## 6 Experiments and Results
### Experiment Environments and Parameter Setting
#### 6.1.4 Data Recovery Parameters Setup  
In the simulated network, to simulate the change of a physical quantity, such as temperature, we use four curves, shown as Figure 4, to represent the sensing data during a period of time corresponding to the four corner nodes in Figure 2a. The sensing data of other nodes are then generated by the bilinear interpolation method. A snap of the simulated measurements of the \(21\times 21\) grid is shown in Figure 5a. These clean sensing data are regarded as the ground truth of the physical quantity. However, the real sensing data usually have some noise, so we add Gaussian noise into the clean sensing data to simulate actual measurement data, shown as Figure 5b. To form the measurement matrix \(\mathbf{X}\) for data recovery, here \(\mathbf{X}\) is set as a square matrix with dimension of \((21\times 21-1)\times 440\), _i.e._, \(N=T=440\). The parameters in Equations (9), (10) and (15) are empirically set as: \(\lambda_{1}=\lambda_{2}=0.01\), \(\lambda_{3}=10\), \(\alpha=0.1\), \(\varepsilon=10^{-5}\). In the real network, the temperature sensing data collected on 53 nodes at March 1 st 2004 are used as the measurement data which has dimension of \(53\times 2880\). As there exist large difference for the dimensions of the row and column, we segment the measurement data into 48 blocks, where each block has same dimension of \(53\times 60\). One sample of these block is shown in Figure 6a. In the data recovery experiments, each block is assigned as the measurements matrix \(\mathbf{X}\), _i.e._, \(N=53\), \(T=60\). The parameters in Equations (9), (10) and (15) are set as: \(\lambda_{1}=\lambda_{2}=0.01\), \(\lambda_{3}=10\), \(\alpha=0.1\), \(\varepsilon=10^{-5}\). The whole recovered data can be obtained by stacking all the recovered blocks. Figure 4: The simulated sensing data of four corner nodes \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\), \(\mathbf{d}\) of the simulated network in Figure 2a. To evaluate the accuracy of the recovered measurement, the recovery error is defined as the average relative error between the recovered data and the clean sensing data (denoted by \(\mathbf{X}^{0}\)) at these unknown element with the following representation:  
\[err=\frac{\sqrt{\sum\limits_{(i,j)\in\{(i,j)|\mathbf{Q}(i,j)=0\}}(\mathbf{X}^{ \ast}(i,j)-\mathbf{X}^{0}(i,j))^{2}}}{\sqrt{\sum\limits_{(i,j)\in\{(i,j)| \mathbf{Q}(i,j)=0\}}(\mathbf{X}^{0}(i,j))^{2}}}. \tag{21}\]  
In real networks experiments, we need real clean sensing data to verify the validity of our propose ST-LRMA method. Unfortunately, the actual sampling sensing data are always corrupted and some values are gone missing. The real and accurate temperature data curve should be smooth and continuous, however, the actual temperature sensing data sampled over real networks are always noisy due to one reason or another. Hence, we simply use the Median filter on the temporal direction of the measurements as the clean sensing data to compute the error, _i.e._, a 1D Median filter with dimension of 5 is used on each  
Figure 5: A snap of the sensing data of the simulated network. (**a**) The clean sensing data; (**b**) The sensing data with noise. Figure 6: The temperature sensing data from Intel Berkeley Research Lab: (**a**) The actual sensing data; (**b**) The filtered sensing data. row of \(\mathbf{X}\). The filtered results of the measurements in Figure 6a are shown in Figure 6b. Additionally, for the 48 segmented blocks in the real network, the recovery error is defined as the mean of the recovery errors of the 48 blocks. To verify the performance of the proposed method on different node sampling rate, defined as the ratio of the number of the valid elements of \(\mathbf{M}\) to the number of the complete elements of the measurement \(\mathbf{X}\), the data recovery experiments are conducted with the node sampling rate changing from 5% to 95%. The rank of the measurements matrix is an important parameter for the LRMA method. However, estimation of the rank from an incomplete matrix is an open problem.