## 3 Case Study
### Data Completion and Analysis  
The experimental results of four sensors numbered SZY-02, SZY-03, SZY-06 and SZY-08 are used as examples to introduce the effect of landslide monitoring data completion. After NM40% processing of the dataset of Shuizhuyuan landslide, the data completion effect of the completed training set of SZY-08 is shown in Figure 4. In Figure 4, the blue curve represents the measured data, the yellow dots represent the missing data, the red curve represents the recovered data, \(X\)-axis 0-260, 260-520, 520-780, 780-1040, 1040-1300 represents data recorded for each day and Y-axis is the displacement data. Considering the large amount of data for each sensor, only SZY-08 shows the complementary results for the entire time series, and the other sensors are the complementary results for the first 260 days. Figure 5 shows the complementary effect of the other sensors for the first 260 days. As can be seen in Figures 4 and 5, in the absence of raw landslide displacement data for many consecutive days, the data have been completely processed into missing data, which is challenging for tensor completion. Missing data for several consecutive days has lost the correlation and trend characteristic information between displacement data. The prediction results show that the MLATC model still achieves a very good complementary  
Figure 3: Landslide displacement monitoring data. effect, which is essentially consistent with the original displacement data and achieves a good effective data recovery in some completely missing data days. This is very helpful to analyze the deformation law and deformation trend of the landslide, and also to provide data reference for understanding the relationship between different deformation areas of the landslide. Table 1 shows the evaluation metrics for data completion in the non-random missing case. The results in Table 1 show that the MLATC model has the best MAPE and RMSE for four different scenarios of NM5%, NM10%, NM20% and NM40%, indicating that the MLATC model has a greater improvement in data recovery performance compared to the HALRTC and TRMF models. From Table 1, it can be concluded that the MLATC model is better than the HALRTC and TRMF models in terms of completion effect, indicating that the tensor autoregressive kernel parametrization can effectively replace the rank function, which enables the low-rank tensor completion model to obtain a more accurate completion effect. In addition, \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **TRMF** & **HALRTC** & **MLATC** \\ \hline NM 5\% & 11.8542/14.0631 & 1.3161/3.8315 & 0.9066/0.9196 \\ NM 10\% & 10.2681/16.5992 & 1.1953/3.6270 & 0.7770/0.8958 \\ NM 20\% & 11.7050/18.1097 & 11.7976/25.2256 & 0.7708/0.9616 \\ NM 40\% & 13.8412/25.9528 & 15.0777/38.6958 & 0.7928/1.0070 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Evaluation metrics for data completion in the NM case (MAPE/RMSE). Figure 4: Completion of displacement data of SZY-08 (NM40%). The blue curve represents the measured data, the yellow dots represent the missing data and the red curve represents the recovered data. the MLATC model introduces an autoregressive norm on the basis of the HALRTC model, which can make full use of the structural correlation and local trend feature information between the high-dimensional multivariate time series data, as well as more clearly correlate the multivariate time series data, which also proves that the addition of the autoregressive norm can further improve the complementary performance and accuracy of the model in the low-rank tensor complementary structure. Since the tensor structure well maintains the structural information of the spatiotemporal data in the time dimension and exploits the correlation between the daily displacements of different sensors in the time series, the tensor completion results of the MLATC model are better than those of TRMF in the matrix form.