## 5 Performance Evaluations
### Accuracy
#### 5.2.2 Standard Deviation of the Localization Error  
The low std of error values in a data set indicates that most of the data set are close to the mean. The std of the localization error is represented as follows:  
\[\text{error}_{\text{std}}=\sqrt{\frac{1}{\text{N}}\sum_{i=1}^{\text{N}}\bigl{(} \text{error}_{\text{(i)}}-\text{error}_{\text{(avg)}}\bigr{)}^{2}} \tag{12}\]  
Figure 13: Average localization errors of the models according to the std of the noise (\(R=1\)) in (a) WCL, and (b) APT.  
where N is the number of localized nodes, error(i) is the localization error for node i, and error(avg) is the average localization error.  
Figure 14a shows the std of the localization error for each path planning model for WCL and Figure 14b for APT. \(R=1\) and \(\sigma=7\) for each simulation. Since the std of localization error performance of the errors according to the number of simulation runs was close, as in Figure 11, statistical characterization is again preferred to facilitate performance interpretation of the considered path planning models. Tables 4 and 5 show the statistical characterization of different static path planning models for WCL and APT, respectively. As can be seen from Table 4, the std of error performance of the H-curve and NHexCurves path planning models for the WCL method is better than the other models, and the performances of these models are similar. According to this performance criterion, the LMAT model's performance is relatively poor compared to the others. When the same evaluation is performed for the APT localization technique, it can be said that the H-curve model stands out compared to other models. The Z-curve model has been the worst-performing method in this scenario.  
Figure 14: Std of errors of all path planning models in **(a)** WCL, and **(b)** APT.  
The average of the 50 simulations is then calculated using different \(R\) values, as shown in Figure 15. In Figure 15, \(\sigma\) = 7 and \(R\) varies between 0.5 and 2.5. These figures show the std of the localization error for different resolution values. When the performances of the models for the WCL method are evaluated, the error performances of the models at \(R\) = 1 and smaller resolution values are generally close to their average. However, at larger resolutions, the error performance of the models is slightly different from the average. As the resolution increases, the std of the error value of the models increases according to the previous resolution. The std of error performance of the proposed path planning model is higher at lower resolutions. For the WCL method, the LMAT and H-curve models show the best std of the error performance. According to the same evaluation criteria, HILBERT again shows the worst result. When the same error performance evaluation of the models for the APT localization method is made, it can be said that the proposed NHexCurves model shows the best performance at resolution values less than \(R\) = 1. It should be noted here that the LMAT model, which is one of the models that gives the results of the std of the best error for the WCL method, gives poor performance, especially at low resolutions (\(R\) < 1) in the APT method. As shown in Figure 12b, the LMAT show high error results for \(R\) < 1. After \(R\) = 1, it can be said that the H-curve is the best performing model and the Z-curve model is the worst performing model, although there are no significant differences between the performance of the models in general.  
method observed that for \(R=0.5\) and \(R=0.75\), the LMAT method localized very few nodes, and showed worse results than other methods. For \(R>1\), although the localization ratio performance of all models is high and close to each other, it can be said that the M-curves model stands out, albeit a little, here.  
Figure 15: Std of errors of the path planning models according to resolution (\(\sigma=7\)) (**a**) WCL, and (**b**) APT.