## 4 The ST-LRMA Based Data Collection Method
Therefore, it is a straight forward way to use the spatial relation of the nodes to model the spatial correlation of the sensing data. In the ideal case, such as the simulated network in Section 6, the measurement of a node can be represented as the linear combination of its neighbors according to its adjacent relation. Thus, we can construct \(\mathbf{S}\) from the combination coefficients directly. But in practice, the accurate positions of sensor nodes are usually not available. Additionally, in a complex sensing scenario, the variation of the physical quantities is not smoothly distributed in the spatial domain and the measurements are often interfered by the complicated environment. Hence, in most cases, a feasible way inferring the spatial correlation of the sensing data is to utilize the sensing data themselves. Although the traditional clustering methods, such as KNN, can be used to obtain the relation of a set of data in Euclid Space, it is difficult to get good result for high-dimension sensing data. Given that manifold learning methods [32, 33, 34] have good performance in learning the hidden structure in high dimension data, we adopt the manifold clustering method, namely sparse manifold clustering (SMC) method [35], to get the spatial correlation of sensing data. Conveniently, we implement the SMC model for sensing data spatial clustering as follows. Similar to the ideal case in the simulated network, the measurement of a node can be represented as the linear combination of the measurements of correlated nodes, _i.e._, for a row \(\mathbf{X}_{i},i=1,...,N\) of \(\mathbf{X}\), there exist \(K\) most correlated rows \(\mathbf{X}_{j_{k}},k=1,..,K\) s.t. \(\mathbf{X}_{i}=\sum_{k=1}^{K}w_{j_{k}}^{i}\mathbf{X}_{j_{k}}\), where \(j_{k}\
eq i\). So the problem of constructing \(\mathbf{S}\) is to find the suitable weights \(w_{j_{k}^{i}},k=1,...,K\) for each \(\mathbf{X}_{i}\), \(i=1,...,N\), which represent the intrinsic structure among the high dimension signals, the rows of the sensing data. According to SMC, for each \(\mathbf{X}_{i}\), \(i=1,...,N\), we firstly construct a new signal matrix \(\mathbf{Y}^{i}\) from the measurement matrix \(\mathbf{X}\) as the following form:  
\[\mathbf{Y}^{i}=[\mathbf{Y}_{1}^{i},...,\mathbf{Y}_{i-1}^{i},\mathbf{Y}_{i+1}^ {i},...,\mathbf{Y}_{N}^{i}]. \tag{8}\]  
where \(\mathbf{Y}_{j}^{i}=\frac{\mathbf{X}_{j}^{T}-\mathbf{X}_{j}^{T}}{||\mathbf{X}_ {j}^{T}-\mathbf{X}_{j}^{T}||_{2}},j=1,...,N\) and \(j\
eq i\). Then we construct a diagonal distance matrix \(\mathbf{Z}^{i}\) with elements \(\frac{||\mathbf{X}_{j}^{T}-\mathbf{X}_{j}^{T}||_{2}}{\sum_{j\
eq i}||\mathbf{ X}_{j}^{T}-\mathbf{X}_{j}^{T}||_{2}}\), \(j=1,...,N\) and \(j\
eq i\). Therefore, we can get an optimized sparse coefficient \(\mathbf{c}_{i}\) from the following optimization problem:  
\[\text{min}\quad\alpha\|\mathbf{Z}_{i}\mathbf{c}_{i}\|_{1}+\frac{1}{2}\| \mathbf{Y}_{i}\mathbf{c}_{i}\|_{2}^{2},\quad\text{s.t.}\quad\mathbf{1}^{T} \mathbf{c}_{i}=1. \tag{9}\]  
From the optimized \(\mathbf{c}_{i}\), we can get the spatial correlation weights \(w_{j_{k}^{i}}\), \(k=1,...,K\) by omitting some small elements less than the given threshold \(\varepsilon\):  
\[w_{j_{k}^{i}}=\frac{c_{j_{k}}/\|\mathbf{X}_{j_{k}}^{T}-\mathbf{X}_{i}^{T}\|_{ 2}}{\sum\limits_{ \begin{subarray}{c}j\
eq i\\ |c_{ij}|>\varepsilon\end{subarray} }(c_{ij}/||\mathbf{X}_{j}^{T}-\mathbf{X}_{i} ^{T}||_{2})}. \tag{10}\]  
Based on the spatial correlation weights \(w_{j_{k}^{i}}\), \(k=1,...,K\), the spatial correlation matrix \(\mathbf{S}\) can be formulated by the following element representation:  
\[\mathbf{S}(i,j)= \begin{cases}1,&\text{if }i=j.\\ -w_{j_{k}^{i}},&\text{if }j\
eq i\text{ and }j=j_{k}^{i},k=1,...,K.\\ 0,&\text{otherwise}.\end{cases}  \tag{11}\]  
One should note that \(\mathbf{S}\) constructed in such a way is based on the complete measurement matrix \(\mathbf{X}\). However, at the very beginning, we could only obtain the incomplete sensing data.