Papers

The rapid ascent of artificial intelligence (AI) is often portrayed as a revolution born from computer science and engineering. This narrative, however, obscures a fundamental truth: the theoretical and methodological core of AI is, and has always been, statistical. This paper systematically argues that the field of statistics provides the indispensable foundation for machine learning and modern AI. We deconstruct AI into nine foundational pillars-Inference, Density Estimation, Sequential Learning, Generalization, Representation Learning, Interpretability, Causality, Optimization, and Unification-demonstrating that each is built upon century-old statistical principles. From the inferential frameworks of hypothesis testing and estimation that underpin model evaluation, to the density estimation roots of clustering and generative AI; from the time-series analysis inspiring recurrent networks to the causal models that promise true understanding, we trace an unbroken statistical lineage. While celebrating the computational engines that power modern AI, we contend that statistics provides the brain-the theoretical frameworks, uncertainty quantification, and inferential goals-while computer science provides the brawn-the scalable algorithms and hardware. Recognizing this statistical backbone is not merely an academic exercise, but a necessary step for developing more robust, interpretable, and trustworthy intelligent systems. We issue a call to action for education, research, and practice to re-embrace this statistical foundation. Ignoring these roots risks building a fragile future; embracing them is the path to truly intelligent machines. There is no machine learning without statistical learning; no artificial intelligence without statistical thought.

Hastie et al. (2022) found that ridge regularization is essential in high dimensional linear regression $y=\beta^Tx + \epsilon$ with isotropic co-variates $x\in \mathbb{R}^d$ and $n$ samples at fixed $d/n$. However, Hastie et al. (2022) also notes that when the co-variates are anisotropic and $\beta$ is aligned with the top eigenvalues of population covariance, the "situation is qualitatively different." In the present article, we make precise this observation for linear regression with highly anisotropic covariances and diverging $d/n$. We find (both theoretically and empirically) that simply scaling up (or inflating) the minimum $\ell_2$ norm interpolator by a constant greater than one can improve the generalization error. This is in sharp contrast to traditional regularization/shrinkage prescriptions. Moreover, we use a data-splitting technique to produce consistent estimators that achieve generalization error comparable to that of the optimally inflated minimum-norm interpolator. Our proof relies on matching upper and lower bounds for expectations of Gaussian random projections for a general class of anisotropic covariance matrices when $d/n\rightarrow \infty$.

Quantifying and predicting rare and extreme events is challenging because such events are infrequent, severe, and expensive to simulate. Existing data-driven methods often require multiple extremes in the training data or sampling process, leading to accurate predictions in quiescent regimes but high epistemic uncertainty in extreme-event regions. To overcome this limitation, we introduce Extreme Event Aware ($\eta$-) Learning, which does not require extreme events in the available data. The method reduces uncertainty even in uncharted extreme regimes by enforcing during training the statistics of an observable indicative of extremeness, obtained from qualitative knowledge or unlabeled data. This statistical regularization results in models that fit observed data while remaining consistent with prescribed observable statistics, enabling the generation of unprecedented extreme events. Optimal-transport-based theoretical results offer rigorous justification and establish key optimality properties. Numerical experiments on prototype systems and real-world precipitation downscaling problems demonstrate the effectiveness of the $\eta$-learning framework.

Longitudinal data are commonly encountered in biomedical research, including randomized trials and retrospective cohort studies. Subjects are typically followed over a period of time and may be scheduled for follow-up at pre-determined time points. However, subjects may miss their appointments or return at non-specified times, leading to irregularity in the visit process. IIW-GEEs have been developed as one method to account for this irregularity, whereby estimates from a visit intensity model are used as weights in a GEE model with an independent correlation structure. We show that currently available methods can be biased for situations in which the health outcome of interest may influence a subject's dropout from the study. We have extended the IIW-GEE framework to adjust for informative dropout and have demonstrated via simulation studies that this bias can be significantly reduced. We have illustrated this method using the STAR*D clinical trial data, and observed that the disease trajectory was generally overestimated when informative dropout was not accounted for.

