Papers

Time-warping estimation is a fundamental problem in signal processing with applications in bioacoustics, radar, and biomedical analysis. This paper introduces a Time-Warping Estimation Trainable (TWET) model for estimating timewarping functions from a single observation. The proposed approach formulates time-warping estimation as a stationarization problem in the wavelet domain and leverages a hierarchical dilated convolutional architecture to estimate the time-warping functions. A differentiable stationarity criterion is introduced for end-to-end optimization. TWET is compared with existing approaches. Experimental results show improved deformation reconstruction accuracy together with significantly reduced computation time, making the framework compatible with low-latency applications.

Degree-sequence realizability is the combinatorial basis of configuration models, but degree constraints alone do not ensure the preservation of graph dynamics. Hence, configuration models are unable to recover centrality measures, unless these are strongly correlated with the degree sequence. To address this matter, we introduce EP-realizability, the analogue problem induced by an equitable partition (EP): given the EP of a graph, decide whether the partition is realized by a simple undirected loopless graph and therefore construct such a graph. After defining the problem, we solve it by reducing it to sub-problems related to Havel–Hakimi and the Gale–Ryser theorem. We also face the challenge of solving the problem with an Approximate Equitable Partition ($\varepsilon$-EP), so that it is possible to reconstruct a network starting from partial and more privacy-preserving information. We evaluate privacy with edge overlap, deriving also, for our proposed $\varepsilon$-EP-realizability solution, a predictor for this metric. Experiments on Karate, Cora, CiteSeer and PubMed datasets show that our algorithm achieves a favourable and tunable privacy–utility trade-off, comparing the results with Havel–Hakimi algorithm, Newman's configuration model and a stochastic block model. Finally, both with real data and random graphs, we show that our algorithm has approximately linear time complexity with respect to the number of edges.

Longitudinal continuous outcomes in clinical trials are commonly analyzed using mixed models for repeated measures (MMRM) under normality assumptions. However, many clinical outcomes are skewed, making mean-based treatment effects difficult to interpret and potentially reducing statistical efficiency. The Box–Cox MMRM (BCMMRM) approach accommodates skewness by enabling inference on model-based median differences via inverse transformation. However, BCMMRM typically assumes a common transformation parameter across treatment groups and time points. When distributional shapes differ between groups or evolve over time, this assumption may lead to biased treatment effect. Furthermore, when treatment affects not only central tendency but also distributional shape or tail behavior, treatment effects may not be adequately characterized by a single location summary such as the median. We propose the Box–Cox multivariate regression (BCMVR) framework for longitudinal data with skewed outcomes. BCMVR relaxes this restriction by allowing transformation parameters to vary across groups and time points. The framework enables inference based on interpretable summaries, including median differences and a probability-based treatment effect quantifying the probability that a randomly selected patient in one group has a better outcome than one in another group. This measure integrates information over the entire outcome distribution and provides a complementary summary when distributional shapes differ. Simulation studies demonstrate that BCMMRM can produce biased estimates when distributions differ in shape, whereas BCMVR provides nearly unbiased estimation. The probability-based measure achieves a favorable balance between robustness and statistical efficiency. The proposed framework provides a flexible and interpretable approach to treatment effect inference under distributional heterogeneity.

This work addresses a longstanding gap in the statistical foundations of marginal maximum likelihood estimation for high-dimensional latent variable models. Marginal maximum likelihood estimation is widely used to fit latent variable models across the social sciences, ecology, and machine learning. Despite its broad use, rigorous asymptotic theory for nonlinear models remains limited when both the sample size and the number of observed variables diverge. The gap arises largely from the fact that integration over the latent variables creates a nonlinear objective that tightly couples the high-dimensional model parameters. To address this issue, we first show that the marginal likelihood exhibits multiple nearly flat directions even at the true parameter, in contrast to the behavior in the fixed-dimensional case. Building on this geometric characterization, we develop new techniques to establish consistency and asymptotic normality for the marginal estimator of the high-dimensional parameters. For the latent variables, we provide frequentist and Bayesian uncertainty quantification, proving asymptotic normality of the maximum a posteriori estimator and a Bernstein-von Mises-type result for the plug-in posterior. Together, these results provide rigorous foundations for marginal estimation and latent-variable inference in high-dimensional models.

