# Beyond Correlation: Mutual Information to Detect Damage in Nonlinear Systems
## 5 Example Application
### Calculation of ATC, TMAC, and M1 Scores  
_ATC, TMAC_, and M1 scores between Channel 2 and Channel 5 and those between Channel 3 and Channel 5 were computed as follows: To compute the auto- and cross-power spectra, [PERSON]'s method [PERSON] (1952) was used with a Hann window of 2096 time points and 50% overlap.  
[PERSON]'s method produces an estimate of the power spectral density of a signal by dividing the signal into overlapping segments of equal length, applying a window function to each segment, computing the periodogram of each windowed segment, and averaging the resulting periodograms. The window function acts to mitigate the spectral leakage resulting from analyzing a finite portion of the signal. However, since the window function tends to zero at the boundaries, data points near the boundaries have a smaller influence on the periodogram than those near the midpoint. To equalize the influence of each data point on the resulting PSD, segments are overlapped. The amount of overlap is often selected to be between 50-75%. Specifically, when the Hann window is used, 50% overlap is considered optimal, as it ensures equal contribution of each data point in the resulting PSD estimate ([PERSON], 2018). The choice of window length determines the trade-off between the frequency resolution and the variance of the estimate. A shorter window leads to reduced spectral variance at the expense of frequency resolution. In this study, the window length was set to 2048 to achieve a good balance between spectral variance and frequency resolution. With a sampling frequency of 320 Hz, this window length corresponds to a frequency resolution of 0.156 Hz. Although the Dactron SpectraBOOK system [PERSON] (1952) used in data acquisition has an operational frequency range from 0.1 Hz to 100 kHz, the analysis here is bounded by the sampling frequency of 320 Hz. The frequency resolution of 0.156 Hz is appropriate to analyze the vibration characteristics of the selected window length.  
The transmissibility coherence between two channels \((i,j)\) was computed by normalizing the cross-power spectra using Equation (3). \(ATC\) scores between Channel 2 and Channel 5 and those between Channel 3 and Channel 5 were calculated using Equation (4), between \(f_{min}=20\) Hz and \(f_{max}=80\) Hz. \(TMAC\) scores were calculated using Equation (5), using the first measurement from State 1 as the reference signal. To calculate the \(MI\) scores, joint and marginal probabilities were estimated using a 2D histogram, with 15 bins in each direction. The number of bins \(b\) was selected using the equation derived by [PERSON] et al. (2018):  
\[b=\text{round}\Big{(}\frac{e}{6}+\frac{2}{3}e+\frac{1}{3}\Big{)}, \tag{8}\]  
where \(\text{round}(x)\) indicates the integer closest to \(x\), and \(e\) is given by  
\[e=\sqrt[3]{8+324\,n+12\sqrt{36\,n+729\,n^{2}}}, \tag{9}\]  
for a vector of length \(n\). [PERSON] et al. (2018) show that Equation (8) leads to a low bias in histogram-based estimation of entropy. Mutual information was then computed from the estimated probabilities using Equation (7).