Higher-Order Multivariate Environmental Influences in Structural Health Monitoring

Abstract

System outputs such as eigenfrequencies or strain data, often used in structural health monitoring (SHM), not only react to damage but also depend on environmental conditions. When trying to correct for these confounding effects, it is often (at least implicitly) assumed that only the expected, i.e., mean, output values are affected by environmental conditions. However, the evaluation of real-world SHM data indicates that environmental conditions may influence not only the mean output 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. To address these issues, we discuss two approaches for identifying and quantifying multivariate confounding effects on output covariances and correlations: a random forest and a nonparametric, kernel-based approach. We compare the two competing methods on both artificial and real-world SHM data, finding that the kernel-based approach achieves higher accuracy, but the random forest produces estimates that are more robust and sometimes easier to interpret.

Figures (5)

• Figure 1: Simulation study setup. Right: conditional mean $\mu_1(\mathbf{z})$ (top) and $\mu_2(\mathbf{z})$ (bottom). Middle: conditional variance $\sigma^2_1(\mathbf{z})$ (top) and $\sigma^2_2(\mathbf{z})$ (bottom). Right: conditional covariance $\sigma_{1,2}(\mathbf{z})$ (top) and correlation (bottom).
• Figure 2: Simulation study setup. Left: simulated "temperatures" $z_1$ (green) and $z_2$ (red). Middle: One week of $z_1$ in February (purple), June (pink), and September (brown). Right: irrelevant, faulty measurements of "relative humidity" $z_3$ (top, blue) and "temperature" $z_4$ (bottom, orange)
• Figure 3: The estimated conditional correlations using either the Nadaraya-Watson approach (top left) or the random forest (bottom left) for one simulation run with $q=2$, as well as the root mean square error (RMSE) of the estimates for the Nadaraya-Watson estimator (middle) and the random forest (right) for $q=2,3,4$ covariates and 50 simulation runs.
• Figure 4: Railway bridge KW51 in September 2024 (left), natural frequency data (middle), steel surface temperature (top-right), and relative humidity (bottom-right) over the monitoring period before the retrofitting. [increase fontsize]
• Figure 5: Conditional correlation of selected mode pairs as a function of steel temperature and relative humidity using the Nadaraya-Watson kernel estimator (top) and the random forest approach (bottom).