Parameter estimation in finite mixture models can exhibit highly heterogeneous convergence behavior: locally isolated components may be estimated substantially faster than groups of competing components. Existing analyses based on Wasserstein distances typically characterize only the worst-case rate and therefore do not fully capture this local heterogeneity. In this paper, we introduce a Voronoi-based partial optimal transport (VPOT) framework for obtaining refined local and global convergence guarantees for the maximum likelihood estimator of the mixing measure. The key geometric idea is to localize the comparison of two mixing measures to extended Voronoi neighborhoods and use partial optimal transport to accommodate the unequal masses of their local restrictions. Within each neighborhood, the first-order POT discrepancy is raised to a power determined by the number of locally competing atoms, allowing the resulting loss to adapt to the local degree of singularity. Under suitable regularity and strong identifiability conditions, we establish uniform local and global upper bounds for a maximum likelihood estimator under the VPOT loss. These bounds reveal a configuration-dependent form of parameter estimation: less singular local configurations admit faster convergence, whereas the most singular configuration recovers the classical worst-case behavior characterized by Wasserstein-based analyses. We further establish a minimax lower bound showing that the convergence rate for estimating the mixing measure under the VPOT loss is optimal. Our results hold in arbitrary fixed dimension without requiring mixing proportions to be uniformly bounded away from zero or prior knowledge of the true number of mixture components. Overall, VPOT provides a configuration-adaptive framework for capturing heterogeneous parameter-estimation behavior in finite mixture models.

Neural PDE solvers provide efficient surrogates for time-dependent physical systems, but autoregressive prediction over long horizons remains challenging because local errors can induce distribution shift and accumulate under recursive deployment. We develop a variational approach to this problem by introducing latent Markov dynamics in which physical states are represented by latent distributions and evolved through probabilistic transitions. The framework is formulated directly on function spaces and specialized to functional Gaussian models, where structured latent perturbations induce a spectral geometry and variational transition alignment regularizes the learned dynamics. We further analyze how these mechanisms affect autoregressive error propagation, providing a theoretical connection between variational training and long-horizon prediction. We instantiate the framework as the Variational Autoencoding Markov Operator (VAMO), which combines spatially resolved latent fields, structured Gaussian perturbations, and a neural-operator transition. Empirically, we demonstrate the effectiveness of VAMO on several fluid-dynamics benchmarks with prediction horizons extending substantially beyond those represented during training, where it consistently reduces error accumulation and improves rollout stability over several deterministic and noise-injection baselines. Overall, these results highlight variational modeling as a complementary approach to robust long-horizon neural PDE dynamics.

Model robustness analysis estimates an effect across a multiverse of specifications that pools control sets identifying the declared estimand with sets that condition on mediators or colliders. We propose stating rival assumptions about contested controls as a small set of candidate causal graphs, enumerating the adjustment sets each graph licenses, and reporting robustness metrics conditional on each graph. A finite-mixture identity splits the licensed multiverse's dispersion into within-graph and between-graph components; the between-graph share is a conditional descriptive summary whose reading depends on the candidate set, the weights, and a common estimand. Simulations examine misleading pooled robustness assessments and the limits of the decomposition. Applications to hurricane fatalities, job training, and union wages show fragility that survives every graph, instability produced by unlicensed specifications, and a fragility verdict concealing a significant premium in each adjustment-identified candidate world. An R package implements the workflow.

Causal discovery methods aim to determine the causal direction between variables using observational data. Functional causal discovery methods rely on structural and distributional assumptions to determine directionality but typically lack statistical inference. This paper reviews the statistical guarantees of existing functional causal discovery methods in theory, software, and applied use, highlighting a key gap: the absence of a unified inferential framework that applies broadly across model classes. As a first step toward addressing this gap, we formalize a test-based approach for bivariate causal discovery by repurposing goodness-of-fit and independence tests within a hypothesis-testing framework. Because directionality is determined through two disjoint hypothesis tests, corresponding to four causal discovery outcomes, the approach provides explicit uncertainty quantification and diagnostic insight into assumption violations. Resampling is further used to estimate the rates of causal discovery outcomes, offering an additional layer of inference. We demonstrate the use and behavior of our inferential framework through simulations that vary the degree of assumption violation, as well as through real-data applications. We conclude with practical lessons and recommendations for advancing statistical guarantees in functional causal discovery.

