Papers

The convergence of statistical learning and molecular physics is transforming our approach to modeling biomolecular systems. Physics-informed machine learning (PIML) offers a systematic framework that integrates data-driven inference with physical constraints, resulting in models that are accurate, mechanistic, generalizable, and able to extrapolate beyond observed domains. This review surveys recent advances in physics-informed neural networks and operator learning, differentiable molecular simulation, and hybrid physics-ML potentials, with emphasis on long-timescale kinetics, rare events, and free-energy estimation. We frame these approaches as solutions to the "biomolecular closure problem", recovering unresolved interactions beyond classical force fields while preserving thermodynamic consistency and mechanistic interpretability. We examine theoretical foundations, tools and frameworks, computational trade-offs, and unresolved issues, including model expressiveness and stability. We outline prospective research avenues at the intersection of machine learning, statistical physics, and computational chemistry, contending that future advancements will depend on mechanistic inductive biases, and integrated differentiable physical learning frameworks for biomolecular simulation and discovery.

Machine learning systems exhibit diverse failure modes: unfairness toward protected groups, brittleness to spurious correlations, poor performance on minority sub-populations, which are typically studied in isolation by distinct research communities. We propose a unifying theoretical framework that characterizes when different bias mechanisms produce quantitatively equivalent effects on model performance. By formalizing biases as violations of conditional independence through information-theoretic measures, we prove formal equivalence conditions relating spurious correlations, subpopulation shift, class imbalance, and fairness violations. Our theory predicts that a spurious correlation of strength $\alpha$ produces equivalent worst-group accuracy degradation as a sub-population imbalance ratio $r \approx (1+\alpha)/(1-\alpha)$ under feature overlap assumptions. Empirical validation in six datasets and three architectures confirms that predicted equivalences hold within the accuracy of the worst group 3%, enabling the principled transfer of debiasing methods across problem domains. This work bridges the literature on fairness, robustness, and distribution shifts under a common perspective.

This study presents a novel framework for counterfactual user behavior forecasting that combines structural causal models with transformer-based generative artificial intelligence. To model fictitious situations, the method creates causal graphs that map the connections between user interactions, adoption metrics, and product features. The framework generates realistic behavioral trajectories under counterfactual conditions by using generative models that are conditioned on causal variables. Tested on datasets from web interactions, mobile applications, and e-commerce, the methodology outperforms conventional forecasting and uplift modeling techniques. Product teams can effectively simulate and assess possible interventions prior to deployment thanks to the framework improved interpretability through causal path visualization.

Link prediction is a fundamental task in graph machine learning with applications, ranging from social recommendation to knowledge graph completion. Fairness in this setting is critical, as biased predictions can exacerbate societal inequalities. Prior work adopts a dyadic definition of fairness, enforcing fairness through demographic parity between intra-group and inter-group link predictions. However, we show that this dyadic framing can obscure underlying disparities across subgroups, allowing systemic biases to go undetected. Moreover, we argue that demographic parity does not meet desired properties for fairness assessment in ranking-based tasks such as link prediction. We formalize the limitations of existing fairness evaluations and propose a framework that enables a more expressive assessment. Additionally, we propose a lightweight post-processing method combined with decoupled link predictors that effectively mitigates bias and achieves state-of-the-art fairness-utility trade-offs.

In this paper, we propose Random Forests by Random Weights (RF-RW), a theoretically grounded and practically effective alternative RF modelling for nonlinear time series data, where existing RF-based approaches struggle to adequately capture temporal dependence. RF-RW reconciles the strengths of classic RF with the temporal dependence inherent in time series forecasting. Specifically, it avoids the bootstrap resampling procedure, therefore preserves the serial dependence structure, whilst incorporates independent random weights to reduce correlations among trees. We establish non-asymptotic concentration bounds and asymptotic uniform consistency guarantees, for both fixed- and high-dimensional feature spaces, which extend beyond existing theoretical analyses of RF. Extensive simulation studies demonstrate that RF-RW outperforms existing RF-based approaches and other benchmarks such as SVM and LSTM. It also achieves the lowest error among competitors in our real-data example of predicting UK COVID-19 daily cases.

