A time series based dynamic prediction model for loess mudstone landslides

Abstract

This study targets the poor dynamic adaptability of traditional prediction models for loess mudstone landslides, their neglect of time‑series patterns, and weak noise handling. Using the Zhangjiawan landslide as a case study, we propose a dynamic prediction model for loess mudstone landslides based on time‑series analysis. The model employs VMD to adaptively decompose the GNSS‑monitored cumulative displacement sequence into three physically meaningful components—trend, periodic and random. This decomposition reveals complex features caused by geological structures, rainfall, freeze–thaw cycles, and other factors. We design special prediction strategies for each component: the trend component is predicted using a sliding‑window improved quadratic least‑squares method that removes early data influence and improves responsiveness to deformation trends; the periodic component is further decomposed with the CEEMDAN–K‑means–VMD method and then predicted using an SSA‑Transformer‑BiLSTM model with rainfall and freeze–thaw factors; the random component is tested for white noise, and non‑white sequences are predicted using an SSR‑LSTM model based on historical deformation data. Applications show that VMD reduces data complexity, and the combined predictions agree well with measured values (MAE = 1.98, MAPE = 0.88%, R² = 0.93). The model improves the prediction accuracy and applicability for loess mudstone landslides and provides a reference for dynamic forecasting.

Introduction

Landslides are among the most destructive geological disasters in mainland China; their suddenness and unpredictability often cause major casualties and economic losses, especially in geologically sensitive areas^1–3. Traditional landslide displacement prediction models can be divided into empirical models (such as the Saito method)⁴ and statistical models⁵; however, they adapt poorly to complex dynamic systems and easily exhibit overfitting or underfitting problems^6,7. Therefore, nonlinear time‑series prediction models have gradually become a common means of predicting landslide displacement⁸. Machine learning methods show significant advantages in this field and are currently a focus of research. Models such as backpropagation neural networks (BP)^9,10, support vector machines (SVM)^11,12, artificial neural networks (ANN)^13,14, convolutional neural networks (CNN)¹⁵, recurrent neural networks (RNN)¹⁶ and long short‑term memory neural networks (LSTM)¹⁷ have been widely applied because of their powerful nonlinear mapping ability and generalization performance. These models can not only effectively capture the nonlinear characteristics of landslide displacement but also integrate various influencing factors for multivariate prediction¹⁸, thus revealing the intrinsic mechanisms of landslide deformation more accurately. In recent years, to further enhance the accuracy of model predictions. Cao X Y et al.¹⁹ optimized BP neural networks based on bird swarm optimization for landslide deformation prediction, and these studies achieved good results. Qin S W et al.²⁰ used PCA for data dimensionality reduction and Bayesian optimization for hyperparameter tuning of a CNN‑LSTM combined multivariate neural network model, showing certain advantages in prediction accuracy. Dong V D et al.²¹ used a spatially explicit deep learning (DL) neural network model of landslide susceptibility to study 217 landslide events in the Mong Lai area of Vietnam. Ashok Dahal et al.²² implemented a physics‑informed neural network (PINN) approach. However, for landslides that are more affected by their own geological conditions and less affected by available influencing factors, some scholars have decomposed cumulative displacement into time series of trend displacement and periodic displacement²³; based on component prediction, they then form combined prediction models^24,25. Combined models usually combine the strengths of linear and nonlinear models, further improving prediction accuracy. Even so, most existing models still rely on static training of historical data and fail to fully incorporate the information weighting rules in time‑series data, resulting in lagging prediction of sudden deformation. To address this problem, landslide prediction research is gradually evolving towards dynamic prediction. Dynamic prediction models can use real‑time data updates, capture changes in time‑series characteristics and integrate time‑series decomposition to more accurately depict the nonlinear evolutionary characteristics of landslides^26–28. However, in the face of noisy sequences generated by interference from monitoring equipment caused by harsh plateau climates, models have difficulty effectively identifying noise terms and decoupling their corresponding influencing factors, limiting prediction accuracy.