Reserving models increasingly rely on loss-based estimation, where the loss function encodes the assumed error structure. Mack demonstrated this for the chain ladder, showing that the volume-weighted average estimator minimises a volume-weighted squared-error loss function that is additive in successive claim developments. This paper instead considers a multiplicative error structure and proposes the corresponding volume-weighted squared log-error loss function. Adapting Mack's distribution-free framework, we show that this loss function is minimised by the volume-weighted geometric average of the individual development ratios. This provides practitioners with an alternative estimator of development ratios for chain-ladder-based models, and a candidate loss function for machine-learning-based reserving models. We further show that the same estimator arises from two independent arguments: a stability requirement on successive ultimate loss projections, and maximum-likelihood estimation under a log-normal model. The estimator thus admits three complementary justifications: loss minimisation, reserve stability, and parametric likelihood. An out-of-sample study of 362 Schedule P company-line datasets supports the use of the proposed estimator in place of the volume-weighted average for the chain ladder, by showing that it improves predictive accuracy and reduces a slight over-prediction bias.

The chain ladder's volume-weighted pattern minimises an explicit loss function, yet is rarely booked as such. Practitioners adjust the pattern and record the final adjusted ratios. This paper treats the chain ladder's pattern selection as a supervised-learning problem. Judgement on pattern adjustments becomes a framework of defined penalties and hyperparameters on the chain ladder's loss function, treated here as an objective function in machine learning. Data weights are generalised with a decay and a power parameter for recency and volume weighting. Benchmark shaping and smoothness enter through a reference penalty and Whittaker-Henderson smoothing. The assembled objective is strictly convex and minimised by a single linear system. Each hyperparameter becomes an interpretable adjustment in its own right, declarable by judgement and categorised as an experience or a prospective adjustment. Experience adjustments can be set more objectively by a proposed training loop and a reserve validation score on held-out calendar diagonals. Further hyperparameter-based adjustments are written as almost-everywhere differentiable penalties that re-time or reshape the pattern. A worked example carries one real Schedule P triangle through an incurred and then a paid training stage, demonstrating the workflow.

Single-cell perturbation experiments provide causal information on gene regulation, whereas population-scale single-cell studies characterize gene expression and phenotypes in human populations. We develop a framework that integrates these complementary data sources for causal path analysis. Rather than assuming that a perturbational gene network transfers directly to the target population, we use externally learned ancestral relationships to constrain the network topology and re-estimate its direct edges and effects from population data. To address latent heterogeneity and measurement error in multiscale single-cell measurements, we develop a surrogate-variable procedure operating at both the cell and subject levels, combined with errors-in-variables correction for network and outcome regressions. We establish theoretical guarantees for confounder recovery and high-dimensional estimation of network and gene-outcome effects. Simulations demonstrate the importance of jointly correcting confounding and measurement error. An application to acute myeloid leukemia identifies distinct regulatory pathways linking transcriptional regulators to blast count.

Anthropogenic forcing components follow different long-run paths, while persistent temperature change can involve distributional changes beyond the mean. Scalar regressions aggregate these components and retain only mean temperature, obscuring how distinct forcing paths relate to persistent distributional change. We develop new testing, estimation, and inference methods for long-run relations between an integrated predictor vector and a density-valued response. These comprise a residual-based test of between-cointegration (whether predictor trends account for all stochastic trends in the response density), a fully modified least-squares estimator of predictor-specific functional responses, and simulation-based inference for interpretable projections. We apply the methods to densities of observed local temperature anomalies and anthropogenic effective radiative forcing divided into CO$_2$ and non-CO$_2$ portfolios. The test results are consistent with persistent movements in these portfolios statistically accounting for the persistent evolution of the anomaly distribution, with no additional stochastic trend detected in the residual. A joint test rejects the common-response restriction imposed by aggregating the two portfolios. The fitted CO$_2$ response mainly shifts mass toward warmer anomalies and increases central concentration, whereas the non-CO$_2$ response produces a smaller shift but greater dispersion and off-center reshaping. Positive fitted mean responses for both portfolios conceal these contrasts, demonstrating the information lost through scalar aggregation.

