FCPCA: Fuzzy clustering of high-dimensional time series based on common principal component analysis

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.

CSTS: A Benchmark for the Discovery of Correlation Structures in Time Series Clustering

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.

Accurate and Efficient Multivariate Time Series Forecasting via Offline Clustering

Accurate and efficient multivariate time series (MTS) forecasting is essential for applications such as traffic management and weather prediction, which depend on capturing long-range temporal dependencies and interactions between entities. Existing methods, particularly those based on Transformer architectures, compute pairwise dependencies across all time steps, leading to a computational complexity that scales quadratically with the length of the input. To overcome these challenges, we introduce the Forecaster with Offline Clustering Using Segments (FOCUS), a novel approach to MTS forecasting that simplifies long-range dependency modeling through the use of prototypes extracted via offline clustering. These prototypes encapsulate high-level events in the real-world system underlying the data, summarizing the key characteristics of similar time segments. In the online phase, FOCUS dynamically adapts these patterns to the current input and captures dependencies between the input segment and high-level events, enabling both accurate and efficient forecasting. By identifying prototypes during the offline clustering phase, FOCUS reduces the computational complexity of modeling long-range dependencies in the online phase to linear scaling. Extensive experiments across diverse benchmarks demonstrate that FOCUS achieves state-of-the-art accuracy while significantly reducing computational costs.

Early Detection of Multidrug Resistance Using Multivariate Time Series Analysis and Interpretable Patient-Similarity Representations

Background and Objectives: Multidrug Resistance (MDR) is a critical global health issue, causing increased hospital stays, healthcare costs, and mortality. This study proposes an interpretable Machine Learning (ML) framework for MDR prediction, aiming for both accurate inference and enhanced explainability. Methods: Patients are modeled as Multivariate Time Series (MTS), capturing clinical progression and patient-to-patient interactions. Similarity among patients is quantified using MTS-based methods: descriptive statistics, Dynamic Time Warping, and Time Cluster Kernel. These similarity measures serve as inputs for MDR classification via Logistic Regression, Random Forest, and Support Vector Machines, with dimensionality reduction and kernel transformations improving model performance. For explainability, patient similarity networks are constructed from these metrics. Spectral clustering and t-SNE are applied to identify MDR-related subgroups and visualize high-risk clusters, enabling insight into clinically relevant patterns. Results: The framework was validated on ICU Electronic Health Records from the University Hospital of Fuenlabrada, achieving an AUC of 81%. It outperforms baseline ML and deep learning models by leveraging graph-based patient similarity. The approach identifies key risk factors -- prolonged antibiotic use, invasive procedures, co-infections, and extended ICU stays -- and reveals clinically meaningful clusters. Code and results are available at \https://github.com/oscarescuderoarnanz/DM4MTS. Conclusions: Patient similarity representations combined with graph-based analysis provide accurate MDR prediction and interpretable insights. This method supports early detection, risk factor identification, and patient stratification, highlighting the potential of explainable ML in critical care.

Spatiotemporal Forecasting in Climate Data Using EOFs and Machine Learning Models: A Case Study in Chile

Effective resource management and environmental planning in regions with high climatic variability, such as Chile, demand advanced predictive tools. This study addresses this challenge by employing an innovative and computationally efficient hybrid methodology that integrates machine learning (ML) methods for time series forecasting with established statistical techniques. The spatiotemporal data undergo decomposition using time-dependent Empirical Orthogonal Functions (EOFs), denoted as \(\phi_{k}(t)\), and their corresponding spatial coefficients, \(\alpha_{k}(s)\), to reduce dimensionality. Wavelet analysis provides high-resolution time and frequency information from the \(\phi_{k}(t)\) functions, while neural networks forecast these functions within a medium-range horizon \(h\). By utilizing various ML models, particularly a Wavelet - ANN hybrid model, we forecast \(\phi_{k}(t+h)\) up to a time horizon \(h\), and subsequently reconstruct the spatiotemporal data using these extended EOFs. This methodology is applied to a grid of climate data covering the territory of Chile. It transitions from a high-dimensional multivariate spatiotemporal data forecasting problem to a low-dimensional univariate forecasting problem. Additionally, cluster analysis with Dynamic Time Warping for defining similarities between rainfall time series, along with spatial coherence and predictability assessments, has been instrumental in identifying geographic areas where model performance is enhanced. This approach also elucidates the reasons behind poor forecast performance in regions or clusters with low spatial coherence and predictability. By utilizing cluster medoids, the forecasting process becomes more practical and efficient. This compound approach significantly reduces computational complexity while generating forecasts of reasonable accuracy and utility.

