逆问题背景下状态变量轨迹的不确定性量化：一种基于贝叶斯推断与函数型数据分析的方法

中文摘要 AI 辅助

本文针对逆问题背景下状态变量的不确定性量化问题展开研究。逆问题与可通过常微分方程或偏微分方程描述的现象相关，这类问题虽有观测数据可用，但方程的参数、初始条件或边界条件的值未知。当前文献中分析状态变量不确定性传播的方案极少，且仅局限于构造伪可信区域或在孤立点量化其不确定性。本文提出一种结合贝叶斯推断工具的方法，采用哈密顿蒙特卡洛采样进行高效后验探索，并运用函数型数据分析（FDA）中的改进带深度（Modified Band Depth）方法，以确定感兴趣时间范围内状态变量轨迹的可信区域。本文通过模拟研究验证该方法，结果显示本文方案捕获的状态变量轨迹比例高于传统逐点分析方法：本文方案生成的可信区域在96.4%的情况下包含状态变量的真实轨迹，而逐点方法的区域仅为80%。此外，本文还展示了该方法在与神经现象相关的非平凡模型中的应用，该方法能有效捕获状态变量的时间依赖动态。

英文摘要

In this article, we address the problem of uncertainty quantification of state variables in the context of inverse problems. Inverse problems are associated with phenomena that can be represented through ordinary or partial differential equations, for which observations or data are available, but the values of the parameters that characterize the equations, the initial or boundary conditions are not known. Currently, the literature offers very few alternatives for analyze the propagation of uncertainty of state variables, which are limited to constructing pseudo-credible regions or quantifying their uncertainty at isolated points. We propose a methodology that combines tools of Bayesian inference, with Hamiltonian Monte Carlo sampling employed for efficient posterior exploration, and functional data analysis, specifically the Modified Band Depth method, to determine credible regions for the trajectories of the state variables throughout the time horizon of interest. We expose a methodology validation through a simulation study, which shows that our proposal captures a higher proportion of state variable trajectories than traditional pointwise analysis methods, our proposal generated credible regions that contained the true trajectory of the state variables 96.4\% of the times, versus 80\% that the regions of the pointwise method did. Furthermore, we demonstrate its application to a non-trivial model associated with a neurological phenomenon, for which the methodology effectively captures the time-dependent dynamics of the state variables.

发表机构

• School of Applied Sciences and Engineering, Universidad EAFIT（埃阿菲特大学应用科学与工程学院）
• Faculty of Economic and Administrative Sciences, Institución Universitaria ITM（ITM大学经济与行政科学学院）

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