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We propose a general approach for encouraging fairness in survival analysis models based on minimizing a worst-case error across all subpopulations that occur with at least a user-specified probability. This approach can be used to convert many existing survival analysis models into ones that simultaneously encourage fairness, without requiring the user to specify which attributes or features to treat as sensitive in the training loss function. From a technical standpoint, our approach applies recent developments of distributionally robust optimization (DRO) to survival analysis. The complication is that existing DRO theory uses a training loss function that decomposes across contributions of individual data points, i.e., any term that shows up in the loss function depends only on a single training point. This decomposition does not hold for commonly used survival loss functions, including for the Cox proportional hazards model, its deep neural network variants, and many other recently developed models that use loss functions involving ranking or similarity score calculations. We address this technical hurdle using a sample splitting strategy. We demonstrate our sample splitting DRO approach by using it to create fair versions of a diverse set of existing survival analysis models including the Cox model (and its deep variant DeepSurv), the discrete-time model DeepHit, and the neural ODE model SODEN. We also establish a finite-sample theoretical guarantee to show what our sample splitting DRO loss converges to. For the Cox model, we further derive an exact DRO approach that does not use sample splitting. For all the models that we convert into DRO variants, we show that the DRO variants often score better on recently established fairness metrics (without incurring a significant drop in accuracy) compared to existing survival analysis fairness regularization techniques.

Numerical simulation is powerful to study nonlinear solid mechanics problems. However, mesh-based or particle-based numerical methods suffer from the common shortcoming of being time-consuming, particularly for complex problems with real-time analysis requirements. This study presents a clustering adaptive Gaussian process regression (CAG) method aiming for real-time prediction for nonlinear structural responses in solid mechanics. It is a data-driven machine learning method featuring a small sample size, high accuracy, and high efficiency, leveraging nonlinear structural response patterns. Similar to the traditional Gaussian process regression (GPR) method, it operates in offline and online stages. In the offline stage, an adaptive sample generation technique is introduced to cluster datasets into distinct patterns for demand-driven sample allocation. This ensures comprehensive coverage of the critical samples for the solution space of interest. In the online stage, following the divide-and-conquer strategy, a pre-prediction classification categorizes problems into predefined patterns sequentially predicted by the trained multi-pattern Gaussian process regressor. In addition, dimension reduction and restoration techniques are employed in the proposed method to enhance its efficiency. A set of problems involving material, geometric, and boundary condition nonlinearities is presented to demonstrate the CAG method's abilities. The proposed method can offer predictions within a second and attain high precision with only about 20 samples within the context of this study, outperforming the traditional GPR using uniformly distributed samples for error reductions ranging from 1 to 3 orders of magnitude. The CAG method is expected to offer a powerful tool for real-time prediction of nonlinear solid mechanical problems and shed light on the complex nonlinear structural response pattern.

Quantitative measurement of ageing across systems and components is crucial for accurately assessing reliability and predicting failure probabilities. This measurement supports effective maintenance scheduling, performance optimisation, and cost management. Examining the ageing characteristics of a system that operates beyond a specified time $t > 0$ yields valuable insights. This paper introduces a novel metric for ageing, termed the Variance Residual Life Ageing Intensity (VRLAI) function, and explores its properties across various probability distributions. Additionally, we characterise the closure properties of the two ageing classes defined by the VRLAI function. We propose a new ordering, called the Variance Residual Life Ageing Intensity (VRLAI) ordering, and discuss its various properties. Furthermore, we examine the closure of the VRLAI order under coherent systems.

Data dispersed across multiple files are commonly integrated through probabilistic linkage methods, where even minimal error rates in record matching can significantly contaminate subsequent statistical analyses. In regression problems, we examine scenarios where the identifiers of predictors or responses are subject to an unknown permutation, challenging the assumption of correspondence. Many emerging approaches in the literature focus on sparsely permuted data, where only a small subset of pairs ($k << n$) are affected by the permutation, treating these permuted entries as outliers to restore original correspondence and obtain consistent estimates of regression parameters. In this article, we complement the existing literature by introducing a novel generalized robust Bayesian formulation of the problem. We develop an efficient posterior sampling scheme by adapting the fractional posterior framework and addressing key computational bottlenecks via careful use of discrete optimal transport and sampling in the space of binary matrices with fixed margins. Further, we establish new posterior contraction results within this framework, providing theoretical guarantees for our approach. The utility of the proposed framework is demonstrated via extensive numerical experiments.

