Time-To-Event Data

The digitization of healthcare has generated vast, longitudinal, and multimodal patient records over a lifetime, yet fully exploiting these data to represent and predict patient state trajectories remains a critical challenge. Current AI models often struggle to capture the complex, irregular temporal dynamics and inherent stochasticity of real-world multimodal patient data. Existing AI approaches for modeling longitudinal patient records are predominantly discriminative, limited to a few modalities, constrained by closed categorical vocabularies, treating time as a monotonic inductive bias, or they are limited in forecasting future patient states. We introduce NOAH, a time-aware, task-agnostic, generative transformer model representing and forecasting the full multimodal patient journey. NOAH features a novel bidirectional time integration and a variational latent space to capture the continuous evolution of patient states and the stochasticity of clinical trajectories. Built from over 559 million clinical events from 431,000 hospital visits of 299,000 patients across the MIMIC dataset family, NOAH natively processes medical images, time-series and numeric signals, categorical events, as well as structured and unstructured clinical records. NOAH is the first truly holistic generative model in its field, enabling autoregressive forecasting with optional time control, zero-shot classification, and counterfactual intervention simulation. It generates highly informative and predictive patient state representations that demonstrate strong performance in probing for clinical outcomes, 15 ICD chapters, and 29 comorbidities, as well as in time-to-event prediction. Seamlessly handling diverse modalities and complex temporal dynamics, NOAH provides a versatile, task-agnostic, scalable foundation for intelligent predictive systems in personalized clinical care and digital medicine.

Kolmogorov-Arnold Networks (KANs) place learnable B-spline activations on network edges rather than fixed activations on nodes. The standard Cox-de Boor recursion evaluates these activations through $k$ sequential passes for degree-$k$ splines, consuming over 90% of forward-pass time. InKAN replaces this recursion with the truncated power form, a classical result from approximation theory that expresses each uniform cubic B-spline as five $(x)_+^3$ terms at shifted knot positions. This paper makes three contributions: (1) a torch.compile-fused implementation that collapses these operations into a single GPU kernel, eliminating all recursion, span lookup, and scatter-gather operations; (2) a bounded-coordinate stabilization that clamps the normalized input to $[0, k{+}1]$, preventing the catastrophic cancellation that historically motivated the Cox-de Boor recursion; and (3) a production-ready, open-source package (pip install inkan) that serves as a drop-in replacement for existing KAN layers.

Clinical data sources such as electronic health records and wearable sensors record patient status repeatedly over follow-up, often at irregular times and on different schedules for different measurements. These data create opportunities for continuously updated, individualized risk prediction. Existing approaches, however, often simplify the temporal structure for model fitting. We introduce Random Hazard Forests (RHF), a survival tree ensemble that estimates how a patient's hazard changes in continuous time as new measurements become available. The method formulates the estimation problem directly through a nonparametric hazard likelihood for predictable covariate processes. An efficient working model guides tree construction, after which flexible time-varying hazards are estimated for each terminal node. Given any predictable covariate path, each tree follows the path through its terminal nodes over time and assembles the corresponding node-level hazards into a trajectory. Averaging these trajectories across trees yields the pathwise hazard estimate. Because routing at each time uses only the covariate state available immediately beforehand, the construction accommodates internal longitudinal covariates without lookahead. Simulations and an intensive care application show that the forest accurately estimates changing risk under irregular and asynchronous covariate updates.