Categorical predictors are omnipresent in everyday regression practice: in fact, most regression data involve some categorical predictors, and this tendency is increasing in modern applications with more complex structures and larger data sizes. However, including too many categories in a regression model would seriously hamper accuracy, as the information in the data is fragmented by the multitude of categories. In this paper, we introduce a systematic method to reduce the complexity of categorical predictors by adaptively collapsing categories in regressions, so as to enhance the performance of regression estimation. Our method is based on the pairwise vector fused LASSO, which automatically fuses the categories that bear a similar regression relation with the response. We develop our method under a wide class of regression models defined by a general loss function, which includes linear models and generalized linear models as special cases. We rigorously established the category collapsing consistency of our method, developed an Inexact Proximal Gradient Descent algorithm to implement it, and proved the feasibility and convergence of our algorithm. Through simulations and an application to Spotify music data, we demonstrate that our method can effectively reduce categorical complexity while improving prediction performance, making it a powerful tool for regression with mixed predictors.

We introduce a new dependence order, termed the conditional convex order, whose minimal and maximal elements characterize independence and perfect dependence. Moreover, it characterizes conditional independence, satisfies information monotonicity, and exhibits several invariance properties. Consequently, it is an ordering for the strength of functional dependence of a random variable Y on a random vector X. As we show, various recently studied dependence measures – including Chatterjee's rank correlation, Wasserstein correlations, and rearranged dependence measures – are increasing in this order and inherit their fundamental properties from it. We characterize the conditional convex order by the Schur order and by the concordance order, and we verify it in settings such as additive error models, the multivariate normal distribution, and various copula-based models. Our results offer a unified perspective on the behavior of dependence measures across statistical models.

We propose and investigate probabilistic guarantees for the adversarial robustness of classification algorithms. While traditional formal verification approaches for robustness are intractable and sampling-based approaches do not provide formal guarantees, our approach is able to efficiently certify a probabilistic relaxation of robustness. The key idea is to sample an $\epsilon$-net and invoke a local robustness oracle on the sample. Remarkably, the size of the sample needed to achieve probably approximately global robustness guarantees is independent of the input dimensionality, the number of classes, and the learning algorithm itself. Our approach can, therefore, be applied even to large neural networks that are beyond the scope of traditional formal verification. Experiments empirically confirm that it characterizes robustness better than state-of-the-art sampling-based approaches and scales better than formal methods.

Supply chain disruptions and volatile demand pose significant challenges to the UK automotive industry, which relies heavily on Just-In-Time (JIT) manufacturing. While qualitative studies highlight the potential of integrating Artificial Intelligence (AI) with traditional optimization, a formal, quantitative demonstration of this synergy is lacking. This paper introduces a novel stochastic learning-optimization framework that integrates Bayesian inference with inventory optimization for supply chain management (SCM). We model a two-echelon inventory system subject to stochastic demand and supply disruptions, comparing a traditional static optimization policy against an adaptive policy where Bayesian learning continuously updates parameter estimates to inform stochastic optimization. Our simulations over 365 periods across three operational scenarios demonstrate that the integrated approach achieves 7.4% cost reduction in stable environments and 5.7% improvement during supply disruptions, while revealing important limitations during sudden demand shocks due to the inherent conservatism of Bayesian updating. This work provides mathematical validation for practitioner observations and establishes a formal framework for understanding AI-driven supply chain resilience, while identifying critical boundary conditions for successful implementation.

Small sample sizes in clinical studies arises from factors such as reduced costs, limited subject availability, and the rarity of studied conditions. This creates challenges for accurately calculating confidence intervals (CIs) using the normal distribution approximation. In this paper, we employ a quadratic-form based statistic, from which we derive more accurate confidence intervals, particularly for data with small sample sizes or proportions. Based on the study, we suggest reasonable values of sample sizes and proportions for the application of the quadratic method. Consequently, this method enhances the reliability of statistical inferences. We illustrate this method with real medical data from clinical trials.

