Calculation of Hessian under constraints with applications to Bayesian system identification
Calculation of Hessian under constraints with applications to Bayesian system identification
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DOI：
10.1016/j.cma.2017.05.021
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发表时间：
2017-08
影响因子：
7.2
通讯作者：
S. Au;Yan-Long Xie
中科院分区：
文献类型：
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作者：
S. Au;Yan-Long Xie
在具有全局可识别模型的贝叶斯系统识别中，后验（即， 给定数据）模型参数的概率密度函数（PDF）可以由高斯PDF近似。参数的最可能值（MPV）等于高斯PDF的平均值。它最大化后验PDF，或者等效地最小化后验PDF的对数负（NL）。高斯PDF的协方差矩阵等于MPV处NL的Hessian。模型参数可能受到约束，这必须在后验协方差矩阵的计算中考虑。在模态识别等应用中，现有的策略定义了一组自由参数，并将它们映射到模型参数，以便始终满足约束。得到NL关于自由参数的Hessian，然后进行变换以给出模型参数的后验协方差矩阵，其中考虑了约束。由于NL和映射的复合作用，该Hessian的解析表达式是复杂的;这在计算机编码中造成了显著的负担。在这项工作中，一个理论框架的开发，用于评估的Hessian函数的约束下，在一个系统的方式。应用该方法得到了贝叶斯操作模态分析中后验协方差矩阵的新的解析表达式。由此产生的表达式比现有的直接微分的基础上更简单。它们允许具有类似数学结构的问题以连贯的方式被计算机编码。给出了数值例子来说明一致性和计算方面。
In Bayesian system identification with globally identifiable models, the posterior (i.e., given data) probability density function (PDF) of model parameters can be approximated by a Gaussian PDF. The most probable value (MPV) of the parameters is equal to the mean of the Gaussian PDF. It maximises the posterior PDF, or equivalently, minimises the negative of logarithm (NL) of the posterior PDF. The covariance matrix of the Gaussian PDF is equal to the Hessian of the NL at the MPV. Model parameters can be subjected to constraints, which must be accounted for in the calculation of the posterior covariance matrix. In applications such as modal identification, existing strategies define a set of free parameters and map them to the model parameters so that the constraints are always satisfied. The Hessian of the NL with respect to the free parameters is obtained and then transformed to give the posterior covariance matrix of the model parameters where constraints are accounted for. Analytical expressions for this Hessian are complicated because of the composite actions of the NL and the mapping; and this creates significant burden in computer coding. In this work, a theoretical framework is developed for evaluating the Hessian of a function under constraints in a systematic manner. It is applied to obtain new analytical expressions for evaluating the posterior covariance matrix in Bayesian operational modal analysis. The resulting expressions are simpler than existing ones based on direct differentiation. They allow problems with similar mathematical structures to be computer-coded in a coherent manner. Numerical examples are presented to illustrate consistency and computational aspects.