In this paper we propose a procedure for robust estimation in the context of generalized linear models based on the maximum Lq-likelihood method. Alongside this, an estimation algorithm that represents a natural extension of the usual iteratively weighted least squares method in generalized linear models is presented. It is through the discussion of the asymptotic distribution of the proposed estimator and a set of statistics for testing linear hypothesis that it is possible to define standardized residuals using the mean-shift outlier model. In addition, robust versions of deviance function and the Akaike information criterion are defined with the aim of providing tools for model selection. Finally, the performance of the proposed methodology is illustrated through a simulation study and analysis of a real dataset.
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In many applied sciences a popular analysis strategy for high-dimensional data is to fit many multivariate generalized linear models in parallel. This paper presents a novel approach to address the resulting multiple testing problem by combining a recently developed sign-flip test with permutation-based multiple-testing procedures. Our method builds upon the univariate standardized flip-scores test which offers robustness against misspecified variances in generalized linear models, a crucial feature in high-dimensional settings where comprehensive model validation is particularly challenging. We extend this approach to the multivariate setting, enabling adaptation to unknown response correlation structures. This approach yields relevant power improvements over conventional multiple testing methods when correlation is present.
In this paper the accuracy and robustness of quality measures for the assessment of machine learning models are investigated. The prediction quality of a machine learning model is evaluated model-independent based on a cross-validation approach, where the approximation error is estimated for unknown data. The presented measures quantify the amount of explained variation in the model prediction. The reliability of these measures is assessed by means of several numerical examples, where an additional data set for the verification of the estimated prediction error is available. Furthermore, the confidence bounds of the presented quality measures are estimated and local quality measures are derived from the prediction residuals obtained by the cross-validation approach.
In this paper, we develop a multi-step estimation procedure to simultaneously estimate the varying-coefficient functions using a local-linear generalized method of moments (GMM) based on continuous moment conditions. To incorporate spatial dependence, the continuous moment conditions are first projected onto eigen-functions and then combined by weighted eigen-values, thereby, solving the challenges of using an inverse covariance operator directly. We propose an optimal instrument variable that minimizes the asymptotic variance function among the class of all local-linear GMM estimators, and it outperforms the initial estimates which do not incorporate the spatial dependence. Our proposed method significantly improves the accuracy of the estimation under heteroskedasticity and its asymptotic properties have been investigated. Extensive simulation studies illustrate the finite sample performance, and the efficacy of the proposed method is confirmed by real data analysis.
In this article, we propose novel boundary treatment algorithms to avoid order reduction when implicit-explicit Runge-Kutta time discretization is used for solving convection-diffusion-reaction problems with time-dependent Di\-richlet boundary conditions. We consider Cartesian meshes and PDEs with stiff terms coming from the diffusive parts of the PDE. The algorithms treat boundary values at the implicit-explicit internal stages in the same way as the interior points. The boundary treatment strategy is designed to work with multidimensional problems with possible nonlinear advection and source terms. The proposed methods recover the designed order of convergence by numerical verification. For the spatial discretization, in this work, we consider Local Discontinuous Galerkin methods, although the developed boundary treatment algorithms can operate with other discretization schemes in space, such as Finite Differences, Finite Elements or Finite Volumes.
In this work, the infinite GMRES algorithm, recently proposed by Correnty et al., is employed in contour integral-based nonlinear eigensolvers, avoiding the computation of costly factorizations at each quadrature node to solve the linear systems efficiently. Several techniques are applied to make the infinite GMRES memory-friendly, computationally efficient, and numerically stable in practice. More specifically, we analyze the relationship between polynomial eigenvalue problems and their scaled linearizations, and provide a novel weighting strategy which can significantly accelerate the convergence of infinite GMRES in this particular context. We also adopt the technique of TOAR to infinite GMRES to reduce the memory footprint. Theoretical analysis and numerical experiments are provided to illustrate the efficiency of the proposed algorithm.