The boundary discontinuity (BD) design is a non-experimental method for identifying causal effects that exploits a thresholding rule based on a bivariate score and a boundary curve. This widely used method generalizes the univariate regression discontinuity design but introduces unique challenges arising from its multidimensional nature. We synthesize over 80 empirical papers that use the BD design, tracing the method's application from its formative stages to its implementation in modern research. We also overview ongoing theoretical and methodological research on identification, estimation, and inference for BD designs employing local polynomial regression, and offer recommendations for practice.

In many applications, the variables that characterize a stochastic system are measured along a second dimension, such as time. This results in multivariate functional data and the interest is in describing the statistical dependencies among these variables. It is often the case that the functional data are only partially observed. This creates additional challenges to statistical inference, since the functional principal component scores, which capture all the information from these data, cannot be computed. Under an assumption of Gaussianity and of partial separability of the covariance operator, we develop an Expectation-Maximization (EM)-type algorithm for penalized inference of a functional graphical model from multivariate functional data which are only partially observed. A simulation study and an illustration on environmental, social and governance (ESG) data show the potential of the proposed method.

Interpretable representation learning is a central challenge in modern machine learning, particularly in high-dimensional settings such as neuroimaging, genomics, and text analysis. Current methods often struggle to balance the competing demands of interpretability and model flexibility, limiting their effectiveness in extracting meaningful insights from complex data. We introduce Non-negative Stiefel Approximating Flow (NSA-Flow), a general-purpose matrix estimation framework that unifies ideas from sparse matrix factorization, orthogonalization, and constrained manifold learning. NSA-Flow enforces structured sparsity through a continuous balance between reconstruction fidelity and column-wise decorrelation, parameterized by a single tunable weight. The method operates as a smooth flow near the Stiefel manifold with proximal updates for non-negativity and adaptive gradient control, yielding representations that are simultaneously sparse, stable, and interpretable. Unlike classical regularization schemes, NSA-Flow provides an intuitive geometric mechanism for manipulating sparsity at the level of global structure while simplifying latent features. We demonstrate that the NSA-Flow objective can be optimized smoothly and integrates seamlessly with existing pipelines for dimensionality reduction while improving interpretability and generalization in both simulated and real biomedical data. Empirical validation on the Golub leukemia dataset and in Alzheimer's disease demonstrate that the NSA-Flow constraints can maintain or improve performance over related methods with little additional methodological effort. NSA-Flow offers a scalable, general-purpose tool for interpretable ML, applicable across data science domains.

While zero-shot diffusion-based compression methods have seen significant progress in recent years, they remain notoriously slow and computationally demanding. This paper presents an efficient zero-shot diffusion-based compression method that runs substantially faster than existing methods, while maintaining performance that is on par with the state-of-the-art techniques. Our method builds upon the recently proposed Denoising Diffusion Codebook Models (DDCMs) compression scheme. Specifically, DDCM compresses an image by sequentially choosing the diffusion noise vectors from reproducible random codebooks, guiding the denoiser's output to reconstruct the target image. We modify this framework with Turbo-DDCM, which efficiently combines a large number of noise vectors at each denoising step, thereby significantly reducing the number of required denoising operations. This modification is also coupled with an improved encoding protocol. Furthermore, we introduce two flexible variants of Turbo-DDCM, a priority-aware variant that prioritizes user-specified regions and a distortion-controlled variant that compresses an image based on a target PSNR rather than a target BPP. Comprehensive experiments position Turbo-DDCM as a compelling, practical, and flexible image compression scheme.