Bayesian aggregation lets election forecasters combine diverse sources of information, such as state polls and economic and political indicators: as in our collaboration with The Economist magazine. However, the demands of real-time posterior updating, model checking, and communication introduce practical methodological challenges. In particular, sensitivity and scenario analysis help trace intricate dependencies and understand model behavior. Yet, under standard Markov chain Monte Carlo, even small tweaks to the model (e.g., in priors, data, hyperparameters) require full refitting, making it computationally expensive in real time. We introduce a meta-modeling strategy paired with a sequential sampling scheme; by traversing posterior meta-models, we enable real-time structured analyses without repeated refitting. In a back-test of the model, we demonstrate substantial computational gains, and show how seemingly innocuous data-wrangling choices can unintentionally introduce partisan asymmetries into the model's mechanisms for handling systematic polling error. Code is available at https://github.com/geonhee619/SMC-Sense.

Different transformer architectures implement identical linguistic computations via distinct connectivity patterns, yielding model imprinted “computational fingerprints” detectable through spectral analysis. Using graph signal processing on attention induced token graphs, we track changes in algebraic connectivity (Fiedler value, $\Delta\lambda_2$) under voice alternation across 20 languages and three model families, with a prespecified early window (layers 2–5). Our analysis uncovers clear architectural signatures: Phi-3-Mini shows a dramatic English specific early layer disruption ($\overline{\Delta\lambda_2}_{[2,5]}\!\approx\!-0.446$) while effects in 19 other languages are minimal, consistent with public documentation that positions the model primarily for English use. Qwen2.5-7B displays small, distributed shifts that are largest for morphologically rich languages, and LLaMA-3.2-1B exhibits systematic but muted responses. These spectral signatures correlate strongly with behavioral differences (Phi-3: $r=-0.976$) and are modulated by targeted attention head ablations, linking the effect to early attention structure and confirming functional relevance. Taken together, the findings are consistent with the view that training emphasis can leave detectable computational imprints: specialized processing strategies that manifest as measurable connectivity patterns during syntactic transformations. Beyond voice alternation, the framework differentiates reasoning modes, indicating utility as a simple, training free diagnostic for revealing architectural biases and supporting model reliability analysis.

Large language models achieve impressive results but distinguishing factual reasoning from hallucinations remains challenging. We propose a spectral analysis framework that models transformer layers as dynamic graphs induced by attention, with token embeddings as signals on these graphs. Through graph signal processing, we define diagnostics including Dirichlet energy, spectral entropy, and high-frequency energy ratios, with theoretical connections to computational stability. Experiments across GPT architectures suggest universal spectral patterns: factual statements exhibit consistent "energy mountain" behavior with low-frequency convergence, while different hallucination types show distinct signatures. Logical contradictions destabilize spectra with large effect sizes ($g>1.0$), semantic errors remain stable but show connectivity drift, and substitution hallucinations display intermediate perturbations. A simple detector using spectral signatures achieves 88.75% accuracy versus 75% for perplexity-based baselines, demonstrating practical utility. These findings indicate that spectral geometry may capture reasoning patterns and error behaviors, potentially offering a framework for hallucination detection in large language models.

Modern weather forecasting has increasingly transitioned from numerical weather prediction (NWP) to data-driven machine learning forecasting techniques. While these new models produce probabilistic forecasts to quantify uncertainty, their training and evaluation may remain hindered by conventional scoring rules, primarily MSE, which are designed for single time point predictions and ignore the highly correlated data structures present in weather behaviour. This work introduces the signature kernel scoring rule to the domain of weather forecasting, which reframes weather variables as continuous paths to encode temporal and spatial dependencies through iterated integrals. Validated as strictly proper through the use of path augmentations to guarantee uniqueness, the signature kernel provides a theoretically robust metric for forecast verification and model training. Empirical evaluations through weather scorecards on WeatherBench 2 models demonstrate the signature kernel scoring rule's high discriminative power and unique capacity to capture path-dependent interactions. Following previous demonstration of successful adversarial-free probabilistic training, we train sliding window generative neural networks using a predictive-sequential scoring rule on ERA5 reanalysis weather data. Using a lightweight model, we demonstrate that signature kernel based training outperforms climatology for forecast paths of up to fifteen timesteps.

