# Beyond Correlation: Mutual Information to Detect Damage in Nonlinear Systems
## 5 Example Application
### Results  
Because \(ATC\), \(TMAC\), and \(MI\) scores do not measure the same quantity; we convert these scores to percent deviation scores to facilitate their comparison. To do this, we first compute a reference score for each metric by computing the average score from the 10 baseline measurements (State #1):  
\[x_{ref}=\frac{1}{10}\sum_{k=1}^{10}x(k), \tag{10}\]  
where \(x(k)\) is the score (\(ATC\), \(TMAC\), or \(MI\)) for the \(k\)th measurement. Next, the percent deviation of the \(k\)th measurement from the reference was calculated using  
\[q_{(k)}=\frac{x(k)-x_{ref}}{x_{ref}}\times 100,\quad\text{for }k=1,2,\dots,170 \tag{11}\]  
Noting that \(q(k)\) where \(10(s-1)\leq k\leq 10s\) belong to State #s, percent deviations corresponding to each structural state were presented using a box plot. The bottom and top edges of each box indicate the lower and upper quartile, respectively, and the mark between the two edges is the median (50 th percentile). The symbol 'o' shows the outliers, and the whiskers extend to the lowest and highest values not considered outliers.  
Figure 4(a) shows the percent deviation in \(ATC(5,2)\) and Figure 4(b) shows the same for \(ATC(5,3)\). Figure 5(a) and Figure 5(b) show the percent deviation in \(TMAC(5,2)\) and \(TMAC(5,3)\), respectively. Figure 6(a) and Figure 6(b) show the percent deviation in \(M1(5,2)\) and \(MI(5,3)\), respectively. These figures present a series of side-by-side box plots summarizing the distribution of the three metrics across all seventeen states. Each graphical element encodes the box outlines, showing the interquartile range for each state. The thick horizontal line inside each box marks the median value for that state. Whiskers extend to the most extreme non-outlier points, while any measurement beyond those bounds is plotted individually as an outlier dot. To highlight differences between groups of states, three vertical lines divide the 17 boxes into Category I (States #2-#9), Category II (States #10-#14), and Category III (States #15-#17).  
Based on these figures, the following observations can be made:  
* **Category I (States #2-#9):**\(ATC\) and \(TMAC\) show less than 0.5% deviation from the baseline. Thus, \(ATC\) and \(TMAC\) are not sensitive to changes in stiffness and mass when the structure remains linear. In contrast, \(MI\) scores show a significant reduction compared to the reference state.
* **Category II (States #10-#14):** All three metrics show significant reductions from the baseline when nonlinearity is induced using the adjustable gap. \(ATC\) appears to be the most sensitive, although the monotonous drop in \(ATC\) as the adjustable gap is decreased is unexpected. However, \(TMAC\) scores for State #10 show very high variability and an unexpectedly large drop in State #11, compared to the other states in the same category. In contrast, \(MI\) scores for State #11 show high variability like \(TMAC\) scores, although not as extreme.
* **Category III (States #15-#17):** In all three figures, State #15 and State #16 appear to be on the same level as State #10. These three states are the ones where the adjustable gap was set to its maximum value (0.20 mm). \(ATC\) and \(TMAC\) scores of these states are close to the baseline, suggesting that these states may not be identified as damaged. In contrast, \(MI\) scores indicate a significant difference between these states and the baseline state. The observed behavior is discussed in more detail in Section 6.1.  
Figure 6: Percent deviation in \(TMAC(5,2)\) and \(TMAC(5,3)\).  
Figure 7: Percent deviation in \(MI(5,2)\) and \(MI(5,3)\).