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 of time series based on their underlying dynamics is keeping attracting researchers due to its impacts on assisting complex system modelling. Most current time series clustering methods handle only scalar time series, treat them as white noise, or rely on domain knowledge for high-quality feature construction, where the autocorrelation pattern/feature is mostly ignored. Instead of relying on heuristic feature/metric construction, the system identification approach allows treating vector time series clustering by explicitly considering their underlying autoregressive dynamics. We first derive a clustering algorithm based on a mixture autoregressive model. Unfortunately it turns out to have significant computational problems. We then derive a `small-noise' limiting version of the algorithm, which we call k-LMVAR (Limiting Mixture Vector AutoRegression), that is computationally manageable. We develop an associated BIC criterion for choosing the number of clusters and model order. The algorithm performs very well in comparative simulations and also scales well computationally.

Spatial labeling assigns labels to specific spatial locations to characterize their spatial properties and relationships, with broad applications in scientific research and practice. Measuring the similarity between two spatial labelings is essential for understanding their differences and the contributing factors, such as changes in location properties or labeling methods. An adequate and unbiased measurement of spatial labeling similarity should consider the number of matched labels (label agreement), the topology of spatial label distribution, and the heterogeneous impacts of mismatched labels. However, existing methods often fail to account for all these aspects. To address this gap, we propose a methodological framework to guide the development of methods that meet these requirements. Given two spatial labelings, the framework transforms them into graphs based on location organization, labels, and attributes (e.g., location significance). The distributions of their graph attributes are then extracted, enabling an efficient computation of distributional discrepancy to reflect the dissimilarity level between the two labelings. We further provide a concrete implementation of this framework, termed Spatial Labeling Analogy Metric (SLAM), along with an analysis of its theoretical foundation, for evaluating spatial labeling results in spatial transcriptomics (ST) \textit{as per} their similarity with ground truth labeling. Through a series of carefully designed experimental cases involving both simulated and real ST data, we demonstrate that SLAM provides a comprehensive and accurate reflection of labeling quality compared to other well-established evaluation metrics. Our code is available at https://github.com/YihDu/SLAM.

Research on Graph Structure Learning (GSL) provides key insights for graph-based clustering, yet current methods like Graph Neural Networks (GNNs), Graph Attention Networks (GATs), and contrastive learning often rely heavily on the original graph structure. Their performance deteriorates when the original graph's adjacency matrix is too sparse or contains noisy edges unrelated to clustering. Moreover, these methods depend on learning node embeddings and using traditional techniques like k-means to form clusters, which may not fully capture the underlying graph structure between nodes. To address these limitations, this paper introduces DeSE, a novel unsupervised graph clustering framework incorporating Deep Structural Entropy. It enhances the original graph with quantified structural information and deep neural networks to form clusters. Specifically, we first propose a method for calculating structural entropy with soft assignment, which quantifies structure in a differentiable form. Next, we design a Structural Learning layer (SLL) to generate an attributed graph from the original feature data, serving as a target to enhance and optimize the original structural graph, thereby mitigating the issue of sparse connections between graph nodes. Finally, our clustering assignment method (ASS), based on GNNs, learns node embeddings and a soft assignment matrix to cluster on the enhanced graph. The ASS layer can be stacked to meet downstream task requirements, minimizing structural entropy for stable clustering and maximizing node consistency with edge-based cross-entropy loss. Extensive comparative experiments are conducted on four benchmark datasets against eight representative unsupervised graph clustering baselines, demonstrating the superiority of the DeSE in both effectiveness and interpretability.

Accurately predicting customers' purchase intentions is critical to the success of a business strategy. Current researches mainly focus on analyzing the specific types of products that customers are likely to purchase in the future, little attention has been paid to the critical factor of whether customers will engage in repurchase behavior. Predicting whether a customer will make the next purchase is a classic time series forecasting task. However, in real-world purchasing behavior, customer groups typically exhibit imbalance - i.e., there are a large number of occasional buyers and a small number of loyal customers. This head-to-tail distribution makes traditional time series forecasting methods face certain limitations when dealing with such problems. To address the above challenges, this paper proposes a unified Clustering and Attention mechanism GRU model (CAGRU) that leverages multi-modal data for customer purchase intention prediction. The framework first performs customer profiling with respect to the customer characteristics and clusters the customers to delineate the different customer clusters that contain similar features. Then, the time series features of different customer clusters are extracted by GRU neural network and an attention mechanism is introduced to capture the significance of sequence locations. Furthermore, to mitigate the head-to-tail distribution of customer segments, we train the model separately for each customer segment, to adapt and capture more accurately the differences in behavioral characteristics between different customer segments, as well as the similar characteristics of the customers within the same customer segment. We constructed four datasets and conducted extensive experiments to demonstrate the superiority of the proposed CAGRU approach.

