🤖 AI Summary
本文针对逆问题中状态变量的不确定性量化问题，提出了一种结合贝叶斯推断和功能数据分析的方法，通过模拟研究验证了该方法的有效性。
📝 Abstract
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.