This work presents a novel attention-based framework for estimating the Individual Probability of Treatment Benefit (IPTB) in survival analysis contexts. The proposed model, called Surv-IPTB, directly quantifies the probability that a specific patient will experience extended survival time under treatment versus control. We reformulate IPTB estimation as a binary classification problem, leveraging pairwise patient comparisons across treatment and control cohorts. The framework incorporates a principled handling of right-censored observations through imprecise probability representations, where uncertain treatment effects are characterized by interval-valued probabilities. An attention mechanism with learnable query-key transformations enables flexible, data-driven aggregation of pairwise comparisons, while simultaneously learning soft class probabilities for censored cases. Through extensive experiments on synthetic datasets with complex nonlinear structures, including spiral, bell-shaped, and circular feature spaces, we demonstrate that our approach maintains robust performance across varying censoring rates and treatment effect strengths. The model consistently outperforms meta-learner baselines (T-learner and S-learner) equipped with random survival forests, Cox proportional hazards, and Beran estimators, particularly in challenging nonlinear scenarios where conventional methods exhibit significant degradation. The results establish the proposed attention-based framework as a scalable and statistically principled solution for personalized treatment benefit assessment in survival settings. The code implementing the model is publicly available.

Mean-reverting dynamics are pervasive in finance, and the Cox--Ingersoll--Ross (CIR) process is a standard model for the time series they produce, from short rates to credit default swap (CDS) spreads. Yet CIR models capture only \emph{correlated} co-movement, not \emph{causal} influence between series, so they cannot answer the system's response when one series is externally shocked, which observational conditionals confound with historical co-movement. We make two contributions. First, an amortized model for distributional causal effect estimation that frames trajectories as time-stamped observations and predicts the calibrated multi-horizon shock response without retraining per scenario. Second, a causal multivariate CIR data-generating process that supplies the paired observational and interventional ground truth that real markets cannot. We instantiate and calibrate the framework on CDS spreads as a testbed. CIR-ACTIVA's validity is established on synthetic ground truth, independent of how well the simulator matches reality, while practical grounding is assessed by backtesting the generated traces against real CDS data. Against observational and amortized causal-inference baselines, CIR-ACTIVA leads on both causal selectivity in the joint distribution and horizon-resolved calibration, retaining its selectivity once the interventional law varies over the horizon, with gains concentrating at short horizons. This opens up a class of what-if queries on coupled spread systems, CDS stress testing among them, that observational forecasters cannot answer.

Longitudinal tumor measurements, dropout information, and genetic covariates provide complementary information about treatment response, but integrating these data sources within a single population modeling framework remains challenging. We extend the empirical Bayes variational autoencoder (EB-VAE) framework to joint longitudinal and time-to-event modeling and evaluate it on tumor growth data. The framework represents inter-individual variability using latent individual effects regularized by a covariate-conditioned empirical Bayes prior, while a decoder maps these latent effects to tumor-volume trajectories. To account for informative dropout, the decoder was augmented with a hazard model, yielding joint predictions of tumor growth and time to dropout. We further compared fully neural and hybrid semi-mechanistic decoder formulations and incorporated genomic covariates through a genetics-conditioned prior adaptation. The hybrid decoder recovered treatment-effect parameters broadly consistent with previously reported nonlinear mixed-effects estimates, while achieving prior predictive performance comparable to the neural decoder. The joint model reproduced both tumor-volume distributions and dropout patterns in held-out individuals, and genetic conditioning improved individual-level prior predictions in both cutaneous melanoma and breast cancer experiments. Stability selection identified several biologically plausible genetic indicators, including alterations in BRAF, NRAS, NF1, and MDM2. These results demonstrate that EB-VAE provides a flexible probabilistic framework for combining neural dynamics, mechanistic structure, time-to-event modeling, and high-dimensional covariates in pharmacometric applications.

