## 3 Methodology  
### Prediction Model  
Since the deep displacement monitor sensor consists of several identical sensing units, there is a strong correlation among the units, so the multivariate grey prediction model is more effective than the univariate grey prediction model in predicting the displacement of landslides. The traditional GM(1, N) model [46] is often used to analyze the influences of several influencing factors on the system behavior variables. Here is the definition of it. Assuming that \(X_{1}{}^{(0)}\) is the system characteristic sequence while the \(X_{i}{}^{(0)}\)(\(i\) = 2, 3,..., N) is the series of explanatory variables having high correlations with sequence \(X_{1}{}^{(0)}\). \[ \begin{array}{l}X_{1}^{(0)}=(x_{1}^{(0)}(1),x_{1}^{(0)}(2),\ldots,x_{1}^{(0) }(m))\\ X_{2}^{(0)}=(x_{2}^{(0)}(1),x_{2}^{(0)}(2),\ldots,x_{2}^{(0)}(m))\\ \vdots\\ X_{\rm N}^{(0)}=(x_{\rm N}^{(0)}(1),x_{\rm N}^{(0)}(2),\ldots,x_{\rm N}^{(0) }(m))\end{array}  \tag{1}\]  
The first-order accumulative generation operation (1-AGO), defined as Equation (2), generates the first-order cumulative sequence \(X_{j}^{(1)}\)(\(j\) = 1, 2,..., N). \[ \begin{array}{l}X_{j}^{(0)}=(x_{j}^{(0)}(1),x_{j}^{(0)}(2),\ldots,x_{j}^{(0) }(m))\\ x_{j}^{(0)}(k)=\sum_{g=1}^{k}x_{j}^{(0)}(g),k=1,2,\ldots,m\end{array}  \tag{2}\]  
\(Z_{1}{}^{(1)}\) is the sequence of immediately adjacent mean generation of \(X_{1}{}^{(1)}\), defined as follows. \[ \begin{array}{l}Z_{1}^{(1)}=(z_{1}^{(1)}(2),z_{1}^{(1)}(3),\ldots,z_{1}^{(1) }(m))\\ z_{1}^{(1)}(k)=0.5\times(x_{1}^{(1)}(k)+x_{1}^{(1)}(k-1)),k=2,3,\ldots,m\end{array}  \tag{3}\]  
Then Equation (4) is the discrete grey model with multiple variables where \(a\) and \(b_{i}\) can be calculated by least-squares estimation. \[x_{1}^{(0)}(k)+az_{1}^{(1)}(k)=\sum_{i=2}^{N}b_{i}x_{1}^{(1)}(k) \tag{4}\]  
According to previous research, the GM(1, N) model has some serious flaws. First, the GM(1, N) model is a first-order grey system prediction model with N variables, however, when N equals 1, this model cannot be converted to GM(1, 1) model equivalently. Second, the final expression is derived by an ideal simplification method, which may make a mismatch with the actual situation [47]. Finally, the default background value of 0.5 is used in the adjacent mean generation, resulting in large errors in fitting and prediction. [PERSON] [48] had proposed an improved background value method to compensate for this flaw. [PERSON] proposed an optimizing grey model [49] to solve part of the first and second drawbacks mentioned above. Let \(X_{1}{}^{(0)}\) be the system behavior characteristic's original non-negative sequence. \[X_{1}^{(0)}=(x_{1}^{(0)}(1),x_{1}^{(0)}(2),\ldots,x_{1}^{(0)}(m)) \tag{5}\]\(X_{i}^{(0)}(i=2,3,\ldots,\) N) are the original positive sequence containing N \(-\) 1 related factors, \(X_{j}^{(1)}\) is the 1-AGO sequence of \(X_{j}^{(0)}(j=1,2,\ldots,\) N) and \(Z_{1}^{(1)}\) is the sequence of immediately adjacent mean generation of \(X_{1}^{(1)}\)  
\[ \begin{array}{l}X_{i}^{(0)}=(x_{i}^{(0)}(1),x_{i}^{(0)}(2),\ldots,x_{i}^{(0)} (m))\\ X_{i}^{(1)}=(x_{i}^{(1)}(1),x_{i}^{(1)}(2),\ldots,x_{i}^{(1)}(m))\\ Z_{1}^{(1)}=(z_{1}^{(1)}(1),z_{1}^{(1)}(2),\ldots,z_{1}^{(1)}(m))\end{array}  \tag{6}\]  
The linear correction term \(h_{1}(k-1)\) and the grey action \(h_{2}\) are shown in the following model, which is the first-order differential optimizing grey model with N variables, abbreviated as OGM(1, N). \[ \begin{array}{l}\chi_{i}^{(0)}(k)+az_{1}^{(1)}(k)=\sum_{i=2}^{N}b_{i}x_{i}^{ (1)}(k)+h_{1}(k-1)+h_{2}\end{array}  \tag{7}\]  
Based on the previous research and combined with the research object of this article, a further optimization of the grey model has been made. A system often consists of several interrelated factors, and the development trend of the system behavior factor is fed by the other influencing variables in the system.