Confidence Intervals for Conditional Covariances of Natural Frequencies

In structural health monitoring (SHM), sensor measurements are collected, and damage-sensitive features such as natural frequencies are extracted for damage detection. However, these features depend not only on damage but are also influenced by various confounding factors, including environmental conditions and operational parameters. These factors must be identified, and their effects must be removed before further analysis. However, it has been shown that confounding variables may influence the mean and the covariance of the extracted features. This is particularly significant since the covariance is an essential building block in many damage detection tools. To account for the complex relationships resulting from the confounding factors, a nonparametric kernel approach can be used to estimate a conditional covariance matrix. By doing so, the covariance matrix is allowed to change depending on the identified confounding factor, thus providing a clearer understanding of how, for example, temperature influences the extracted features. This paper presents two bootstrap-based methods for obtaining confidence intervals for the conditional covariances, providing a way to quantify the uncertainty associated with the conditional covariance estimator. A proof-of-concept Monte Carlo study compares the two bootstrap versions proposed and evaluates their effectiveness. Finally, the methods are applied to the natural frequency data of the KW51 railway bridge near Leuven, Belgium. This real-world application highlights the practical implications of the findings. It underscores the importance of accurately accounting for confounding factors to generate more reliable diagnostic values with fewer false alarms.