Amyotrophic lateral sclerosis (ALS) is a progressive and heterogeneous neurodegenerative disease in which predicting clinically meaningful milestones, such as assistive device use, remains challenging. We developed a time-to-event, digital-twin-inspired framework that integrates longitudinal ALS Functional Rating Scale-Revised (ALSFRS-R) trajectories with survival modeling to support individualized prediction of functional decline and assistive device utilization. We constructed a harmonized longitudinal dataset by integrating diagnosis records, ALSFRS-R assessments, activities of daily living, and demographic information, followed by preprocessing to ensure data quality, temporal alignment, and cohort consistency. Correlation-based clustering identified coherent functional domains spanning bulbar, upper limb, axial, lower limb, and respiratory systems. Generalized additive mixed models characterized nonlinear, domain-specific functional decline across all domains. In addition, a temporal machine learning model was developed to predict longitudinal functional decline and capture stage-dependent disease progression. Cox proportional hazards modeling further identified lower limb function, particularly walking and stair climbing, as the strongest predictors of earlier wheelchair access. Building on these results, we implemented a digital twin-inspired temporal machine learning-based time-to-event (TTE) model that generates individualized survival curves and dynamically predicts wheelchair-free survival. This framework provides a scalable, interpretable, and clinically actionable approach for linking ALS progression with personalized decision support, with applications in proactive care planning, clinical trial stratification, and precision medicine.

Spatio-temporal point-process models must often generalise across space when local event histories are sparse. We study whether exogenous spatial context can compensate in such regimes. Using a fixed log-Gaussian Cox process backbone, we compare an event-only model with the same model augmented by AlphaEarth embeddings as linear spatial context. We evaluate spatial transfer on emergency medical services (EMS) forecasting across eight held-out regions, fixed forecast anchors, and a sweep over history length $w$, using only AlphaEarth (AE) embeddings available strictly before each anchor. AE improves out-of-region predictive performance across all history regimes, with the largest gains under scarce histories: approximately $2$--$6\times$ multiplicative improvements at $1-2$ weeks, tapering to roughly $10$--$20\%$ at $w=20$--$104$ weeks. These results show that contextual information can substantially stabilise spatially transferred point-process forecasts when event history is limited.

Spatiotemporal point processes (STPPs) model event data in continuous time and space, with applications in mobility, epidemiology, and public safety. Recent neural STPPs span expressive intensity models, conditional density models, continuous-time latent dynamics, normalizing-flow spatial decoders, and score-based generative mechanisms. Yet comparison remains fragile because implementations differ in preprocessing, coordinate normalization, splits, likelihood conventions, and evaluation protocols. We present SEAHORSE, a unified framework for reproducible STPP experimentation. SEAHORSE formalizes neural STPPs through a common encode-evolve-decode interface and trains, tunes, and evaluates every model family under a single executable benchmark protocol with raw-coordinate likelihood reporting. This enables fair comparisons but, more importantly, controlled diagnostic studies. We pair SEAHORSE with HawkesNest, a synthetic stress-test suite, and show that increasing event-pattern complexity exposes each family's inductive bias, degrading some models sharply and leaving others stable. Code: https://github.com/YahyaAalaila/seahorse.

Parkinson's disease (PD) selectively impairs distinct stages of motor control. Using backspace events as natural error-correction episodes in the public neuroQWERTY MIT-CSXPD dataset (n=57 subjects, 27 PD with UPDRS-III scores), we test whether passively-collected keystroke timing dissociates a variability-based pre-error monitoring signal from a speed-based post-error recovery signal. Pre-error typing instability does not track PD severity (r=-0.072, p=0.721), while post-error pause duration does (r=+0.656, p=0.0002; subject-level OLS p=1.2x10^-4, n=27, primary analysis given within-subject event clustering). A mixed-effects log-normal model with a random subject intercept confirms this while retaining full event-level power (coef=0.0250, p=1.2x10^-4, n=1,563 events), closely matching the subject-level OLS despite an unrelated estimation strategy. Error-detection latency also correlates with UPDRS-III (r=+0.660), confirming the dissociation is between variability and speed, not detection and correction per se. A log-normal accelerated failure time model yields a coefficient of 0.0255 (bootstrap 95% CI [0.0146, 0.0357]), a 2.6% increase per UPDRS-III point. After controlling the immediately thecoefficient attenuates to 0.0133 (p=3.1x10^-21),consistent w timingincrement above general bradykinesia. Results reacross both S2:r=+0.645), survive jackknife exclusion of every subject, andnatingfinger-tapping and mPower smartphone tapping (group AUC=0.836). cellent(ICC(2,1)=0.945, n=20).

