**1. Introduction**

Landslides are one of the many common natural disasters in the world [1,2]. A frequent occurrence of extreme weather increases the likelihood of landslides. Rapid economic and social development further aggravates the loss of landslide disasters [3,4]. Therefore, accurate landslide displacement prediction is becoming increasingly important in order to prevent and mitigate the damage caused by landslides [5]. In the process of automated realtime monitoring of landslides, data acquisition and transmission are generally performed using different types of sensors and other electronic devices [6]. Automated monitoring equipment is always in the open-air environment, and most of them inevitably suffer from tear, aging, power loss and other phenomena, all of which can lead to missing monitoring data. In addition, most landslide disasters are located in a relatively harsh geological environment, such as heavy rainfall, hail, dense fog, electromagnetic interference, etc., and the installation and deployment of geological hazard monitoring equipment in open fields will inevitably be affected by the abovementioned harsh environment. Randomness or prolonged interruptions in the operation of the monitoring device can cause the monitoring device to fail to properly send monitoring data to the server, which leads to the problem of missing monitoring time series in the server. Time series forecasting is a valid basis for

**Citation:** Wang, C.; Zhao, Y. Time Series Prediction Model of Landslide Displacement Using Mean-Based Low-Rank Autoregressive Tensor Completion. *Appl. Sci.* **2023**, *13*, 5214. https://doi.org/10.3390/ app13085214

Academic Editors: Daniel Dias, Yuzhu Wang, Jinrong Jiang and Yangang Wang

Received: 6 February 2023 Revised: 7 April 2023 Accepted: 19 April 2023 Published: 21 April 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

making accurate discriminations. To enhance the accuracy of landslide prediction, it is necessary to construct a corresponding accurate data completion method.

For the problem of completing and predicting missing landslide data, most traditional time series models have focused on models such as regression analysis and exponential smoothing. The problem of incompleteness for missing time series data can be broadly classified as either deletion or padding. Data deletion is used as anomalous data to remove some objective abnormal monitoring data, which is primarily used for anomaly detection and feature analysis, while filling is utilized to find the long-term time series change pattern of monitoring data and to supplement the missing monitoring data. The main methods include missing value filling algorithm based on nearest neighbor method, cyclic neural network, random forest and matrix decomposition, but more data are required for machine learning training [7,8]. Statistical filling is more effective for data series with less dimensions that can establish a maximum model, provided that the relationship between missing eigenvalues and existing eigenvalues can be established through observation. The main methods of machine learning include missing value filling algorithm based on the nearest neighbor method, cyclic neural network, and matrix decomposition. Matrix decomposition can effectively explore the correlation between different time series for different dimensions of long time series problems. The matrix decomposition method is used to learn the overall characteristics of the time series matrix, which can be used to approximate the matrix with time characteristics in low-rank, and then complete the missing data [9].

Large-scale time series data are always accompanied by the missing problem. Therefore, the tensor completion method has been introduced into this field to complement the traditional data completion method based on probability and statistics [10–12]. The data completion scheme based on simple quantitative statistics has a relatively simple and efficient processing effect for small datasets and simple regression models, but it is not feasible for massive data in the era of data explosion. Modern research not only has many kinds of data variables and long time series, but also requires fast processing speed, high universality and portability. Multiple variables and long time series can better describe the complex causal relationship between each other [13,14]. Now, neural network technology is often used to deal with the above situations and delete the missing data in order to form a complex intelligent model, but it is not the best solution for missing data because deletion may strengthen or weaken the connection degree of a causal relationship. Based on this, scholars have explored the application of tensor decomposition technology to data completion in multivariate long time series, trying to improve the resolution speed and data missing problems.

Recent studies have found that low-rank matrices have certain advantages in the analysis of multivariate long time series data [11,12], including the sequence tensor completion method [15]. The sequence tensor complement restores the potential tensor from the sampling structure of the time series, allocates the position of the missing items as needed, seamlessly integrates the future value of the time series into the framework of the missing data and improves the data completion accuracy [15]. The low-rank matrix completion method performs singular spectrum analysis and singular value decomposition on the time series in order to complete the low-rank completion of the missing data of the time series, although its calculation is large [16]. Therefore, by adding a time dimension to transform the status time series into a high-dimensional tensor, the cost of computing complexity is better solved. This is also in line with the law of human activities, both short and long term activities, so there are studies using tensors (sensors × 1 day × 24 h) which indicate the above activity mode [17,18]. The dependency between sensors is preserved, providing a new feasible scheme for capturing local and global time patterns [16]. More scholars have combined the autoregressive moving average model with the tensor model to propose the low-rank matrix autoregressive tensor completion model and have achieved good results in the completion and prediction of financial time series data [19].

The mean-based low-rank autoregressive tensor completion mainly includes completing low-rank matrix decomposition/tensor completion and constructing time series autoregressive models, as well as processing the missing data with the neighboring data mean instead of zero before the operation. The low-rank matrix completion model uses the underlying low-rank structure to recover incomplete matrices (assuming that the long-term landslide data sequence is incomplete) [20]. Considering that the deformation displacement of landslide has a great correlation with the previous deformation, the autoregressive model is constructed to represent the deformation law of landslide displacement with time. The autoregressive regularizer is introduced in the low-rank matrix decomposition to characterize the temporal dynamics in landslide displacement deformation, and the learned autoregressive regularizer is implemented to predict the temporal factor matrix, thus realizing the landslide displacement monitoring data completion and predictive modeling [12].

The purpose of this study is to establish a new method for completing and predicting landslide displacement data based on MLATC. In this paper, the causes of data loss of landslide displacement are analyzed. Taking the Shuizhuyuan landslide in the Three Gorges Reservoir area as an example, the data completion and prediction algorithm are designed by using MLATC. Then, the landslide displacement data are divided into training set and test set, and the random missing and non-random missing are selected for corresponding data completion and prediction. The designed model can achieve an accurate completion and prediction of landslide displacement. Finally, a comparative analysis with existing models verifies the effectiveness of the model.