For the displacement prediction of loess mudstone landslides, this paper uses a VMD‑based dynamic prediction model for cumulative landslide displacement. Using the idea of combining variational mode decomposition (VMD)²⁹ with component prediction³⁰, the cumulative displacement series at the maximum displacement point DB06 of the Zhangjiawan landslide is adaptively decomposed into three components: trend, periodic and random³¹, to fully reveal the internal structure and complexity of the original displacement series. Prediction strategies are adopted according to the characteristics of different components. The trend component uses an improved quadratic least squares method with sliding intervals for extrapolation. The periodic component uses an SSA‑Transformer‑BiLSTM model to capture and predict features of the re‑decomposed periodic component. Significant influencing factors such as rainfall and freeze–thaw are used as input factors to build a model that adapts to its complex periodic fluctuations³². For the random component, a white‑noise test is first performed; if the series is non‑white noise, an SSA‑LSTM model predicts it based on historical data. This study takes the Zhangjiawan landslide as the research object to verify the model’s fitting and extrapolation prediction performance on single loess mudstone landslide time‑series data. By integrating GNSS monitoring data and rainfall, temperature and other data, the above VMD‑based component prediction model is constructed to optimize the handling of complex time‑series data and improve the accuracy of loess mudstone landslide displacement prediction.

Component-based dynamic prediction model

Time series decomposition

The evolution process of landslide cumulative displacement is regarded as a complex time series that varies with time, and its mathematical expression is:

1

Among them, represents the cumulative displacement at time . This sequence is decomposed into three components to better capture and analyze its intrinsic characteristics: is the trend term component, describing the overall upward trend of landslide cumulative displacement, typically exhibiting monotonically increasing characteristics. It originates primarily from the combined effects of fundamental factors such as lithological conditions, geological structures, and topographic features, which determine the basic direction and velocity of landslide displacement. is the periodic term component. Its cyclical fluctuations reflect regular variations in the data, presenting an annual cyclical pattern. This component is mainly induced by external triggering factors, such as atmospheric rainfall and freeze-thaw cycles. is the random term component, primarily representing fluctuations that cannot be fully explained by trend or periodic models. It is closely associated with unpredictable stochastic factors like wind loads and human activities. The introduction of this component helps describe noise and abrupt changes in the data.

Based on the Variational Mode Decomposition (VMD)-based cumulative displacement decomposition model, this study utilizes programming software for automatic decomposition of cumulative displacement. The model employs an adaptive decomposition mechanism to decouple nonlinear, non-stationary cumulative displacement time series signals into physically meaningful modal components, addressing the limitations of conventional methods in the incomplete separation of trend and periodic terms. The VMD algorithm decomposes the original displacement signal into a linear combination of K Intrinsic Mode Functions (IMFs) through the introduction of Hilbert transform and frequency aliasing correction techniques:

2

Each modal component is constrained near a specific centre frequency. By adjusting the regularisation parameter α, the modal bandwidth is controlled to ensure that the decomposition process reflects the physical properties of the signal. For non-stationary signals such as cumulative displacement, which have a pronounced trend, the model adopts a quadratic penalty function and the alternating direction method of multipliers (ADMM) for iterative optimisation and sets a convergence tolerance to guarantee decomposition accuracy. In this paper, the VMD model is used for decomposition. Because the cumulative displacement needs to be decomposed into three components, the fixed parameter K = 3 (the number of modes). The key parameter α (penalty factor) is adjusted to tune the weight of the modal bandwidth. By comparing the decomposition results, the trend and periodic components are ensured to be free of noise. Finally, K = 3 and α = 2000 are selected, and the cumulative displacement is decomposed into a trend component (a low-frequency slow-varying component), a periodic component (a medium-frequency oscillating component) and a random component (a high-frequency noise component). This provides a data foundation for subsequent neural network training.

Trend term prediction

The trend term of monitoring data refers to the long-term, directional component of landslide displacement over time, reflecting the overall movement trend of the landslide. By decomposing the original displacement data, the trend term isolates periodic fluctuations and random noise, thereby more clearly revealing the evolutionary direction of the landslide. In prediction, the cumulative displacement trend term typically exhibits a monotonically increasing pattern, serving as a critical basis for landslide stability assessment and early warning. A sliding window is applied to the monitoring data, where, after fitting and predicting each value, the predicted value is incorporated into the baseline dataset while the first value of the original sequence is removed to maintain a constant cardinality of baseline data for subsequent predictions. Defining the sliding window size as , he initial window comprises the data points:

3

After each prediction, the window slides forward by one step: the new predicted point , is incorporated, while the oldest data point is removed. The updated window then becomes:

