## 5 Simulation and Analysis
### Experiment with Synthetic 2-D Data  
In this subsection, the KNN and the sparsity-based interpolation methods are compared on the synthetic data. Sensor data are smooth if they have a strong spatial correlation. By applying the 2-D DCT transform on the spatially deployed sensor data, the major energy of these data will concentrate on low frequency domain. When the data change rapidly in local regions, it has strong high frequency components in the DCT domain. So, it is meaningful to discuss the performance of missing data recovery when the sparsity of sensor data is represented in the high frequency, low frequency, and mixed high and low frequency DCT coefficients, respectively. In addition, what the recovered data look like for different missing patterns and missing rates is also very useful. The effect of the missing-block-size on the RMSE is also discussed. Figure 5: 2-D synthetic low frequency data with size 64 \(\times\)64 is generated from 64 nonzero coefficients in low frequency domain of DCT. **(a)** 2-D synthetic data. The color bar denotes the sample value of each spatial node. **(b)** 64 nonzero coefficients in DCT dictionary. The size of the DCT dictionary is 64 \(\times\)64. The color bar denotes the coefficient value of each atom in DCT dictionary. A set of 64 \(\times\) 64 2-D synthetic data, shown in Figure 5(a), is generated from 64 nonzeros in low frequency DCT domain as shown in Figure 5(b). It is clear that the low frequency DCT coefficients can provide a smooth representation of spatially 2-D sensor data. Figure 6 shows the recovery performance of KNN and the proposed approach for spatially smooth data. KNN results in a large RMSE which means KNN fails to recover the missing samples. When a block of samples are missing, KNN has to choose the nodes beyond the block as its neighbors whose values may differ significantly from the interpolated node, indicating that KNN is sensitive to the size of missing patterns. Our method produces very low RMSE if the missing-block-size is smaller than 8 \(\times\) 8 and the missing rate is smaller than 0.5. This missing rate is promising since we can recover the missing samples when half of sensor data are missing. As the missing rate increases, the RMSE of the proposed method remains nearly the same within certain intervals until the missing rate reaches a turning point. As shown in Figure 6(b), for example, if the acceptable value of RMSE is at 10\({}^{-4}\), the turning point of missing rates are 0.3, 0.6, and 0.8 for 4 \(\times\)4, 2 \(\times\)2, and 1 \(\times\)1 missing-block-sizes, respectively. It is a very appealing characteristic that in these stable ranges, the estimation quality is still good and nearly independent of the missing rate. This can be explained by Equation (7), which states that when the number of samples is large enough, the missing samples can be well recovered with overwhelming probability. The stable range shortens as the missing-block-size increases because the increase of block-size introduces less randomness to the sensing matrix \(\mathbf{\Phi}\). When a large block of sensor samples are missing, e.g., 8 \(\times\) 8, the proposed method cannot guarantee a low RMSE. However, 8 \(\times\) 8 missing-block is a very extreme case for the 64 \(\times\) 64 sensor network. Even in this situation, our method performs better than KNN in term of the RMSE. Figure 7 shows the recovered data by KNN and the proposed approach under different missing-block-sizes. The missing rate is fixed at 0.3. We can see that the estimation quality is much  
Figure 6: Effect of missing rate and missing block size on estimation quality with spatially smooth sensor data. **(a)** and **(b)** shows the RMSE curve of KNN and the proposed approach, respectively. Error bar stands for the standard deviation with aspect to the repeated 100 times of simulations for the same size of missing blocks and same missing rate. This can help eliminating and understanding the influence of randomness of each simulation. better by our sparsity-based recovery method than KNN.