Survival regression is widely used to model time-to-events data, to explore how covariates may influence the occurrence of events. Modern datasets often encompass a vast number of covariates across many subjects, with only a subset of the covariates significantly affecting survival. Additionally, subjects often belong to an unknown number of latent groups, where covariate effects on survival differ significantly across groups. The proposed methodology addresses both challenges by simultaneously identifying the latent sub-groups in the heterogeneous population and evaluating covariate significance within each sub-group. This approach is shown to enhance the predictive accuracy for time-to-event outcomes, via uncovering varying risk profiles within the underlying heterogeneous population and is thereby helpful to device targeted disease management strategies.

Missing data is a common issue in medical, psychiatry, and social studies. In literature, Multiple Imputation (MI) was proposed to multiply impute datasets and combine analysis results from imputed datasets for statistical inference using Rubin's rule. However, Rubin's rule only works for combined inference on statistical tests with point and variance estimates and is not applicable to combine general F-statistics or Chi-square statistics. In this manuscript, we provide a solution to combine F-test statistics from multiply imputed datasets, when the F-statistic has an explicit fractional form (that is, both the numerator and denominator of the F-statistic are reported). Then we extend the method to combine Chi-square statistics from multiply imputed datasets. Furthermore, we develop methods for two commonly applied F-tests, Welch's ANOVA and Type-III tests of fixed effects in mixed effects models, which do not have the explicit fractional form. SAS macros are also developed to facilitate applications.

We introduce the BMRMM package implementing Bayesian inference for a class of Markov renewal mixed models which can characterize the stochastic dynamics of a collection of sequences, each comprising alternative instances of categorical states and associated continuous duration times, while being influenced by a set of exogenous factors as well as a 'random' individual. The default setting flexibly models the state transition probabilities using mixtures of Dirichlet distributions and the duration times using mixtures of gamma kernels while also allowing variable selection for both. Modeling such data using simpler Markov mixed models also remains an option, either by ignoring the duration times altogether or by replacing them with instances of an additional category obtained by discretizing them by a user-specified unit. The option is also useful when data on duration times may not be available in the first place. We demonstrate the package's utility using two data sets.

Calibration ensures that predicted uncertainties align with observed uncertainties. While there is an extensive literature on recalibration methods for univariate probabilistic forecasts, work on calibration for multivariate forecasts is much more limited. This paper introduces a novel post-hoc recalibration approach that addresses multivariate calibration for potentially misspecified models. Our method involves constructing local mappings between vectors of marginal probability integral transform values and the space of observations, providing a flexible and model free solution applicable to continuous, discrete, and mixed responses. We present two versions of our approach: one uses K-nearest neighbors, and the other uses normalizing flows. Each method has its own strengths in different situations. We demonstrate the effectiveness of our approach on two real data applications: recalibrating a deep neural network's currency exchange rate forecast and improving a regression model for childhood malnutrition in India for which the multivariate response has both discrete and continuous components.

We propose a novel cointegrated autoregressive model for matrix-valued time series, with bi-linear cointegrating vectors corresponding to the rows and columns of the matrix data. Compared to the traditional cointegration analysis, our proposed matrix cointegration model better preserves the inherent structure of the data and enables corresponding interpretations. To estimate the cointegrating vectors as well as other coefficients, we introduce two types of estimators based on least squares and maximum likelihood. We investigate the asymptotic properties of the cointegrated matrix autoregressive model under the existence of trend and establish the asymptotic distributions for the cointegrating vectors, as well as other model parameters. We conduct extensive simulations to demonstrate its superior performance over traditional methods. In addition, we apply our proposed model to Fama-French portfolios and develop a effective pairs trading strategy.