4

Generally, the window after the k-th slide is represented as:

5

Where is the prediction result of the k-th iteration. The sliding window approach is particularly effective for real-time time series prediction, especially in trend term analysis, as it dynamically adapts to data variations while balancing the utilization of historical information and the capture of emerging trends. In landslide monitoring, fitting the trend term through a sliding window enables iterative refinement of the prediction model, thereby enhancing the accuracy of medium-to-long-term displacement forecasting. After establishing the sliding window, a quadratic least squares method is used to fit the trend term in each window. The model takes the following form:

6

Here, , , are the quadratic regression coefficients for the window. To determine these three coefficients, one must first compute the power sums and weighted sums of each order within the window, defined as:

7

8

According to the quadratic least squares principle, the system of normal equations formed by these sums is as follows:

9

By solving the above 3 × 3 system of linear equations, one obtains the quadratic regression coefficients for the trend term in the current window. Then, substituting the next time point into this quadratic equation yields the predicted value.

10

The predicted point is then added to the data series, the earliest data point is removed, and a new window is formed to continue the prediction. To avoid recalculating all sums from scratch during the sliding-window process, recursion formulas can be used to update these sums: assume the window slides from to , removing the old point and adding the new point then there is::

11

12

In this relation, remains constant. Using these recurrence formulas, the power sums and the weighted sums can be updated quickly. One then solves Eq. (9) again to obtain the new coefficients and the predicted value .

In summary, the quadratic polynomial least-squares fitting method based on sliding intervals offers certain advantages for extracting the trend component of landslide displacement. It better describes the nonlinear trend of displacement and can dynamically update with new data through a sliding-window strategy, so that changes in the trend are reflected promptly. Because it fits only a finite interval of data, the computational cost is relatively low. Solving through matrix equations makes the algorithm robust and simple, providing an effective approach for predicting the trend component of landslides.

However, to avoid error distortion caused by long-term dependence on predicted values during recursive prediction, this study imposes a constraint on the sliding-window recursion: the maximum number of consecutive prediction steps is strictly less than the number of observed data points in the window. This means that after each extrapolation step, the model must wait for a new set of observed displacement data to enter the window before continuing the rolling prediction. This mechanism aligns more closely with the periodic data updates in landslide monitoring. It also tests the model’s extrapolation ability during short periods without data and ensures the authenticity and stability of the predictions, making it more consistent with the real engineering scenario of regular data collection.

Periodic and random terms prediction

The prediction of the periodic displacement component is an important part of landslide forecasting. It reflects the repeated evolution of landslide displacement under external periodic disturbances such as rainfall and freeze–thaw cycles. Compared with the long‑term evolution represented by the trend component, the periodic component typically has strong nonstationarity, complex frequency composition and a magnitude that changes significantly over time. Its multi‑scale and nonlinear characteristics need to be considered simultaneously. Therefore, it is necessary to introduce multilevel decomposition and efficient nonlinear modelling methods into the prediction of the periodic component to describe its internal structure more precisely, thereby improving the reliability and stability of landslide displacement prediction.

This paper further decomposes the periodic component using the CEEMDAN‑K‑means‑VMD method. This method applies complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) to perform time‑frequency decomposition on the original signal. CEEMDAN is an improvement of classical empirical mode decomposition; by introducing adaptive noise and averaging multiple iterations, it reduces mode mixing and improves the stability and accuracy of the decomposition³³. This process obtains a set of intrinsic mode functions (IMFs) representing oscillating components in different frequency bands. To measure the complexity of each IMF, the code computes the sample entropy of each IMF. Sample entropy is a non‑parametric metric for evaluating the randomness and complexity of a time series and is more robust than approximate entropy when handling boundary effects and variations in data length. After using the magnitude of sample entropy to reflect the frequency characteristics of each mode, K‑means clustering performs an unsupervised classification of the sample entropy, dividing it into high‑, medium‑ and low‑frequency categories.

After clustering, the program merges the IMFs in the same category to obtain combined components for different frequency bands and applies variational mode decomposition (VMD) to the high‑frequency components. VMD formulates a constrained variational problem and decomposes the signal into several modes with a central frequency and finite bandwidth. It effectively suppresses the end effects and mode mixing of empirical mode decomposition. The specific process is shown in Fig. 1.