Alternating recurrent events -- event-times of a specific nature that trigger a secondary refractory period -- occur in a wide-range of fields, including behavioral science, criminal justice, and biostatistics. Analysis of these events requires careful attention to the statistical nuance, including correlated observations and repeated outcomes subject to potential censoring. We develop an online dynamic prediction framework appropriate for predicting subsequent alternating recurrent events, by developing neural network theory for a statistical audiences and applying inverse probability weighted pseudo-observations. The proposed model is applied to dynamically predict alternating recurrent event-free time, showing good performance in simulation, and outstanding capability in application to predicting periods of low mood for first-year medical residents. We close with a discussion.

Survival analysis is central to clinical decision-making, yet reliable time-to-event models require large, diverse cohorts that are rarely available at a single institution, while privacy regulations restrict the centralization of patient data. Federated learning (FL) offers a privacy-preserving alternative by training shared models without exchanging raw data, but its effectiveness for survival modeling under realistic, heterogeneous conditions remains insufficiently understood. This paper presents a systematic, multi-model evaluation of federated survival analysis on a cross-institutional breast cancer cohort with naturally heterogeneous distributed clients. Three representative survival models, the Cox Proportional Hazards model, DeepSurv, and Random Survival Forest (RSF), are compared across centralized, local, and federated training, and three federated optimization strategies (FedAvg, FedProx, and FedAdam) are assessed for the gradient-based models. Results show that FL consistently outperforms local training and approaches, and occasionally exceeds, centralized performance, while RSF offers the best overall balance of discrimination, calibration, and robustness across heterogeneous clients. We further find that performance depends on the diversity of client distributions, and that FedAvg and FedProx are stronger and more stable than FedAdam. Based on these findings, we derive practical, decision-oriented guidelines mapping data, privacy, interpretability, and resource constraints to recommended model and training-paradigm choices for federated survival modeling in healthcare.

Competing risks are commonly observed in engineering fields and can bring challenges to time-to-event data modeling when the application scenarios are complicated. Recently, deep neural networks have received great attention for prediction with competing risks, due to their flexibility and high learning capability. However, the complexity of neural network structure brings extra difficulty in hyperparameter tuning based on different data inputs. Additionally, when an engineered system has complex physical structures with multiple hierarchical levels, treating all structural levels as a single group of inputs may fail to capture critical information. To address the issues, we propose a Structured Segmented Hazard Deep Neural Network (SSH-Net) for failure time prediction under cause-specific competing risks framework. Our approach associates neural network structure with data structures, and allows different covariate groups to impact the failure prediction through separate sub-networks. The neural network is constructed based on a cause-specific competing risks model. The SSH-Net outputs cause-specific hazard functions, and utilizes the penalized log-likelihood as the loss function. The prediction accuracy of SSH-Net is validated through simulation studies by evaluating the Brier score, the area under receiver operating characteristic curves (AUC), and the root mean square error (RMSE) of the predicted cause-specific cumulative incident function. We further demonstrate the model's ability to predict failure time distribution functions using the Titan GPU failure time data.

Accurate time-to-event (TTE) prediction from multimodal clinical data remains challenging due to modality imbalance and distribution shift. We introduce a foundation model-driven framework for cross-modal representation alignment between CT imaging and longitudinal EHR data, designed to generalize across tasks and institutions. CT and EHR modalities are encoded independently using domain-specific foundation models and aligned in a shared latent space through four principled fusion strategies: late fusion, contrastive alignment, cross-attention, and co-attention. We evaluate two clinically distinct TTE tasks: pulmonary embolism (PE) mortality and cardiovascular disease (CVD) outcomes, on large-scale multi-institutional cohorts (PE: N=3,099 train; 1,098 internal; 435 external; CVD: N=2,951 train; 837 internal; 682 external). Fusion consistently improves concordance index by 1.5-5.4% over unimodal baselines when modalities contribute comparably. Overall, contrastive multimodal fusion, particularly with CLMBR representations, provided the most consistent and statistically robust improvements, especially for PE mortality prediction. For MACE, cross-attention (one-hot) achieved the highest internal performance and image-guided co-attention achieved the best external performance. We therefore introduce a generalizable foundation model-based cross-modal alignment framework and provide the first systematic analysis of fusion behavior under modality imbalance in TTE prediction. Our results establish task-aware multimodal alignment as a necessary design principle for robust generalization and scalable clinical deployment.