Self-exciting point processes are widely used to model the contagious effects of crime events living within continuous geographic space, using their occurrence time and locations. However, in urban environments, most events are naturally constrained within the city's street network structure, and the contagious effects of crime are governed by such a network geography. Meanwhile, the complex distribution of urban infrastructures also plays an important role in shaping crime patterns across space. We introduce a novel spatio-temporal-network point process framework for crime modeling that integrates these urban environmental characteristics by incorporating self-attention graph neural networks. Our framework incorporates the street network structure as the underlying event space, where crime events can occur at random locations on the network edges. To realistically capture criminal movement patterns, distances between events are measured using street network distances. We then propose a new mark for a crime event by concatenating the event's crime category with the type of its nearby landmark, aiming to capture how the urban design influences the mixing structures of various crime types. A graph attention network architecture is adopted to learn the existence of mark-to-mark interactions. Extensive experiments on crime data from Valencia, Spain, demonstrate the effectiveness of our framework in understanding the crime landscape and forecasting crime risks across regions.

We establish empirical risk minimization principles for active learning by deriving a family of upper bounds on the generalization error. Aligning with empirical observations, the bounds suggest that superior query algorithms can be obtained by combining both informativeness and representativeness query strategies, where the latter is assessed using integral probability metrics. To facilitate the use of these bounds in application, we systematically link diverse active learning scenarios, characterized by their loss functions and hypothesis classes to their corresponding upper bounds. Our results show that regularization techniques used to constraint the complexity of various hypothesis classes are sufficient conditions to ensure the validity of the bounds. The present work enables principled construction and empirical quality-evaluation of query algorithms in active learning.

Shape graphs are complex geometrical structures commonly found in biological and anatomical systems. A shape graph is a collection of nodes, some connected by curvilinear edges with arbitrary shapes. Their high complexity stems from the large number of nodes and edges and the complex shapes of edges. With an eye for statistical analysis, one seeks low-complexity representations that retain as much of the global structures of the original shape graphs as possible. This paper develops a framework for reducing graph complexity using hierarchical clustering procedures that replace groups of nodes and edges with their simpler representatives. It demonstrates this framework using graphs of retinal blood vessels in two dimensions and neurons in three dimensions. The paper also presents experiments on classifications of shape graphs using progressively reduced levels of graph complexity. The accuracy of disease detection in retinal blood vessels drops quickly when the complexity is reduced, with accuracy loss particularly associated with discarding terminal edges. Accuracy in identifying neural cell types remains stable with complexity reduction.

We show that for any family of distributions with support on [0,1] with strictly monotonic cumulative distribution function (CDF) that has no jumps and is quantile-identifiable (i.e., any two distinct quantiles identify the distribution), knowing the first moment and c-statistic is enough to identify the distribution. The derivations motivate numerical algorithms for mapping a given pair of expected value and c-statistic to the parameters of specified two-parameter distributions for probabilities. We implemented these algorithms in R and in a simulation study evaluated their numerical accuracy for common families of distributions for risks (beta, logit-normal, and probit-normal). An area of application for these developments is in risk prediction modeling (e.g., sample size calculations and Value of Information analysis), where one might need to estimate the parameters of the distribution of predicted risks from the reported summary statistics.

While the classic off-policy evaluation (OPE) literature commonly assumes decision time points to be evenly spaced for simplicity, in many real-world scenarios, such as those involving user-initiated visits, decisions are made at irregularly-spaced and potentially outcome-dependent time points. For a more principled evaluation of the dynamic policies, this paper constructs a novel OPE framework, which concerns not only the state-action process but also an observation process dictating the time points at which decisions are made. The framework is closely connected to the Markov decision process in computer science and with the renewal process in the statistical literature. Within the framework, two distinct value functions, derived from cumulative reward and integrated reward respectively, are considered, and statistical inference for each value function is developed under revised Markov and time-homogeneous assumptions. The validity of the proposed method is further supported by theoretical results, simulation studies, and a real-world application from electronic health records (EHR) evaluating periodontal disease treatments.