Clustering is a widely used and powerful machine learning technique, but its effectiveness is often limited by the need to specify the number of clusters, k, or by relying on thresholds that implicitly determine k. We introduce k*-means, a novel clustering algorithm that eliminates the need to set k or any other parameters. Instead, it uses the minimum description length principle to automatically determine the optimal number of clusters, k*, by splitting and merging clusters while also optimising the standard k-means objective. We prove that k*-means is guaranteed to converge and demonstrate experimentally that it significantly outperforms existing methods in scenarios where k is unknown. We also show that it is accurate in estimating k, and that empirically its runtime is competitive with existing methods, and scales well with dataset size.

We introduce a framework for robust uncertainty quantification in situations where labeled training data are corrupted, through noisy or missing labels. We build on conformal prediction, a statistical tool for generating prediction sets that cover the test label with a pre-specified probability. The validity of conformal prediction, however, holds under the i.i.d assumption, which does not hold in our setting due to the corruptions in the data. To account for this distribution shift, the privileged conformal prediction (PCP) method proposed leveraging privileged information (PI) -- additional features available only during training -- to re-weight the data distribution, yielding valid prediction sets under the assumption that the weights are accurate. In this work, we analyze the robustness of PCP to inaccuracies in the weights. Our analysis indicates that PCP can still yield valid uncertainty estimates even when the weights are poorly estimated. Furthermore, we introduce uncertain imputation (UI), a new conformal method that does not rely on weight estimation. Instead, we impute corrupted labels in a way that preserves their uncertainty. Our approach is supported by theoretical guarantees and validated empirically on both synthetic and real benchmarks. Finally, we show that these techniques can be integrated into a triply robust framework, ensuring statistically valid predictions as long as at least one underlying method is valid.

Graph structure learning is a core problem in graph-based machine learning, essential for uncovering latent relationships and ensuring model interpretability. However, most existing approaches are ill-suited for large-scale and dynamically evolving graphs, as they often require complete re-learning of the structure upon the arrival of new nodes and incur substantial computational and memory costs. In this work, we propose GraphFLEx: a unified and scalable framework for Graph Structure Learning in Large and Expanding Graphs. GraphFLEx mitigates the scalability bottlenecks by restricting edge formation to structurally relevant subsets of nodes identified through a combination of clustering and coarsening techniques. This dramatically reduces the search space and enables efficient, incremental graph updates. The framework supports 48 flexible configurations by integrating diverse choices of learning paradigms, coarsening strategies, and clustering methods, making it adaptable to a wide range of graph settings and learning objectives. Extensive experiments across 26 diverse datasets and Graph Neural Network architectures demonstrate that GraphFLEx achieves state-of-the-art performance with significantly improved scalability.

Port berthing sites are regions of high interest for monitoring and optimizing port operations. Data sourced from the Automatic Identification System (AIS) can be superimposed on berths enabling their real-time monitoring and revealing long-term utilization patterns. Ultimately, insights from multiple berths can uncover bottlenecks, and lead to the optimization of the underlying supply chain of the port and beyond. However, publicly available documentation of port berths, even when available, is frequently incomplete - e.g. there may be missing berths or inaccuracies such as incorrect boundary boxes - necessitating a more robust, data-driven approach to port berth localization. In this context, we propose an unsupervised spatial modeling method that leverages AIS data clustering and hyperparameter optimization to identify berthing sites. Trained on one month of freely available AIS data and evaluated across ports of varying sizes, our models significantly outperform competing methods, achieving a mean Bhattacharyya distance of 0.85 when comparing Gaussian Mixture Models (GMMs) trained on separate data splits, compared to 13.56 for the best existing method. Qualitative comparison with satellite images and existing berth labels further supports the superiority of our method, revealing more precise berth boundaries and improved spatial resolution across diverse port environments.

Anomaly detection is a fundamental problem in domains such as healthcare, manufacturing, and cybersecurity. This thesis proposes new unsupervised methods for anomaly detection in both structured and streaming data settings. In the first part, we focus on structure-based anomaly detection, where normal data follows low-dimensional manifolds while anomalies deviate from them. We introduce Preference Isolation Forest (PIF), which embeds data into a high-dimensional preference space via manifold fitting, and isolates outliers using two variants: Voronoi-iForest, based on geometric distances, and RuzHash-iForest, leveraging Locality Sensitive Hashing for scalability. We also propose Sliding-PIF, which captures local manifold information for streaming scenarios. Our methods outperform existing techniques on synthetic and real datasets. We extend this to structure-based clustering with MultiLink, a novel method for recovering multiple geometric model families in noisy data. MultiLink merges clusters via a model-aware linkage strategy, enabling robust multi-class structure recovery. It offers key advantages over existing approaches, such as speed, reduced sensitivity to thresholds, and improved robustness to poor initial sampling. The second part of the thesis addresses online anomaly detection in evolving data streams. We propose Online Isolation Forest (Online-iForest), which uses adaptive, multi-resolution histograms and dynamically updates tree structures to track changes over time. It avoids retraining while achieving accuracy comparable to offline models, with superior efficiency for real-time applications. Finally, we tackle anomaly detection in cybersecurity via open-set recognition for malware classification. We enhance a Gradient Boosting classifier with MaxLogit to detect unseen malware families, a method now integrated into Cleafy's production system.