Current-status data arise when an event time is observed only through an indicator of whether it occurred before an examination time. This paper studies a nonparametric neural-network sieve maximum likelihood estimator of the conditional cumulative distribution function of the event time. Under Hölder smoothness assumptions, we establish an explicit convergence rate by combining approximation theory for rectified linear unit neural networks with empirical-process arguments. This result provides theoretical support for neural-network estimation and subsequent inference under current-status observation.

Biochemical recurrence (BCR) after radical prostatectomy is a critical endpoint in prostate cancer, yet risk stratification relies almost entirely on variables dominated by Gleason grade. Whether H&E whole slide images (WSIs) carry prognostic signal beyond grade, and whether multiple instance learning (MIL) can recover it, remains unsettled. A key obstacle is that many pipelines select model checkpoints on the evaluation fold, artificially inflating concordance. We construct a rigorous benchmark on TCGA-PRAD (487 patients, 101 BCR events) using strict out-of-fold scoring over five-fold cross-validation repeated across five seeds. The choice of MIL aggregator (ABMIL, CLAM, TransMIL, PatchGCN) has little effect (C-index 0.61-0.64 with UNI2-h), while the feature extractor is the dominant factor (ResNet50 0.566 versus pathology foundation models up to 0.639). A clinical Cox model on grade, stage, and age reaches 0.687; no imaging-only model significantly outperforms it (p > 0.10). We introduce Grade-Disentangled MIL (GD-MIL), a gated-attention MIL encoder trained with a gradient-reversal grade adversary that encourages the slide representation to be invariant to Gleason grade before late fusion with clinical variables. GD-MIL achieves C-index 0.704, significantly outperforming both the clinical baseline (delta-c = +0.029, p = 0.0005) and the best imaging-only model (delta-c = +0.062, p = 0.039), suggesting H&E morphology contains prognostic information complementary to grade. A median risk split yields log-rank p < 0.0001 separation in BCR-free survival (~20% vs ~70% at five years).

We investigate whether information about time-to-event risk estimated by a Cox proportional hazards model can be transferred into a generative large language model. We propose a text-based survival modelling pipeline in which structured clinical covariates are converted into text prompts and a Qwen-based large language model is fine-tuned to generate patient-specific survival risk using Cox model predictions as a training target. Across GBSG2, ACTG320, and WHAS500, the model achieves competitive held-out discrimination and calibration despite being trained as a text-generation task rather than with a conventional survival-analysis loss. We further analyse the geometry of the model's hidden states, where t-SNE visualisations reveal smooth risk gradients in latent space, suggesting that the model represents survival risk as a continuous structure rather than isolated risk categories. Together, these findings suggest that large language models can internalise survival-risk structure while supporting calibrated prediction, providing a route towards time-to-event reasoning in language models.

Externally controlled survival trials are increasingly used when concurrent randomized controls are infeasible, particularly in oncology and rare-disease settings with time-to-event endpoints. We target an average-treatment-effect-on-the-treated (ATT)-type marginal hazard-ratio estimand, comparing treatment with counterfactual control in the treated trial population, and estimate it using inverse-probability-weighted (IPW) Cox regression. Valid inference is challenging because IPW Cox regression depends on the weights through both event contributions and risk-set averages, making flexible machine-learning nuisance estimation difficult to incorporate directly. Building on machine-learning-assisted generalized entropy calibration (MEC) by Lee and Kim (2026), we propose MEC-Cox for ATT-weighted IPW Cox regression. The method begins with normalized source-propensity-score odds weights for external controls and then applies Bregman calibration to balance cross-fitted prognostic summaries between external controls and treated trial patients. The calibration basis may include control-survival predictions, Cox linear predictors, penalized-survival-model predictions, or other prognostic-score summaries. MEC-updated weights therefore play a dual role as source-transport and prognostic-score balancing weights. We establish consistency, characterize a calibration-induced efficiency gain, and develop a stacked sandwich variance estimator. Simulations show that MEC-Cox can reduce bias, increase efficiency, and improve coverage through flexible machine-learning-assisted adjustment.

