fit linear response models

This paper addresses the problem of locally linearizing nonlinear systems within experimentally constrained initial-state regions, moving beyond conventional linear assumptions and reliance on a single long trajectory. We propose a finite-sample identification framework that integrates multi-trajectory deterministic sampling with regularized least squares, and establish—for the first time—an explicit error bound quantifying the trade-off between nonlinear approximation error and measurement noise. Theoretically, we prove that the estimated linearization model converges consistently under finite data. Numerical experiments demonstrate that classical i.i.d. single-trajectory excitation methods suffer significant failure risks in nonlinear settings, whereas our approach remains robust and effective. Key contributions are: (1) a localized linearization modeling paradigm tailored to nonlinear systems; (2) a synergistic design of multi-trajectory deterministic sampling and regularized estimation; and (3) the first finite-sample theoretical analysis jointly accounting for both nonlinear model mismatch and statistical estimation error.

Existing mathematical modeling lacks a rigorous, unambiguous ontological foundation, hindering a unified characterization of the mapping between models and real-world phenomena. This paper introduces, for the first time, an axiomatic definition of mathematical models grounded in Hilbert-space operator theory: a model is formalized as a computable operator acting on random variables, systematically unifying theoretical derivation, experimental implementation, and statistical identification. We further establish a geometric correspondence between the model manifold and the prediction surface, exposing intrinsic structural properties and the fundamental nature of model computability. This framework fills a critical gap in the formal ontology of modeling, providing a unified mathematical foundation for interdisciplinary model construction. It significantly enhances the logical rigor of theoretical inference and the reliability of empirical validation.

Linear modeling instruction often struggles to balance theoretical rigor with practical reproducibility, particularly for advanced undergraduate and graduate students. Method: This paper develops an intermediate pedagogical framework integrating formal mathematical derivations with intuitive, heuristic explanations, comprehensively covering classical linear regression, generalized linear models (GLMs), and modern extensions. It introduces a novel “teach–simulate–validate” paradigm: every theoretical result is accompanied by Monte Carlo simulations and real-world case studies, supported by fully documented, modular R code. Contribution/Results: The framework has been refined over seven consecutive years of classroom deployment at the University of California, Berkeley, yielding a mature, open-source lecture note series and code repository. Empirical evaluation demonstrates substantial improvements in students’ conceptual understanding of model assumptions, diagnostic reasoning, and capacity to implement and extend linear models in practice.

Existing pedagogical treatments of linear regression often lack rigorous theoretical unification across geometric, algebraic, and statistical perspectives, and seldom establish formal optimality guarantees or bridge frequentist and Bayesian interpretations. Method: This paper constructs a rigorous theoretical framework for linear regression targeting readers familiar with ordinary least squares (OLS), integrating geometric projection, matrix algebra, and statistical inference. It introduces Gaussian noise assumptions to derive maximum likelihood estimation, exact sampling distributions (t- and F-tests, confidence intervals), and Bayesian linear regression with closed-form posterior inference. Contribution/Results: We provide the first rigorous proof that OLS achieves the Cramér–Rao lower bound among unbiased linear estimators—establishing its theoretical optimality. The work unifies frequentist and Bayesian paradigms, yielding analytically tractable inference tools. The resulting verifiable statistical pipeline serves as an interpretable linear foundation and error-analysis benchmark for nonlinear models, including deep learning.

In high-dimensional linear regression, existing methods struggle to simultaneously address model selection uncertainty and ensure reliable inference under finite-sample settings. This paper proposes a reproducible-sample-based simulation inference framework that, for the first time, unifies inference for model selection, individual or multiple regression coefficients, and joint parameters—while rigorously guaranteeing finite-sample confidence coverage probability. The method constructs confidence sets via reproducible-sample generation and simulation-based inference, achieving both finite-sample validity and asymptotic optimality. Theoretically, it attains superior coverage accuracy and interval tightness compared to state-of-the-art debiased estimators and bootstrap methods. Empirical evaluations across diverse high-dimensional scenarios confirm its more accurate coverage rates and tighter confidence sets. This work bridges two critical theoretical gaps: (i) valid inference under model selection uncertainty and (ii) finite-sample guarantees in high-dimensional settings.

This work proposes a unified inference framework based on parametric programming to address the bias in statistical inference for regression coefficients following Lasso variable selection in generalized linear models. By locally linearizing the maximum likelihood estimator, the method constructs a linear relationship between pseudo-responses and covariates, thereby extending parametric programming—previously limited to Gaussian settings—to non-Gaussian response distributions. This extension enables valid post-selection inference across a range of exponential family models, including logistic and Poisson regression. The approach maintains computational efficiency while substantially improving inferential accuracy. Simulation studies demonstrate that, in non-Gaussian settings, the proposed method effectively corrects the naive inference that ignores the selection process and achieves higher statistical efficiency compared to existing approaches such as the polyhedral method.

该研究针对参数ODE控制模型，提出一种计算可观测函数生成集的算法，通过Lie导数和可识别参数组合提高效率。

This paper addresses the data-driven robust model predictive control (MPC) problem for systems with linear-noise dynamics, quadratic costs, and convex state, input, and disturbance constraints. The proposed method first generates high-fidelity state-action data via exact solution of a convex semi-infinite program on a gridded domain; it then constructs feedback mappings using polynomial or piecewise-affine approximations with certified uniform error bounds. Crucially, the approximation error is explicitly incorporated into controller synthesis to rigorously ensure closed-loop recursive feasibility and input-to-state stability. Unlike conventional approximation-based MPC schemes, the approach avoids conservatism, eliminates online optimization, and provides provable robustness and performance guarantees. Evaluated on two benchmark numerical examples, the learned policy satisfies all constraints and achieves stable closed-loop control.

This work addresses the lack of intuitive, immediate feedback on fitting errors in existing model-fitting approaches. It proposes an interactive fitting framework that integrates visual and auditory feedback: as users manipulate parametric curves, the system synthesizes audio in real time, with greater model-data discrepancies producing louder and more dissonant sounds. This is the first approach to incorporate auditory cues into model exploration, enabling multisensory assessment of fit quality. Combining interactive visualization, real-time audio synthesis, and Gaussian process regression, the method demonstrates effectiveness and generalizability across four diverse case studies—golf putting, dilution experiments, cosmological parameter estimation, and temperature data fitting—significantly enhancing users’ intuitive perception of model misfit.

This study addresses the problem of testing for the existence of non-negative solutions to a system of linear equations when all parameters, including slope coefficients, are unknown. The authors propose a novel sample-splitting test that, for the first time, characterizes the closure of the null hypothesis under total variation distance, eliminating the need for simulated critical values and enabling applicability in high-dimensional settings with rapidly growing numbers of variables. By integrating total variation distance, sample splitting, and asymptotic theory under weak identification conditions, the method demonstrates strong power in both theoretical analysis and simulations. It combines computational simplicity with high-dimensional scalability and can be employed to construct confidence sets for partially identified parameters in nonparametric instrumental variable models.