## 4 Performance Evaluations
### Simulation Results
#### 4.2.1 Regular Deployment  
In the case of regular distribution, the sensor nodes and anchors are deployed uniformly along with the grids within the network. The size of the grid is set to 45 m \(\times\) 45 m. We consider five levels of communication range \(R\): 50 m, 100 m, 150 m, 200 m, and 250 m. Table 3 shows us the ALE results of the C-shaped network with 44 training anchor nodes. For example, the result of DV-hop is 283.8/92.6% in the case of \(R\) = 100 m, which means that the ALE of DV-hop is 283.8 and LSRR-LA improves the ALE by 92.6%. The ALE value 283.8 is a percentage, and if we turn this value to an absolute form, it will be 283.8 m (i.e., \(R^{*}\)ALE/100). By contrast, the ALE value of LSRR-LA is 21.0, which means the absolute localization error is 21.0 m.  
We can see from the simulation results in Table 3 that, as the communication range becomes larger, all of the competing methods demonstrate a performance improvement. However, all of the methods exhibit a substantial performance improvement except the proposed LSRR-LA method, which keeps a lower ALE as the communication range \(R\) increases gradually. It is clear that the proposed LSRR-LA method outperforms the previous methods, for it is rather stable and not sensitive to communication range. We also considered five different anchor populations: 13.7% of the total nodes (\(M\) = 30 anchors), 18.3% (\(M\) = 40 anchors), 22.8% (\(M\) = 50 anchors), 27.4% (\(M\) = 60 anchors), and 31.9% (\(M\) = 70 anchors). Table 3 gives the ALE results of the C-shaped topology when the communication radius \(R\) = 100 m. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Compared methods & DV-hop, Amorphous, MDS-MAP, PDM, LSVR \\ Size & 1000 m \(\times\) 1000 m \\ Type of distribution & regular and random \\ Type of network & C-shaped, L-shaped, and D-shaped \\ Anchor node ratio (\%) & 13.7–31.9\% \\ Communication radius (m) & 50–300 \\ \hline DV-hop: Distance Vector hop; MDS-MAP: Multidimensional scaling MAP; PDM: Proximity-Distance Map; LSVR: \\ Localized Support Vector Regression \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The simulation parameters. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(R\) = 50 m/5\% & \(R\) = 100 m/10\% & \(R\) = 150 m/15\% & \(R\) = 200 m/20\% & \(R\) = 250 m/25\% \\ \hline \multicolumn{5}{c}{**Number of Anchors \(M\) = 44/20\%**} \\ \hline DV-hop & 613.1/93.2\% & 283.8/92.6\% & 189.9/78.4\% & 138.6/71.3\% & 103.9/65.6\% \\ Amorphous & 766.8/94.5\% & 400.4/94.8\% & 301.1/86.2\% & 275.7/85.6\% & 222.8/83.9\% \\ MDS-MAP & 651.8/93.6\% & 341.8/93.9\% & 231.1/82.2\% & 176.2/77.4\% & 143.8/75.1\% \\ PDM & 138.3/69.8\% & 73.5/71.4\% & 62.6/34.3\% & 53.4/25.6\% & 46.2/22.5\% \\ LSVR & 142.7 /70.7\% & 76.2/72.4\% & 61.8/32.5\% & 49.3/19.4\% & 41.5/13.7\% \\ LSRR-LA & 41.8 & 21.0 & 41.9 & 39.7 & 35.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of the average localization error (ALE) under different radio ranges in the regular distribution network. From the Table 4, it is obvious that the proposed LSRR-LA greatly outperforms the previous methods for the C-shaped anisotropic network. All of the competing methods demonstrate a performance improvement as the network gets denser; however, only the proposed LSRR-LA exhibits a substantial performance improvement while the previous methods do not exhibit satisfactory improvement. Figure 2 shows a plot of Tables 3 and 4. Figure 1(a) depicts the ALE results in terms of Table 2, where the communication radius varies from 50 m to 250 m. All of them generally conform to the trend that ALE decreases as the communication radius increases. When the communication radius \(R\) is smaller (\(R\) = 50 m), the positioning accuracy of DV-Hop, Amophos, and MDS-MAP becomes very poor. As the \(R\) increases, their localization errors begin to decrease. The LSRR-LA method always maintains the best positioning accuracy, whether \(R\) is smaller (\(R\) = 50 m) or \(R\) is larger (\(R\) = 250 m).