# Spatial Data Reconstruction via ADMM and Spatial Spline Regression +
## 5 Performance Evaluation
### Performance Evaluation  
#### 5.3.1 Comparison of Different Algorithms  
We show the cumulative distribution function (CDF) of Relative Errors given by different approaches in Figures 3 and 4. In addition, we compare the estimation's Mean Relative Error of different approaches in Figure 5. It is quite clear that our proposed algorithms outperform other three baseline approaches significantly in all cases (\(n_{\text{BS}}=200\) and \(n_{\text{BS}}=100\), data aggregated in November and in December). Figure 3: Comparison of the CDFs of estimation relative errors given by different methods when \(n_{\text{BS}}=200\). The legends follow the same order as the curves at relative error \(=0.5\). (**a**) compares the CDFs based on the data aggregated in November; (**b**) compares the CDFs based on the data aggregated in December. By comparing Patched Estimation + SSR 1 with Patched Estimation approach, we can see that using spatial smoothing based on only base station squares' observations leads to worse performance than patched estimation. This can be explained by the smoothing property of SSR and how we set the values of base station squares. As we described, we set the activity value of base stations by averaging the total activity amount of each base station on all squares it covers. Thus, given the activity \(\frac{z_{i}}{|\Omega_{B_{i}}|}\), (\(|\Omega_{B_{i}}|\) denotes the number of squares within region \(\Omega_{B_{i}}\)) of a base station \(B_{i}\), the true activities of itself and its surrounding squares within region \(B_{i}\) are distributed with a mean of \(\frac{z_{i}}{|\Omega_{B_{i}}|}\). Given two base stations \(B_{1}\) and \(B_{2}\) that are close to each other, with aggregated activities of \(z_{1}\) and \(z_{2}\) respectively, the Spatial Smoothing approach will fit a smooth surface between the two base stations. Suppose  
Figure 4: Comparison of the CDFs of estimation relative errors given by different methods when \(n_{\text{BS}}=100\) for stress-testing. The legends follow the same order as the curves at relative error \(=0.5\). (**a**) compares the CDFs based on the data aggregated in November; (**b**) compares the CDFs based on the data aggregated in December. Figure 5: Comparison of the estimation’s Mean Relative Error of different methods when \(n_{\text{BS}}=200\) or \(n_{\text{BS}}=100\) for stress-testing. In each figure, the bars from left to right stands for Patched Estimation, Patched Estimation + SSR 1, Patched Estimation + SSR 2, Constrained Spatial Smoothing, and Constrained Spatial Smoothing + Features respectively. (**a**) we use the data aggregated in November, and set number of base stations to be 200, similarly for (**b–d**). \(z_{1}>z_{2}\), in this case, overall, the activities of \(B_{1}\)'s neighbour squares will be underestimated, and that of \(B_{2}\) will be overestimated. Therefore, Patched Estimation + SSR 1's performance is not as good as Patched Estimation. By comparing Patched Estimation + SSR 2 with Patched Estimation and Patched Estimation + SSR 1, we can observe that applying spatial smoothing on the results of patched estimation improves the performance. This proves the rationality and effectiveness of introducing smoothness into the estimated cell phone activity distribution surface. Our proposed approach achieves much better performance compared with the three baseline methods. By using Constrained Spatial Smoothing instead of applying Spatial Spline Regression directly, we are able to fit a smooth activity distribution while forcing it to match the observations of base station squares (the aggregated activity volumes) at the same time. By comparing Constrained Spatial Smoothing that incorporates additional features of each square with the version without features, we can see that the performance is further improved. The reason is that the heterogeneity of different locations will influence the telecommunication activity distribution, therefore making the distribution not smooth everywhere.