Irregular Time-Series Modeling

This paper introduces a framework specifically designed for sparse and irregular time series {risk estimation}. It is based on a Transformer Autoencoder with local attention, which leverages the powerful pattern identification capabilities of transformers complemented by traditional data cleaning and normalization methods. It efficiently captures relevant patterns within irregular sequences suffering from sparse data collection, benefiting from the discriminative ability of the local attention mechanism. The proposed framework is applied to a real-world case study, on the risk estimation of non-technical losses in electrical power systems in a wide area in Greece. Non-technical losses in electrical power systems, primarily stemming from electricity theft, pose significant economic and operational challenges. Detecting these anomalies is particularly challenging due to the inherent sparse and irregular nature of real-world data collection practices. Traditional risk estimation methods struggle with effectively capturing long-range dependencies and robustly handling such data characteristics. We demonstrate that our approach effectively yields highly discriminative latent features, which results in more consistent risk estimation compared with existing state-of-the-art and widely used methods. It achieves high recall and precision, meeting the critical objectives of the problem. As such, our solution offers a robust and effective tool for risk detection in irregular time series datasets.

We propose a new class of waveform foundation models that departs from conventional sequence based representations by modeling physiological time series as realizations of latent event processes. Rather than treating signals as collections of local tokens or patches, our approach assumes that clinically meaningful structure arises from temporally extended, interacting events whose boundaries and dynamics are not directly observed. To capture this structure, we introduce a self supervised learning framework that enforces consistency across stochastic segmentations and time frequency projections of the same waveform, encouraging representations that are invariant to signal level perturbations while preserving event level organization. The resulting model combines a segmentation aware encoder with a latent interaction operator that captures dependencies among inferred events, and naturally extends to multimodal settings by aligning modalities through shared event representations. Across a range of physiological benchmarks, including arrhythmia classification, hemodynamic prediction, and waveform retrieval, the proposed method improves performance, robustness, and label efficiency relative to strong sequence based baselines. These results suggest that shifting from signal centric to event centric representations provides a more appropriate inductive bias for modeling physiological dynamics and offers a complementary path to scaling foundation models in healthcare.

Optical satellite image time series are extensively used in many Earth observation applications, including agriculture, climate monitoring, and land surface analysis. However, clouds and swath edges result in irregular sampling along the temporal dimension, limiting continuous monitoring. To address this issue, a growing body of work has focused on temporal densification and reconstruction of satellite image time series, with the objective of filling missing or cloud-contaminated observations within the temporal extent of the available data. While these approaches improve temporal continuity, they are inherently restricted to the reconstruction of the gaps within the observed time periods, and do not address the prediction of future observations. This work proposes a probabilistic deep learning framework for the densification and forecasting of Sentinel-2 time series by generating optical images at arbitrary past or future dates. The approach leverages multimodal satellite data by jointly exploiting Sentinel-2 optical and Sentinel-1 SAR observations. Unlike most existing works, we propose to focus on the uncertainty of the generated images. Experimental results demonstrate effective densification and forecasting, on sparse and temporally misaligned time series.

Medium-horizon Alzheimer's disease progression prediction is difficult because future clinical scores can remain tied to baseline severity, while biomarker histories are irregular and incompletely observed. We develop an anchor-based analysis of 24-month Clinical Dementia Rating Sum of Boxes (CDR-SB) change using harmonized Alzheimer's Disease Neuroimaging Initiative (ADNI) tables. Each labeled sample is anchored at a mild cognitive impairment visit, uses only clinical and biomarker history observed at or before that anchor, and defines the response as CDR-SB at the future visit closest to 24 months within an 18--30 month window minus anchor CDR-SB. The analytic cohort contains 2,600 labeled anchors from 858 participants and 7,276 longitudinal rows. We propose a residual gap-aware transformer that combines a mixed-effects statistical reference with transformer-based residual learning from pre-anchor clinical and biomarker histories. The model uses participant-level random intercepts in the mixed-effects reference, observation-level triplet tokenization for irregular histories, and a learned nonnegative time-gap penalty inside self-attention. We compare the proposed model with a Bayesian-information-criterion-selected linear mixed-effects baseline, GRU-D, and STraTS under repeated participant-level train--test splits. Across five participant-level random seeds, the proposed model achieves the best mean test performance across all reported metrics, reducing MSE by 13.1% and increasing prediction--observation correlation by 26.4% relative to the mixed-effects baseline. It also improves over both GRU-D and STraTS in mean error and correlation. These results show that statistical anchoring and gap-aware residual learning provide a useful structure for medium-horizon Alzheimer's disease progression prediction.