Survival Analysis (SA) is a statistical framework that models the time span until some event of interest occurs. Widely used in several domains, including healthcare and churn prediction, a central challenge in its applicability stems from the time of the event being partially observed or \emph{right-censoring}. Tabular Foundation Models (TFM) have attracted significant interest in recent years due to their ability to perform prediction tasks in a single forward pass, requiring no dataset-specific parameter fitting. Despite their success, their application to prediction tasks on time-to-event data remains difficult due to right censoring. In this work, we present a training-free method to survival regression by leveraging TFMs to both predict the time of the event and iteratively impute right-censored data. Our method uses a TFM to construct an Accelerated Failure Time (AFT) model requiring no training beyond fitting a single scalar parameter. Subsequently, by building on the Buckley-James estimator, we introduce a non-parametric in-context estimator for right-censored data. Our experiments on standard survival analysis benchmarks show that our method is competitive with several parametric and semi-parametric survival regression models that require training, including Cox regression and parametric AFT models.

A mobile robot following a graph of known routes can make costly navigation errors when a temporary obstacle blocks a critical edge: waiting too long behind a parked cart wastes time, but immediately rerouting around a person who would move in a few seconds is also inefficient. Standard reactive obstacle avoidance addresses local motion around obstacles, while fixed wait-or-reroute rules ignore how long different obstacle types tend to persist. We propose OSCAR: an adaptive survival-modeling framework for graph-based navigation with temporary blockages. Assuming obstacle class labels are available at encounter time, the robot learns class-conditioned residual clearance-time distributions from online experience, including right-censored observations when it reroutes before observing clearance. These survival models are integrated into a time-dependent graph planner that maintains obstacle memory and computes a patience threshold at each blocked edge: how long to wait before taking an alternate route. The method continuously updates its clearance estimates across episodes and uses them to balance waiting against rerouting. We evaluate the approach in simulation and on a real mobile robot in a university atrium with obstacles including people, chairs, bins, and tubes. In simulation, the learned policy's time-to-goal converges to within 1% of an oracle with access to ground-truth clearance distributions after fewer than 20 observations per obstacle class, outperforming all heuristic baselines. Real-world deployment confirms that the policy improves online, adapting its patience thresholds from experience across 50 navigation episodes.

Survival analysis aims to estimate a time-to-event distribution from data with censored observations. Many existing methods either impose structural assumptions on the hazard function or discretize the time axis, which may limit flexibility and introduce approximation errors. We propose the Survival Diffusion Probabilistic Model (SDPM), a generative approach to continuous-time survival analysis. SDPM models the conditional distribution of the survival outcome, represented by the pair of observed time and censoring indicator, $\mathbb{P}(T,δ\mid \mathbf{x})$, using a denoising diffusion model. Under the assumption of conditionally independent censoring, conditional samples generated by the model can be transformed into survival function estimates using the Kaplan-Meier estimator. This formulation avoids parametric assumptions on the event-time distribution and does not require a discretization of the output time space. The model operates in a transformed target space, using standardized log-times and a continuous Gaussian-mixture representation of the censoring indicator. We evaluate SDPM on ten real survival datasets and compare it with five strong baselines, including tree-based, boosting-based, and neural survival models. Results show that SDPM achieves competitive predictive performance across C-index, integrated time-dependent AUC, and integrated Brier score. A study on synthetic Cox-Weibull data demonstrates that SDPM can recover the shape of an underlying continuous survival distribution more accurately than a strong nonparametric baseline when sufficiently many samples are generated. An ablation study confirms the importance of the proposed target-space transformations, which improve event-rate calibration, reduce invalid generated times, and provide consistent gains in predictive discrimination. Codes implementing the proposed model are publicly available.