Local inference for spatial point distributions is dominated by Monte Carlo calibration. We develop an alternative based on the Tweedie–Miyasawa identities of empirical Bayes, which relate locally weighted moments of a point distribution under a Gaussian kernel to derivatives of its log-intensity in scale space. We first establish a rigidity theorem showing that the structure of these identities forces the Gaussian kernel. Under complete spatial randomness, we derive a closed-form null distribution for a bounded inter-scale contrast, yielding a calibrated simulation-free pointwise test. In experiments, the measured type I error is 0.070 at a nominal level of 0.05, with a calibration cost 199 times smaller than Monte Carlo for the same local statistic. We then derive a contamination law for structures of dimension m and width w embedded in a uniform background, together with an explicit detectability threshold. For a filament in three dimensions, the threshold is 16 pi. The resulting scale-resolved local dimension estimator has no free parameters. Applications to California seismicity, a trefoil knot, and 10,071 SDSS galaxies show that the method separates local structures across scales without a spatial mesh and reproduces published cosmic web fractions. All experiments are reproducible from a single public script.

Neighborhoods are multifaceted entities whose attributes span sociocultural, economic, and environmental dimensions. Clustering neighborhoods based on social determinants of health (SDoH) can inform the allocation of public health resources and interventions tailored to community needs. Continuous and bounded data, as seen in aggregated neighborhood-level SDoH data, can pose challenges for widely used clustering methods (e.g., k-means or Gaussian mixture models) that rely on Euclidean distances or normality assumptions. Moreover, not all SDoH variables contribute equally to meaningfully characterize neighborhood clusters, and the inclusion of irrelevant or weakly informative determinants can obscure the underlying structure. To address this issue, we propose a Bayesian Salient Multivariate Beta mixture (SAMBA-Mix) model that flexibly models proportional data with embedded feature saliency to identify relevant variables for clustering. Our model showed improved accuracy in cluster identification and structure, compared to existing models. Using data from the U.S. Agency for Healthcare Research and Quality(AHRQ) SDoH database, we demonstrate the utility of our model to derive neighborhood-level SDoH profiles in Massachusetts and identify key determinants that best characterize differences amongst the different profiles.

We develop a certified continuation framework for inference and training in deep equilibrium networks (DEQs), with training posed as interpolation to accuracy $2^{-b}$. For inference, input homotopy selects an equilibrium branch from a supplied start root, and a rounded tracker follows it under quantitative conditioning, derivative, boundary, and tube-radius certificates. The framework includes structured factorized certificates, sequential block elimination, inheritance of contraction guarantees in adapted coordinates, and bordered continuation through simple folds. For smooth multidimensional DEQs, including tanh networks, rational local tests can construct and validate oriented continuation charts under explicit geometric promises. For training, programmable dormant bilinear rank-one channels provide output-preserving residual-aligned repairs. Loaded Tikhonov solves diagnose insufficient parameter-to-output directions, while certified gate realization, column stability, well-posed inference, and finite-update error budgets control each pass. Under polynomially bounded certificate, encoding, precision, and backend costs, both inference and training have bit complexity $O(\mathrm{poly}(L+b))$, where $L$ is the encoded instance length; training uses $O(b+\ell)$ passes and reserve channels from an initial residual bounded by $2^\ell$. A budgeted implementation returns either certified success or inconclusive termination. The quantitative core and local certificate machinery are machine-checked in Lean 4, while numerical experiments illustrate the training mechanism.

