An efficient EM algorithm for both element-wise and structural missingness in matrix-variate normal mixture models

Matrix-variate data with missing entries arise frequently in applications where observations are naturally organized as two-dimensional arrays. Although the matrix normal distribution provides a parsimonious model through its Kronecker covariance structure, standard EM estimation can be computationally expensive because arbitrary missingness patterns typically destroy this separability in the E-step. In this paper, we propose an efficient partial EM algorithm for matrix-variate normal data with missing entries. The proposed method updates the conditional mean and covariance of the missing component through coordinate-wise approximations, avoiding repeated inversion of pattern-specific covariance matrices and avoiding construction of the full vectorized covariance matrix. We further develop a specialized update for submatrix missingness, where the missing-block precision retains a Kronecker product structure, and the covariance update can be carried out independently in the row and column directions. Simulation studies show that the proposed methods substantially reduce computation time compared with exact EM while preserving nearly identical observed-data likelihood across a range of dimensions and missing proportions. A real-data application to hyperspectral image patches demonstrates that the proposed imputation strategy can be embedded within a matrix-variate mixture model for simultaneous imputation and clustering.

A Deep Latent Variable Framework for Jointly Modeling Missingness, Measurement Error, and Heterogeneity

Missing data, measurement error, and population heterogeneity are pervasive challenges in analyzing data arising from modern observational studies and machine learning applications. Although these problems frequently coexist and interact, they are often treated separately in existing works. We propose a unified probabilistic framework that jointly addresses these issues utilizing deep latent variable representation. The proposed method integrates a novel hierarchical tree-routed variational autoencoder with pattern-aware latent representations and calibration-based denoising. The framework accommodates missing data mechanisms, including MCAR, MAR, and MNAR, while simultaneously learning subgroup-specific and globally shared latent structure. The introduced reconvergent routing mechanism enables selective parameters to be shared across related subpopulations, which offers flexibility as well as improved statistical efficiency. Simulation studies demonstrate substantial improvements over existing deep generative imputation approaches under complex heterogeneous missingness and measurement-error settings. The proposed framework provides a principled approach for learning from noisy and incomplete data in modern healthcare and other high-dimensional applications.

Evaluating Deep Multivariate Imputation Models on Wearable Device Data

Wearable device data enables continuous health monitoring, but suffers from structured missingness: features sharing a physical sensor drop out together. Deep imputation methods such as BRITS and SAITS have seen limited evaluation on multimodal physiological data under realistic missingness, and existing benchmarks use random-point holdout protocols that incorrectly assume missingness is independent across features and time. Using data from a person with epilepsy recorded on a Garmin smartwatch, we develop an evaluation protocol that mines contiguous missing-run templates from training data, stratifies them by per-feature gap-length quantiles, and injects them as block masks with preserved co-missingness structure. A matched training protocol exposing models to the same missingness distribution reduces BRITS's severe-gap MAE by 43%, demonstrating the potential benefit of the proposed evaluation and training protocol within this single-participant dataset. We further extend BRITS with time-of-day encoding and a circadian harmonic channel. No single model dominates: linear interpolation is optimal for slow-moving features over short gaps; extended BRITS achieves lower MAE on dynamic cardiac features in moderate and severe gaps; and SAITS better preserves the ground-truth distribution by Jensen-Shannon distance despite higher MAE. Ultimately, model rankings strongly depend on evaluation designs. By exposing how traditional evaluation methods obscure true model capabilities, our transferable protocol establishes critical steps towards developing better imputation strategies for future multi-sensor wearable datasets.