Spatial transcriptomics measures the expression of thousands of genes in a tissue sample while preserving its spatial structure. This class of technologies has enabled the investigation of the spatial variation of gene expressions and their impact on specific biological processes. Identifying genes with similar expression profiles is of utmost importance, thus motivating the development of flexible methods leveraging spatial data structure to cluster genes. Here, we propose a modeling framework for clustering observations measured over numerous spatial locations via Gaussian processes. Rather than specifying their covariance kernels as a function of the spatial structure, we use it to inform a generalized Cholesky decomposition of their precision matrices. This approach prevents issues with kernel misspecification and facilitates the estimation of a non-stationarity spatial covariance structure. Applied to spatial transcriptomic data, our model identifies gene clusters with distinctive spatial correlation patterns across tissue areas comprising different cell types, like tumoral and stromal areas.

Estimating the causal dose-response function is challenging, particularly when data from a single source are insufficient to estimate responses precisely across all exposure levels. To overcome this limitation, we propose a data fusion framework that leverages multiple data sources that are partially aligned with the target distribution. Specifically, we derive a Neyman-orthogonal loss function tailored for estimating the dose-response function within data fusion settings. To improve computational efficiency, we propose a stochastic approximation that retains orthogonality. We apply kernel ridge regression with this approximation, which provides closed-form estimators. Our theoretical analysis demonstrates that incorporating additional data sources yields tighter finite-sample regret bounds and improved worst-case performance, as confirmed via minimax lower bound comparison. Simulation studies validate the practical advantages of our approach, showing improved estimation accuracy when employing data fusion. This study highlights the potential of data fusion for estimating non-smooth parameters such as causal dose-response functions.

Background: Target trial emulation (TTE) that applies trial design principles to improve the analysis of non-randomized studies is increasingly being used. Applications of TTE to emulate cluster randomized trials (RCTs) have been limited. This study explored how to integrate simulation-guided design into the TTE framework to inform planning of a non-randomized cluster trial. Methods: We performed simulations to prospectively plan data collection of a non-randomized study emulating a village-level cluster RCT when cluster-randomization was infeasible. The planned study will assess the impact of mass distribution of nutritional supplements embedded within an existing immunization program to improve pentavalent vaccination rates among children 12-24 months old in Niger. The design included covariate-constrained random selection of villages for outcome ascertainment at follow-up. Simulations used baseline census data on pentavalent vaccination rates and cluster-level covariates to compare the type I error rate and power of four statistical methods: beta-regression; quasi-binomial regression; inverse probability of treatment weighting (IPTW); and naive Wald test. Results: Of the four analytic methods considered, only IPTW and beta-regression controlled the type I error rate at 0.05, but IPTW yielded poor statistical power. Beta-regression that showed adequate statistical power was chosen as our primary analysis. Conclusions: Adopting simulation-guided design principles within TTE can enable robust planning of a group-level non-randomized study emulating a cluster RCT. Lessons from this study also apply to TTE planning of individually-RCTs.

Solar flares, email exchanges, and many natural or social systems exhibit bursty dynamics, with periods of intense activity separated by long inactivity. These patterns often follow power- law distributions in inter-event intervals or event rates. Existing models typically capture only one of these features and rely on non-local memory, which complicates analysis and mechanistic interpretation. We introduce a novel self-reinforcing point process whose event rates are governed by local, Markovian nonlinear dynamics and post-event resets. The model generates power-law tails for both inter-event intervals and event rates over a broad range of exponents observed empirically across natural and human phenomena. Compared to non-local models such as Hawkes processes, our approach is mechanistically simpler, highly analytically tractable, and also easier to simulate. We provide methods for model fitting and validation, establishing this framework as a versatile foundation for the study of bursty phenomena.

We propose a new method for statistical inference in generalized linear models. In the overparameterized regime, Principal Component Regression (PCR) reduces variance by projecting high-dimensional data to a low-dimensional principal subspace before fitting. However, PCR incurs truncation bias whenever the true regression vector has mass outside the retained principal components (PC). To mitigate the bias, we propose Calibrated Principal Component Regression (CPCR), which first learns a low-variance prior in the PC subspace and then calibrates the model in the original feature space via a centered Tikhonov step. CPCR leverages cross-fitting and controls the truncation bias by softening PCR's hard cutoff. Theoretically, we calculate the out-of-sample risk in the random matrix regime, which shows that CPCR outperforms standard PCR when the regression signal has non-negligible components in low-variance directions. Empirically, CPCR consistently improves prediction across multiple overparameterized problems. The results highlight CPCR's stability and flexibility in modern overparameterized settings.