Our understanding of how geographical and social segregation interact remains limited, as relatively few studies investigate them jointly, and existing approaches often lack a framework distinguishing geographical, social, and total segregation. Additionally, large-scale individually resolved geo-social network data are rarely publicly available. We address both. Conceptually, we develop a unified framework that measures segregation in geosocial networks by comparing network models to appropriate null models and recovers the Theil index, dissimilarity index, and network modularity as special cases. Empirically, we turn to privacy-preserving aggregated relational data (ARD): we combine the Facebook Social Connectedness Index for the US with US Census and Pew data, and introduce an ARD-compatible joint geosocial intervening-opportunities model to infer link probabilities between region–group cell pairs. Applying our segregation framework, we find that social segregation predominates over geographical segregation, with notable separation for White–Black, college-degree–no-degree, and high-income–low/middle-income across both segregation types. We find increasing social homophily with geographical distance and group-specific geographical connectivity patterns, suggesting that geographical segregation may affect cross-group connectivity not only directly but also by amplifying social segregation.

Contrail cirrus represents a critical component of aviation's non-CO2 climate impact, but its net radiative forcing, the balance between longwave warming and shortwave cooling, remains poorly constrained by direct observations. As a result, current assessments rely almost exclusively on microphysical models such as CoCiP and global climate simulations. Existing empirical estimates are largely restricted to young, linear tracks, because satellite detection masks have a poor recall of contrails once they spread and merge with natural cirrus, leaving a structural gap in our understanding of long-lived, non-linear contrail cirrus. To address this we use a causal framework that isolates the net radiative contrail effect of flight traffic over the Americas. Building on recent progress that quantified the longwave warming contrail effect using advected flight paths as a proxy for contrails, we expand this continuous treatment approach to capture the highly skewed shortwave cooling impact, delivering a 12-hour lifespan net observational radiative forcing. Our analysis reveals a statistically significant net warming energy forcing of 33.7 (95% CI: 20.8, 47.8) GJ per km flown from April 2019 to April 2020, providing a large-scale empirical quantification of long contrail lifespan impact of the same order as, though somewhat larger than, previous bottom-up simulation estimates. This observational benchmark offers an independent line of evidence on the sign and magnitude of the climate impact of contrails.

Sentiment analysis is a common tool for converting unstructured text into quantitative signals in finance and risk management. Yet most applications reduce the output to a discrete polarity label or a single predictive feature, overlooking the distributional structure of sentiment intensity in consumer complaint narratives. In this paper we treat negative sentiment in consumer complaints as a bounded continuous variable and study its full distribution rather than a single label. We score each narrative with a transformer classifier, model the scores with Beta distributions, and compare the fitted distributions of meritorious and non-meritorious complaints through the Kullback Leibler divergence and the squared Hellinger distance. The fitted distributions are then linked with dollar amounts and company response outcomes to construct anomaly diagnostics that flag complaints whose textual severity is inconsistent with the recorded relief. We find that the two groups have strongly overlapping distributions, so negative sentiment intensity is not a sharp classifier of outcomes on its own; combined with monetary and categorical attributes, it isolates unusually severe complaints for operational risk monitoring. Treating sentiment analysis as continuous distributional measurement, this study links sentiment extraction, bounded response modeling, and anomaly detection in a unified framework for consumer complaint assessment.

Neural networks are becoming popular in solving mathematical problems, but stochastic models do not provide mathematical exactness by themselves. This study introduces a discovery-to-certification framework in which high-dimensional variational trial functions are learned in a compact separable representation, spectrally diagnosed and pruned, and then certified exactly through multimodular evaluation. Exact certification makes the numerical proofs fully explicit and independently verifiable. This framework can be run on a standard personal computer. Across three extremal problems, we show that neural optimization can contribute to rigorous mathematics in three distinct ways: by discovering improved constructions, by exposing empirical invariants that lead to proofs, and by revealing optimization barriers whose geometry motivates new analytic or numerical representations. More broadly, these results suggest a path toward AI-assisted mathematics in which flexible computational discovery and exact certification become complementary components of a single rigorous workflow.

