# Beyond Correlation: Mutual Information to Detect Damage in Nonlinear Systems
## 6 Discussion of Results
### Statistical Significance  
It was shown in Section 5 that all three metrics are able to differentiate the nonlinear damage states from the baseline state. In this section, we focus on the performance of the three metrics for the linear damage states (States #2-#9). Specifically, for each pair of damage states, we test the statistical significance of the difference between the population means of the damage metric corresponding to these states. We use nonparametric bootstrapping [56] to obtain the p-values, as bootstrapping does not require assuming an underlying probability distribution, as opposed to commonly used tests such as the two-sample t-test. The \(p\)-value of a test indicates the probability of observing results as extreme as or more extreme than the results observed in the study, if the null hypothesis holds true. The null hypothesis is that the population mean of the damage metric is the same in the two states being compared. A \(p\)-value exceeding a preselected threshold (e.g., 0.01 or 0.05) can be interpreted as a lack of evidence to reject the null hypothesis, indicating a lack of statistical significance. In the context of the damage detection problem addressed in this paper, a large \(p\)-value indicates that the damage metric obtained for one damage state is not significantly different from the other damage state, thus indicating that the two damage states cannot be reliably distinguished using the damage metric selected. Bootstrapping was performed using 10,000 repetitions, resulting in a \(9\times 9\) matrix of \(p\)-values. The \((i,j)\)th entry of this matrix quantifies the probability that the observed difference between damage states \(i\) and \(j\) could have occurred by random chance, if in fact there was no difference in the average damage metric associated with the two states. Considering that the matrix of \(p\)-values is symmetric, the likelihood of damage can be assessed by setting \(i\) or \(j\) to the baseline (undamaged) state. Figure 8 shows the results of significance testing for the quantities ATC(5,2) and ATC(5,3). In each of the two windows, the bottom row indicates the \(p\)-values in distinguishing damage states 2 through 9 from the baseline (undamaged) state. The largest \(p\)-values in the bottom row of the ATC(5,2) matrix is 0.436, and that in ATC(5,3) is 0.028. Thus, it can be concluded that the observed differences in the ATC metric between the damaged and undamaged states are not statistically significant at the 0.01 level, while the ATC(5,3) case is statistically significant at the 0.05 level. The \(p\)-values above the bottom line have values exceeding 0.8, indicating that the ATC metric is unable to differentiate between different damage states. Figure 9 shows the results of significance testing for the quantities MTAC(5,2) and TMAC(5,3). The largest \(p\)-value in the bottom row of both MI(5,2) and MI(5,3) is less than 0.01, indicating that the observed differences between the MI values in each damage state and those in the undamaged state are statistically significant at the 0.01 level. The \(p\)-values above the bottom line have values exceeding 0.8, indicating that the TMAC metric is unable to differentiate between two damage states. Figure 10 shows the \(p\)-values for the quantities MI(5,2) and MI(5,3). The largest \(p\)-value in the bottom row of both MI(5,2) and MI(5,3) is less than 0.01, indicating that the observed differences between the MI values in each damage state and those in the undamaged state are statistically significant at the 0.01 level. The \(p\)-values above the bottom line, in general,  
Figure 8: \(p\)-values in ATC. Figure 9: \(p\)-values in TMAC. are smaller than both the ATC and TMAC values, showing that the MI metric has a better ability to distinguish between different damage states, compared to the other two metrics, although the MI metric is unable to differentiate certain pairs of damage states.