## 4 Results and Discussion
### Cumulative Instantaneous Entropy, Counts, and Energy
#### 4.1.2 Comparing Damage Trends  
Rather than attempting to compare the final values of cumulative AE parameters that have different units, comparisons between the normalized, unitless parameters with respect to MDD can be drawn. The parameters are normalized by the following equation:  
\[p_{\text{normalized}}=\frac{p_{i}-p_{\text{min}}}{p_{\text{max}}-p_{\text{min}}} \tag{5}\]  
where \(p\) is any AE parameter so that variabilities between experiments are reduced and metrics are no longer scale-dependent. Based on the cyclic trends, it is expected that each cumulative AE parameter  
Figure 8: Modulus degradation damage (MDD) plotted versus cycles until specimen fracture for all experiments (S1 to S8).  
\begin{table}
\begin{tabular}{c c c c} \hline
**Cumulative AE Parameter** & **Mean (\(\mu\))** & **Standard Deviation (\(\sigma\))** & **Coefficient of Variation (CV, \%)** \\ \hline Instantaneous entropy & 4221.6 bits & 945.2 bits & 22.4 \\ Counts & 12,360 & 2245 & 18.2 \\ Energy & 9.21 \(\times\) 10\({}^{8}\) aJ & 1.87 \(\times\) 10\({}^{8}\) aJ & 20.2 \\ \hline \end{tabular}
\end{table}
Table 3: Statistics for final cumulative values of instantaneous entropy, counts, and energy.  
increases more at the beginning and end of damage and increases less for the majority of damage. However, normalized cumulative AE parameters with a constant relationship with MDD would be ideal and suggest \"perfect correlation\". Thus, it is assumed that cumulative AE features are best to predict damage if they have a one-to-one relationship on normalized scales.  
These graphs until specimen fracture along with the desired one-to-one relationship are plotted in Figure 9. All features show a greater increase at the beginning and end of damage compared to the middle damage values and deviate from the ideal relationship. More specifically, about 60% of the total increase in counts occurs near the point of fracture, while about 50% of the cumulative energy occurs at initial damage. This result suggests that counts may be more responsive at the end of fatigue life and energy may be more responsive during initial damage. In contrast, instantaneous entropy seems to be more equally responsive for all fatigue damage stages, meaning instantaneous entropy may be better correlated to damage than counts and energy.  
The differences between the cumulative AE parameters damage trends can be quantified by measuring the deviations from the one-to-one ideal relationship. The deviations are measured at each instance of an AE signal. The summation of these deviations, referred to as the deviation factor, can then be used as a goodness of fit metric of the one-to-one model and compared between the AE parameters for each test. The AE parameter with the lowest deviation factor is assumed to be a better representation of damage. It should be noted that only the deviation factors can be compared because all AE features are normalized to the same scale and the number of deviations are equal between parameters for each experiment. The deviation factors for each of the experiments and for each of the cumulative AE parameters were calculated and are plotted in Figure 10. Results show the deviation factor for instantaneous entropy is lower than the deviation factors for counts and energy for all but one experiment. This result further suggests instantaneous entropy may be a better statistic of damage compared to counts and energy.  
Figure 9: Normalized cumulative trend for three parameters with respect to measured damage or MDD where a one-to-one relationship is desired: (**a**) instantaneous entropy; (**b**) energy; and (**c**) counts.