Computational Filtering Methods for Time-Varying Parameter Estimation in Nonlinear Systems

现代科学中的许多应用涉及未知的系统参数，这些参数必须使用很少或根本没有先验信息来估计。在分析和预测这类系统行为的数学模型中，估计和量化模型参数中的不确定性仍然是一个挑战。这对于其中参数值的知识对于获得可信的模型输出至关重要的系统尤其如此，例如在用于个性化药物的患者特定的模型中。这些问题的子集包括已知随时间变化但没有已知演化模型的类型的参数。例如，用于模拟传染病传播的季节性传输参数和用于模拟神经元放电动态的外部电压参数。在某些情况下，参数可能具有某些已知的结构特征(如周期性)，这些结构特征可在整个估计过程中使用和维护。然而，估计时变参数的主要挑战在于在没有关于其时间动力学的详细信息的情况下准确地解释它们的时间演化。本项目的目标是设计和分析用于估计这种时变参数的新的计算方法。本研究的目的是设计和分析使用非线性滤波来估计时变参数的新的计算方法。利用贝叶斯统计过滤框架的优势，其中先验信念自然被纳入，这项工作将涉及开发参数演变模型，该模型考虑到与参数随时间的结构或行为有关的先验知识，而无需定义明确的函数来描述动态。对于可能没有任何先验已知的参数结构特征的更困难的问题，也将开发方法。这项研究中开发的算法和计算工具将应用于各种非线性系统的数据，这可能会进一步启发方法学进步的新方向。具体的应用领域包括工程和生命科学，特别是涉及基于激光的显微手术的组织热响应的外科机器人。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Many applications in modern science involve unknown system parameters that must be estimated using little to no prior information. In mathematical models to analyze and predict the behavior of such systems, the problem of estimating and quantifying uncertainty in model parameters remains a challenge. This is particularly true for systems where knowledge of parameter values is critical in obtaining trustworthy model output, as in patient-specific models for personalized medicine, for example. A subset of these problems includes parameters of the type that are known to vary with time but do not have known evolution models. Examples include the seasonal transmission parameter in modeling the spread of infectious diseases and the external voltage parameter in modeling the spiking dynamics of neurons. In certain cases, the parameters may have some known structural characteristics (such as periodicity) that can be utilized in and maintained throughout the estimation process. However, the main challenge in estimating time-varying parameters lies in accurately accounting for their time evolution without detailed information regarding their temporal dynamics. The goal of this project is to design and analyze novel computational methods for estimating such time-varying parameters.The aim of this study is to design and analyze novel computational methods for estimating time-varying parameters through use of nonlinear filtering. Leveraging the strengths of the Bayesian statistical filtering framework, where prior beliefs are naturally incorporated, this work will involve developing models for parameter evolution that take into account prior knowledge relating to the structure or behavior of the parameter over time without defining explicit functions to describe the dynamics. Methods will also be developed for more difficult problems where there may not be any parameter structural characteristics known a priori. The algorithms and computational tools developed in this study will be applied to data for a variety of nonlinear systems, which may further inspire new directions for methodological advancement. Specific areas of application include engineering and the life sciences, with particular application to surgical robotics involving tissue thermal response to laser-based microsurgery.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.