# Detection and Localization of Multiple Damages through Entropy in Information Theory
## 5 Results
### Applications for Unsupervised Learning  
The results presented so far demonstrate how Wiener entropy can define a reliable damage-sensible feature. Its feasibility for machine learning-based SHM can be demonstrated by using it as a training dataset for a pattern recognition technique. Specifically, a  
Figure 12: Increasing stiffness reduction (50%, 60%, and 80%) for damage in #1 and #3: (**a**) Noise-free measurements; (**b**) noise = 10%. Graphs are limited between 7 m and 13 m to show the increase of the intensity of damage for increasing stiffness reduction. simple method for detecting and localizing damage based on outlier detection analysis has been chosen for this aim. The 3-sigma method defines a threshold value for the 'normality' model at 3\(\sigma\), i.e., corresponding to a 99.73% confidence interval if data are Gaussian distributed. This classic approach is commonly used to test newly-proposed DSFs (as done, e.g., in [56]) due to both its simplicity, its statistically principled bases, but also its well-known limitations. It is known that the 3-sigma rule is an empirical method [57]; in addition, if data deviate from a Gaussian distribution, the basic assumption of this method is lost. Thus, any valid result obtained by training a damage detection/localization algorithm with the proposed feature can only be improved by applying more recent, sophisticated options. This, however, falls beyond the aim of this paper, that instead takes advantage of the use of non-robust outlier classification methods (i.e., 3-sigma method), which intrinsically define the robustness and the noise rejection capability of the WE as a damage-sensitive feature, estimated starting from random ambient vibrations data. The results are here statistically described in terms of precision and recall, calculated from the total numbers of false positives and negatives (\(f_{p}\) and \(f_{n}\)) and true positives \(t_{p}\) as follows:  
\[\text{Precision}=t_{p}/(t_{p}+f_{p}) \tag{5a}\] \[\text{Recall}=t_{p}/(t_{p}+f_{n}) \tag{5b}\]  
It is possible to note from the results that all of the damage scenarios have relatively high precision and recall. For the precision it is worth underling that the results are quite variable based on the meaning that is given to the class \"False positive\", as can be perceived by Table 3. This is mainly because all of the false positives automatically detected are associated with a point located close to a damaged area. For example, when the damage occurs at \(z=8\) m (\(\theta=90^{\circ}\)), and \(z=12\) m (\(\theta=0^{\circ}\)), the false positives are related to the effect of damage that occurs at \(z=12\) m (\(\theta=0^{\circ}\)), which is perceived by sensors located at \(z=12\) m, and \(\theta=90^{\circ},\ 180^{\circ},\ 270^{\circ}\), or vice versa. The same applies to damage at \(z=8\) m. This was found not only in the damage scenario 40%/60%/80% but also for all the investigated damage scenarios, as shown in Figure 13. In fact, in a real SHM scenario, such \"false positives\" could be a benefit as they should help to strengthen the localization of the damage thanks to considerations on the amplitude of variation of WE. This because all of the \"false positives\" have a value of the health state indicator \(D_{WE}\) that is very close to zero. If the damage detected by the sensors placed near to the damaged area is not considered as a \"false positive\", then the number of false positives is zero (with the 3-sigma outliers detection method), as reported in Table 3. Regarding the recall, it is possible to conclude that a relatively high value, 90.69%, is obtained by considering all damage scenarios.