Transmission Electron Microscopy enables high-resolution imaging of materials, but the resulting images are difficult to interpret directly. One way to address this is exit wave reconstruction, i.e., the recovery of the complex-valued electron wave at the specimen's exit plane from intensity-only measurements. This is an inverse problem with a nonlinear forward model. We consider a simplified forward model, making the problem equivalent to phase retrieval, and propose a discretized regularized variational formulation. To solve the resulting non-convex problem, we employ the proximal gradient algorithm (PGA) and unfold its iterations into a neural network, where each layer corresponds to one PGA step with learnable parameters. This unrolling approach, inspired by LISTA, enables improved reconstruction quality, interpretability, and implicit dictionary learning from data. We analyze the effect of parameter perturbations and show that they can accumulate exponentially with the number of layers $L$. Building on proof techniques of Behboodi et al., originally developed for LISTA, i.e., for a linear forward model, we extend the analysis to our nonlinear setting and establish generalization error bounds of order $\mathcal{O}(\sqrt{L})$. Numerical experiments support the exponential growth of parameter perturbations.

Hierarchical Bayesian models based on Gaussian processes are considered useful for describing complex nonlinear statistical dependencies among variables in real-world data. However, effective Monte Carlo algorithms for inference with these models have not yet been established, except for several simple cases. In this study, we show that, compared with the slow inference achieved with existing program libraries, the performance of Riemannian-manifold Hamiltonian Monte Carlo (RMHMC) can be drastically improved by optimising the computation order according to the model structure and dynamically programming the eigendecomposition. This improvement cannot be achieved when using an existing library based on a naive automatic differentiator. We numerically demonstrate that RMHMC effectively samples from the posterior, allowing the calculation of model evidence, in a Bayesian logistic regression on simulated data and in the estimation of propensity functions for the American national medical expenditure data using several Bayesian multiple-kernel models. These results lay a foundation for implementing effective Monte Carlo algorithms for analysing real-world data with Gaussian processes, and highlight the need to develop a customisable library set that allows users to incorporate dynamically programmed objects and finely optimises the mode of automatic differentiation depending on the model structure.

The one-epoch overfitting problem has drawn widespread attention, especially in CTR and CVR estimation models in search, advertising, and recommendation domains. These models which rely heavily on large-scale sparse categorical features, often suffer a significant decline in performance when trained for multiple epochs. Although recent studies have proposed heuristic solutions, the fundamental cause of this phenomenon remains unclear. In this work, we present a theoretical explanation grounded in Rademacher complexity, supported by empirical experiments, to explain why overfitting occurs in models with large-scale sparse categorical features. Based on this analysis, we propose a regularization method that constrains the norm budget of embedding layers adaptively. Our approach not only prevents the severe performance degradation observed during multi-epoch training, but also improves model performance within a single epoch. This method has already been deployed in online production systems.

A popular method for likelihood-free inference is approximate Bayesian computation sequential Monte Carlo (ABC-SMC) algorithms. These approximate the posterior using a population of particles, which are updated using Markov kernels. Several such kernels have been proposed. In this paper we review these, highlighting some less well known choices, and proposing some novel options. Further, we conduct an extensive empirical comparison of kernel choices. Our results suggest using a one-hit kernel with a mixture proposal as a default choice.

The widespread adoption of online randomized controlled experiments (A/B Tests) for decision-making has created ongoing capacity constraints which necessitate interim analyses. As a consequence, platform users are increasingly motivated to use ad-hoc means of optimizing limited resources via peeking. Such processes, however, are error prone and often misaligned with end-of-experiment outcomes (e.g., inflated type-I error). We introduce a system based on Bayesian Predictive Probabilities that enable us to perform interim analyses without compromising fidelity of the experiment; This idea has been widely utilized in applications outside of the technology domain to more efficiently make decisions in experiments. Motivated by at-scale deployment within an experimentation platform, we demonstrate how predictive probabilities can be estimated without numerical integration techniques and recommend systems to study its properties at scale as an ongoing health check, along with system design recommendations - all on experiment data from Instagram - to demonstrate practical benefits that it enables.