Fig. 1.

Schematic Diagram of CEEMDAN-K-means-VMD.

After the prediction data are decomposed, neural network models are used for training, testing and forecasting. This section employs three neural network models for prediction: BiLSTM, Transformer‑BiLSTM and SSA‑Transformer‑BiLSTM. The bidirectional long short‑term memory network (Bi‑LSTM) is a time‑series model developed from the long short‑term memory network (LSTM). Bi‑LSTM builds forward and backward LSTM subnetworks so that the input sequence propagates both forward and backward in time. The connection of the two directional outputs allows the model to use past and future context information simultaneously. Therefore, Bi‑LSTM is more suitable for tasks that require a complete understanding of the sequence.

The Transformer‑BiLSTM is a hybrid neural network that combines the Transformer architecture with a bidirectional LSTM to compensate for their respective weaknesses. The Transformer uses a multi‑head self‑attention mechanism to weight the importance of positions within a sequence, achieving global feature modelling and parallel computation. However, its dot‑product self‑attention is insensitive to local information and tends to accumulate error when used for time‑series prediction. Bi‑LSTM excels at capturing bidirectional temporal dependencies but lacks the ability to model global context. Combining the Transformer with Bi‑LSTM balances global and local information: self‑attention models long‑term dependencies, while Bi‑LSTM captures local temporal dynamics, so the hybrid model can account for global semantics and short‑term fluctuations³⁴.

The SSA‑Transformer‑BiLSTM model improves on Transformer‑BiLSTM by introducing SSA to automatically optimise key hyperparameters. The Sparrow Search Algorithm (SSA) simulates the cooperative foraging behaviour of “discoverers and followers” and the anti‑predator strategy of “vigilantes” in a sparrow population³⁵. It performs global optimisation of key model parameters to overcome the subjectivity of manual tuning. Using SSA’s population search capability, each “sparrow” represents a candidate hyperparameter set. By iteratively updating the sparrows’ positions to minimise prediction error, the algorithm ultimately finds the network configuration with the best performance. This strategy fully exploits SSA’s global search and local refinement capabilities, making the Transformer‑BiLSTM more robust and more accurate when processing non‑stationary time series such as landslide displacement.

For the residual component, because monitoring methods are limited and factors such as wind loads and human activities cannot be obtained, the sequence is first tested to determine whether it is white noise. If it is white noise, subsequent predictions use a random sequence. If it is non‑white noise, the SSA‑LSTM neural network model is used for subsequent prediction based on historical data patterns. The specific process of component prediction is shown in Fig. 2.

Fig. 2.

Flowchart of the component prediction model structure.

Engineering case validation

Basic Characteristics of the Landslide.

The Zhangjiawan landslide is located in Zhangjiawan Village, Pengjiazhai Town, Chengxi District, Xining City, Qinghai Province. The geographic location is shown in Fig. 3. It lies on the transitional slope zone between the low mountainous hilly area and the alluvial plain along the south bank of the Huangshui River. With a main sliding direction of 20°, it represents a typical binary-structure landslide composed of loess and mudstone. The landslide area features higher terrain in the south and lower elevations in the north, with elevations ranging from 2,124 to 2,630 m and an average slope gradient of 30°. It covers an area of 4.64 × 10⁵ m² with a total volume of 1.229 × 10⁷ m³; the average thickness of the sliding mass is 28 m, extending 400-1,100 m north-south and 200–600 m east-west. The landslide exhibits an irregular tongue-shaped planform and concave cross-section profile, characterized by gently inclined stratified rock-soil composites. The geological structure consists primarily of Neogene mudstone interbedded with sandstone, overlain by Late Pleistocene aeolian loess. Intense gully erosion has fragmented the topography, resulting in rounded hilltops.

Fig. 3.

Geographic location map of the Zhangjiawan landslide. The satellite imagery data were obtained from the publicly available platform GSCloud (https://www.gscloud.cn/). The map was generated and post-processed using ArcMap 10.8 (https://desktop.arcgis.com/).

