Fig. 19: Percentage of localization error vs. Average Hop Counts vs. anchor nodes.
\begin{equation} \beta_{1}=\ \frac{\left(\sum{\text{Hop}_{\text{count}}}^{2}\right)\left(\sum mLE_{o}\right)-\left(\sum{\text{m\ Ho}p_{\text{count}}}\right)\left(\sum{\text{Ho}p_{\text{count}}\text{\ L}E_{o}}\right)}{\left(\sum m^{2}\right)\left(\sum{\text{Hop}_{\text{count}}}^{2}\right)-\left(\sum{\text{m\ Ho}p_{\text{count}}}\right)^{2}},\ and\nonumber \\ \end{equation}\begin{equation} \beta_{2}=\ \frac{\left(\sum m^{2}\right)\left(\sum\text{Hop}_{c\text{ount}}LE_{o}\right)-\left(\sum{\text{m\ Ho}p_{\text{count}}}\right)\left(\sum{\text{m\ L}E_{o}}\right)}{\left(\sum m^{2}\right)\left(\sum{\text{Hop}_{\text{count}}}^{2}\right)-\left(\sum{\text{m\ Ho}p_{\text{count}}}\right)^{2}};\nonumber \\ \end{equation}
where \(LE_{o}\) is LE observed from Table 4.
The localization error LE is analyzed by a linear equation only just to reduce the complexity. By using equation (24), the localization error is summarized as equation (25)-
\begin{equation} LE=\left\{\begin{matrix}0.21851\ m\ -\ 149.80688\ Hop_{\text{count}}\ +\ 711.76707;for\ DV-Hop\\ 0.09028\ m\ -\ 38.57935\ Hop_{\text{count}}\ +\ 193.82679;for\ IDV\\ 0.06698\ m\ -\ 7.41643\ Hop_{\text{count}}\ +\ 42.53416;for\ ODR\\ \end{matrix}(25)\right.\ \nonumber \\ \end{equation}
From equation (25) it is marked that the localization error is pulled down by the hop counts. The equation (25) needs analysis along with Table 4. The first two columns of Table 4 shows that an increment in the number of anchor nodes, the \(\text{Ho}p_{\text{count}}\) also, increases but the equation (25) shows the effect of\(\text{Ho}p_{\text{count}}\) is several times more than the \({}^{\prime}m^{\prime}\) to reduce the localization error. At the same time, the equation (25) shows that the effect of the contributory factors \({}^{\prime}m^{\prime}\),\(\text{Ho}p_{\text{count}}\) and residual value \({}^{\prime}\eta^{\prime}\) in localization error is very less for ODR in comparison to DV-Hop and IDV, which establishes the claim of robust performance of the proposed model ODR.
Similarly, the effect of communication range on localization error is studied analytically. Consider Fig. 20 obtain through a simulation carried (in reference to Fig. 11 collected) in Table 5.