## 3 Analysis of Experimental Data and Results
### Probabilistic Model Development
#### 3.3.2.1 Model Error Estimation  
After establishing a linear relationship between the explanatory variable, \(dc/dN\), and the response variable, \(da/dN\), the relationship can be used to make predictions of the next value of crack growth rate given the next value of AE count rate. Predicting future observations for certain \(dc/dN\) values is one of the main goals of this linear regression modeling. Better prediction can be made considering the uncertainties and errors in the model. True values for model parameters are unknown and using estimated values in the prediction adds to the uncertainty that should be captured by the error term. The error term showed by \(\varepsilon\) in Equation (2) includes all the uncertainties discussed in the previous section and follows a normal distribution with the mean of zero and standard deviation that needs to be captured (s). The results of this study showed that parameter s of the error term can be described by another normal distribution: \(s\) -\(N(\mu_{s},\sigma_{s})\). The point estimates of the model parameters, \(\alpha_{1}\) and \(\alpha_{2}\), and the model error, \(\varepsilon\), described by Equations (2) and (3) were computed for Al7075-T6 and Ti-6-4 experiments separately. As discussed, the error term is considered to follow a normal distribution with the mean of zero and a standard deviation of \(s\) which accounts for the model uncertainties after the data were normalized. The model parameters and error term distributions were determined through Bayesian analysis for each set of experiments. The software WinBUGS was used to capture the error term distribution in the model in which uninformative, or uniform, prior distributions were chosen for all variables. The posterior distributions of the error term and the corresponding standard deviations were determined and are listed in Table 3.  
The error estimation for Al7075-T6 tests was performed by comparing experimental measurements with estimated model predictions obtained through Bayesian analysis (using parameter estimation for Al7075-T6 in Table 3). Subsequently, it was found that for Al7075-T6, the parameter s of the error term follows a normal distribution as well and was estimated as \(s\sim\)N(0.86, 0.057), where \(\varepsilon\sim\)N(0, \(s\)). The results are shown in Figure 6.  
The result of the posterior predictive distribution for \(da/dN\) as a function of \(dc/dN\) for Al7075-T6 is plotted in Figure 7. The posterior distribution is shown by its median and the 95% confidence level (2.5% and 97.5% prediction bounds). The data used to fit the model is also plotted in Figure 7.  
This finding suggests that since almost all procedures were maintained between the two sets of experiments, the model parameters must be updated to account for material variation from material to material, as expected. In addition, both model parameters are material-dependent since both vary significantly between the material testing. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Material**} & \multirow{2}{*}{\(\mathbf{\kappa_{1}}\)} & \multirow{2}{*}{\(\mathbf{\kappa_{2}}\)} & \(\varepsilon\sim\)N(\(\mu_{e}\),\(s\)) & \(s\sim\)N(\(\mu_{s}\),\(\sigma_{s}\)) \\ \cline{3-5}  & & & \(\mu_{\varepsilon}\) & \(\mu_{s}\) & \(\sigma_{s}\) \\ \hline Al7075-T6 & 0.03 & \(-11.319\) & 0 & 0.86 & 0.057 \\ Ti-6 Al-4V & 0.006 & \(-4.971\) & 0 & 0.24 & 0.010 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Regression parameters and error term distribution for Al7075-T6 and Ti-6 Al-4V tests. Figure 6: Estimated error terms. Figure 7: Posterior predictive AE model with the uncertainty bounds, material: Al7075-T6; WB indicates Bayesian regression analysis results using WinBugs. Each of the three Ti-6-4 experiments demonstrated a good correlation between AE count rate and crack growth rate. These results suggest that the proposed linear model based on AI7075-T6 holds for the Ti-6-4 as well.