Large-Scale Evaluation of Advanced Imputation Methods for Missing Values in Smart Meter Data

Accurate and reliable collection of electricity consumption data through Advanced Metering Infrastructure (AMI) is of great importance for the operation of smart grids, especially for the detection of non-technical losses (NTL). However, real-world datasets frequently suffer from missing values due to communication failures. This paper presents an empirical evaluation of three advanced algorithms for large-scale data imputation: the Optimally Weighted Average (OWA) method, Low-Rank Matrix Completion via SoftImpute, and a Shape-Modeling Autoencoder. Existing studies on missing value imputation in electricity consumption data often lack validation on larger datasets. Therefore, the goal of this paper is to validate the selected algorithms on a large-scale real-world electricity consumption dataset from North Macedonia that includes 17,428 commercial smart meters over two years. The robustness of each algorithm is evaluated by simulating continuous gaps in the data ranging from 1 to 168 hours. The results indicate that OWA provides the lowest overall reconstruction error across the evaluated gap sizes and strong stability in worst-case scenarios for gaps of up to one week. In contrast, the autoencoder exhibits higher variance, while SoftImpute has stable but inferior accuracy. These findings suggest that imputation methods should be selected based on the characteristics of load curve data and highlight the potential for hybrid algorithmic architectures in future grid management systems.

Sparse Separable Factor Analysis in the Complex Domain with an Application to Local Field Potential Data

Complex-valued arrays arise in signal processing, where scientific interpretation depends on retaining amplitude and phase information. Existing covariance estimation methods either ignore the multiway organization of such data or rely on real-domain embeddings that do not directly exploit their complex structure. We develop sparse separable factor analysis (SSFA), a latent factor model for complex-valued arrays with a separable covariance structure across modes. Each mode-specific covariance matrix is modeled through a low-rank Hermitian factor structure and a diagonal residual covariance matrix. To obtain interpretable estimates, we impose elementwise lasso penalties on the complex loading matrices and estimate the SSFA parameters using a mode-wise parameter-expanded expectation-maximization procedure. The resulting loading updates admit closed-form complex soft-thresholding solutions, which shrink the modulus of each loading while preserving its phase. A separate balancing step resolves the scale nonidentifiability of the separable covariance structure. Simulation studies show that SSFA improves covariance estimation relative to vectorization-based methods, including complex principal component analysis. We apply SSFA to local field potential recordings from mice, where we compare separability structures induced by different groupings of brain region, frequency, and time and perform model-based imputation of recordings missing because of electrode misplacement.

Population Health-Based Machine Learning Reveals Associations Between Psychosocial Factors and Chronic Kidney Disease

Chronic kidney disease (CKD) progresses silently and severely undermines quality of life, making early detection critical for improving patient outcomes. We present a two-part study that combines large-scale telehealth data with advanced machine learning to both classify self-reported CKD status and identify key drivers of disease. Using selected features from the Behavioral Risk Factor Surveillance System (BRFSS 2021: 438,693 samples; BRFSS 2019: 418,268 samples) and the National Health Interview Survey (NHIS 2021: 29,482 samples; NHIS 2020: 31,568 samples), we addressed missing data with nine state-of-the-art imputation methods and mitigated class imbalance via sampling strategies. Our customized stacked ensemble model achieved balanced accuracy of 72.56-76.12%, with corresponding AUROC scores of 79.59-82.29%. SHapley Additive exPlanations (SHAP) analysis, followed by clinical review, highlighted critical predictors, including regular medical check-ups, age, blood pressure, and indicators of mental health stress. These findings deliver a robust and interpretable framework for CKD risk stratification and provide actionable insights into its associated factors.

Deep Learning Imputation of Missing Radius of Maximum Winds (Rmax) Values in Tropical Cyclone Best-Track Data

Probabilistic coastal hazard assessments require accurate characterization of tropical cyclone (TC) parameters, yet datasets often contain missing records for the radius of maximum winds (Rmax), a key variable in Joint Probability Method analyses. This study evaluates data-driven approaches for Rmax imputation, including one-dimensional Convolutional Neural Networks (1DCNNs), Long Short-Term Memory (LSTM) networks, and conventional machine learning models. We examine physics-informed input augmentation, temporal modeling, and transfer learning using synthetic RAFT and STORM datasets for pre-training and observational IBTrACS data for fine-tuning. Including the radius of 34-knot winds (R34) substantially improves performance across all model types. Temporal models achieve higher average correlations than non-temporal models despite using approximately an order of magnitude fewer samples, indicating better preservation of relative Rmax variability across storms. This advantage is more pronounced when R34 is unavailable, suggesting temporal information can partially compensate for missing storm-size predictors. Transfer learning does not improve performance, likely because synthetic datasets have lower and less variable Rmax distributions than IBTrACS. These findings demonstrate the potential of temporal deep learning for reconstructing incomplete TC records and highlight the importance of physics-informed inputs, observational data availability, and distributional consistency in coastal hazard assessment.