Radon-Nikodým Derivative: Re-imagining Anomaly Detection from a Measure Theoretic Perspective

Which principle underpins the design of an effective anomaly detection loss function? The answer lies in the concept of \rnthm{} theorem, a fundamental concept in measure theory. The key insight is -- Multiplying the vanilla loss function with the \rnthm{} derivative improves the performance across the board. We refer to this as RN-Loss. This is established using PAC learnability of anomaly detection. We further show that the \rnthm{} derivative offers important insights into unsupervised clustering based anomaly detections as well. We evaluate our algorithm on 96 datasets, including univariate and multivariate data from diverse domains, including healthcare, cybersecurity, and finance. We show that RN-Derivative algorithms outperform state-of-the-art methods on 68\% of Multivariate datasets (based on F-1 scores) and also achieves peak F1-scores on 72\% of time series (Univariate) datasets.

Anomalous Agreement: How to find the Ideal Number of Anomaly Classes in Correlated, Multivariate Time Series Data

Detecting and classifying abnormal system states is critical for condition monitoring, but supervised methods often fall short due to the rarity of anomalies and the lack of labeled data. Therefore, clustering is often used to group similar abnormal behavior. However, evaluating cluster quality without ground truth is challenging, as existing measures such as the Silhouette Score (SSC) only evaluate the cohesion and separation of clusters and ignore possible prior knowledge about the data. To address this challenge, we introduce the Synchronized Anomaly Agreement Index (SAAI), which exploits the synchronicity of anomalies across multivariate time series to assess cluster quality. We demonstrate the effectiveness of SAAI by showing that maximizing SAAI improves accuracy on the task of finding the true number of anomaly classes K in correlated time series by 0.23 compared to SSC and by 0.32 compared to X-Means. We also show that clusters obtained by maximizing SAAI are easier to interpret compared to SSC.

Multivariate Time Series Clustering for Environmental State Characterization of Ground-Based Gravitational-Wave Detectors

Gravitational-wave observatories like LIGO are large-scale, terrestrial instruments housed in infrastructure that spans a multi-kilometer geographic area and which must be actively controlled to maintain operational stability for long observation periods. Despite exquisite seismic isolation, they remain susceptible to seismic noise and other terrestrial disturbances that can couple undesirable vibrations into the instrumental infrastructure, potentially leading to control instabilities or noise artifacts in the detector output. It is, therefore, critical to characterize the seismic state of these observatories to identify a set of temporal patterns that can inform the detector operators in day-to-day monitoring and diagnostics. On a day-to-day basis, the operators monitor several seismically relevant data streams to diagnose operational instabilities and sources of noise using some simple empirically-determined thresholds. It can be untenable for a human operator to monitor multiple data streams in this manual fashion and thus a distillation of these data-streams into a more human-friendly format is sought. In this paper, we present an end-to-end machine learning pipeline for features-based multivariate time series clustering to achieve this goal and to provide actionable insights to the detector operators by correlating found clusters with events of interest in the detector.

DUET: Dual Clustering Enhanced Multivariate Time Series Forecasting

Multivariate time series forecasting is crucial for various applications, such as financial investment, energy management, weather forecasting, and traffic optimization. However, accurate forecasting is challenging due to two main factors. First, real-world time series often show heterogeneous temporal patterns caused by distribution shifts over time. Second, correlations among channels are complex and intertwined, making it hard to model the interactions among channels precisely and flexibly. In this study, we address these challenges by proposing a general framework called \textbf{DUET}, which introduces \underline{DU}al clustering on the temporal and channel dimensions to \underline{E}nhance multivariate \underline{T}ime series forecasting. First, we design a Temporal Clustering Module (TCM) that clusters time series into fine-grained distributions to handle heterogeneous temporal patterns. For different distribution clusters, we design various pattern extractors to capture their intrinsic temporal patterns, thus modeling the heterogeneity. Second, we introduce a novel Channel-Soft-Clustering strategy and design a Channel Clustering Module (CCM), which captures the relationships among channels in the frequency domain through metric learning and applies sparsification to mitigate the adverse effects of noisy channels. Finally, DUET combines TCM and CCM to incorporate both the temporal and channel dimensions. Extensive experiments on 25 real-world datasets from 10 application domains, demonstrate the state-of-the-art performance of DUET.

A Multi-Task Learning Approach to Linear Multivariate Forecasting

Accurate forecasting of multivariate time series data is important in many engineering and scientific applications. Recent state-of-the-art works ignore the inter-relations between variates, using their model on each variate independently. This raises several research questions related to proper modeling of multivariate data. In this work, we propose to view multivariate forecasting as a multi-task learning problem, facilitating the analysis of forecasting by considering the angle between task gradients and their balance. To do so, we analyze linear models to characterize the behavior of tasks. Our analysis suggests that tasks can be defined by grouping similar variates together, which we achieve via a simple clustering that depends on correlation-based similarities. Moreover, to balance tasks, we scale gradients with respect to their prediction error. Then, each task is solved with a linear model within our MTLinear framework. We evaluate our approach on challenging benchmarks in comparison to strong baselines, and we show it obtains on-par or better results on multivariate forecasting problems. The implementation is available at: https://github.com/azencot-group/MTLinear