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.