There has been an increasing interest in using cell and gene therapy (CGT) to treat/cure difficult diseases. The hallmark of CGT trials are the small sample size and extremely high efficacy. Due to the innovation and novelty of such therapies, when there is missing data, more scrutiny is exercised, and regulators often request for missing data handling strategy when missing data occurs. Often, multiple imputation (MI) will be used. MI for continuous endpoint is well established but literature of MI for binary endpoint is lacking. In this work, we compare and develop 3 new methods to handle missing data using MI for binary endpoints when the sample size is small and efficacy extremely high. The parameter of interest is population proportion of success. We show that our proposed methods performed well and produced good 95% coverage. We also applied our methods to an actual clinical study, the Clinical Islet Transplantation (CIT) Protocol 07, conducted by National Institutes of Health (NIH).

We present Anchored Direct Preference Optimization (ADPO), a policy alignment method derived from first principles of KL-regularized reinforcement learning. Unlike standard approaches that treat the reference policy merely as a regularizer, we show that the optimal policy in reinforcement learning from human feedback inherently operates in a differential coordinate system, optimizing relative advantage in the form of log ratios rather than absolute probabilities. ADPO explicitly parameterizes this optimal structure through anchored logits, effectively decoupling response quality from prior popularity and creating an implicit trust region through curvature scaling. We show that this formulation unifies supervised fine-tuning, reinforcement learning, and ranking-based objectives under a single geometric perspective. Theoretically, ADPO resolves the probability smearing problem of supervised fine-tuning while avoiding the mode-seeking instability characteristic of reverse-KL methods. Empirically, the listwise ranking variant of ADPO achieves state-of-the-art performance on reasoning tasks, outperforming GRPO by 30.9 percent on Qwen3-1.7B and demonstrating superior robustness under distribution shift.

We provide an inferential framework to assess variable importance for heterogeneous treatment effects. This assessment is especially useful in high-risk domains such as medicine, where decision makers hesitate to rely on black-box treatment recommendation algorithms. The variable importance measures we consider are local in that they may differ across individuals, while the inference is global in that it tests whether a given variable is important for any individual. Our approach builds on recent developments in semiparametric theory for function-valued parameters, and is valid even when statistical machine learning algorithms are employed to quantify treatment effect heterogeneity. We demonstrate the applicability of our method to infectious disease prevention strategies.

In clinical studies with paired organs, binary outcomes often exhibit intra-subject correlation and may include a mixture of unilateral and bilateral observations. Under Donner's constant correlation model, we develop three likelihood-based test statistics (the likelihood ratio, Wald-type, and score tests) for assessing the risk difference between two proportions. Simulation studies demonstrate good control of type I error and comparable power among the three tests, with the score test showing slightly better stability. Applications to otolaryngologic and ophthalmologic data illustrate the methods. An online calculator is also provided for power analysis and risk difference testing. The score test is recommended for practical use and future studies with combined unilateral and bilateral binary data.

Value-based algorithms are a cornerstone of off-policy reinforcement learning due to their simplicity and training stability. However, their use has traditionally been restricted to discrete action spaces, as they rely on estimating Q-values for individual state-action pairs. In continuous action spaces, evaluating the Q-value over the entire action space becomes computationally infeasible. To address this, actor-critic methods are typically employed, where a critic is trained on off-policy data to estimate Q-values, and an actor is trained to maximize the critic's output. Despite their popularity, these methods often suffer from instability during training. In this work, we propose a purely value-based framework for continuous control that revisits structural maximization of Q-functions, introducing a set of key architectural and algorithmic choices to enable efficient and stable learning. We evaluate the proposed actor-free Q-learning approach on a range of standard simulation tasks, demonstrating performance and sample efficiency on par with state-of-the-art baselines, without the cost of learning a separate actor. Particularly, in environments with constrained action spaces, where the value functions are typically non-smooth, our method with structural maximization outperforms traditional actor-critic methods with gradient-based maximization. We have released our code at https://github.com/USC-Lira/Q3C.

