## 5 Simulation and Analysis
### Experiment with Real 2-D Data  
To validate the performance of the sparsity-based missing data recovery in a sensor network, a mean monthly surface climate over global land areas, excluding Antarctica [2] is employed as the data set for simulation. The climatology data includes eight climate elements--precipitation, wet-day frequency, temperature, diurnal temperature range, relative humidity, sunshine duration, ground frost frequency and wind speed--and was interpolated from a data set covering the period from 1961 to 1990. The data are available through the International Water Management Institute World Water and Climate Atlas ([[http://www.iwmi.org](http://www.iwmi.org)]([http://www.iwmi.org](http://www.iwmi.org))) and the Climatic Research Unit ([[http://www.cru.uea.ac.uk](http://www.cru.uea.ac.uk)]([http://www.cru.uea.ac.uk](http://www.cru.uea.ac.uk))). This data set consists of the monthly averaging surface sunshine duration in June over global land areas from 1961 to 1990. The final measurement points in the data set formed a regular grid of 10' latitude/longitude over the region under study. We select a subset of 64 \(\times\) 64 data that has no missing values, shown in Figure 15, as the original data without missing samples. Since the data are the average values from 1961 to 1990, it is very smooth and should be highly compressible in the DCT domain. When applying the real data set to simulate the sparsity-based signal processing, [PERSON] _et al._[27] suggest preserving the \(S\) largest coefficients in a transform domain. Let \(\boldsymbol{a}=[\alpha_{1}\,\alpha_{2}\cdots\,\alpha_{N}]^{T}\) be a vector to represent the DCT coefficient of the real data, we define \(\boldsymbol{a}_{S}\) as the vector of coefficients (\(\alpha_{i}\)) with all but the largest \(S\) set to zero. By calculating the normalized energy loss that is smaller than \(10^{-5}\):  
Figure 15: A snapshot of mean monthly surface sunshine duration in June over global land areas, excluding Antarctica. Our proposed method results in very small RMSE when the missing rate is smaller than 0.5 for 1 \(\times\) 1 and 2 \(\times\) 2 missing-blocks. The improvement over KNN holds for 1 \(\times\) 1, 2 \(\times\) 2 and 4 \(\times\) 4 blocks until too many sensor samples are missing, _i.e._, when the missing rate is larger than 0.8. Both our proposed method and KNN have very large RMSE for 8 \(\times\) 8 missing-blocks because the missing-block-size is too large for the 64 \(\times\)64 sensor network. Figure 18 shows the recovered sensor network data with a missing rate 0.4. Compared with the non-missing data in Figure 16(a), KNN failed to recover some features and introduce the blocking artifacts when the missing block becomes large. As shown in Figure 18(d-f), this disadvantage of KNN becomes serious when the missing-block-size increases. In contrast, our method shows the ability to recover the data without significant information loss. In addition, the change of missing-block-size nearly does not affect the recovered data. Thus, the proposed method is robust to missing block sizes. As demonstrated from Figures 5 to 14, the proposed method can give much better performance in simulations. However, for the real dataset captured from a snapshot of mean monthly surface sunshine,  
Figure 18: Effect of the missing block size on the estimation quality of Figure 16(a) for missing rate at 0.4. **(a)**, **(b)** and **(c)** show sensor data with 1 \(\times\) 1, 2 \(\times\) 2 and 4 \(\times\) 4 missing block size, respectively; **(d)**, **(e)** and **(f)** are the recovered data by KNN corresponding to (a), (b) and (c), respectively; **(g)**, **(h)** and **(i)** are the recovered data by the proposed approach corresponding to (a), (b) and (c), respectively. these advantages are not so evident, as shown in Figures 17 and 18.