A 'Winner's Curse' arises in large-scale online experimentation platforms when the same experiments are used to both select treatments and evaluate their effects. In these settings, classical difference-in-means estimators of treatment effects are upwardly biased and conventional confidence intervals are rendered invalid. The bias scales with the magnitude of sampling variability and the selection threshold, and inversely with the treatment's true effect size. We propose a new Bayesian approach that incorporates experiment-specific 'local shrinkage' factors that mitigate sensitivity to the choice of prior and improve robustness to assumption violations. We demonstrate how the associated posterior distribution can be estimated without numerical integration techniques, making it a practical choice for at-scale deployment. Through simulation, we evaluate the performance of our approach under various scenarios and find that it performs well even when assumptions about the sampling and selection processes are violated. In an empirical evaluation, our approach demonstrated superior performance over alternative methods, providing more accurate estimates with well-calibrated uncertainty quantification.

Since 2010, Kaggle has been a platform where data scientists from around the world come together to compete, collaborate, and push the boundaries of Data Science. Over these 15 years, it has grown from a purely competition-focused site into a broader ecosystem with forums, notebooks, models, datasets, and more. With the release of the Kaggle Meta Code and Kaggle Meta Datasets, we now have a unique opportunity to explore these competitions, technologies, and real-world applications of Machine Learning and AI. And so in this study, we take a closer look at 15 years of data science on Kaggle - through metadata, shared code, community discussions, and the competitions themselves. We explore Kaggle's growth, its impact on the data science community, uncover hidden technological trends, analyze competition winners, how Kagglers approach problems in general, and more. We do this by analyzing millions of kernels and discussion threads to perform both longitudinal trend analysis and standard exploratory data analysis. Our findings show that Kaggle is a steadily growing platform with increasingly diverse use cases, and that Kagglers are quick to adapt to new trends and apply them to real-world challenges, while producing - on average - models with solid generalization capabilities. We also offer a snapshot of the platform as a whole, highlighting its history and technological evolution. Finally, this study is accompanied by a video (https://www.youtube.com/watch?v=YVOV9bIUNrM) and a Kaggle write-up (https://kaggle.com/competitions/meta-kaggle-hackathon/writeups/kaggle-chronicles-15-years-of-competitions-communi) for your convenience.

Accurate estimation of Aerosol Optical Depth (AOD) is crucial for understanding climate change and its impacts on public health, as aerosols are a measure of air quality conditions. AOD is usually retrieved from satellite imagery at coarse spatial and temporal resolutions. However, producing high-resolution AOD estimates in both space and time can better support evidence-based policies and interventions. We propose a spatio-temporal disaggregation model that assumes a latent spatio–temporal continuous Gaussian process observed through aggregated measurements. The model links discrete observations to the continuous domain and accommodates covariates to improve explanatory power and interpretability. The approach employs Gaussian processes with separable or non-separable covariance structures derived from a diffusion-based spatio-temporal stochastic partial differential equation (SPDE). Bayesian inference is conducted using the INLA-SPDE framework for computational efficiency. Simulation studies and an application to nowcasting AOD at 550 nm in India demonstrate the model's effectiveness, improving spatial resolution from 0.75{\deg} to 0.25{\deg} and temporal resolution from 3 hours to 1 hour.

In observational studies, exposures are often continuous rather than binary or discrete. At the same time, sensitivity analysis is an important tool that can help determine the robustness of a causal conclusion to a certain level of unmeasured confounding, which can never be ruled out in an observational study. Sensitivity analysis approaches for continuous exposures have now been proposed for several causal estimands. In this article, we focus on the average derivative effect (ADE). We obtain closed-form bounds for the ADE under a sensitivity model that constrains the odds ratio (at any two dose levels) between the latent and observed generalized propensity score. We propose flexible, efficient estimators for the bounds that are compatible with machine learning algorithms as well as point-wise and simultaneous (over the sensitivity parameter) confidence intervals. We examine the finite sample performance of the methods through simulations and illustrate the methods on a study assessing the effect of parental income on educational attainment and a study assessing the price elasticity of petrol.