Background: Timely, uncertainty-aware forecasting from irregular electronic health records (EHR) can support critical-care decisions, yet most approaches either impute to a grid or sacrifice interpretability. We introduce StructGP, a continuous-time multi-task Gaussian process that couples process convolutions with differentiable structure learning to uncover a sparse, ordered directed acyclic graph (DAG) of inter-variable dependencies while preserving principled uncertainty. We further propose LP-StructGP, which augments StructGP with latent pathways-shared, temporally shifted trajectories inferred via subject-specific coupling filters and a softmax gating mechanism-to capture cross-patient progression patterns. Both models are trained under sparsity and acyclicity constraints (augmented Lagrangian, Adam) using scalable low-rank updates. Results: In simulations, the approach reliably recovers ground-truth graphs (Structural Hamming Distance approaching 0 as cohorts grow) and pathway assignments (high Adjusted Rand Index). On a MIMIC-IV septic shock cohort (n=1,008; norepinephrine, creatinine, mean arterial pressure), StructGP improves short-horizon (6 h) forecasting over independent-task baselines (average RMSE 0.68 [95%CI: 0.63--0.74] vs. 0.88 [0.83-0.94]) and, with 15 additional inputs, markedly outperforms unstructured kernels (0.63 [0.58-0.69] vs. 3.02 [2.85-3.18]) with superior calibration (coverage 0.96 vs. 0.84). On the PhysioNet Challenge (12k patients, 41 variables), StructGP attains competitive accuracy (MAE 3.72e-2) relative to a state-of-the-art graph neural model while maintaining calibrated uncertainty. Conclusion: These results show that structured process convolutions with latent pathways deliver interpretable, scalable, and well-calibrated forecasting for irregular clinical time series.

Joint probabilistic modeling is essential for forecasting irregular multivariate time series (IMTS) to accurately quantify uncertainty. Existing approaches often struggle to balance model expressivity with consistent marginalization, frequently leading to unreliable or contradictory forecasts. To address this, we propose CircuITS, a novel architecture for probabilistic IMTS forecasting based on probabilistic circuits. Our model is flexible in capturing intricate dependencies between time series channels while structurally guaranteeing valid joint distributions. Experiments on four real world datasets demonstrate that CircuITS achieves superior joint and marginal density estimation compared to state of the art baselines.

Partially-observed time series (POTS) is ubiquitous in real-world applications, yet most existing toolchains separate missing-value handling from downstream learning, which limits reproducibility and overall performance. This tutorial introduces PyPOTS, an open-source Python ecosystem for end-to-end data mining and machine learning on POTS. We present practical workflows spanning missingness simulation, data preprocessing, model training, and evaluation across core tasks, including imputation, forecasting, classification, clustering, and anomaly detection. The tutorial consists of two parts: Part I emphasizes hands-on application for practitioners through unified APIs and benchmark-oriented experiments. Part II targets developers and researchers, focusing on extending PyPOTS with custom models, domain-specific constraints, and contribution-ready engineering practices. Participants will gain both conceptual understanding and implementation experience for building robust, transparent, and reusable POTS pipelines in research and production settings. PyPOTS is publicly available at https://github.com/WenjieDu/PyPOTS