Predictive modelling is important for health data analysis and data-driven clinical decision-making. However, predictive studies are challenging to design optimally by hand when tens or even hundreds of features require selection, transformation, or interaction modelling. While complex machine learning models offer high performance, their "black-box" nature limits the clinical trust, transparency, and interpretability required for decision-making. We developed and evaluated an Exploratory AI Recommender that provides data-driven recommendations to improve predictive performance of existing interpretable statistical models. The developed framework uses flexible AI modelling to capture complex data patterns and explainable AI techniques to translate the patterns into three recommendation types: feature exclusion, non-linear terms, and feature interactions. We evaluated the framework by comparing predictive performance of a baseline (i.e., no interactions or non-linear terms) Cox Proportional Hazards (CPH) model against an augmented CPH incorporating recommendations suggested by our method. The primary analysis predicts the time to the first occurrence of a fall or related injury in 245,614 patients. Our method recommended excluding 23 features, including non-linear terms for two features, and including 221 suggested feature interactions. The C-index improved from 0.805 (95% CI 0.798-0.812) to 0.815 (95% CI 0.809-0.822), and so did calibration (intercept: -0.006 to 0.003; slope: 1.063 to 0.950). All recommendations were supported by existing literature. The method also proved effective on two additional public datasets, demonstrating wider applicability. The proposed Exploratory AI Recommender demonstrates the potential of explainable AI and data-driven study design to improve the process of developing, and the performance of high-dimensional transparent predictive models.

Infrastructure deterioration poses significant challenges for asset management, yet existing approaches rely on population-averaged models that overlook equipment-specific heterogeneity. We present a novel framework that combines Bayesian hierarchical hazard modeling with causal discovery to identify operational patterns that drive heterogeneous deterioration rates in pump equipment. Our approach first estimates pump-specific random effects $u_i$ using GPU-accelerated No-U-Turn Sampling (NUTS), achieving 3--5$\times$ speedup over CPU implementations. We then employ DirectLiNGAM to discover causal relationships between 22 engineered time-series features and deterioration rates, stratified by positive ($u_i > 0$, faster deterioration) versus negative ($u_i \leq 0$, slower deterioration) random effects. Analyzing 112 pumps with 92,861 observations over 650 days, we uncover striking heterogeneity: the negative group exhibits causal effects 400$\times$ larger than the positive group, with standard deviation (std) showing a strong positive causal effect ($+1.515$) on deterioration rates in low-risk equipment. We validate linearity assumptions through NonlinearLiNGAM comparison and demonstrate practical scalability through GPU acceleration. Our findings enable targeted maintenance strategies by revealing that different operational regimes require fundamentally distinct management approaches, advancing predictive maintenance from population-averaged to heterogeneity-aware decision making.

Time-to-event data is widespread across the life sciences and engineering, but it is typically encountered together with censoring, which complicates the application of standard machine learning methods. Deep Cox models have emerged as a popular method for analyzing time-to-event data because they gracefully handle censoring and can be used with unstructured data such as clinical text reports, genomic sequences, and pathology images. However, their predicted survival probabilities are often poorly calibrated, thus limiting their practical utility. In this paper, we propose a novel post hoc calibration method for Deep Cox models that uses isotonic regression to refine predicted survival probabilities without affecting discriminative power. We establish favorable theoretical guarantees, including a double-robustness property and asymptotic calibration. Experiments on synthetic and real-world clinical data demonstrate the empirical effectiveness of our method.