Implicit Neural Representations for Multimodal Longitudinal Image Imputation and Interpolation

Longitudinal multiparametric MRI is central to follow-up imaging in oncology, yet real-world clinical data are characterised by missing sequences, heterogeneous acquisition protocols, and varying spatial resolutions across time points. We propose a patient-specific conditional implicit neural representation (INR) that models multimodal longitudinal MRI as a continuous function of world coordinates, time, and modality conditioning. The model is trained with stochastic modality dropout to handle incomplete data, and its continuous coordinate-space formulation enables both spatial and temporal interpolation without resampling to a fixed voxel grid. A self-consistency-based confidence estimator is derived from cross-modal reconstruction performance at inference time. We evaluate the framework on longitudinal MRI from paediatric brain tumour patients, demonstrating statistically significant improvements over linear interpolation for T1CE and FLAIR (p < 0.05), with mean MS-SSIM of 0.95 $\pm$ 0.02 for T1CE. Predicted confidence correlates strongly with true reconstruction quality (Pearson r up to 0.996), suggesting reliable deployment potential in heterogeneous clinical settings.

Transformer-based Diffusion models for Hydrological Time Series Probabilistic Imputation and Forecasting

The modeling of hydrometeorological time series with limited observations is a key challenge in the monitoring of hydro-systems and water resources, as well as for flood or drought risk assessment. Due to the high variability of the underlying processes and the sparsity of available measurements, traditional statistical approaches often struggle to accurately represent their dynamics. In this context, recent advances in deep learning offer a promising direction for improving the representation and generation of complex temporal processes sampled at several observation sites. This study investigates the application of transformer-based diffusion models to the simulation and reconstruction of hydrological time series. The proposed framework is applied to the joint modeling of water quantity and quality at six sites spread across three adjacent headwater catchments located in North-East France on a limestone plateau covered by forests and field crops. The model is calibrated and validated using available observational data, which has been quality controlled and corrected for sensor drift and malfunction through collaborative efforts by LNE metrology expertise and Andra monthly quality control over more than 15 years. Its performance is compared with several established baseline approaches commonly used for time series modeling. Quantitative evaluation metrics are employed to assess the ability of the proposed method to reproduce key temporal characteristics of the observed signals in two settings: the imputation of incomplete time series and the forecasting of upcoming hydrological conditions. Results support the effectiveness of the transformer-based approach and highlight its capacity to capture and simulate the complex patterns present in hydrological data. In particular, the results indicate that diffusion models can efficiently sample realistic time series distributions under observation settings with variable missing data for both forecasting and imputation.

Zero-Shot Heart Rate Variability Forecasting from Consumer Wearables Using Time Series Foundation Models

Short-term Heart Rate Variability (HRV) forecasting could provide clinicians with actionable lead time for detecting autonomic dysfunction and adverse cardiac events. Consumer wearable devices generate fragmented, artifact-rich HRV signals that challenge conventional forecasting approaches. In this study, we evaluated the forecasting ability of three Time Series Foundation Models (TSFMs), TimesFM, Chronos, and MOIRAI, against traditional baselines (Mean, Exponential Smoothing, and Exponentially Weighted Moving Average) on real-world wearable data collected from 49 healthy individuals. To address data fragmentation, we introduce a variability-preserving imputation method that augments linear interpolation with locally adaptive stochastic noise, retaining physiological dynamics essential for accurate forecasting. The results show that TSFMs outperformed all baselines without fine-tuning, achieving average Mean Absolute Scaled Error (MASE) between 0.81 and 0.87 across TSFMs and both context lengths (32 and 64 time steps), with Chronos and TimesFM as the top models, though MOIRAI showed limited gains over baselines. With up to a 2-hour forecast horizon, the results establish a baseline for TSFMs' performance on a real-world dataset, highlighting domain-specific fine-tuning as a promising direction for clinical deployment.