Learning-based methods for sampling from the Gibbs distribution in finite-dimensional spaces have progressed quickly, yet theory and algorithmic design for infinite-dimensional function spaces remain limited. This gap persists despite their strong potential for sampling the paths of conditional diffusion processes, enabling efficient simulation of trajectories of diffusion processes that respect rare events or boundary constraints. In this work, we present the adjoint sampler for infinite-dimensional function spaces, a stochastic optimal control-based diffusion sampler that operates in function space and targets Gibbs-type distributions on infinite-dimensional Hilbert spaces. Our Functional Adjoint Sampler (FAS) generalizes Adjoint Sampling (Havens et al., 2025) to Hilbert spaces based on a SOC theory called stochastic maximum principle, yielding a simple and scalable matching-type objective for a functional representation. We show that FAS achieves superior transition path sampling performance across synthetic potential and real molecular systems, including Alanine Dipeptide and Chignolin.

This paper presents a comprehensive analysis of hyperparameter estimation within the empirical Bayes framework (EBF) for sparse learning. By studying the influence of hyperpriors on the solution of EBF, we establish a theoretical connection between the choice of the hyperprior and the sparsity as well as the local optimality of the resulting solutions. We show that some strictly increasing hyperpriors, such as half-Laplace and half-generalized Gaussian with the power in $(0,1)$, effectively promote sparsity and improve solution stability with respect to measurement noise. Based on this analysis, we adopt a proximal alternating linearized minimization (PALM) algorithm with convergence guaranties for both convex and concave hyperpriors. Extensive numerical tests on two-dimensional image deblurring problems demonstrate that introducing appropriate hyperpriors significantly promotes the sparsity of the solution and enhances restoration accuracy. Furthermore, we illustrate the influence of the noise level and the ill-posedness of inverse problems to EBF solutions.

Sparse linear regression is one of the most basic questions in machine learning and statistics. Here, we are given as input a design matrix $X \in \mathbb{R}^{N \times d}$ and measurements or labels ${y} \in \mathbb{R}^N$ where ${y} = {X} {w}^* + {\xi}$, and ${\xi}$ is the noise in the measurements. Importantly, we have the additional constraint that the unknown signal vector ${w}^*$ is sparse: it has $k$ non-zero entries where $k$ is much smaller than the ambient dimension. Our goal is to output a prediction vector $\widehat{{w}}$ that has small prediction error: $\frac{1}{N}\cdot \|{X} {w}^* - {X} \widehat{{w}}\|^2_2$. Information-theoretically, we know what is best possible in terms of measurements: under most natural noise distributions, we can get prediction error at most $\epsilon$ with roughly $N = O(k \log d/\epsilon)$ samples. Computationally, this currently needs $d^{\Omega(k)}$ run-time. Alternately, with $N = O(d)$, we can get polynomial-time. Thus, there is an exponential gap (in the dependence on $d$) between the two and we do not know if it is possible to get $d^{o(k)}$ run-time and $o(d)$ samples. We give the first generic positive result for worst-case design matrices ${X}$: For any ${X}$, we show that if the support of ${w}^*$ is chosen at random, we can get prediction error $\epsilon$ with $N = \text{poly}(k, \log d, 1/\epsilon)$ samples and run-time $\text{poly}(d,N)$. This run-time holds for any design matrix ${X}$ with condition number up to $2^{\text{poly}(d)}$. Previously, such results were known for worst-case ${w}^*$, but only for random design matrices from well-behaved families, matrices that have a very low condition number ($\text{poly}(\log d)$; e.g., as studied in compressed sensing), or those with special structural properties.

Popular technologies for generating spatially resolved transcriptomic data measure gene expression at the resolution of a "spot", i.e., a small tissue region 55 microns in diameter. Each spot can contain many cells of different types. In typical analyses, researchers are interested in using these data to identify and profile discrete spatial domains in tissue. In this paper, we propose a new method, DUET, which simultaneously identifies discrete spatial domains and estimates each spot's expected cell-type proportion. This allows the identified spatial domains to be characterized in terms of the underlying expected cell-type proportions, which affords interpretability and biological insight. DUET utilizes a constrained version of model-based convex clustering, and as such, can accommodate Poisson, negative binomial, normal, and other types of expression data. Moreover, our convex clustering-type criterion allows for both the number of clusters and degree of spatial smoothness to be controlled by a single tuning parameter, which can be chosen in a data-driven fashion. Through simulation studies and a real data application, we show that DUET can achieve better clustering and deconvolution performance than some existing methods.

