# Beyond Correlation: Mutual Information to Detect Damage in Nonlinear Systems
## 7 Conclusions  
A new damage metric based on the concept of mutual information was proposed, and its performance was compared to two existing damage metrics that use output-only vibration data. The major advantage of mutual information over correlation as a similarity measure is that mutual information measures the amount of information shared between a pair of measurements while correlation measures only the strength of linear associations. In addition, because marginal and joint probabilities are estimated directly from the data as part of mutual information estimation, assumptions regarding the underlying distribution of the data (e.g., Gaussian) are not needed. A standard data test was used to demonstrate the method. The dataset contains acceleration time series measured at each level, collected under 17 different structural states representing undamaged, linear damage, and nonlinear damage conditions. The linear damage states represent the case where the structure remains linear, but with modified stiffness and/or mass. The nonlinear damage is simulated using an adjustable gap to mimic the opening and closing of fatigue cracks under load reversals. The statistical significance of the proposed method was assessed using nonparametric bootstrapping. Damage detection results obtained using the proposed damage metric were compared to those obtained using ATC and TMAC. The following conclusions can be drawn from the results of this study: (1) the MI-based damage metric proposed in this study is able to detect the existence of damage in all structural states of the model frame; (2) compared to ATC and TMAC, the proposed metric has a better ability to distinguish linear damage states from the baseline state; (3) the proposed metric shows some potential in differentiating between different damage states, although the observed differences are not statistically significant for all pairs of damage states. In terms of noise sensitivity, addition of zero-mean Gaussian white noise up to a signal-to-noise ratio of 20 dB does not appear to cause noticeable differences in the \(p\)-values in any of the three metrics. An important limitation of the method described in this study, which is shared by a large class of existing output-only damage detection methods, is that it requires data from the baseline condition of the structure, which may not be available. Further, because of the inherent assumption of stationarity in evaluating transmissibility coherences, abrupt damages that cause transient changes in the vibration characteristics, such as those induced by wind or earthquake, cannot be reliably detected. Although the method used in this paper, in its current form, is not intended for real-time SHM applications, it can be extended to such applications by using wavelet-based transmissibility instead of its Fourier-based counterpart. The computational complexity of such an approach will be governed by the transmissibility calculations, rather than that of the mutual information calculations. In addition to extending this approach to the nonstationary case, future research will focus on the problem of selecting the locations where the responses are to be measured and the interpretation of the damage indices. While simulations using a finite element model of the structure can provide significant insight into how different damage scenarios affect a particular damage index, the inverse problem of characterizing damage and predicting its location based solely on the damage index remains a challenge. Extension of the bivariate mutual information to the multivariate case or conditioning the mutual information between two channels on additional channels can be very helpful in tracking changes in the dynamic properties of complex structures. Time-delayed mutual information computed using a time-delayed version of one of the signals can be used to infer the direction of nonlinear interactions between the responses collected at different channels, which can indicate the type and location of the damage.