiment illustrates the potential of the approach.
Traumatic Brain Injury (TBI) presents a broad spectrum of clinical presentations and outcomes due to its inherent heterogeneity, leading to diverse recovery trajectories and varied therapeutic responses. While many studies have delved into TBI phenotyping for distinct patient populations, identifying TBI phenotypes that consistently generalize across various settings and populations remains a critical research gap. Our research addresses this by employing multivariate time-series clustering to unveil TBI's dynamic intricates. Utilizing a self-supervised learning-based approach to clustering multivariate time-Series data with missing values (SLAC-Time), we analyzed both the research-centric TRACK-TBI and the real-world MIMIC-IV datasets. Remarkably, the optimal hyperparameters of SLAC-Time and the ideal number of clusters remained consistent across these datasets, underscoring SLAC-Time's stability across heterogeneous datasets. Our analysis revealed three generalizable TBI phenotypes ({\alpha}, \b{eta}, and {\gamma}), each exhibiting distinct non-temporal features during emergency department visits, and temporal feature profiles throughout ICU stays. Specifically, phenotype {\alpha} represents mild TBI with a remarkably consistent clinical presentation. In contrast, phenotype \b{eta} signifies severe TBI with diverse clinical manifestations, and phenotype {\gamma} represents a moderate TBI profile in terms of severity and clinical diversity. Age is a significant determinant of TBI outcomes, with older cohorts recording higher mortality rates. Importantly, while certain features varied by age, the core characteristics of TBI manifestations tied to each phenotype remain consistent across diverse populations.
Self-supervised learning approaches provide a promising direction for clustering multivariate time-series data. However, real-world time-series data often include missing values, and the existing approaches require imputing missing values before clustering, which may cause extensive computations and noise and result in invalid interpretations. To address these challenges, we present a Self-supervised Learning-based Approach to Clustering multivariate Time-series data with missing values (SLAC-Time). SLAC-Time is a Transformer-based clustering method that uses time-series forecasting as a proxy task for leveraging unlabeled data and learning more robust time-series representations. This method jointly learns the neural network parameters and the cluster assignments of the learned representations. It iteratively clusters the learned representations with the K-means method and then utilizes the subsequent cluster assignments as pseudo-labels to update the model parameters. To evaluate our proposed approach, we applied it to clustering and phenotyping Traumatic Brain Injury (TBI) patients in the Transforming Research and Clinical Knowledge in Traumatic Brain Injury (TRACK-TBI) study. Our experiments demonstrate that SLAC-Time outperforms the baseline K-means clustering algorithm in terms of silhouette coefficient, Calinski Harabasz index, Dunn index, and Davies Bouldin index. We identified three TBI phenotypes that are distinct from one another in terms of clinically significant variables as well as clinical outcomes, including the Extended Glasgow Outcome Scale (GOSE) score, Intensive Care Unit (ICU) length of stay, and mortality rate. The experiments show that the TBI phenotypes identified by SLAC-Time can be potentially used for developing targeted clinical trials and therapeutic strategies.
In the field of psychopathology, Ecological Momentary Assessment (EMA) methodological advancements have offered new opportunities to collect time-intensive, repeated and intra-individual measurements. This way, a large amount of data has become available, providing the means for further exploring mental disorders. Consequently, advanced machine learning (ML) methods are needed to understand data characteristics and uncover hidden and meaningful relationships regarding the underlying complex psychological processes. Among other uses, ML facilitates the identification of similar patterns in data of different individuals through clustering. This paper focuses on clustering multivariate time-series (MTS) data of individuals into several groups. Since clustering is an unsupervised problem, it is challenging to assess whether the resulting grouping is successful. Thus, we investigate different clustering methods based on different distance measures and assess them for the stability and quality of the derived clusters. These clustering steps are illustrated on a real-world EMA dataset, including 33 individuals and 15 variables. Through evaluation, the results of kernel-based clustering methods appear promising to identify meaningful groups in the data. So, efficient representations of EMA data play an important role in clustering.