Current practices for designing cluster-randomized trials (cRCTs) typically rely on closed-form formulas for power calculations. For cRCTs using covariate-constrained randomization, the utility of conventional calculations might be limited, particularly when data is nested. We compared simulation-based planning of a nested cRCT using covariate-constrained randomization to conventional power calculations using OptiMAx-Chad as a case study. OptiMAx-Chad will examine the impact of embedding mass distribution of small-quantity lipid-based nutrient supplements within an expanded programme on immunization on first-dose measles-containing vaccine (MCV1) coverage among children aged 12-24 months in rural villages in Ngouri. Within the 12 health areas to be randomized, a random subset of villages will be selected for outcome collection. 1,000,000 assignments of health areas with different possible village selections were generated using covariate-constrained randomization to balance baseline village characteristics. The empirically estimated intracluster correlation coefficient (ICC) and the World Health Organization (WHO) recommended values of 1/3 and 1/6 were considered. The desired operating characteristics were 80% power at 0.05 one-sided type I error rate. Using conventional calculations target power for a realistic treatment effect could not be achieved with the WHO recommended values. Conventional calculations also showed a plateau in power after a certain cluster size. Our simulations matched the design of OptiMAx-Chad with covariate adjustment and random selection, and showed that power did not plateau. Instead, power increased with increasing cluster size. Planning complex cRCTs with covariate constrained randomization and a multi-nested data structure with conventional closed-form formulas can be misleading. Simulations can improve the planning of cRCTs.

A fundamental challenge in protein design is the trade-off between generating structural diversity while preserving motif biological function. Current state-of-the-art methods, such as partial diffusion in RFdiffusion, often fail to resolve this trade-off: small perturbations yield motifs nearly identical to the native structure, whereas larger perturbations violate the geometric constraints necessary for biological function. We introduce Protein Generation with Embedding Learning (PGEL), a general framework that learns high-dimensional embeddings encoding sequence and structural features of a target motif in the representation space of a diffusion model's frozen denoiser, and then enhances motif diversity by introducing controlled perturbations in the embedding space. PGEL is thus able to loosen geometric constraints while satisfying typical design metrics, leading to more diverse yet viable structures. We demonstrate PGEL on three representative cases: a monomer, a protein-protein interface, and a cancer-related transcription factor complex. In all cases, PGEL achieves greater structural diversity, better designability, and improved self-consistency, as compared to partial diffusion. Our results establish PGEL as a general strategy for embedding-driven protein generation allowing for systematic, viable diversification of functional motifs.

Fractional Gradient Descent (FGD) offers a novel and promising way to accelerate optimization by incorporating fractional calculus into machine learning. Although FGD has shown encouraging initial results across various optimization tasks, it faces significant challenges with convergence behavior and hyperparameter selection. Moreover, the impact of its hyperparameters is not fully understood, and scheduling them is particularly difficult in non-convex settings such as neural network training. To address these issues, we propose a novel approach called Learning to Optimize Caputo Fractional Gradient Descent (L2O-CFGD), which meta-learns how to dynamically tune the hyperparameters of Caputo FGD (CFGD). Our method's meta-learned schedule outperforms CFGD with static hyperparameters found through an extensive search and, in some tasks, achieves performance comparable to a fully black-box meta-learned optimizer. L2O-CFGD can thus serve as a powerful tool for researchers to identify high-performing hyperparameters and gain insights on how to leverage the history-dependence of the fractional differential in optimization.

While Variational Inference (VI) is central to modern generative models like Variational Autoencoders (VAEs) and Denoising Diffusion Models (DDMs), its pedagogical treatment is split across disciplines. In statistics, VI is typically framed as a Bayesian method for posterior approximation. In machine learning, however, VAEs and DDMs are developed from a Frequentist viewpoint, where VI is used to approximate a maximum likelihood estimator. This creates a barrier for statisticians, as the principles behind VAEs and DDMs are hard to contextualize without a corresponding Frequentist introduction to VI. This paper provides that introduction: we explain the theory for VI, VAEs, and DDMs from a purely Frequentist perspective, starting with the classical Expectation-Maximization (EM) algorithm. We show how VI arises as a scalable solution for intractable E-steps and how VAEs and DDMs are natural, deep-learning-based extensions of this framework, thereby bridging the gap between classical statistical inference and modern generative AI.

