# Factorial Design Analysis for Localization Algorithms
## 4 Results
### Anchor Nodes  
This is the number of nodes with a known position closest to the NOI, which are necessary to estimate the NOI's position. For our purpose, 5 and 20 RNs are used as critical values. This experiment considered 5 RNs as the lower critical level, because experiments considering 4 or 3 RNs do not provide enough information for obtaining a precise localization, especially with range-based algorithms, which are more affected by Gaussian noise. Thus, 5 RNs were used as the lower critical level and 20 RNs as the upper critical level, since higher numbers of RNs have not been observed to increase the impact of this factor on localization. Table 3 show the results obtained from the performance evaluation of the localization techniques using the MSE and CDF, respectively. To obtain the CDF values, a localization error of 10 m was considered, since for higher values of localization error a very high probability of obtaining said parameter is obtained. To obtain the MSE and CDF of the localization techniques being analyzed, \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbol** & **Factor** & **Level \(-\)1** & **Level 1** \\ \hline A & Path-loss exponent (\(\eta\)) & 1.5 & 5 \\ B & Level noise (\(\sigma_{dB}\)) & 2 dB & 10 dB \\ C & Node density (\(\rho\)) & \(1\times 10^{-4}m\) & \(9\times 10^{-4}m\) \\ D & Anchor nodes (\(N_{rx}\)) & 5 & 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Characterization of the study factors. four study factors are used; it therefore, 16 testing cases are required for each experiment and each case represents a specific combination of the critical levels of the study factors (A, B, C and D) as shown in Table 3.  
Table 4 shows the percentage of variation of the performance metrics being studied for each of the study factors. A very high percentage of variation indicates that the study factor has a very high impact on the performance metric. The results obtained from the \(2^{k}\) factorial analysis show that: \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Factors**} & \multicolumn{8}{c}{**Variation Explained (\%)**} \\ \cline{2-10}  & \multicolumn{4}{c}{**MSE (m)**} & \multicolumn{4}{c}{**CDF (\%)**} \\ \cline{2-10}  & **CL** & **REWL** & **Hyperbolic** & **WLS** & **CL** & **REWL** & **Hyperbolic** & **WLS** \\ \hline A & 22.47 & 32.37 & 29.07 & 29.74 & 11.07 & 15.01 & 22.69 & 22.71 \\ \hline B & 30.85 & 30.62 & 52.22 & 45.84 & 25.8 & 19.84 & 30.07 & 38.49 \\ \hline C & 23.66 & 13.17 & 4.773 & 4.347 & 45.56 & 45.18 & 14.01 & 13.76 \\ \hline D & 3.019 & 0.03 & 0.156 & 0.277 & 1.451 & 0.011 & 0.146 & 0.587 \\ \hline AB & 17.68 & 21.48 & 13.18 & 18.25 & 2.622 & 1.821 & 13.86 & 11.14 \\ \hline AC & 0.111 & 1.31 & 0.015 & 0.028 & 3.697 & 6.177 & 8.213 & 3.483 \\ \hline AD & 0.04 & 0.046 & 0.048 & 0.489 & 0.006 & 0.854 & 0.211 & 1.147 \\ \hline BC & 0.659 & 0.801 & 0.475 & 0.739 & 9.299 & 11.08 & 10.53 & 7.261 \\ \hline BD & 0.000 & 0.113 & 0.021 & 0.26 & 0.006 & 0.022 & 0.215 & 1.425 \\ \hline CD & 1.512 & 0.061 & 0.04 & 0.027 & 0.486 & 0.011 & 0.062 & 0.000 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Impact of the study factors on the MSE and the CDF.