## 2 Theory and Method
### Construction of MLATC
#### 2.3.2 Mean-Based Low-Rank Autoregressive Tensor Completion  
The time series matrix \(Y\in\mathbb{R}^{M\times(N=I\times J)}\) with partially missing data can be compressed into a tensor \(Y\in\mathbb{R}^{M\times I\times J}\), thus transforming the matrix completion problem into a tensor completion problem. The traditional tensor completion method assigns the missing data to zero. Considering the time accumulation and continuity of landslide displacement and deformation, the average value of the data before and after the missing data are taken instead of assigning the value to zero when completing the landslide displacement data.  
The specific difference is that the projection of matrix \(Z\) in the domain \(\Omega\) is further required to obtain a useful non-orthogonal projection:  
\[\left[\mathcal{P}_{\Omega}(Z^{\prime})\right]_{m,n}{}^{\prime}=\left\{\begin{array} []{ll}\left[\mathcal{P}_{\Omega}(Z)\right]_{m,n}&if(m,n)\in\Omega\\ \frac{1}{2!}\sum\limits_{i=n-t}^{n+t}[\mathcal{P}_{\Omega}(Z)]_{m,i}&if(m,n) \
otin\Omega\end{array}\right. \tag{11}\]  
where \(t\) is a hyperparameter representing the time sequence length of taking the mean value, in this case, 10 is taken. The matrix \(Z^{\prime}\) conforming to such constraints is called a close orthogonal matrix of \(Y\) in the field \(\Omega\) (recorded as \(Z^{\prime}\mathop{\sim}\limits^{t}\gamma_{\Omega}\)). For ease of understanding, \(Z^{\prime}\) is still noted as \(Z\).  
MLATC is a simple linear combination of LRMC and vector auto-regressive (VAR) models. The first term of the objective function is the LRMC objective function, and the second term is the VAR objective function. The Lagrange function in matrix form is constructed according to the optimization problem:  
\[L(\mathcal{X},Z,A)=\sum\limits_{k}a_{k}\parallel\mathcal{X}_{k(k)}\parallel_{s }+\lambda\parallel\mathcal{Z}\parallel_{A,\mathcal{H}}+\left\langle\mathcal{ Q}^{-1}(\mathcal{X})-Z,\mathcal{Q}^{-1}(\tau_{k})\right\rangle+\frac{\rho}{2} \parallel\mathcal{Q}^{-1}(\mathcal{X})-Z\parallel_{F}^{2} \tag{12}\]  
Then, Question (12) is transformed into \(arg\min_{\mathcal{X}_{k(k)}Z,A}L(\mathcal{X},Z,A,\tau_{k})\), where \(X\) and \(Z\) are variables, and \(\tau_{k}\) is the parameters to be learned. When solving the tensor \(\mathcal{X}_{k}\), \(\frac{\partial L(\mathcal{X},Z,A)}{\partial\mathcal{X}}=0\) should hold. According to Formula (9), since \(\frac{\partial\parallel Z\parallel_{A,\mathcal{H}}}{\partial\mathcal{X}}=0\), \(arg\min_{\mathcal{X}}L(\mathcal{X},Z,A)\Leftrightarrow arg\min_{\mathcal{X}} L(\mathcal{X},Z,A)-\parallel Z\parallel_{A,\mathcal{H}}\). Thus, \(\mathcal{X}_{k}^{l+1}\) can be resolved by Formula (13):  
\[\mathcal{X}_{k}^{l+1}=arg\min_{\mathcal{X}}\ a_{k}\parallel\mathcal{X}_{(k)} \parallel_{s}+\frac{\rho}{2}\parallel\mathcal{Q}^{-1}(\mathcal{X})-Z^{l} \parallel_{F}^{2}+\left\langle\mathcal{Q}^{-1}(\mathcal{X})-Z^{l},\mathcal{Q} ^{-1}\left(\tau_{k}^{l}\right)\right\rangle \tag{13}\]  
In Equation (13), the number \(l\) of iterations of the algorithm is expressed, and the VAR model is used to solve the matrix \(Z\)[12; 21; 26].  
The MLATC achieves a more accurate completion of the original data than the zero-filled algorithm, and then the prediction of landslide displacement data can be completed on this basis.