论文速递 | 基于可识别性修正的相关未知模型变量统计模型校准

文摘   2024-10-04 19:00   山东  
Statistical model calibration of correlated unknown model variables through identifiability improvement

基于可识别性修正的相关未知模型变量统计模型校准

引用格式 | Cited by
Choo JH, Jung YS, Jo HS, Lee IJ, 2024. Statistical model calibration of correlated unknown model variables through identifiability improvement. Probabilistic Engineering Mechanics, 77: 103670.
DOI: 10.1016/j.probengmech.2024.103670
摘要 | Abstract
统计模型校准问题由于其病态反问题特性,往往具有不稳定或非唯一的最优解,因时间和成本限制导致的测试数据有限使得这一问题更加复杂。为克服这些挑战并提高校准参数的可识别性,本研究提出了一类新的统计模型校准框架。该方法结合了未知模型变量的输入测试数据和系统响应的输出测试数据,采用二元连接函数对概率分布建模,同时考虑未知模型变量的相关性。此外,使用样本平均对数似然作为校准度量,假设条件独立性,以便在单一度量中均匀反映输入和输出测试数据。采用优化模型校准 (optimization-based model calibration, OBMC) 识别使校准度量最大化的概率模型,基于一组给定的输入和输出测试数据,从备选模型中选边缘概率分布和连接函数。因此,该方法在统计模型校准过程中考虑未知模型变量的观测数据,提高了校准参数的可识别性,并克服了数据不足的问题。通过数值算例验证了所提框架。
关键词统计模型校准, 可识别性, 条件独立, Fisher 信息矩阵, 连接函数, 样本平均对数似然
A statistical model calibration problem is known to have unstable or non-unique optimal solutions due to its ill-posed inverse nature, which is further complicated by limited test data availability due to time and cost constraints. To overcome these challenges and improve the identifiability of calibration parameters, this study proposes a novel statistical model calibration framework. The proposed method integrates input test data for unknown model variables and output test data for a system response, employing a bivariate form of copula function to model the probability distribution while accounting for the correlations between unknown model variables. Furthermore, a sample-averaged log-likelihood is used as a calibration metric, assuming conditional independence to reflect input and output test data evenly in a single metric. Optimization-based model calibration (OBMC) is performed to identify the probability models that maximize the calibration metric for a given set of input and output test data, among candidates of marginal probability distributions and copula functions. Consequently, this proposed method enhances the identifiability of calibration parameters and overcomes insufficient data issues by taking observations of unknown model variables into account in the statistical model calibration procedure. The proposed framework is validated using numerical examples.
KeywordsStatistical model calibration; Identifiability; Conditional independent; Fisher information matrix; Copula function; Sample-averaged log-likelihood.

图 1: 所提统计模型校准方法总体过程的流程图

Fig. 1. Flowchart representing the overall process of the proposed statistical model calibration method

图 2: 基于现有优化模型校准方法仅采用 10 与 30 个输出测试数据估计的概率分布

Fig. 2. Probability distributions estimated by the existing OBMC method using only (a) 10 and (b) 30 output test data: both true and estimated distributions represent the same contour levels of the PDF, which are presented at 0.05 intervals from 0.02 to 0.5

图 3: 将 10 与 30 个输出测试数据和 3 与 10 个输入测试数据结合的估计结果

Fig. 3. Estimation results from integrating (a) 10 and (b) 30 output test data with 3 input test data; similarly, estimation results from integrating (c) 10 and (d) 30 output test data with 10 input test data: Both true and estimated distributions represent the same contour levels of the PDF, which are presented at 0.05 intervals from 0.02 to 0.5

图 4: 仅基于图 3 中结果每一输入测试数据的校正估计结果

Fig. 4. Corrected estimation results using only each input test data from the results in Fig. 3