The Zhangjiawan landslide exhibits a complex formation mechanism, representing a composite geological hazard resulting from the reactivation of an ancient landslide through multi-stage and multi-phase sliding. Based on morphological characteristics, failure surfaces, shear exits, and overriding mass features, the landslide can be divided into four distinct sliding phases. The landslide area experiences a plateau semi-arid continental climate characterized by long, cold seasons, short warm periods, significant temperature variations, low but concentrated precipitation, and high evaporation rates. The mean annual evaporation is 1,442.6 mm, while the mean annual rainfall measures 414.5 mm, with uneven distribution concentrated primarily from June to September. Complex hydrological conditions involve both surface water and groundwater in the landslide activity.

Based on the principle of “across to the edge, lengthwise to the base and vertically to the sliding zone,” the Zhangjiawan landslide uses surface displacement deformation monitoring (GNSS) as the main monitoring method. The arrangement of monitoring points is shown in Fig. 4 in combination with the overall deformation activity of the Zhangjiawan landslide. In this study, the monitoring point DB06—which exhibits the largest deformation among the Zhangjiawan landslide monitoring points—is selected for analysis. This point is located in the eastern slide zone. The data monitoring period runs from 29 July 2020 to 13 July 2024, with one sample taken each day. The cumulative displacement–rainfall–temperature relationship is shown in Fig. 5. The evolution of cumulative displacement at DB06 is controlled by cyclic fluctuations in temperature and rainfall. During hot and rainy seasons, the deformation rate increases. During cold seasons, because of freeze–thaw cycles on the plateau, the cumulative displacement shows fluctuations. The final cumulative displacement reaches 229.42 mm.

Fig. 4.

Engineering geological map of the study area. The DEM data were obtained from the publicly available platform GSCloud (https://www.gscloud.cn/). The geological map was generated using AutoCAD Student Version (https://www.autodesk.com.cn).

Fig. 5.

Relationship diagram of cumulative displacement-rainfall-temperature at monitoring point DB06.

Landslide displacement time series decomposition

VMD decomposition, when processing cumulative displacement data, uses the constraint mechanism of intrinsic mode functions to effectively separate high‑frequency noise from physically meaningful low‑frequency trends and periodic fluctuations. As shown in Fig. 6, the decomposition of the cumulative displacement at monitoring point DB06 clearly reveals three core components in the displacement evolution: a trend component, a periodic component and a random component. The trend component dominates the entire process and reflects a long‑term, nearly linear increase in displacement. The periodic component shows significant periodic variations in the displacement, but it accounts for only a small proportion, indicating that periodic factors have a limited influence at this landslide point. The random component has the lowest proportion and the smallest amplitude, representing unpredictable noise or non‑systematic changes. It is noteworthy that this component exhibits abnormal fluctuations from each November to the following March.

Fig. 6.

Decomposition diagram of cumulative displacement at monitoring point DB06.

Landslide displacement prediction

Prediction of the trend displacement component

The displacement trend component was modeled using a sliding-window second-order least-squares linear fitting method implemented in programming software. The software first reads the data to be fitted and sets the sliding window length, which is 200 in this paper. The model then extrapolates 100 steps into the future and compares the predictions with actual displacement trend points, as shown in Fig. 7. During the fitting phase (historical data back-testing), the model exhibits very high accuracy: the overall mean absolute percentage error (MAPE) is 1.08%, the mean absolute error (MAE) is 1.67 mm, and the coefficient of determination (R²) is 0.99. In the prediction phase (future trend extrapolation), the accuracy remains stable: the MAPE is 0.87%, the MAE rises to 11.96 mm, and the R² becomes 0.96. These results verify the core advantage of the sliding-window second-order least-squares method. The sliding mechanism removes distant historical data and captures the gradual change trend in recent data. It is noteworthy that the MAE increases from 1.67 mm to 11.96 mm during the prediction phase, reflecting the accumulation of absolute errors when extrapolating over long ranges without updating the base data with new information. The MAPE decreases from 1.08% to 0.87% because, as the trend component displacement increases, the denominator in the MAPE calculation increases. Although R² decreases from 0.99 to 0.96, it still indicates high precision. Finally, the nearly overlapping shapes of the two curves in the figure visually reflect the strong fitting and predictive ability of this method for the trend component, providing a reliable basis for judging the evolution stage of the landslide trend component.

Fig. 7.

Comparison diagram of trend term displacement prediction at monitoring point DB06.