Predicting customers' long-term revenue from sparse and irregular transaction data is central to marketing resource allocation in non-contractual settings, yet existing approaches face a trade-off. Traditional probabilistic customer base models deliver robust long-horizon forecasts by imposing strong structural assumptions, while flexible machine-learning models often require substantial training data and careful tuning. We propose a variational-autoencoder-based model that preserves the process-based likelihood of established attrition-transaction-spend models conditional on customer heterogeneity, but replaces the restrictive parametric mixing distribution with a flexible latent representation learned by encoder-decoder networks. The resulting approach (i) provides a single model for customer attrition, transactions and spending, (ii) remains reliable when contextual covariates are unavailable, and (iii) flexibly incorporates rich covariates and nonlinear effects when they are available. This design balances structural stability with the flexibility needed to capture complex purchase dynamics. Across multiple real-world datasets and prediction horizons, the proposed model improves upon the latest benchmarks. Businesses benefit directly, as a better assessment of customers' future revenues improves the efficiency of campaign targeting. For research, this work provides guidance on how to embed domain-specific models into the variational autoencoder framework, enabling flexible representation learning while retaining an econometrically meaningful process structure.

Longitudinal neuroimaging is essential for modeling disease progression in Alzheimer's disease (AD), yet irregular sampling and missing visits pose substantial challenges for learning reliable temporal representations. To address this challenge, we propose SDE-HGNN, a stochastic differential equation (SDE)-driven spatio-temporal hypergraph neural network for irregular longitudinal fMRI connectome modeling. The framework first employs an SDE-based reconstruction module to recover continuous latent trajectories from irregular observations. Based on these reconstructed representations, dynamic hypergraphs are constructed to capture higher-order interactions among brain regions over time. To further model temporal evolution, hypergraph convolution parameters evolve through SDE-controlled recurrent dynamics conditioned on inter-visit intervals, enabling disease-stage-adaptive connectivity modeling. We also incorporate a sparsity-based importance learning mechanism to identify salient brain regions and discriminative connectivity patterns. Extensive experiments on the OASIS-3 and ADNI cohorts demonstrate consistent improvements over state-of-the-art graph and hypergraph baselines in AD progression prediction. The source code is available at https://anonymous.4open.science/r/SDE-HGNN-017F.

Short-term forecasting of vegetation dynamics is a key enabler for data-driven decision support in precision agriculture. Normalized Difference Vegetation Index (NDVI) forecasting from satellite observations, however, remains challenging due to sparse and irregular sampling caused by cloud masking, as well as the heterogeneous climatic conditions under which crops evolve. In this work, we propose a probabilistic forecasting framework for field-level NDVI prediction under sparse, irregular clear-sky acquisitions. The architecture separates the encoding of historical NDVI and meteorological observations from future exogenous covariates, fusing both representations for multi-step quantile prediction. To address irregular revisit patterns and horizon-dependent uncertainty, we introduce a temporal-distance weighted quantile loss that aligns the training objective with the effective forecasting horizon. In addition, we incorporate cumulative and extreme-weather feature engineering to capture delayed meteorological effects relevant to vegetation response. Experiments on European satellite data show that the proposed approach outperforms statistical, deep learning, and time-series baselines on both pointwise and probabilistic evaluation metrics. Ablation studies confirm that target history is the primary driver of performance, with meteorological covariates providing additional gains in the full multimodal setting. The code is available at https://github.com/arco-group/ndvi-forecasting.

Time series generation (TSG) is widely used across domains, yet most existing methods assume regular sampling and fixed output resolutions. These assumptions are often violated in practice, where observations are irregular and sparse, while downstream applications require continuous and high-resolution TS. Although Neural Controlled Differential Equation (NCDE) is promising for modeling irregular TS, it is constrained by a single dynamics function, tightly coupled optimization, and limited ability to adapt learned dynamics to newly generated samples from the generative model. We propose Diff-MN, a continuous TSG framework that enhances NCDE with a Mixture-of-Experts (MoE) dynamics function and a decoupled architectural design for dynamics-focused training. To further enable NCDE to generalize to newly generated samples, Diff-MN employs a diffusion model to parameterize the NCDE temporal dynamics parameters (MoE weights), i.e., jointly learn the distribution of TS data and MoE weights. This design allows sample-specific NCDE parameters to be generated for continuous TS generation. Experiments on ten public and synthetic datasets demonstrate that Diff-MN consistently outperforms strong baselines on both irregular-to-regular and irregular-to-continuous TSG tasks. The code is available at the link https://github.com/microsoft/TimeCraft/tree/main/Diff-MN.

