# Beyond Correlation: Mutual Information to Detect Damage in Nonlinear Systems
###### Abstract
Recent research also highlights significant efforts to automate the damage-detection process. A key challenge is determining the level of deviation from the baseline state that constitutes damage. [PERSON] et al. [13] proposed a threshold-selection scheme that combines Bayesian inference with Monte Carlo discordancy testing, building on the work of [PERSON] et al. [14]. They measure the deviation of the transmissibility distribution at a potentially damaged state from its baseline counterpart using a modified Kullback-Leibler divergence [15], thereby accounting for measurement errors and uncertainties in the transmissibility function. [PERSON] et al. [16] combined a deep convolutional autoencoder with an unsupervised clustering algorithm to detect structural damage automatically. The autoencoder reconstructs wavelet-transmissibility power spectra from the intact state, and reconstruction errors are used to identify damaged states via OPTICS clustering [17]. [PERSON] et al. [18] pro posed a deep neural network consisting of a feature generator, two classifiers, and a discriminator; unlike existing deep domain-adaptation techniques (e.g., [19; 20; 21]), their method is designed to work with limited unlabeled datasets. This paper proposes a novel alternative to traditional coherence-based damage detection by introducing Mutual Information (MI) as a damage-sensitive index. The ability of the proposed metric to quantify the similarity between two signals with no assumptions on the nature of their interactions or the underlying dynamics of the system makes it particularly well-suited for damage identification in nonlinear systems. In this study, we test the performance of the MI-based damage metric in identifying both linear (mass/stiffness) and nonlinear (e.g., gap-induced) damages. Collectively, vibration-based damage detection methods operate under one or both of the following assumptions: (1) the system is linear, or weakly nonlinear; (2) the probability distribution of the excitation and response processes are fully characterized by the first- and second-order statistics (e.g., Gaussian). These assumptions are difficult to justify in practice, especially in scenarios involving structural damage. The fact that the use of inadequate methods may bias the estimated structural response parameters in unpredictable ways underscores a need for a new approach that does not rely on these assumptions. To address this need, this study introduces a model-free and distribution-free approach for detecting structural damages. To this end, a novel damage metric is derived based on the concept of mutual information. Mutual information has emerged as a nonlinear extension of the correlation coefficient. The premise of the proposed approach is that structural damage induces a change in the amount of information shared between the responses measured at a pair of locations on the structure. Although mutual information has been used as a measure of similarity in statistical pattern recognition and machine learning applications in the last couple of decades, the damage detection approach described in this study is a novel contribution to the damage detection literature. The novelty of this study is characterized by the following desirable qualities: (1) as a model-free approach, it eliminates model mis-specification concerns; (2) as a distribution-free approach, it does not assume data to follow a particular probability distribution; (3) because mutual information is a metric with relatively low computational complexity, the proposed approach can be adapted to use in real-time SHM applications. We demonstrate the validity of the proposed approach using a standard dataset. In addition, we perform a rigorous analysis of the statistical significance of our findings via nonparametric bootstrapping. Figure 1 shows a flowchart for detecting damage using the proposed Mutual Information (MI)-based metric. We start with raw acceleration or displacement traces from each level of the structure.