# Missing Data Imputation in the Internet of Things Sensor Networks
## 5 Results
### Imputation  
We compared the accuracy of the different imputation techniques. First, we simulated artificial missing values on the \(\mathrm{C_{6}}H_{6}\) sensor variable at different rates (10%, 20%, 30%, 40%). The different techniques (\(k\)NN, RI, missForest, EM and BEMVI) were used to impute the missing values and the accuracy of these techniques at the different missing data rates were calculated. Table 1 shows the performance of the different techniques. From the results, our approach recorded the best imputation accuracy across the different rates of missing data. This could be explained by the use of strong predictors through clustering the data before imputation was carried out. The results of the RF and RI imputation methods also showed similar imputation performance, owing to the effect of the auxiliary variables (i.e., CO, NMHC, NO\({}_{x}\) and O\({}_{3}\)) that showed strong correlation with the missing sensor variable and were added to the imputation model.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Imputation Method** & **Avg. Computational Complexity (s)** & **Missing Rate** & **RMSE** & \(\mathbf{R^{2}}\) & **MAE** \\ \hline KNN & 0.82 & 10\% & 0.941595 & 0.984187 & 0.216405 \\  & & 20\% & 1.410407 & 0.964461 & 0.452831 \\  & & 30\% & 1.831899 & 0.964461 & 0.694035 \\  & & 40\% & 1.930442 & 0.933555 & 0.861461 \\ \hline RI & 0.85 & 10\% & 0.802989 & 0.98831 & 0.193356 \\  & & 20\% & 1.206899 & 0.97327 & 0.939866 \\  & & 30\% & 1.630078 & 0.949827 & 0.61114 \\  & & 40\% & 1.722670 & 0.943538 & 0.771931 \\ \hline EM & 2.28 & 10\% & 3.083324 & 0.824947 & 0.719108 \\  & & 20\% & 4.507154 & 0.608994 & 1.435791 \\  & & 30\% & 5.445520 & 0.405416 & 2.189204 \\  & & 40\% & 3.899858 & 0.152979 & 0.914241 \\ \hline missForest & 1.71 & 10\% & 0.835237 & 0.987369 & 0.197484 \\  & & 20\% & 1.231309 & 0.972118 & 0.392628 \\  & & 30\% & 1.631028 & 0.949937 & 0.60595 \\  & & 40\% & 1.746202 & 0.941935 & 0.77397 \\ \hline BEMVI & 3.39 & 10\% & **0.011758** & 0.9999985 & 0.000599 \\  & & 20\% & **0.029012** & 0.999985 & 0.001917 \\  & & 30\% & **0.215160** & 0.999165 & 0.006739 \\  & & 40\% & **0.169418** & 0.999483 & 0.006136 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Performance and computational complexity of imputation techniques.  
We also assessed the performance of the imputation techniques on missing data occurring over a consecutive period of time {1 day, 1 week and 1 month} as seen in Table 2 and Figure 3. In general, across all imputation tasks examined in this paper, BFMVI illustrates a reliable performance in handling the different rates and characteristics of missing data considered in this study as opposed to the other imputation techniques.  
The average computational complexity \(T_{avg}(n)\) of the imputation techniques were also computed based on the function \(T(n)=cn\) where \(T\) represents the time, \(n\) represents the input size and \(c\) represents some constant [48]. We computed the complexity of the imputation techniques on each missing data rate (10-40%) and obtained the average complexity based on \(T_{avg}(n)=T(n)/m\), where \(m=4\) simulations. Overall, KNN imputation showed the best computational performance among all other imputation techniques considered with an averege of 0.82. Our proposed BFMVI technique however showed a trade-off between accuracy and complexity with a higher average computational complexity of 3.39 owing to the imputation process which computes multiple imputation options for the final process.