Deep spline superposition networks face a tension between approximation order and stability across depth. We study approximation under a hard layerwise Lipschitz budget, and organise it around two quantities: the factorisation stability complexity of a given deep factorisation, and the budget-compatible approximation complexity of a discretisation operator. First, we solve exactly the finite-depth diagonal balancing problem for a fixed chain of nonnegative envelope matrices: the optimal uniform layer budget equals $\|M_{L-1}\cdots M_0\|_{\infty\to\infty}^{1/L}$, attained by an explicit one-pass minimiser, for rectangular layers, with a complete treatment of degeneracies and non-attainment. The optimum can be arbitrarily larger than the Lipschitz constant of the network itself, because passing to envelopes destroys sign cancellation. Second, we give a constructive spline discretisation theorem preserving the budget up to a controlled slack, with an explicit grid threshold. Conversely, for linear spline-valued operators that preserve the budget exactly, we prove budget-compatible minimax lower bounds on classes constrained simultaneously in the first and third derivative norms – a constraint pair that is forced by the problem and that rules out the usual scaling escapes. Finally, we show that the corresponding layer errors need not cancel under composition: for every operator of the class there is a stable depth-$L$ tower realising a constant fraction of the accumulated error, so the linear-in-depth accumulation of the upper bound is not a proof artefact.

Classification and regression trees are typically constructed using a greedy splitting rule that maximizes the immediate reduction in prediction error at each node. Although this strategy is computationally efficient, it can miss splits that yield small short-term gains but create substantial downstream improvements after further partitioning. We propose a look-ahead tree-building method that evaluates each candidate split by the prediction error reduction achieved after growing a conventional CART subtree below that split. Because the full look-ahead procedure can be computationally expensive, we also describe a smart look-ahead algorithm that learns downstream split values using node-level features. The proposed framework preserves the interpretability of recursive partitioning while improving split selection in hierarchical or interaction-driven settings. We conduct a simulation study comparing conventional, full look-ahead, and smart look-ahead methods under several settings and apply the proposed methods to analyze two real data examples demonstrating the merit of the new methods.

Bayesian clinical trials require a well-specified prior distribution, and regulators increasingly expect that specification to be documented, diagnosed for prior-data conflict, and shown to be robust to reasonable alternatives, yet few R tools connect the full workflow of prior construction, validation, and regulatory justification into a single package. We introduce bayprior, an R package and Shiny application that implements structured expert elicitation via quantile matching, moment matching, and the SHELF roulette method across six distribution families; linear and logarithmic expert opinion pooling with Bhattacharyya agreement diagnostics; derivation of meta-analytic-predictive (MAP) priors from historical trial data via random-effects meta-analysis; prior-data conflict assessment using the Box p-value, S-value (surprisal) surprise index, information divergence, Bhattacharyya overlap, and multivariate Mahalanobis distance; hyperparameter sensitivity analysis with tornado and influence heatmap visualisations; and robust, sceptical, and power prior alternatives. A regulatory reporting module generates self-contained HTML, PDF, or Word documents addressing expectations in the FDA's 2026 draft Bayesian methods guidance and in EMA guidance on incorporating external and historical information. The package is demonstrated on a synthetic oncology Phase II trial. bayprior is available on CRAN and includes a fully modular Shiny application for interactive use.

Class-balanced learning and label noise create a coupled failure mode: frequency correction prevents majority classes from dominating the decision rule, but can amplify incorrectly labeled minority examples. We introduce BARGE (Bounded Adjustment with Reliability-Guided Embeddings), a single-stage objective combining a bounded, prior-adjusted density-power score with reliability-guided angular geometry. Its classification score is strictly proper in the adjusted probability space and recovers balanced Bayes ordering under clean supervision and the true class prior. Under label contamination, its finite range bounds classification-risk perturbation at a fixed predictor, while its logit gradient redescends when the model confidently contradicts the supplied label. The adjusted target probability also weights class-equal feature compactness, and a one-sided separation term discourages aligned class directions. BARGE requires neither a noise rate nor a transition matrix, uses one network, and leaves inference unchanged. We evaluate it on CIFAR-10, CIFAR-100, and Tiny ImageNet under long-tail and step imbalance, clean labels, and 20% and 40% random incorrect-label replacement. Across 12 clean settings, BARGE ranks second overall and attains the lowest error in four. Under corruption, it achieves the lowest mean balanced error in all six dataset-corruption settings, reducing the six-setting average from 72.32% for the strongest competitor to 70.00%. It also obtains the highest macro-F1 and macro-AUPRC in every corrupted-label setting. Ablations show that class-equal angular compactness improves on the bounded score alone. These results support bounded predictive influence and reliability-guided geometry as complementary mechanisms for imbalanced learning with uncertain labels.

