We extend prior work comparing linear multilevel models (MLM) and fixed effect (FE) models to the generalized linear model (GLM) setting, where the coefficient on a treatment variable is of primary interest. This leads to three key insights. (i) First, as in the linear setting, MLM can be thought of as a regularized form of FE. This explains why MLM can show large biases in its treatment coefficient estimates when group-level confounding is present. However, unlike the linear setting, there is not an exact equivalence between MLM and regularized FE coefficient estimates in GLMs. (ii) Second, we study a generalization of "bias-corrected MLM" (bcMLM) to the GLM setting. Neither FE nor bcMLM entirely solves MLM's bias problem in GLMs, but bcMLM tends to show less bias than does FE. (iii) Third, and finally, just like in the linear setting, MLM's default standard errors can misspecify the true intragroup dependence structure in the GLM setting, which can lead to downwardly biased standard errors. A cluster bootstrap is a more agnostic alternative. Ultimately, for non-linear GLMs, we recommend bcMLM for estimating the treatment coefficient, and a cluster bootstrap for standard errors and confidence intervals. If a bootstrap is not computationally feasible, then we recommend FE with cluster-robust standard errors.
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Dynamical theories of speech use computational models of articulatory control to generate quantitative predictions and advance understanding of speech dynamics. The addition of a nonlinear restoring force to task dynamic models is a significant improvement over linear models, but nonlinearity introduces challenges with parameterization and interpretability. We illustrate these problems through numerical simulations and introduce solutions in the form of scaling laws. We apply the scaling laws to a cubic model and show how they facilitate interpretable simulations of articulatory dynamics, and can be theoretically interpreted as imposing physical and cognitive constraints on models of speech movement dynamics.
In this paper, we proposed a monotone block coordinate descent method for solving absolute value equation (AVE). Under appropriate conditions, we analyzed the global convergence of the algorithm and conduct numerical experiments to demonstrate its feasibility and effectiveness.
High-dimensional parabolic partial differential equations (PDEs) often involve large-scale Hessian matrices, which are computationally expensive for deep learning methods relying on automatic differentiation to compute derivatives. This work aims to address this issue. In the proposed method, the PDE is reformulated into a martingale formulation, which allows the computation of loss functions to be derivative-free and parallelized in time-space domain. Then, the martingale formulation is enforced using a Galerkin method via adversarial learning techniques, which eliminate the need of computing conditional expectations in the margtingale property. This method is further extended to solve Hamilton-Jacobi-Bellman (HJB) equations and the associated Stochastic optimal control problems, enabling the simultaneous solution of the value function and optimal feedback control in a derivative-free manner. Numerical results demonstrate the effectiveness and efficiency of the proposed method, capable of solving HJB equations accurately with dimensionality up to 10,000.
In Coevolving Latent Space Networks with Attractors (CLSNA) models, nodes in a latent space represent social actors, and edges indicate their dynamic interactions. Attractors are added at the latent level to capture the notion of attractive and repulsive forces between nodes, borrowing from dynamical systems theory. However, CLSNA reliance on MCMC estimation makes scaling difficult, and the requirement for nodes to be present throughout the study period limit practical applications. We address these issues by (i) introducing a Stochastic gradient descent (SGD) parameter estimation method, (ii) developing a novel approach for uncertainty quantification using SGD, and (iii) extending the model to allow nodes to join and leave over time. Simulation results show that our extensions result in little loss of accuracy compared to MCMC, but can scale to much larger networks. We apply our approach to the longitudinal social networks of members of US Congress on the social media platform X. Accounting for node dynamics overcomes selection bias in the network and uncovers uniquely and increasingly repulsive forces within the Republican Party.
This study focuses on addressing the challenge of solving the reduced biquaternion equality constrained least squares (RBLSE) problem. We develop algebraic techniques to derive both complex and real solutions for the RBLSE problem by utilizing the complex and real forms of reduced biquaternion matrices. Additionally, we conduct a perturbation analysis for the RBLSE problem and establish an upper bound for the relative forward error of these solutions. Numerical examples are presented to illustrate the effectiveness of the proposed approaches and to verify the accuracy of the established upper bound for the relative forward errors.
