## 5 Simulation and Analysis
### Experiment with Synthetic 2-D Data
For KNN, due to the missing blocks of samples, the recovered data suffer from blocking effects, _i.e._, the edge is not smooth for the missing-block. This effect becomes worse when the missing-block-size increases. Conversely, the proposed approach recovers the missing data almost equally well under different missing-block-sizes. Now, we discuss the performance of recovery for fast changes in the sensor data network. The 64 \(\times\) 64 2-D synthetic data, shown in Figure 8(a), is generated from 64 nonzero high frequency DCT coefficients as shown in Figure 8(b). In this case, fast oscillations are presented. Figure 9 shows the recovery performance of KNN and the proposed approach for oscillating data. In Figure 9(a), KNN performs poorly in term of the RMSE. Because KNN has the tendency to smoothen the data, it is not suitable for high frequency data. Under different block sizes, the RMSE curves of KNN for high frequency data are all much higher than that for low frequency data. On the contrary, our method approaches very low RMSE if the block-size is smaller than 8 \(\times\) 8 and the missing rate is smaller than 0.5. This result is very similar to the recovery of smooth data in Figure 6(b). So, if the sparsity is satisfied and the block-size is not too large, the sparsity-based recovery is robust to smooth or oscillating sensor data. According to the compressive sensing theory, sparsity-based recovery mainly  
Figure 7: Effect of missing block size on the estimation quality of Figure 5(a) for missing rate = 0.3. **(a–c)** show sensor data with 1 \(\times\) 1, 2 \(\times\) 2 and 4 \(\times\) 4 missing block size, respectively; **(d–f)** Recovered data by KNN under (a–c) respectively; **(g–i)** Recovered data by the proposed approach under (a–c), respectively. depends on the global sparsity of data but not too much on whether this data is sparse in low or high frequencies. Meanwhile, the proposed approach can get good estimation results as long as the missing rate is lower than a certain value, e.g., 0.3, 0.6, and 0.8 for 4 \(\times\) 4, 2 \(\times\) 2, and 1 \(\times\) 1 missing-block-size, respectively. Thus, the proposed approach is robust with different sizes of missing blocks. For example, with 4 \(\times\) 4 missing-blocks, the RMSE is still low if the missing rate is smaller than 0.3. It means one quarter of sensors samples can be missed although the block-size is a little large for a 64 \(\times\)64 sensor network. However, an interesting phenomenon in Figure 9(a) is that larger missing blocks lead to lower RMSEs of KNN. Due to the periodic oscillation of the synthetic high frequency data, when missing block is larger than one period of cosine wave, at least one missing sample is recovered correctly. Figure 8: 2-D synthetic high frequency data with size 64 \(\times\) 64 is generated from 64 nonzeros coefficients in high 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. Figure 9: Effect of missing rate and missing block size on estimation quality with oscillating 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. While missing block is small, it is hard or even impossible to recover any missing samples, e.g., each of them cannot be represented via the linear combination of its nearest neighbors. An intuitive explanation is shown in Figure 10 for the 1-D high-frequency component which is generated from high-frequency DCT coefficients.