Training-Free Off-Screen Player Imputation for Broadcast-Based Spatial Football Analytics

Spatial football metrics such as pitch control assume access to the positions of all 22 players, yet the most widely available source of positional data -- the broadcast main camera -- shows only 10-16 of them at any moment. We quantify the resulting distortion with an open, reproducible benchmark: a simulated broadcast viewport applied to open full-pitch tracking data (Metrica Sports; three matches, one held out from method development). Ignoring off-screen players -- the visible-only baseline implied whenever a video-based game-state-reconstruction (GSR) pipeline adds no imputation layer -- inflates hidden-zone pitch-control error to 25.1-26.9 percentage points and a mean absolute control-share error of 11.1-13.4 points across the three matches. We then evaluate a ladder of training-free, online imputation baselines that use only observations from the match being analysed. The best overall on these decision-relevant metrics, role-anchored centroid voting (each visible player votes for the full-team centroid by subtracting its running role offset, attenuating the viewport-induced subset bias), roughly halves hidden-zone error (to 12.2-13.8 points) and cuts control-share error to 28-48% of the ignore policy at every viewport width from 36 m to 60 m in all three matches. For occlusions <=9.6 s -- the regime of the closest learned prior work -- it reaches binwise median position errors of 3.3-8.9 m; but 50-57% of hidden-player observations lie beyond that regime. Integrated end-to-end into a broadcast-video GSR pipeline, imputation moves a downstream possession-quality score (Space-Creation Index) by 15.6 and 17.2 points on two real World Cup broadcast windows, flipping the verdict class in one.

Pattern-Aware Graph Neural Networks for Handling Missing Data

Missing data is ubiquitous in real-world datasets. Traditional methods either discard incomplete samples or apply imputation techniques that ignore potentially informative missingness patterns, implicitly assuming that missingness occurs randomly. However, missingness patterns might provide additional information. We propose pattern-aware graph neural networks that explicitly encode which features are missing alongside observed values. We used four encoding strategies -- learned embeddings, frozen random embeddings, statistical features, and hierarchical representations -- across seven UCI datasets with naturally occurring missingness. Our Pattern-aware methods achieve substantial improvements over baselines, with an average improvement of 17\% in balanced accuracy and 22\% in F1-macro across all datasets. The benefits vary significantly by dataset: annealing shows dramatic improvement (+80\% balanced accuracy), while hepatitis and soybean show minimal gains (+4--5\%). Notably, even simple random pattern embeddings perform comparably to learned embeddings (0.650 vs 0.663 balanced accuracy), suggesting that distinguishing between patterns may be more important than task-specific optimization. Our ablation study reveals that attention mechanisms, while helpful, are not critical when pattern information is available -- simple mean aggregation with pattern awareness achieves 0.640 balanced accuracy compared to 0.645 for attention-based variants.

ImputeViz: A Visual Analytics Dashboard for Diagnosing Missing Data and Comparing Imputation Methods

Missing data is a persistent obstacle in scientific, social science, and public health research, often biasing analyses and placing accountability on analysts for how they handle missing values. We introduce ImputeViz, an integrated visual analytics dashboard that supports diagnosing missingness, configuring imputation models, and evaluating results. The system brings together widely used methods, including MICE, Random Forest, XGBoost, and kNN, within an interactive environment that makes missingness patterns explicit. To support geospatial reasoning, we introduce gKNN, a geographically informed kNN variant that blends socioeconomic and spatial distances and exposes donor contributions, enabling provenance-based visual accountability by showing which regions drive each estimate. Our primary contribution is a method-agnostic visual analytics environment that makes cross-method comparison a first-class visual task and integrates gKNN alongside standard methods. Coordinated views reveal missingness structure through heatmaps, co-missingness summaries, and distributional diagnostics that help analysts reason about missingness patterns (MCAR/MAR) and cases where missingness may be non-random (MNAR). Users can compare and tune models and interrogate results via distributional overlays, a Method Comparison Summary reporting MAE, RMSE, Delta RMSE, and runtime for each algorithm on the current target and mask, along with variable-level discrepancy views. Cached per-method results and locked axis scales reduce cognitive overhead from shifting ranges during method switching. These comparisons highlight where methods disagree, which variables are sensitive, and how imputation choices affect downstream summaries. Case studies demonstrate how ImputeViz helps analysts select effective strategies, surface sensitive variables, and assess model robustness.