The probability of causation (PC) is often used in liability assessments. In a legal context, for example, where a patient suffered the side effect after taking a medication and sued the pharmaceutical company as a result, the value of the PC can help assess the likelihood that the side effect was caused by the medication, in other words, how likely it is that the patient will win the case. Beyond the issue of legal disputes, the PC plays an equally large role when one wants to go about explaining causal relationships between events that have already occurred in other areas. This article begins by reviewing the definitions and bounds of the probability of causality for binary outcomes, then generalizes them to ordinal outcomes. It demonstrates that incorporating additional mediator variable information in a complete mediation analysis provides a more refined bound compared to the simpler scenario where only exposure and outcome variables are considered.

A novel method is proposed for the exact posterior mean and covariance of the random effects given the response in a generalized linear mixed model (GLMM) when the response does not follow normal. The research solves a long-standing problem in Bayesian statistics when an intractable integral appears in the posterior distribution. It is well-known that the posterior distribution of the random effects given the response in a GLMM when the response does not follow normal contains intractable integrals. Previous methods rely on Monte Carlo simulations for the posterior distributions. They do not provide the exact posterior mean and covariance of the random effects given the response. The special integral computation (SIC) method is proposed to overcome the difficulty. The SIC method does not use the posterior distribution in the computation. It devises an optimization problem to reach the task. An advantage is that the computation of the posterior distribution is unnecessary. The proposed SIC avoids the main difficulty in Bayesian analysis when intractable integrals appear in the posterior distribution.

Linear model prediction with a large number of potential predictors is both statistically and computationally challenging. The traditional approaches are largely based on shrinkage selection/estimation methods, which are applicable even when the number of potential predictors is (much) larger than the sample size. A situation of the latter scenario occurs when the candidate predictors involve many binary indicators corresponding to categories of some categorical predictors as well as their interactions. We propose an alternative approach to the shrinkage prediction methods in such a case based on mixed model prediction, which effectively treats combinations of the categorical effects as random effects. We establish theoretical validity of the proposed method, and demonstrate empirically its advantage over the shrinkage methods. We also develop measures of uncertainty for the proposed method and evaluate their performance empirically. A real-data example is considered.

This paper introduces a novel framework for tensor eigenvalue analysis in the context of multi-modal data fusion, leveraging topological invariants such as Betti numbers. While traditional approaches to tensor eigenvalues rely on algebraic extensions of matrix theory, this work provides a topological perspective that enriches the understanding of tensor structures. By establishing new theorems linking eigenvalues to topological features, the proposed framework offers deeper insights into the latent structure of data, enhancing both interpretability and robustness. Applications to data fusion illustrate the theoretical and practical significance of the approach, demonstrating its potential for broad impact across machine learning and data science domains.

The identification of surrogate markers is motivated by their potential to make decisions sooner about a treatment effect. However, few methods have been developed to actually use a surrogate marker to test for a treatment effect in a future study. Most existing methods consider combining surrogate marker and primary outcome information to test for a treatment effect, rely on fully parametric methods where strict parametric assumptions are made about the relationship between the surrogate and the outcome, and/or assume the surrogate marker is measured at only a single time point. Recent work has proposed a nonparametric test for a treatment effect using only surrogate marker information measured at a single time point by borrowing information learned from a prior study where both the surrogate and primary outcome were measured. In this paper, we utilize this nonparametric test and propose group sequential procedures that allow for early stopping of treatment effect testing in a setting where the surrogate marker is measured repeatedly over time. We derive the properties of the correlated surrogate-based nonparametric test statistics at multiple time points and compute stopping boundaries that allow for early stopping for a significant treatment effect, or for futility. We examine the performance of our testing procedure using a simulation study and illustrate the method using data from two distinct AIDS clinical trials.

In many data-driven applications, higher-order relationships among multiple objects are essential in capturing complex interactions. Hypergraphs, which generalize graphs by allowing edges to connect any number of nodes, provide a flexible and powerful framework for modeling such higher-order relationships. In this work, we introduce hypergraph diffusion wavelets and describe their favorable spectral and spatial properties. We demonstrate their utility for biomedical discovery in spatially resolved transcriptomics by applying the method to represent disease-relevant cellular niches for Alzheimer's disease.