In recent years, there has been growing interest in building closed-loop supply chains (SCs). However, many of the current methods struggle when it comes to fully embracing circular economy principles. Enhancing the design and management of these networks holds significant potential to promote stronger collaboration among supply chain partners, ultimately fostering more sustainable and efficient operational practices. For this purpose, this study addresses a new circular economy model based on a real case study of a mask SC in the healthcare sector in British Columbia, Canada. The objective is to show that implementing circular practices can lead to considerable financial and environmental benefits, as well as the creation of new job opportunities. To achieve this, a multi-objective mixed-integer linear programming model is developed to identify the most efficient trade-off among sustainable objectives, while adhering to imposed constraints. The proposed closed-loop SC model outperforms the existing linear model in all three aspects of sustainability, namely economic, environmental and social. The improvement leads to significant economic and environmental benefits by preventing the disposal of used masks, giving them a second chance for disinfection and reprocessing, and reintroducing them into the SC as new ones. While the results of the circular economy model demonstrate profit gains and environmental recovery, the linear SC model showed negative profit with higher carbon emissions. Regarding the social aspects, compared to the current system, our approach not only nearly doubles the number of jobs created but also significantly reduces shortages, highlighting sustainable development aspects related to equity and social welfare. The findings offer valuable insights for researchers and practitioners seeking to implement sustainability within a circular economy framework in SCs.

We study online preference-based reinforcement learning (PbRL) with the goal of improving sample efficiency. While a growing body of theoretical work has emerged-motivated by PbRL's recent empirical success, particularly in aligning large language models (LLMs)-most existing studies focus only on pairwise comparisons. A few recent works (Zhu et al., 2023, Mukherjee et al., 2024, Thekumparampil et al., 2024) have explored using multiple comparisons and ranking feedback, but their performance guarantees fail to improve-and can even deteriorate-as the feedback length increases, despite the richer information available. To address this gap, we adopt the Plackett-Luce (PL) model for ranking feedback over action subsets and propose M-AUPO, an algorithm that selects multiple actions by maximizing the average uncertainty within the offered subset. We prove that M-AUPO achieves a suboptimality gap of $\tilde{O}\left( \frac{d}{T} \sqrt{ \sum_{t=1}^T \frac{1}{|S_t|}} \right)$, where $T$ is the total number of rounds, $d$ is the feature dimension, and $|S_t|$ is the size of the subset at round $t$. This result shows that larger subsets directly lead to improved performance and, notably, the bound avoids the exponential dependence on the unknown parameter's norm, which was a fundamental limitation in most previous works. Moreover, we establish a near-matching lower bound of $\Omega \left( \frac{d}{K \sqrt{T}} \right)$, where $K$ is the maximum subset size. To the best of our knowledge, this is the first theoretical result in PbRL with ranking feedback that explicitly shows improved sample efficiency as a function of the subset size.

E-values have gained prominence as flexible tools for statistical inference and risk control, enabling anytime- and post-hoc-valid procedures under minimal assumptions. However, many real-world applications fundamentally rely on sensitive data, which can be leaked through e-values. To ensure their safe release, we propose a general framework to transform non-private e-values into differentially private ones. Towards this end, we develop a novel biased multiplicative noise mechanism that ensures our e-values remain statistically valid. We show that our differentially private e-values attain strong statistical power, and are asymptotically as powerful as their non-private counterparts. Experiments across online risk monitoring, private healthcare, and conformal e-prediction demonstrate our approach's effectiveness and illustrate its broad applicability.

Risk assessment in casualty insurance, such as flood risk, traditionally relies on extreme-value methods that emphasizes rare events. These approaches are well-suited for characterizing tail risk, but do not capture the broader dynamics of environmental variables such as moderate or frequent loss events. To complement these methods, we propose a modelling framework for estimating the full (daily) distribution of environmental variables as a function of time, that is a distributional version of typical climatological summary statistics, thereby incorporating both seasonal variation and gradual long-term changes. Aside from the time trend, to capture seasonal variation our approach simultaneously estimates the distribution for each instant of the seasonal cycle, without explicitly modelling the temporal dependence present in the data. To do so, we adopt a framework inspired by GAMLSS (Generalized Additive Models for Location, Scale, and Shape), where the parameters of the distribution vary over the seasonal cycle as a function of explanatory variables depending only on the time of year, and not on the past values of the process under study. Ignoring the temporal dependence in the seasonal variation greatly simplifies the modelling but poses inference challenges that we clarify and overcome. We apply our framework to daily river flow data from three hydrometric stations along the Fraser River in British Columbia, Canada, and analyse the flood of the Fraser River in early winter of 2021.

