# Factorial Design Analysis for Localization Algorithms
## 2 Related Work
### Localization Techniques Analyzed  
For this paper, four localization techniques were selected to determine the impact of the study factors on the performance metrics of accuracy and precision, using the \(2^{k}\) factorial methodology. The CL and REWL range-free localization techniques were selected, because the CL technique yields the worst performance and the REWL technique yields the best performance in terms of accuracy and precision, respectively, in the proposed evaluation scenario. Range-based localization techniques were also selected, such as the hyperbolic positioning algorithm and the WLS multilateration algorithm, which are the techniques that yield the worst and best performance, respectively, in terms of accuracy and precision in the proposed evaluation scenario. The purpose of selecting range-free and range-based techniques is to determine the localization techniques where the study factors show the greatest impact on their performance in terms of accuracy and precision. Below are brief descriptions of the selected localization algorithms. * **CL Algorithm** Given a set of \(N\) RNs within the range of transmission of the NOI, the localization of the NOI is calculated by way of the average of the positions of the RNs [22]. The centroid \(\mathbf{p}\) of a set of \(N\) RNs is calculated by \[\mathbf{p}=\frac{1}{N}\sum_{i=1}^{N}p_{i}\] (1) where \(p_{i}\) is the position with coordinates \((x_{i},y_{i})\) of the RN \(i\). * **REWL Algorithm** This algorithm is a variation of the WCL algorithm [22]. The weighting factor increases exponentially as the RSS given in dBm increases. Considering a positive real \(\lambda\) constant factor between 0 and 1, the weighting factor \(\omega_{i}\) of each RN \(i\) is given by \[\omega_{i}=(1-\lambda)^{v_{max}-v_{i}}\] (2) where \(v_{i}\) is the RSS in dBm between the NOI and the RN \(i\) and \(v_{max}\) is the maximum RSS of the set of RNs. Therefore, the centroid \(\mathbf{p}\) of a set of RNs is calculated by \[\mathbf{p}=\frac{\sum_{i=1}^{N}p_{i}(1-\lambda)^{v_{max}-v_{i}}}{\sum_{i=1}^{N }(1-\lambda)^{v_{max}-v_{i}}}.\] (3)* **Hyperbolic Positioning Algorithm**  
The hyperbolic positioning algorithm converts the non-linear problem into a linear problem by using a minimum-squares estimator [22]. The distance between a mobile node and the RN \(i\) is calculated by the Pythagorean theorem  
\[{d_{i}}^{2}=\ (x_{i}-x)^{2}+(y_{i}-y)^{2}. \tag{4}\]  
Developing the subtraction \({d_{i}}^{2}-d_{1}^{2}\), Equation (4) becomes a linear problem, leading to the following:  
\[2 xx_{i}+2 yy_{i}-2 xx_{1}-2 yy_{1}={x_{i}}^{2}+{y_{i}}^{2}-{x_{1}}^{2}-{y_{1}}^{2} -{d_{i}}^{2}+{d_{1}}^{2}. \tag{5}\]  
Expressing Equation (5) in matrix form for \(i=2,\ldots,N\)  
\[\left[ \begin{array}{cc}x_{2}-x_{1}&y_{2}-y_{1}\\ \vdots&\vdots\\ x_{N}-x_{1}&y_{N}-y_{1}\end{array} \right]\left[ \begin{array}{c}x\\ y\end{array} \right]=\frac{1}{2}\left[ \begin{array}{cc}{x_{2}}^{2}+{y_{2}}^{2 }-{x_{1}}^{2}-{y_{1}}^{2}-{d_{2}}^{2}+{d_{1}}^{2}\\ \vdots\\ {x_{N}}^{2}+{y_{N}}^{2}-{x_{1}}^{2}-{y_{1}}^{2}-{d_{N}}^{2}+{d_{1}}^{2}\end{array}  \right]. \tag{6}\]  
Thus, the linear problem can be formulated by  
\[\mathbf{H}\mathbf{p}=\mathbf{b}. \tag{7}\]  
where \(\mathbf{H}=\left[ \begin{array}{cc}x_{2}-x_{1}&y_{2}-y_{1}\\ \vdots&\vdots\\ x_{N}-x_{1}&y_{N}-y_{1}\end{array} \right]\), \(\mathbf{p}=\left[ \begin{array}{c}x\\ y\end{array} \right]\) and \(\mathbf{b}\) is a random vector given by  
\[\mathbf{b}=\frac{1}{2}\left[ \begin{array}{cc}{x_{2}}^{2}+{y_{2}}^{2}-{x_{1} }^{2}-{y_{1}}^{2}-{d_{2}}^{2}+{d_{1}}^{2}\\ \vdots\\ {x_{N}}^{2}+{y_{N}}^{2}-{x_{1}}^{2}-{y_{1}}^{2}-{d_{N}}^{2}+{d_{1}}^{2}\end{array}  \right]. \tag{8}\]  
Finally, position \(\mathbf{p}\) of the mobile node can be calculated by the following expression:  
\[\mathbf{p}=\left(\mathbf{H}^{T}\mathbf{H}\right)^{-1}\mathbf{H}^{T}\mathbf{b} \tag{9}\]  
where \(\mathbf{p}\) is the estimated position of the NOI.