Modeling continuous-time dynamics from sparse and irregularly-sampled time series remains a fundamental challenge. Neural controlled differential equations provide a principled framework for such tasks, yet their performance is highly sensitive to the choice of control path constructed from discrete observations. Existing methods commonly employ fixed interpolation schemes, which impose simplistic geometric assumptions that often misrepresent the underlying data manifold, particularly under high missingness. We propose FlowPath, a novel approach that learns the geometry of the control path via an invertible neural flow. Rather than merely connecting observations, FlowPath constructs a continuous and data-adaptive manifold, guided by invertibility constraints that enforce information-preserving and well-behaved transformations. This inductive bias distinguishes FlowPath from prior unconstrained learnable path models. Empirical evaluations on 18 benchmark datasets and a real-world case study demonstrate that FlowPath consistently achieves statistically significant improvements in classification accuracy over baselines using fixed interpolants or non-invertible architectures. These results highlight the importance of modeling not only the dynamics along the path but also the geometry of the path itself, offering a robust and generalizable solution for learning from irregular time series.

We introduce Temporal Variational Implicit Neural Representations (TV-INRs), a probabilistic framework for modeling irregular multivariate time series that enables efficient and accurate individualized imputation and forecasting. By integrating implicit neural representations with latent variable models, TV-INRs learn distributions over time-continuous generator functions conditioned on signal-specific covariates. Unlike existing INR approaches that require extensive training, fine-tuning or meta-learning, our method achieves accurate individualized predictions through a single forward pass. Our experiments demonstrate that with a single TV-INRs instance, we can accurately solve diverse imputation and forecasting tasks, offering a computationally efficient and scalable solution for real-world applications. TV-INRs performs particularly well in low-data regimes, where on several datasets it achieves substantially lower imputation error, including order-of-magnitude improvements.

We propose a framework for generative modeling of continuous-time processes from irregularly and asynchronously recorded data. It is based on the matching of generators and accommodates discontinuous trajectories. Analytical formulas for diffusion and jump bridges yield a family of reference generators that a neural network is trained to match. The key ingredient is that, for our constructed jump bridge, a parametrization of the jump kernel densities by scaled Gaussians admits closed-form expressions for the Kullback-Leibler divergence, allowing simulation-free training.

Irregular sampling intervals and missing values in real-world time series data present challenges for conventional methods that assume consistent intervals and complete data. Neural Ordinary Differential Equations (Neural ODEs) offer an alternative approach, utilizing neural networks combined with ODE solvers to learn continuous latent representations through parameterized vector fields. Neural Stochastic Differential Equations (Neural SDEs) extend Neural ODEs by incorporating a diffusion term, although this addition is not trivial, particularly when addressing irregular intervals and missing values. Consequently, careful design of drift and diffusion functions is crucial for maintaining stability and enhancing performance, while incautious choices can result in adverse properties such as the absence of strong solutions, stochastic destabilization, or unstable Euler discretizations, significantly affecting Neural SDEs' performance. In this study, we propose three stable classes of Neural SDEs: Langevin-type SDE, Linear Noise SDE, and Geometric SDE. Then, we rigorously demonstrate their robustness in maintaining excellent performance under distribution shift, while effectively preventing overfitting. To assess the effectiveness of our approach, we conduct extensive experiments on four benchmark datasets for interpolation, forecasting, and classification tasks, and analyze the robustness of our methods with 30 public datasets under different missing rates. Our results demonstrate the efficacy of the proposed method in handling real-world irregular time series data.

Real-world time series analysis faces significant challenges when dealing with irregular and incomplete data. While Neural Differential Equation (NDE) based methods have shown promise, they struggle with limited expressiveness, scalability issues, and stability concerns. Conversely, Neural Flows offer stability but falter with irregular data. We introduce 'DualDynamics', a novel framework that synergistically combines NDE-based method and Neural Flow-based method. This approach enhances expressive power while balancing computational demands, addressing critical limitations of existing techniques. We demonstrate DualDynamics' effectiveness across diverse tasks: classification of robustness to dataset shift, irregularly-sampled series analysis, interpolation of missing data, and forecasting with partial observations. Our results show consistent outperformance over state-of-the-art methods, indicating DualDynamics' potential to advance irregular time series analysis significantly.