Transformer architectures can solve unseen tasks based on input-output pairs in a given prompt due to in-context learning (ICL). Existing theoretical studies on ICL have mainly focused on linear regression tasks, often with i.i.d. inputs. To understand how transformers express ICL when modeling dynamics-driven functions, we investigate Markovian function learning through a structured ICL setup, where we characterize the loss landscape to reveal underlying optimization behaviors. Specifically, we (1) provide the closed-form expression of the global minimizer (in an enlarged parameter space) for a single-layer linear self-attention (LSA) model; (2) prove that recovering transformer parameters that realize the optimal solution is NP-hard in general, revealing a fundamental limitation of one-layer LSA in representing structured dynamical functions; and (3) supply a novel interpretation of a multilayer LSA as performing preconditioned gradient descent to optimize multiple objectives beyond the square loss. These theoretical results are numerically validated using simplified transformers.

Volatility clustering and spillovers are key features of real-world financial time series when there are a lot of cross-sectional financial assets. While network analysis helps connect stocks that are 'similar' or 'correlated', which is effective to link volatility spillovers between stocks, contemporary multivariate ARCH-GARCH formulations struggle to represent structured network dependence and remain parsimonious. We introduce the Generalised Network GARCH (GNGARCH) model as a network volatility model that embeds the GARCH dynamics within the Generalised Network Autoregressive (GNAR) framework, to capture the dynamic volatility of financial asset return by both the asset itself and its 'neighbouring' assets from the constructed virtual network. The proposed volatility model GNGARCH also addresses the limitations for current studies of network GARCH by adapting neighbouring volatility persistence, dynamic conditional covariance updates, and allowing higher-order neighbouring effects rather than only immediate neighbours. This paper provides the model derivation, vectorisation and conversion, stationarity conditions, and also an extension by incorporating threshold coefficients to capture leverage effects. We show that the GNGARCH is a valid volatility model satisfying the stylised facts of financial return series through simulation. Parameter estimation is then performed by using squared returns as variance proxy and minimising a loss function that is either mean squared error (MSE) or quasi-likelihood (QLIKE). We apply our model on 75 of the most active US stocks under a virtual network, and highlight the model's ability in volatility estimation and forecast.

Most of the work on checking spherical symmetry assumptions on the distribution of the $p$-dimensional random vector $Y$ has its focus on statistical tests for the null hypothesis of exact spherical symmetry. In this paper, we take a different point of view and propose a measure for the deviation from spherical symmetry, which is based on the minimum distance between the distribution of the vector $\big (\|Y\|, Y/ \|Y\| )^\top $ and its best approximation by a distribution of a vector $\big (\|Y_s\|, Y_s/ \|Y_s \| )^\top $ corresponding to a random vector $Y_s$ with a spherical distribution. We develop estimators for the minimum distance with corresponding statistical guarantees (provided by asymptotic theory) and demonstrate the applicability of our approach by means of a simulation study and a real data example.

Accurate annual average daily traffic (AADT) data are vital for transport planning and infrastructure management. However, automatic traffic detectors across national road networks often provide incomplete coverage, leading to underrepresentation of minor roads. While recent machine learning advances have improved AADT estimation at unmeasured locations, most models produce only point predictions and overlook estimation uncertainty. This study addresses that gap by introducing an interval prediction approach that explicitly quantifies predictive uncertainty. We integrate a Quantile Random Forest model with Principal Component Analysis to generate AADT prediction intervals, providing plausible traffic ranges bounded by estimated minima and maxima. Using data from over 2,000 minor roads in England and Wales, and evaluated with specialized interval metrics, the proposed method achieves an interval coverage probability of 88.22%, a normalized average width of 0.23, and a Winkler Score of 7,468.47. By combining machine learning with spatial and high-dimensional analysis, this framework enhances both the accuracy and interpretability of AADT estimation, supporting more robust and informed transport planning.