In this contribution, we propose a method for statistically evaluating the risk in a deformation monitoring system. When the structure under monitoring moves beyond tolerance, the monitor system should issue an alert. Only a very small probability is acceptable of the system telling us that no change beyond a critical threshold has taken place, while in reality it has. This probability is referred to as integrity risk. We provide a formulation of integrity risk where the interaction between estimation and testing is taken into account, implying the use of conditional probabilities. In doing so, we assumed different scenarios with the alerts being dependent on both the identified hypothesis and the threat that the estimated size of deformations entails. It is hereby highlighted that a correct risk evaluation requires estimation and testing being considered together, as they are typically intimately linked. In practice, one may, however, find it simpler computation-wise to neglect the estimation–testing link. For this case, we provide an approximation of the integrity risk. This approximation may provide a too optimistic or pessimistic description of the integrity risk depending on the testing procedure and tolerances of the structure at hand. Monitoring systems, besides issuing timely alerts, are also required to provide threat estimates together with their corresponding probabilistic properties. As the testing outcome determines how the threat gets estimated, the threat estimator will then inherit the statistical properties of both estimation and testing. We derive the threat estimator b¯j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{b}_{j}$$\end{document} and its probability density function, taking the contributions from combined estimation and testing into account. It is highlighted that although the threat estimator under the identified hypothesis Hj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {H}}_{j}$$\end{document}, i.e., b^j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{b}_{j}$$\end{document}, is normally distributed, the estimator b¯j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{b}_{j}$$\end{document} is not. It is explained that working with b^j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{b}_{j}$$\end{document} instead of b¯j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{b}_{j}$$\end{document}, thus ignoring the estimation–testing link, may provide a too optimistic description of the threat estimator’s quality. The presented method is illustrated by means of two simple deformation examples.

中文翻译：

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一种变形监测系统的风险评估方法

在这篇文章中，我们提出了一种对变形监测系统中的风险进行统计评估的方法。当被监控的结构超出容差时，监控系统应发出警报。系统告诉我们没有发生超过临界阈值的变化，而实际上它发生了，只有很小的概率是可以接受的。这种可能性被称为完整性风险。我们提供了一个完整性风险的公式，其中考虑了估计和测试之间的相互作用，这意味着使用条件概率。在这样做时，我们假设了不同的场景，警报取决于已识别的假设和估计变形大小所带来的威胁。特此强调，正确的风险评估需要同时考虑估计和测试，因为它们通常是密切相关的。然而，在实践中，人们可能会发现忽略估计-测试链接在计算上更简单。对于这种情况，我们提供了完整性风险的近似值。根据测试程序和手头结构的容差，这种近似可能会提供对完整性风险的过于乐观或悲观的描述。除了及时发出警报外，监控系统还需要提供威胁估计及其相应的概率属性。由于测试结果决定了如何估计威胁，威胁估计器将继承估计和测试的统计特性。我们推导出威胁估计量 b¯j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek } \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{b}_{j}$$\end{document} 及其概率密度函数，考虑到组合估计和测试的贡献. 需要强调的是，虽然已识别假设下的威胁估计量 Hj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs } \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {H}}_{j}$$\end{document}，即 据解释，使用 b^j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{ upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{b}_{j}$$\end{document} 而不是 b¯j\documentclass[12pt]{minimal} \usepackage {amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \bar{b}_{j}$$\end{document}，因此忽略了估计-测试链接，可能会提供对威胁估计器质量的过于乐观的描述。通过两个简单的变形示例来说明所提出的方法。