Prediction of the periodic displacement component

The displacement in the periodic component was re-decomposed using the CEEMDAN‑K-means‑VMD method implemented in the software. The original signal was decomposed into several physically meaningful intrinsic mode functions (IMFs). K-means clustering was then used to perform unsupervised classification of the sample entropy, dividing the IMFs into high-, medium-, and low-frequency categories. The high-frequency component was further decomposed using variational mode decomposition (VMD) with the number of modes set to K = 3. This re-decomposition of the periodic component produced three components (IMF1–IMF3), which, together with the medium- and low-frequency components, formed a set of five components (IMF1–IMF5). The decomposition results are shown in Fig. 8.

Fig. 8.

Secondary decomposition diagram of periodic term displacement at monitoring point DB06.

The decomposed results were separately input into three neural network models—BiLSTM, Transformer-BiLSTM, and SSA-Transformer-BiLSTM. Rainfall and freeze‑thaw factors were chosen as input variables, including average temperature, maximum temperature, minimum temperature, daily average rainfall, and the four‑day cumulative rainfall. 70% of the displacement data was used for training and 30% for validation. After training, the neural networks still incorporated the concept of a sliding window for prediction: a fixed amount of data in the window was maintained while sequential predictions were made forward. The prediction results are shown in Fig. 9.

Fig. 9.

prediction comparison chart for different models.

Based on the fit comparisons of the predictive results in Fig. 8, the SSA‑Transformer‑BiLSTM neural network model was ultimately selected as the periodic component prediction model. Figure 10 shows the comparison of testing and prediction effects for the periodic component displacement at the DB06 monitoring point. The two curves in the figure exhibit relatively stable variation, but during the testing phase, the periodic component prediction yields a mean absolute percentage error (MAPE) of 35.25%, a mean absolute error (MAE) of 0.32 mm, and a coefficient of determination (R²) of 0.94, indicating that the model has some deviation in capturing historical periodic fluctuations. In the prediction phase, the MAPE expands to 62.53% while the MAE decreases to 0.079 mm and the R² decreases to 0.90. This phenomenon reflects the complexity of modeling the periodic component: although the model combines rainfall, temperature, and other related periodic factors through SSA‑Transformer‑BiLSTM for deep decomposition and prediction, the actual periodic deformation amplitude in plateau regions is small. Its weak signals are prone to environmental noise, especially disturbances from equipment anomalies in winter, which leads to insufficient extraction of periodic patterns. Overall, when calculating MAPE, although the MAE is small—yielding a small denominator—the MAPE calculation becomes relatively large. However, by introducing the coefficient of determination as an evaluation metric, the overall error remains small and demonstrates good predictive capability.

Fig. 10.

Comparison diagram of periodic term displacement prediction versus measurement at monitoring point DB06.

Prediction of the random displacement component

The random component of displacement represents the noise in cumulative displacement. Before predicting the random component, it is necessary to determine whether it is a white‑noise sequence. The time series of the random component was subjected to the LB (Ljung‑Box) white‑noise test in MATLAB 2018b, with the lag order set to 20 and the significance level set to 0.05. The computed prop (p‑value) is less than 0.001, which is below 0.05, proving that this series is not white noise. Therefore, historical data variations can be used for subsequent prediction. A sliding‑window optimized SSR‑LSTM neural network was adopted for forecasting, using 70% of the data as the training set and 30% as the test set. The sliding window covers 365 days, and the forecast extends 100 days. The following comparison results were obtained.

Figure 11 shows the test and prediction comparison for the random component at the DB06 monitoring point. The figure clearly displays the test‑set curve and the prediction curve alongside the original random series. In the testing phase, the random component of historical data exhibits high precision, and the two curves in the figure follow consistent trends. The mean absolute error (MAE) is 0.02 mm and the coefficient of determination (R²) is 0.96, indicating that the model captures historical random fluctuation characteristics accurately. In the prediction phase, the prediction error of the random component increases slightly, with an MAE of 0.02 mm and an R² of 0.61. Although this error is relatively small compared with the overall error of cumulative displacement, the phenomenon underscores the complexity of predicting the random component. Despite using SSA‑LSTM to identify nonlinear patterns in historical data, the model’s ability to extrapolate unseen random fluctuations remains challenged by the inherent high uncertainty and noise in the random component, particularly disturbances caused by equipment anomalies in winter.

Fig. 11.

Comparison Diagram of Random Term Displacement Prediction versus Measurement at Monitoring Point DB06.

