# Empirical Approach to Defect Detection Probability by Acoustic Emission Testing
## 4 Results and Discussion
### Selection of the AE Testing Parameters Based on Minimizing the Total Error Probability  
The threshold value and the value of the parameter \(N^{clust}\) are the setting constants during the probability calculation. An increase of \(N^{clust}\) and _thr_ leads to a decrease of a type-I error probability and an increase in the probability of a type-II error. A decrease in these parameters, on the contrary, decreases the probability of a type-II error but increases the probability of a type-I error. The correct choice of \(N^{clust}\) and _thr_ allows finding a compromise--to achieve a minimum of the total error probability, which can be estimated from the probabilities of type-I \(\alpha\) and type-II \(\beta\) errors as:  
\[p_{error}=\pi_{0}\alpha+\pi_{1}\beta \tag{6}\]  
where \(\pi_{0}\), \(\pi_{1}\) are a priori probabilities of the events associated with the occurrence of type-I and -II errors. Minimization of the total error probability is one of the criteria for optimal detection that can be used as a criterion for optimal defect detection [32].  
Figure 16: Probability of making a type-I error at different \(N^{clust}\) for uncorrelated (**a**) and correlated (**b**) stationary noise.  
Figure 17: Probability of making a type-I error for a non-stationary friction-induced noise at different \(N^{clust}\).  
To determine the optimal detection parameters, the dependences obtained in Section 4.3 were scaled in accordance with the specified value of the noise standard deviation. Figure 18a-c shows the probability of type-I and -II errors at different \(thr\) for the correlated noise with standard deviation \(std=25\), \(27.5\) and \(30\) dB, respectively. Figure 18d-f illustrates the dependences of the total error probability on the discrimination threshold for various noise levels.  
As follows from Figure 18d-f, the minimum total error probability is achieved at the minimum value of the parameter \(N^{clust}=5\) at a threshold approximately equal to \(6\)\(std\). At a noise \(std=25\) dB, the minimum possible value of the total error probability is \(5\%\), at a \(std=27.5\) dB--14% and at a \(std=30\) dB about \(31\%\). The influence of the parameter \(N^{clust}\) is greater for higher noise levels. At \(std=25\) dB, the total error probability corresponding to \(N^{clust}=5\) is \(5\%\) greater than at \(N^{clust}=20\), and at \(std=30\) dB, increasing \(N^{clust}\) from \(5\) to \(20\) leads to an increase in the minimum value of the total error probability by \(18\%\)  
Figure 19 shows the minimum total probability depending on the threshold for various \(N^{clust}\) for stationary uncorrelated, stationary correlated, and non-stationary frictional noise. For all types of noise, the minimum values of the total error probability are reached at \(N^{clust}=5\). The value of the total error probability significantly depends on the noise type. A total error probability of less than \(0.1\) is achieved for uncorrelated stationary noise with  
Figure 18: Probabilities of making type-I and -II errors and total error probabilities at standard deviation 25 dB (**a**,**d**), \(27.5\) dB (**b**,**e**) and \(30\) dB (**c**,**f**).