In structural health monitoring (SHM) systems, data is collected from a multitude of sensors measuring, for example, vibration or strain in the structure, along with additional features that capture environmental or operational information. It is well known that changes in the measured sensor outputs do not necessarily originate from structural damage but are often induced by environmental changes. One popular approach to account for these effects is regressing the system outputs on the confounding factors, also known as "response surface modeling". Afterward, the predicted values are subtracted from the observed ones to obtain corrected data with the environmental effects (supposedly) removed. However, the evaluation of real-world SHM data shows that environmental conditions may affect not only the expected output values but also higher-order statistical moments, particularly the variances of and the covariances and correlations between the output quantities, such as eigenfrequencies of different modes or strain sensors at different locations. By construction, the (supervised) machine learning techniques commonly used for response surface modeling cannot account for those higher-order effects. To address these issues, we present and discuss several approaches for identifying and quantifying multivariate confounding effects on output covariances and correlations: a nonparametric, kernel-based estimator, a random forest, a semiparametric additive model, and a deep learning approach. Furthermore, we show how the resulting conditional covariance matrices can be used in an SHM pipeline. We compare the competing methods on both artificial data and real-world load test data from the Vahrendorfer Stadtweg bridge in Hamburg, Germany, as well as eigenfrequency data from the railway bridge KW51 near Leuven, Belgium.

翻译：在结构健康监测系统中，从大量传感器采集的数据包括结构振动或应变等测量值，以及捕捉环境或运行信息的附加特征。众所周知，测量传感器输出的变化未必源于结构损伤，而常由环境变化引起。一种流行的处理方法是将系统输出对混淆因子进行回归（即"响应面建模"），随后从观测值中减去预测值，以获取（假定）已去除环境影响的校正数据。然而，对真实结构健康监测数据的评估表明，环境条件不仅影响预期输出值，还会影响高阶统计矩——尤其是输出量之间的方差、协方差及相关性，例如不同模态的特征频率或不同位置的应变传感器数据。由于结构特性，常用于响应面建模的监督式机器学习技术无法处理这些高阶效应。为解决上述问题，我们提出并讨论了多种识别与量化输出协方差和相关性中多变量混淆效应的途径：非参数核估计器、随机森林、半参数加性模型及深度学习方法。此外，我们展示了如何将所得条件协方差矩阵应用于结构健康监测流程中。通过人工数据及德国汉堡Vahrendorfer Stadtweg大桥的真实荷载试验数据，以及比利时鲁汶附近铁路桥KW51的特征频率数据，我们对上述方法进行了比较评估。