Cumulative prediction

By summing the trend, periodic, and random displacement components for both the test and prediction cases, the cumulative test displacement and cumulative predicted displacement were obtained and compared with the original data, as shown in Fig. 12. At the DB06 monitoring point, the cumulative displacement during the testing stage had a mean absolute percentage error (MAPE) of 0.75%, a mean absolute error (MAE) of 1.63 mm, and a coefficient of determination (R²) of 0.99. In the prediction stage, the accuracy increased slightly, with the MAPE rising to 0.88%, the MAE increasing to 1.98 mm, and the R² decreasing to 0.93. This result fully validates the effectiveness of the VMD subcomponent prediction model. By decomposing the displacement sequence into trend, periodic and random components and implementing a dedicated prediction strategy, the model appropriately overcomes the limitations of predicting each component individually and further achieves collaborative error optimization in the displacement reconstruction process. At the same time, a rate comparison chart was added to the cumulative-displacement comparison. By comparing the deformation of the original cumulative displacement with the test and prediction cumulative displacements, it was found that the trends were consistent. The rate comparison and cumulative-displacement comparison corroborated each other, jointly confirming the accuracy of the model. Consistent rates indicate that the model successfully captures the dynamic mechanism of the deformation process, while the agreement of the cumulative displacements ensures the reliability of the long-term trend. This multi-angle verification significantly enhances the model’s credibility and fully demonstrates that the model has a strong capability to analyze the multi-factor coupled deformation characteristics of the Zhangjiawan landslide, providing a reference tool for predicting loess-mudstone landslides.

Fig. 12.

Comparison Diagram of Cumulative Displacement Prediction versus Measurement at Monitoring Point DB06.

Conclusion

(1) The Variational Mode Decomposition (VMD) method is applied to the cumulative landslide displacement time series decomposition, effectively isolating the trend component, periodic fluctuations and noise, thereby greatly reducing data complexity. At the same time, an improved quadratic least squares method with a sliding window is used to carry out linear fitting and prediction of the trend component. The resulting overall error is small: the MAPE is 0.38%, the MAE is 0.85 mm, and the R² is 0.93. The original curve and the predicted curve almost overlap, verifying that this method has good predictive power for the trend component.

(2) The displacement of the periodic component is influenced by cyclic processes such as rainfall and freeze–thaw and contributes relatively little, while the noise component displacement is the weakest. For the periodic and noise components, SSA‑Transformer‑BiLSTM and SSR‑LSTM prediction models were used respectively. By continuously sliding and updating the data window for training and prediction, the results show that both models can accurately capture the actual deformation trend of loess‑mudstone landslides.

(3) Taking the DB06 monitoring point of the Zhangjiawan landslide as an example, the overall prediction results of the model were compared with measured data. The comparison shows that the two trends are essentially consistent: the overall MAPE is 0.88%, the MAE is 1.98 mm, and R² is 0.93. The predicted curve matches the actual displacement curve well, verifying the model’s suitability for predicting displacement in loess‑mudstone landslides.

The time series based dynamic prediction model for loess mudstone landslides showed good accuracy in fitting and extrapolating the deformation time series data of the Zhangjiawan landslide, and it has good applicability to creeping landslides controlled by cyclic factors. However, for predicting sudden instability during the rapid acceleration phase, future research should focus on improving the model’s ability to capture nonlinear responses. This study also relied on a single dataset for a single landslide; subsequent work should combine multi-source data to achieve more comprehensive landslide predictions and further enhance their practical utility.

Author contributions

All authors contributed to the study conception and design. C.T.: Conceptualization and design, writing the outline of the paper, analysis of the results, the first draft of the paper; J.D.: coordinating the team work, technical guidance, revision of the paper; L.P.: providing the basic data, information modification and validation; X.L.: collecting data, data processing, analysis; Y.P.: charting, optimizing pictures and translating content; Z.W.: providing theoretical support and case studies, participating in discussions; M.T: providing theoretical support, participating in discussions; X.Y: revise portions of the manuscript. All authors reviewed the manuscript.

Funding

This work was supported by the Open Fund of State Key Laboratory of Geohazard Prevention and Geoenvironment Protection, Project Code: SKLGP2025K030.

Data availability

All data generated or analysed during this study are included in this published article.

Declarations

Competing interests

The authors declare no competing interests.