Flexible continuous-time survival modeling is critical for capturing complex time-varying hazard dynamics in high-dimensional data; however, training such models remains challenging due to the intractable integral required for likelihood estimation. We introduce QSurv, a scalable deep learning framework that enables nonparametric continuous-time modeling without relying on time discretization or restrictive distributional assumptions. We propose a training objective based on Gauss-Legendre numerical quadrature, which approximates the cumulative hazard with high-order accuracy while facilitating efficient end-to-end training via standard backpropagation. Furthermore, to effectively capture non-stationary hazard dynamics in complex architectures, we introduce time-conditioned low-rank adaptation, a mechanism that conditions general neural backbones on time by dynamically modulating weights via low-rank updates. We provide theoretical analysis establishing approximation error bounds for cumulative-hazard evaluation. Comprehensive experiments across synthetic benchmarks, large-scale real-world tabular datasets, and high-dimensional medical imaging tasks demonstrate that QSurv achieves competitive predictive performance with advantages in instantaneous hazard function estimation, enabling more interpretable characterization of time-varying risk patterns.

Dynamics of interacting systems in engineering, society, and nature often evolve over latent networks that govern which entities can interact. We study the problem of inferring these networks from event-based observations, which arise naturally in finance, seismology, and neuroscience. While there is substantial algorithmic work addressing this important problem, theoretical results are scarce. In this paper we ask the following fundamental question: what is the minimum time that one must observe the dynamics in order to exactly recover the underlying network, as a function of the number $d$ of interacting entities? For a class of stationary Hawkes processes with sparse, weak interactions, we prove that an observation time of order $\log d$ is sufficient and necessary. For the upper bound we construct a two-stage estimator that uses clipped and binned event data for screening, followed by a least-squares refinement, and apply concentration bounds derived from the Poisson cluster representation. For the lower bound we combine Fano's inequality with Jacod's Girsanov formula for point processes on a suitable subclass of networks.

Survival analysis is a widely used statistical framework for modeling time-to-event data under censoring. Classical methods, such as the Cox proportional hazards (Cox PH) model, offer a semiparametric approach to estimating the effects of covariates on the hazard function. Despite its importance, survival analysis has been largely unexplored in online settings, particularly within the bandit framework, where decisions must be made sequentially to optimize treatments as new data arrive over time. In this work, we take an initial step toward integrating survival analysis into a purely online learning setting under the Cox PH model, addressing key challenges including staggered entry, delayed feedback, and right censoring. We adapt three canonical bandit algorithms to balance exploration and exploitation, with theoretical guarantees of sublinear regret bounds. Extensive simulations and semi-real experiments using SEER cancer data demonstrate that our approach enables rapid and effective learning of near-optimal treatment policies.

We revisit Cox's proportional hazards model to improve variable selection in survival analysis. A square-root transformation of the partial likelihood renders the selection of the regularization parameter pivotal, free of the unknown baseline hazard and censoring mechanism. The resulting criterion borrows from information criteria such as BIC and from penalized regression methods such as the lasso, taking the best of both. On simulated and real data, our method substantially improves upon state-of-the-art approaches used daily in support recovery.

Brownian motion is a compact mathematical language for continuous-time uncertainty in biostatistics. This tutorial develops the process from construction and path properties to tools that recur in applied biomedical work: the Markov and strong Markov properties, the Karhunen-Loeve expansion, functional principal component analysis (Functional PCA), reflection principles, local time, stochastic differential equations (SDEs), Brownian bridges, and empirical-process limits. The applications emphasize longitudinal biomarkers, degradation modelling, first-passage endpoints, dynamic frailty, group-sequential monitoring, calibration diagnostics, recurrent-event processes, electronic health records, and wearable streams. A short cross-domain section uses literary and historical archives to make Brownian-bridge thinking concrete without shifting the paper away from biostatistics, and includes a reproducible chapter-level experiment on Frankenstein. The Black-Merton-Scholes model is included as a solved SDE template, not as a finance application in its own right. The aim is to connect rigorous probability with modelling decisions faced by biostatisticians when biological processes evolve between noisy observation times.