Real-world datasets often exhibit long-tailed class distributions, where a few head classes contain a large number of training samples while a large number of tail classes have only a few. The composition of each mini-batch, determined by the sampling strategy, governs which classes contribute to the stochastic gradient estimate, and therefore affects convergence behaviour and generalisation across the whole class spectrum. We provide a systematic theoretical and empirical comparison of four mini-batch sampling strategies for long-tailed image classification: uniform instance sampling, class-balanced sampling, square-root sampling, and progressively balanced sampling. We place all four in a unified bias-variance framework describing their effect on gradient estimation, which exposes the tension between unbiased optimisation of the empirical loss and fair representation of rare classes. We then evaluate them under controlled conditions using ResNet-32 on CIFAR-100-LT at three imbalance ratios (rho = 10, 50, 100), with every strategy sharing the same long-tailed subsets and initialisation within a seed. Progressive sampling improves tail-class accuracy by 25% relative to the uniform baseline at rho = 100 (13.5% versus 10.8%), consistently across all three seeds, while its overall accuracy is not distinguishable from that of uniform sampling given the seed-to-seed variation (40.0% versus 39.7%); the tail-class gain, not the overall gain, is the robust effect. At rho = 100, class-balanced sampling degrades accuracy on every class group, including the tail classes it is designed to help, which we attribute to overfitting caused by extreme oversampling of scarce data; at rho = 50 this failure is confined to head and medium classes. These results indicate that when rebalancing is applied during training matters as much as how much rebalancing is applied.

Fault detection is essential in industrial systems, enabling early identification of abnormal behaviour and improving safety, reliability, and operational efficiency. Modern systems increasingly rely on heterogeneous sensing modalities that capture complementary aspects of the underlying physical process. However, existing data-driven anomaly detection methods often process each modality independently or use simple feature-level fusion, limiting their ability to exploit cross-modal relationships that characterize normal system behaviour. Their performance also commonly assumes similar training and deployment distributions, whereas real-world operation is affected by changing operating conditions, environmental influences, and system degradation that induce distribution shifts and reduce detection performance, especially in unseen regimes. In this work, we propose a multimodal anomaly detection framework based on cross-modal reconstruction of heterogeneous time-series sensor data. Rather than modeling each modality independently, the framework learns system dynamics by reconstructing each modality from the others, thereby exploiting complementary information across modalities. This integrates information across sensing channels without requiring explicit temporal alignment or identical sampling rates, while improving robustness to sensor noise, missing measurements, and modality-specific disturbances. To address distribution shifts during real-world deployment, anomalies are identified using cross-modal reconstruction error and an adaptive test-time thresholding mechanism that adjusts to changing operating conditions. Experiments on three industrial case studies show strong fault detection performance and substantially improved robustness under out-of-distribution conditions, with the largest gains observed in the most challenging operating regimes.

Tracking the energy transition requires reliable statistics on renewable deployment. Rooftop photovoltaics (PV) are especially hard to track, owing to their decentralised nature, and the resulting inaccuracies in official statistics are known but not quantified. Remote sensing offers an independent way to identify rooftop PV systems. We introduce a Bayesian framework to estimate the ground-truth rooftop PV capacity from remote sensing detections, turning an imperfect detector into an uncertainty-aware measurement instrument. Applied to France, the corrected detections estimate a capacity of 4.03 GWp [3.96–4.11] (99% credible interval) of rooftop PV below 36 kWp, matching the transmission system operator's connection data within 3.3% nationally, while identifying local under-reports of up to 61% of local capacity. We also document and quantify a significant truncation bias in French rooftop PV open data. Beyond France, the approach paves the way for more reliable estimates of rooftop PV capacity worldwide.

