linear response analysis

This study addresses the lack of effective methods for quantifying how observables in singular statistical models respond to data perturbations. It introduces susceptibility as a central measure of this response and proposes, for the first time, an estimator for generalized observables grounded in linear response theory. By integrating statistical inference with asymptotic analysis, the estimator is shown to be consistent and asymptotically unbiased in the large-sample limit, relying solely on $n$ observed data points. This work establishes the first theoretically guaranteed framework for sensitivity analysis in singular statistical models, providing rigorous foundations for assessing the stability of model outputs under infinitesimal data perturbations.

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.

Sensitivity analysis of failure probability with respect to input parameters remains challenging for high-dimensional, implicitly nonlinear, and black-box systems—especially under rare-event regimes and large numbers of random variables. Method: This paper proposes a unified Monte Carlo estimator integrating response gradients with respect to sensitive parameters and kernel smoothing. It incorporates the gradient of the system response into Monte Carlo simulation and employs kernel density estimation to handle zero-probability threshold events, enabling simultaneous estimation of reliability sensitivities across multiple failure thresholds from a single sample set. Contribution/Results: The method avoids the high computational cost of conventional finite-difference or resampling approaches while ensuring numerical stability and significantly improving efficiency. Experimental validation on complex engineering systems demonstrates high accuracy, strong robustness, and real-time assessment capability. It provides a novel tool for risk-informed design optimization and decision support.

Standard interpretations of OLS coefficients in multiple linear regression presume linearity of the conditional expectation function (CEF), yet real-world data-generating processes are often nonlinear. Method: We show that when the CEF is nonlinear, OLS coefficients represent weighted averages of its partial derivatives, with bias arising systematically from the nonlinear structure of covariates. Leveraging Taylor expansion of the CEF, weighted average theory, and linear projection, we derive a closed-form expression for this bias—termed the “weighted derivative bias”—and prove its decomposition into functions of covariate nonlinearity. Contribution/Results: We unify this bias within an interpretable framework analogous to classical measurement error and omitted-variable bias, establishing a new theoretical foundation for coefficient interpretation under nonlinearity. Simulation and empirical analyses consistently validate the predicted direction and magnitude of the bias.

When linear regression is used to estimate treatment effects in quasi-experiments, its causal interpretation rests on implicit assumptions—specifically, under what conditions does the regression coefficient represent a comparable contrast of individual potential outcomes? Method: We formally introduce the concept of “latent weights” to characterize regression’s implicit weighting of unobserved counterfactuals; derive necessary linear constraints on treatment assignment for causal interpretability; and define the “implicit causal design set,” unifying and extending existing theoretical frameworks. Our approach integrates design-based inference, counterfactual modeling, and linear constraint analysis. Contribution: We establish a necessary conditions framework for causal interpretation of regression, provide operational transparency diagnostics, and deliver novel theoretical justification for widely used—but previously under-justified—regression specifications, including covariate-adjusted regression.

本文提出了一种表面分解方法，用于分析非高斯风激励下线性系统的首次穿越动态可靠性敏感性，通过分解复杂积分并使用嵌套采样算法有效估计。

This work addresses the persistent challenge in process systems modeling of simultaneously achieving accuracy, simplicity, and physical interpretability—particularly in control applications where nonlinear expressiveness must be balanced against a preference for linear structures. The authors propose a convex hybrid modeling paradigm grounded in operator theory, which constrains models to interpretable subspaces or nonlinearly parameterized interpretable manifolds. By introducing a reparameterization technique based on “canonical features” in an augmented parameter space, the approach effectively integrates kernel methods with convex optimization. This framework enables the construction of kernel-based hybrid surrogate models over families of interpretable static and dynamic systems, significantly enhancing both predictive accuracy and computational efficiency while preserving physical interpretability across diverse process systems modeling scenarios.

This work addresses the limitations of conventional linear control allocation in highly nonlinear flight regimes—namely, degraded accuracy due to model mismatch, high computational cost of high-fidelity models, and poor interpretability of black-box data-driven approaches—by proposing an interpretable control effectiveness learning framework based on Sparse Identification of Nonlinear Dynamics (SINDy). The method identifies an explicit, physics-constrained analytical model directly from flight data, enabling efficient solution of nonlinear control allocation through analytically computable derivatives. An online adaptive mechanism, driven by residual monitoring, facilitates graceful reconfiguration in response to actuator faults and changing operating conditions. High-maneuverability flight tests on a high-fidelity overactuated aircraft benchmark demonstrate that the proposed approach achieves control accuracy comparable to full nonlinear onboard models while substantially reducing computational overhead.

This study addresses the limitations of traditional network calculus, which assumes non-negative service curves and struggles to analyze complex systems with feedback control. By rigorously examining the properties of subadditive functions, the authors reveal that allowing negative service curves in feedback systems often leads to unstable analyses. To overcome this issue while preserving the non-negativity assumption, they develop a refined network calculus framework that integrates network calculus theory, subadditive function analysis, and system stability verification. Applying this approach to the complex feedback system proposed by Hamscher et al., the method achieves accurate modeling and tight performance bounds, effectively circumventing the instability inherent in prior techniques and significantly enhancing the applicability and reliability of network calculus in closed-loop systems.

本文提出一种基于频率响应函数和伪脉冲响应的多变量统计框架，解决了MIMO系统中故障检测受反馈控制器影响及多重比较问题，通过PERMANOVA方法提高了故障诊断准确性。