图 5: 将 10 与 30 个输出测试数据和 3 与 10 个输入测试数据结合的估计结果

Fig. 5. Estimation results from integrating (a) 10 and (b) 30 output test data with 3 input test data; similarly, estimation results from integrating (c) 10 and (d) 30 output test data with 10 input test data: Both true and estimated distributions represent the same contour levels of the PDF, which are presented at 0.05 intervals from 0.02 to 0.5

图 6: 仅基于图 5 中结果每一输入测试数据的校正估计结果

Fig. 6. Corrected estimation results using only each input test data from the results in Fig. 5

图 7: 图 6 中相同 10 个输入测试数据的Bayes 方法估计结果

Fig. 7. Estimation results of adopting the Bayesian method to the identical 10 input test data exhibited in Fig. 6: Maximum likelihood estimation (MLE) results for the unknown model variable (a) X1, (b) X2, and (c) the estimated copula function with the true copula function

图 8: 仅采用 10 和 30 个与之前反映输出测试数据量相同的输入测试数据的估计结果

Fig. 8. Estimation results using only (a) 10 and (b) 30 input test data that are identical to the quantity of output test data reflected in the previously: Both true and estimated distributions represent the same contour levels of the PDF, which are presented at 0.05 intervals from 0.02 to 0.5

图 9: 采用 10 与 30 个输出测试数据和 10 个输入测试数据的疲劳强度参数与疲劳延性参数估计分布

Fig. 9. Estimated distributions of fatigue strength parameter using (a) 10 and (b) 30 output test data with 10 input test data; fatigue ductility parameter using (c) 10 and (d) 30 output test data with 10 input test data: Both true and estimated distributions represent the same contour levels of the PDF

图 10: 仅基于图 9 中结果每一输入测试数据的校正估计结果

Fig. 10. Corrected estimation results using only each input test data from the results in Fig. 9

图 11: 集成动力制动系统中踏板感知模拟器的截面视图与构造

Fig. 11. Cross-sectional view and configuration of the pedal feel simulator in the IDB system

图 12: 模型校准程序的输入与输出试验观测数据: (a) 基于试样测试获得的 21 个 Mooney-Rivlin 材料参数确定的联合概率分布; (b) 通过万能试验机试验获得的 100 个感知阻尼器样本应变能直方图

Fig. 12. Measured data with input and output test for use in model calibration procedure: (a) A joint probability distribution identified from 21 Mooney-Rivlin material parameters obtained from coupon test; (b) A histogram of strain energies obtained through UTM test for the 100 feeling damper samples

图 13: 10 与 30 个输出测试数据和 3 与 10 个输入测试数据下 Mooney-Rivlin 材料参数的估计结果

Fig. 13. Estimation results of Mooney-Rivlin material parameters using (a) 10 and (b) 30 output test data with 3 input test data; similarly, estimation results using (c) 10 and (d) 30 output test data with 10 input test data: Both true and estimated distributions represent the same contour levels of the PDF

图 14: 仅基于图 13 中结果每一输入测试数据的校正估计结果

Fig. 14. Corrected estimation results using only each input test data from the results in Fig. 13

作者信息 | Authors

Jeong-Hwan Choo

韩国 HL 万都公司 (HL Mando Co.)

Yong-Su Jung

韩国科学技术院 (Korea Advanced Institute of Science & Technology机械工程系

Hwi-Sang Jo

韩国科学技术院 (Korea Advanced Institute of Science & Technology机械工程系

Ik-Jin Lee, 通讯作者 (Corresp.)
韩国科学技术院 (Korea Advanced Institute of Science & Technology机械工程系

Email: ikjin.lee@kaist.ac.kr



律梦泽 M.Z. Lyu | 编辑 (Ed) 

P.D. Spanos | 审校 (Rev)

陈建兵 J.B. Chen | 审校 (Rev)

彭勇波 Y.B. Peng | 审校 (Rev)

Probab Eng Mech
国际学术期刊 Probabilistic Engineering Mechanics 创立于 1985 年,SCI 收录,JCR Q1,现任主编是美国工程院院士、中国科学院外籍院士、莱斯大学 Pol D. Spanos 教授。
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