Distributionally Faithful Imputation via Positive Semi-Definite Kernel Density Estimation

Missing values undermine statistical inference and machine learning pipelines, yet most imputation methods rely on heuristics or restrictive parametric assumptions that ignore the joint data distribution. We recast imputation under missing completely at random (MCAR) as density estimation from masked observations: estimate a distribution whose observed marginals exactly match those in the data. Leveraging positive semi definite (PSD) kernel densities we obtain a convex empirical risk problem with closed form marginals, solvable by a Newton interior point method. The resulting PSD Impute model yields both single and multiple imputations from the same fitted density, enjoys statistical consistency with fast adaptive excess risk beating the curse of dimensionality for very regular probabilities. Preliminary experiments on one synthetic and eleven real world datasets already indicate competitive distributional accuracy compared with popular imputation baselines, suggesting strong practical promise.

MNAR-$k$-means: A $k$-means Clustering for Data Missing Not at Random with Magnitude-Decaying Probability

The classical $k$-means clustering, based on distances computed from all data features, cannot be directly applied to incomplete data with missing values. A natural extension of $k$-means to missing data is to involve only the observed positions in clustering, which is equivalent to imputing missing values by corresponding cluster means. However, for data missing not at random (MNAR), since missingness is related to data values, such a mean-imputation-based method may lead to the distortion of estimated cluster centers, resulting in a poor clustering result. Since MNAR mechanisms are very common in reality, it is necessary to improve the performance of $k$-means-based clustering methods for such data. In this paper, we focus on a magnitude-decaying MNAR scenario where data is more likely to be missing at positions with smaller absolute values, and we propose a novel $k$-means clustering method based on the constraint of the size of imputation values, which enjoys a good mathematical interpretation. Moreover, we establish the statistical consistency of the estimated cluster centers of the proposed method to the true cluster centers of fully observed data, and solve the optimization of the proposed loss function via an alternative minimization algorithm. Simulation experiments verify the effect of the proposed method in improving clustering results and reducing the bias of estimated cluster centers. Applications to real-world missing data further show the utility of the proposed method.

Statistical Matching via Schrödinger Bridge beyond Conditional Independence

Statistical matching combines partially overlapping datasets that share covariates $X$ but observe the target $Y$ and auxiliary variables $Z$ separately. Classical approaches typically invoke the conditional independence assumption (CIA), which makes the problem identifiable but fundamentally implies that the imported auxiliary variable provides no additional predictive power for $Y$ once $X$ is known. To capture this latent $Y$--$Z$ dependence, we propose a novel dependency-aware Schrödinger bridge for predictive statistical matching. Our approach couples the two separated databases by tilting the conservative CIA baseline with a transportation-based compatibility cost, recovering an informative joint distribution. The resulting statistical learning framework yields full probabilistic posterior rules for bidirectional imputation. Theoretically, we establish a sufficient condition under which the learned bridge strictly improves over the CIA baseline, alongside an exact joint recovery guarantee in the Gaussian setting under an appropriate cost. Across synthetic benchmarks and real-world datasets (CelebA and Adult), we demonstrate that our dependency-aware completion consistently improves downstream predictive utility, proving especially beneficial in settings like data recoding where the underlying population exhibits strong $Y$--$Z$ dependence.