This paper introduces an algorithm for the detection of change-points and the identification of the corresponding subsequences in transient multivariate time-series data (MTSD). The analysis of such data has become more and more important due to the increase of availability in many industrial fields. Labeling, sorting or filtering highly transient measurement data for training condition based maintenance (CbM) models is cumbersome and error-prone. For some applications it can be sufficient to filter measurements by simple thresholds or finding change-points based on changes in mean value and variation. But a robust diagnosis of a component within a component group for example, which has a complex non-linear correlation between multiple sensor values, a simple approach would not be feasible. No meaningful and coherent measurement data which could be used for training a CbM model would emerge. Therefore, we introduce an algorithm which uses a recurrent neural network (RNN) based Autoencoder (AE) which is iteratively trained on incoming data. The scoring function uses the reconstruction error and latent space information. A model of the identified subsequence is saved and used for recognition of repeating subsequences as well as fast offline clustering. For evaluation, we propose a new similarity measure based on the curvature for a more intuitive time-series subsequence clustering metric. A comparison with seven other state-of-the-art algorithms and eight datasets shows the capability and the increased performance of our algorithm to cluster MTSD online and offline in conjunction with mechatronic systems.
Three robust methods for clustering multivariate time series from the point of view of generating processes are proposed. The procedures are robust versions of a fuzzy C-means model based on: (i) estimates of the quantile cross-spectral density and (ii) the classical principal component analysis. Robustness to the presence of outliers is achieved by using the so-called metric, noise and trimmed approaches. The metric approach incorporates in the objective function a distance measure aimed at neutralizing the effect of the outliers, the noise approach builds an artificial cluster expected to contain the outlying series and the trimmed approach eliminates the most atypical series in the dataset. All the proposed techniques inherit the nice properties of the quantile cross-spectral density, as being able to uncover general types of dependence. Results from a broad simulation study including multivariate linear, nonlinear and GARCH processes indicate that the algorithms are substantially effective in coping with the presence of outlying series (i.e., series exhibiting a dependence structure different from that of the majority), clearly poutperforming alternative procedures. The usefulness of the suggested methods is highlighted by means of two specific applications regarding financial and environmental series.
We propose a new method for clustering multivariate time-series data based on Dynamic Linear Models. Whereas usual time-series clustering methods obtain static membership parameters, our proposal allows each time-series to dynamically change their cluster memberships over time. In this context, a mixture model is assumed for the time series and a flexible Dirichlet evolution for mixture weights allows for smooth membership changes over time. Posterior estimates and predictions can be obtained through Gibbs sampling, but a more efficient method for obtaining point estimates is presented, based on Stochastic Expectation-Maximization and Gradient Descent. Finally, two applications illustrate the usefulness of our proposed model to model both univariate and multivariate time-series: World Bank indicators for the renewable energy consumption of EU nations and the famous Gapminder dataset containing life-expectancy and GDP per capita for various countries.
Residual neural networks can be viewed as the forward Euler discretization of an Ordinary Differential Equation (ODE) with a unit time step. This has recently motivated researchers to explore other discretization approaches and train ODE based networks. However, an important challenge of neural ODEs is their prohibitive memory cost during gradient backpropogation. Recently a method proposed in [8], claimed that this memory overhead can be reduced from O(LN_t), where N_t is the number of time steps, down to O(L) by solving forward ODE backwards in time, where L is the depth of the network. However, we will show that this approach may lead to several problems: (i) it may be numerically unstable for ReLU/non-ReLU activations and general convolution operators, and (ii) the proposed optimize-then-discretize approach may lead to divergent training due to inconsistent gradients for small time step sizes. We discuss the underlying problems, and to address them we propose ANODE, an Adjoint based Neural ODE framework which avoids the numerical instability related problems noted above, and provides unconditionally accurate gradients. ANODE has a memory footprint of O(L) + O(N_t), with the same computational cost as reversing ODE solve. We furthermore, discuss a memory efficient algorithm which can further reduce this footprint with a trade-off of additional computational cost. We show results on Cifar-10/100 datasets using ResNet and SqueezeNext neural networks.
This paper presents a new methodology for clustering multivariate time series leveraging optimal transport between copulas. Copulas are used to encode both (i) intra-dependence of a multivariate time series, and (ii) inter-dependence between two time series. Then, optimal copula transport allows us to define two distances between multivariate time series: (i) one for measuring intra-dependence dissimilarity, (ii) another one for measuring inter-dependence dissimilarity based on a new multivariate dependence coefficient which is robust to noise, deterministic, and which can target specified dependencies.
We present a methodology for clustering N objects which are described by multivariate time series, i.e. several sequences of real-valued random variables. This clustering methodology leverages copulas which are distributions encoding the dependence structure between several random variables. To take fully into account the dependence information while clustering, we need a distance between copulas. In this work, we compare renowned distances between distributions: the Fisher-Rao geodesic distance, related divergences and optimal transport, and discuss their advantages and disadvantages. Applications of such methodology can be found in the clustering of financial assets. A tutorial, experiments and implementation for reproducible research can be found at www.datagrapple.com/Tech.