## 4 The ST-LRMA Based Data Collection Method  
The sequential measurements from a sensor, a row of the measurement matrix, are correlative in time domain and generally behave consistently and smoothly except for some noise and outliers, so it is natural to require the recovered matrix to preserve this property. Based on this observation, we share the similar idea of [15, 16] and introduce a temporal constraint into the B-LRMA model to build a temporal constrained LRMA model (T-LRMA) for maintaining the consistency and smoothness in row direction, the adjacent columns of the recovered measurements. We formulate the T-LRMA model as follows [15, 16]:  
\[ \begin{split}\mathbf{X}^{*}=&\operatorname*{argmin} \limits_{\mathbf{X}}\quad\text{rank}(\mathbf{X})+\lambda_{1}\|\mathbf{X} \mathbf{D}\|_{F}^{2},\\ &\text{subject to}\quad\mathcal{A}(\mathbf{X})=\mathbf{M}.\end{split}  \tag{5}\]  
where \(\lambda_{1}\) is a tunable parameter, and \(\mathbf{D}\) is a matrix with the following form [15, 16]:\[\mathbf{D}=\left( \begin{array}{cccccc}-1&0&0&\cdots&0\\ 1&-1&0&\cdots&0\\ 0&1&-1&\cdots&0\\ \vdots&\vdots&\ddots&\ddots&\vdots\\ 0&0&0&\ddots&-1\\ 0&0&0&\cdots&1\end{array} \right)_{T\times(T-1).} \tag{6}\]  
\(\|\mathbf{XD}\|_{F}^{2}\) represents Frobenius norm of difference matrix of \(\mathbf{X}\) in row direction, and thus enforces the consistency of the columns of the recovered \(\mathbf{X}\). In practice, the sequential consistency is a ubiquitous property for many physical data and its sparse representation is well studied by many researchers. A fused LASSO method was proposed for problems with features that can be ordered in sequence [29]. In this method, the sparsity of regression coefficients and the smoothness constraint of the successive coefficients are considered simultaneously. A spatial subspace clustering method (SpatSC), was proposed for classifying the drill pope spectral data [30]. In this method, each individual regression coefficient vector of the prototypic data is demanded to be sparse and considered to be similar to neighboring. An ordered subspace clustering (OSC) method was proposed to segment data drawn from a sequentially ordered union of subspaces was proposed in [31]. In this method, each sequential nature of sequential data is incorporated by its neighbor penalty term to enforce similarity. The idea of preserving sequential consistency is shared by these methods and our method, but we apply the sequential constraint on the recovered data instead of the sparse coefficients. Except for the temporal correlation of measurements from one node, there also exists spatial correlation among the measurements from different sensor nodes. The fundamental argument for this point is that the physical quantities from natural environment generally have smooth distribution in spatial domain, for example, the temperature distribution in a monitoring area. Another reason for favoring the spatial correlation is that the sensor nodes are generally deployed redundantly, sometimes densely, in order to obtain complete sensing data, which brings the spatial correlation among the measurements, especially for adjacent nodes. The closer nodes usually have similar measurements. To describe this property of the measurements, we incorporate the spatial constraints into the B-LRMA model and propose a spatial constrained LRMA model (S-LRMA) as follows:  
\[\mathbf{X}^{*}= \operatorname*{argmin}_{\mathbf{X}} \text{rank}(\mathbf{X})+\lambda_{2}\|\mathbf{S}\mathbf{X}\|_{F}^{2},\] (7) subject to \[\mathcal{A}(\mathbf{X})=\mathbf{M}.\]  
where \(\lambda_{2}\) is a tunable parameter, and \(\mathbf{S}\) is a matrix representing the relation among the measurements from different nodes, the rows of the measurement matrix \(\mathbf{X}\). It is critical to design a proper \(\mathbf{S}\) to obtain good data recovery results. Generally, the spatial correlation of the sensing data depends on the spatial deployment of the nodes.