Historical trials of fluoride varnish reported large reductions in dental caries, but contemporary trials are often conducted against more intensive background prevention, so the incremental effect may be smaller. For positive continuous outcomes, we modeled the log ratio of treatment to control means and report the prevented fraction, $\mathrm{PF}=1-\exp(y)$. A nine-trial historical dataset was reanalyzed using REML with Hartung–Knapp inference, Bayesian hierarchical models, and sensitivity analyses; completeness was checked against all 13 Cochrane permanent-surface trials. Temporal change was examined in a 14-study dataset combining nine historical and five contemporary trials, with time specifications compared by AICc. The historical pooled PF was $39.7\%$ (95% CI $18.0\%$ to $55.6\%$), with a much wider 95% prediction interval ($-26.5\%$ to $71.2\%$). Expanding to 13 Cochrane trials barely changed the direct estimate ($0.439$ to $0.435$). The calendar coefficient was $0.0986$ log-ratio units per decade (95% CI $-0.0445$ to $0.2417$), with fitted PF declining from $34.8%$ in 2005 to $20.6%$ in 2025. A Bayesian model gave a similar slope (posterior median $0.091$; 95% CrI $-0.036$ to $0.231$). The intercept-only model had the lowest AICc, though the linear-time model was within two units. Adjusting for background prevention reduced the slope to $0.057$ per decade. The historical average effect is stable across modeling choices, whereas prediction for a new setting is much less certain. Contemporary evidence is compatible with attenuation of the incremental varnish effect but does not require a temporal moderator; the pattern is consistent with a changing treatment contrast rather than weakened biological action.

Defensive organization is critical in soccer, particularly during negative transitions when teams are most vulnerable. The back-four defensive line plays a decisive role in preventing goal-scoring opportunities, yet its collective coordination remains difficult to quantify. This study introduces interpretable spatio-temporal indicators namely, space control, stretch index, pressure index, and defensive line height (absolute and relative) to evaluate the effectiveness of the back-four during defensive transitions. Using synchronized tracking and event data from the 2023-24 LaLiga season, 2,413 defensive sequences were analyzed following possession losses by FC Barcelona and Real Madrid CF. Two-way ANOVA revealed significant effects of team, outcome, and their interaction for key indicators, with relative line height showing the strongest association with defensive success. Predictive modeling using XGBoost achieved the highest discriminative performance (ROC AUC: 0.724 for Barcelona, 0.698 for Real Madrid), identifying space score and relative line height as dominant predictors. Comparative analysis revealed distinct team-specific defensive behaviors: Barcelona's success was characterized by higher spatial control and compact line coordination, whereas Real Madrid exhibited more adaptive but less consistent defensive structures. These findings demonstrate the tactical and predictive value of interpretable spatial indicators for quantifying collective defensive performance.

We address the challenge of forecasting counterfactual outcomes in a panel data with missing entries and temporally dependent latent factors – a common scenario in causal inference, where estimating unobserved potential outcomes ahead of time is essential. We propose Forecasting Counterfactuals under Stochastic Dynamics (FOCUS), a method that extends traditional matrix completion methods by leveraging time series dynamics of the factors, thereby enhancing the prediction accuracy of future counterfactuals. Building upon a consistent estimator of the factors, our method accommodates both stochastic and deterministic components within the factors, and provides a flexible framework for various applications. In case of stationary autoregressive factors and under standard conditions, we derive error bounds and establish asymptotic normality of our estimator. Empirical evaluations demonstrate that our method outperforms existing benchmarks when the latent factors have an autoregressive component. We illustrate FOCUS results on HeartSteps, a mobile health study, illustrating its effectiveness in forecasting step counts for users receiving activity prompts, thereby leveraging temporal patterns in user behavior.