Pretraining followed by fine-tuning introduces a compute-allocation problem: under a fixed training budget, compute spent improving the upstream objective reduces the compute available for downstream adaptation. Despite its practical importance, this trade-off is not yet well understood theoretically, even in simple models. In this paper, we cast this allocation as a compute-split problem under a two-stage pretrain–fine-tune procedure with fixed total optimisation budget, using regularised least squares trained by gradient descent as a tractable setting. We characterise the optimal split under data-dependent evaluation geometries induced by the fine-tuning problem. Our results show that the allocation depends on how pretraining directions affect fine-tuning predictions and how fine-tuning shifts are seen through downstream data geometry. In particular, the relevant quantities are determined by prediction-relevant spectral components of the pretraining and fine-tuning empirical covariances. Technically, the analysis relies on a basis-invariant, eigenspace-level spectral decomposition, together with perturbative control of the non-commuting pretraining and fine-tuning dynamics.

Diagnostic errors and mislabeling are common in biomedicine, which compromise the reliability of predictive models and data-driven outcomes. Stratifying unlabeled biomedical data based on complex relationships between features eliminates the need for data labels and overcomes the limitations of supervised learning. Traditional clustering methods assume restrictive data distributions, making them suboptimal for capturing complex dependencies in high-dimensional biomedical data. This paper introduces a novel cluster-friendly data presentation framework that integrates the non-Gaussian and non-linear feature dependence of copula models with an ensemble of causal structure discovery (CSD) methods based on Directed Acyclic Graphs (DAGs). While copulas model flexible multivariate distributions by relaxing assumptions related to multivariate normality, linear dependence, and symmetric relationships, an ensemble of DAG-based CSD methods identifies stable causal relationships between features. When clustered using K-means, the new data representation obtained by the proposed copula-adapted DAG (CopDAG) ranks first among the 12 methods in normalized clustering accuracy and adjusted Rand index across 16 biomedical datasets. Our CopDAG method predicts ground-truth class labels directly from feature relationships without data annotations and supervised learning, while also providing cluster visualizations and explainable causal structures of the biomedical data features.

We develop two complementary Bayesian procedures for analytic inference under informative sampling. The first constructs a coherent likelihood by modeling the sample-level distribution of the design weight given the study variable and recovering the conditional inclusion probability through the identity of Sverchkov and Pfeffermann (2004); it is the Bayesian counterpart of a recently proposed conditional-likelihood estimator, attains the model-based efficiency bound under the correctly specified sample-level joint model, and can be biased otherwise. The second uses the same weight model only as input to a Neyman-orthogonalized analog of the design-weighted score and applies the loss-likelihood bootstrap; the resulting credible intervals are asymptotically sandwich-correct under Poisson sampling, and the posterior remains centered at the truth under misspecification of the weight model, although the working model can still affect the first-order variance. A beta regression on the population-level inclusion probability, inducing a beta prime sample-level weight distribution, serves as the parametric default. A data-adaptive implementation of the second procedure estimates the efficient factor $\bar\pi(x,y)=1/E_p(W\mid x,y)$ by a design-weighted Gamma/log-link regression and uses it in a joint one-step loss-likelihood bootstrap for the analytic parameter; its validity requires no rate conditions on the nuisance fit. We illustrate both procedures with simulations and Canadian Workforce data.

Prototype reduction replaces a training set with a smaller representation, and the established methods return a finite set of points. We propose Skeletal Prototypes on Iterative Nerve Expansions (SPINE). The model for each class is an embedded 1-complex rather than a point set. Its initial edge set is a class-conditional Mapper graph, so the data decide which localized clusters are joined. Later phases fit the vertices under a classification objective, and an observation is assigned to the class whose complex is nearest. The segments therefore enter the decision rule and not only the fitting. We evaluate SPINE on seventeen benchmark datasets under stratified 10-fold cross validation, against seven other prototype reduction methods at a matched budget. SPINE attains the highest mean accuracy and the best average rank. It is significantly better than five of the seven competitors under Wilcoxon signed-rank tests with Holm correction. A budget sweep shows that the decision rule using the entire graph segments contribute most when prototypes are scarce, while the method as a whole competes best at moderate budgets. Construction cost places SPINE with the discriminative methods, and it is faster than generalized learning vector quantization on fourteen of the seventeen datasets.