Unsupervised Learning for Missing Modalities in Multimodal Learning

This paper addresses the missing-modality challenge in multi-modal learning by introducing Unsupervised Learning for Missing Modalities in Multi-Modal Learning (UL4M4), a flexible framework that imputes missing feature embeddings in a task-independent manner before supervised prediction. We propose modality-specific normalization and a novel partial-modality distance metric to enable fair clustering of incomplete observations, capturing cross-modal structures while preserving scale-invariance across varying dimensionalities and modality counts. Cluster centers from this unsupervised stage guide an iterative greedy imputation process for any missing modalities during training or inference, supporting arbitrary numbers of modalities and arbitrary missing patterns per sample. The imputation module is lightweight, uses frozen encoders, and decouples from the downstream task, allowing easy integration with any fusion/prediction architecture. Extensive experiments under diverse and highly incomplete regimes demonstrate UL4M4's robustness, achieving, to the best of our knowledge, the first consistent F1-Micro scores above 0.7 on challenging missing configurations even when more than 50\% of modality slots are missing. Results are also stable across cluster sizes and significantly outperform state-of-the-art baselines. Code is available here: https://github.com/h-ismkhan/Multimodal-Learning-with-Missing-Modalities-via-Unsupervised-Learning.

TS-ICL: A Flexible Time-Indexed Foundation Model for Time Series via In-Context Learning

Foundation models mark a profound paradigm shift in time series modeling, with task-specific models being superseded by general-purpose zero-shot models. Yet, current approaches primarily focus on forecasting, while real-world time series are often irregularly and partially observed, requiring models that can jointly forecast, impute missing values, and handle degraded sampling conditions. To address these challenges, we introduce TS-ICL, a novel probabilistic In-Context Learning encoder--regressor Transformer that unifies forecasting and imputation. TS-ICL formulates time series tasks as timestamp-aligned regression and naturally incorporates covariates by training on synthetic dependency structures generated from a novel causal data prior. Empirically, TS-ICL achieves a new state-of-the-art in imputation, while remaining competitive with leading forecasting foundation models across both univariate and covariate-aware benchmarks. It shows particularly strong performance in forecasting with partially observed look-back windows.

Learning What Not to Impute: An Uncertainty-Aware Diffusion Framework for Meaningful Missingness

Missing value imputation is a fundamental task in machine learning, with most existing methods assuming that all missing entries correspond to unobserved regular values. In many real-world datasets, however, missingness may arise from two distinct sources: some entries are meaningfully missing (intrinsically absent and semantically valid), while others are missing due to the observation process and should be imputed. We formalize this distinction as a selective imputation problem, where the goal is to jointly infer which missing entries should be preserved and which should be recovered. To address this challenge, we propose Diff-Joint, a diffusion-based framework that jointly models tabular data together with a latent missingness mask. The method alternates between conditional sampling and uncertainty-aware aggregation to iteratively refine both imputed values and missingness labels. Empirical results on synthetic and real-world datasets demonstrate that Diff-Joint effectively identifies meaningfully missing entries while achieving competitive imputation accuracy and improved downstream task performance.

Learning Causal Orderings for In-Context Tabular Prediction

In-context learning for tabular data sets strong predictive standards in observational settings; it however primarily relies on correlational structure, which becomes unreliable under distribution shift or intervention. While established methods to discover causal structure exist, they are often focused on structure identifiability and decoupled from the predictive architectures that could benefit from them. To bridge these perspectives, we study how to simultaneously infer and enforce causal structure in the form of topological variable orderings into tabular prediction. Unlike standard architectures, our model TabOrder uses causal order-constrained attention, basing predictions only on features that precede a target under a learned causal order. Similar to causal discovery methods, TabOrder learns the optimal variable ordering in an unsupervised manner through a likelihood-based objective. We justify this choice under standard functional model classes and also study how sample missingness, a common challenge in tabular data, interacts with causal direction identification. Empirically, we confirm that TabOrder recovers accurate variable orderings while addressing prediction and imputation tasks, as well as gives insight into real-world biological data under intervention.

End-to-End Learning for Partially-Observed Time Series with PyPOTS

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