Boundary-preserving weak approximations of some semilinear stochastic partial differential equations
We propose and analyse a boundary-preserving numerical scheme for the weak approximations of some stochastic partial differential equations (SPDEs) with bounded state-space. We impose regularity assumptions on the drift and diffusion coefficients only locally on the state-space. In particular, the drift and diffusion coefficients may be non-globally Lipschitz continuous and superlinearly growing. The scheme consists of a finite difference discretisation in space and a Lie--Trotter splitting followed by exact simulation and exact integration in time. We prove weak convergence of optimal order 1/4 for globally Lipschitz continuous test functions of the scheme by proving strong convergence towards a strong solution driven by a different noise process. Boundary-preservation is ensured by the use of Lie--Trotter time splitting followed by exact simulation and exact integration. Numerical experiments confirm the theoretical results and demonstrate the effectiveness of the proposed Lie--Trotter-Exact (LTE) scheme compared to existing methods for SPDEs.
The construction of loss functions presents a major challenge in data-driven modeling involving weak-form operators in PDEs and gradient flows, particularly due to the need to select test functions appropriately. We address this challenge by introducing self-test loss functions, which employ test functions that depend on the unknown parameters, specifically for cases where the operator depends linearly on the unknowns. The proposed self-test loss function conserves energy for gradient flows and coincides with the expected log-likelihood ratio for stochastic differential equations. Importantly, it is quadratic, facilitating theoretical analysis of identifiability and well-posedness of the inverse problem, while also leading to efficient parametric or nonparametric regression algorithms. It is computationally simple, requiring only low-order derivatives or even being entirely derivative-free, and numerical experiments demonstrate its robustness against noisy and discrete data.
Studying unified model averaging estimation for situations with complicated data structures, we propose a novel model averaging method based on cross-validation (MACV). MACV unifies a large class of new and existing model averaging estimators and covers a very general class of loss functions. Furthermore, to reduce the computational burden caused by the conventional leave-subject/one-out cross validation, we propose a SEcond-order-Approximated Leave-one/subject-out (SEAL) cross validation, which largely improves the computation efficiency. In the context of non-independent and non-identically distributed random variables, we establish the unified theory for analyzing the asymptotic behaviors of the proposed MACV and SEAL methods, where the number of candidate models is allowed to diverge with sample size. To demonstrate the breadth of the proposed methodology, we exemplify four optimal model averaging estimators under four important situations, i.e., longitudinal data with discrete responses, within-cluster correlation structure modeling, conditional prediction in spatial data, and quantile regression with a potential correlation structure. We conduct extensive simulation studies and analyze real-data examples to illustrate the advantages of the proposed methods.
Parameter inference is essential when interpreting observational data using mathematical models. Standard inference methods for differential equation models typically rely on obtaining repeated numerical solutions of the differential equation(s). Recent results have explored how numerical truncation error can have major, detrimental, and sometimes hidden impacts on likelihood-based inference by introducing false local maxima into the log-likelihood function. We present a straightforward approach for inference that eliminates the need for solving the underlying differential equations, thereby completely avoiding the impact of truncation error. Open-access Jupyter notebooks, available on GitHub, allow others to implement this method for a broad class of widely-used models to interpret biological data.
Large-scale eigenvalue problems arise in various fields of science and engineering and demand computationally efficient solutions. In this study, we investigate the subspace approximation for parametric linear eigenvalue problems, aiming to mitigate the computational burden associated with high-fidelity systems. We provide general error estimates under non-simple eigenvalue conditions, establishing the theoretical foundations for our methodology. Numerical examples, ranging from one-dimensional to three-dimensional setups, are presented to demonstrate the efficacy of reduced basis method in handling parametric variations in boundary conditions and coefficient fields to achieve significant computational savings while maintaining high accuracy, making them promising tools for practical applications in large-scale eigenvalue computations.
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