The various limitations of Generative AI, such as hallucinations and model failures, have made it crucial to understand the role of different modalities in Visual Language Model (VLM) predictions. Our work investigates how the integration of information from image and text modalities influences the performance and behavior of VLMs in visual question answering (VQA) and reasoning tasks. We measure this effect through answer accuracy, reasoning quality, model uncertainty, and modality relevance. We study the interplay between text and image modalities in different configurations where visual content is essential for solving the VQA task. Our contributions include (1) the Semantic Interventions (SI)-VQA dataset, (2) a benchmark study of various VLM architectures under different modality configurations, and (3) the Interactive Semantic Interventions (ISI) tool. The SI-VQA dataset serves as the foundation for the benchmark, while the ISI tool provides an interface to test and apply semantic interventions in image and text inputs, enabling more fine-grained analysis. Our results show that complementary information between modalities improves answer and reasoning quality, while contradictory information harms model performance and confidence. Image text annotations have minimal impact on accuracy and uncertainty, slightly increasing image relevance. Attention analysis confirms the dominant role of image inputs over text in VQA tasks. In this study, we evaluate state-of-the-art VLMs that allow us to extract attention coefficients for each modality. A key finding is PaliGemma's harmful overconfidence, which poses a higher risk of silent failures compared to the LLaVA models. This work sets the foundation for rigorous analysis of modality integration, supported by datasets specifically designed for this purpose.
In this paper we generalize the spectral concentration problem as formulated by Slepian, Pollak and Landau in the 1960s. We show that a generalized version with arbitrary space and Fourier masks is well-posed, and we prove some new results concerning general quadratic domains and gaussian filters. We also propose a more general splitting representation of the spectral concentration operator allowing to construct quasi-modes in some situations. We then study its discretization and we illustrate the fact that standard eigen-algorithms are not robust because of a clustering of eigenvalues. We propose a new alternative algorithm that can be implemented in any dimension and for any domain shape, and that gives very efficient results in practice.
This work presents GALAEXI as a novel, energy-efficient flow solver for the simulation of compressible flows on unstructured meshes leveraging the parallel computing power of modern Graphics Processing Units (GPUs). GALAEXI implements the high-order Discontinuous Galerkin Spectral Element Method (DGSEM) using shock capturing with a finite-volume subcell approach to ensure the stability of the high-order scheme near shocks. This work provides details on the general code design, the parallelization strategy, and the implementation approach for the compute kernels with a focus on the element local mappings between volume and surface data due to the unstructured mesh. GALAEXI exhibits excellent strong scaling properties up to 1024 GPUs if each GPU is assigned a minimum of one million degrees of freedom degrees of freedom. To verify its implementation, a convergence study is performed that recovers the theoretical order of convergence of the implemented numerical schemes. Moreover, the solver is validated using both the incompressible and compressible formulation of the Taylor-Green-Vortex at a Mach number of 0.1 and 1.25, respectively. A mesh convergence study shows that the results converge to the high-fidelity reference solution and that the results match the original CPU implementation. Finally, GALAEXI is applied to a large-scale wall-resolved large eddy simulation of a linear cascade of the NASA Rotor 37. Here, the supersonic region and shocks at the leading edge are captured accurately and robustly by the implemented shock-capturing approach. It is demonstrated that GALAEXI requires less than half of the energy to carry out this simulation in comparison to the reference CPU implementation. This renders GALAEXI as a potent tool for accurate and efficient simulations of compressible flows in the realm of exascale computing and the associated new HPC architectures.
This paper introduces an integer-valued generalized autoregressive conditional heteroskedasticity (INGARCH) model based on the novel geometric distribution and discusses some of its properties. The parameter estimation problem of the models are studied by conditional maximum likelihood and Bayesian approach using Hamiltonian Monte Carlo (HMC) algorithm. The results of the simulation studies and real data analysis affirm the good performance of the estimators and the model.
In this paper, a high-order/low-order (HOLO) method is combined with a micro-macro (MM) decomposition to accelerate iterative solvers in fully implicit time-stepping of the BGK equation for gas dynamics. The MM formulation represents a kinetic distribution as the sum of a local Maxwellian and a perturbation. In highly collisional regimes, the perturbation away from initial and boundary layers is small and can be compressed to reduce the overall storage cost of the distribution. The convergence behavior of the MM methods, the usual HOLO method, and the standard source iteration method is analyzed on a linear BGK model. Both the HOLO and MM methods are implemented using a discontinuous Galerkin (DG) discretization in phase space, which naturally preserves the consistency between high- and low-order models required by the HOLO approach. The accuracy and performance of these methods are compared on the Sod shock tube problem and a sudden wall heating boundary layer problem. Overall, the results demonstrate the robustness of the MM and HOLO approaches and illustrate the compression benefits enabled by the MM formulation when the kinetic distribution is near equilibrium.
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