In this paper, we choose the DES method that is adaptive to a small number of data samples, especially when the data has a trend, and can adapt to the changes in the time series data. The DES method, a time series forecasting technique, is particularly suitable for data with a trend but without seasonality. This method separately predicts level changes and trend changes using two smoothing equations. The updated formulas for the level and trend equations are shown in Eqs. (1) and (2): (1)Lt=αYt+(1−α)(Lt−1+Tt−1), (2)Tt=β(Lt−Lt−1)+(1−β)Tt−1,the level component at time t is denoted as Lt, α represents the smoothing coefficient for the level, Yt corresponds to the actual observation at time t, and Tt−1 indicates the trend component from the previous period. The trend component at time t is calculated as Tt, with β being the smoothing coefficient for the trend. The forecasting formula for the test set is expressed as Eq. (3): (3)y^t+1=Lt+Tt,where, y^t+1 is the predicted value of the test set.

NSGA-II is a multi-objective genetic algorithm proposed by Deb et al. in 2002. In multi-objective optimization, NSGA-II does not rely on explicit weighting or weighted combinations after normalization. Instead, it achieves an “implicit balance” among multiple objectives through the Pareto dominance relationship. As shown in Fig. 4a, for any two solutions A and B, if both the RMSE and MAE of A are smaller than those of B, and the R² of A is greater than that of B, then A strictly dominates B, and B will be eliminated by the algorithm. If A is better in some indicators (e.g., lower RMSE) but worse in other indicators (e.g., higher MAE), then A and B do not dominate each other and will be jointly retained in the Pareto front. Solutions like A and B are Pareto optimal solutions. The goal of the multi-objective optimization algorithm is to find these Pareto optimal solutions [36].

As shown in Fig. 4b, when combining the generated offspring population with the parent population to form a new population, in addition to considering the non-dominated rank, the crowding distance also needs to be considered within each non-dominated layer. Each objective function will be normalized before calculating the crowding distance. The calculation of all crowding distance values is the sum of the absolute values of the differences between adjacent individuals of each individual across all objective functions. Individuals with larger crowding distances are more likely to be retained, such as Z1, Z2, and the trimmed Z3. This can prevent the individuals in the population from being overly concentrated in the objective space, ensuring that the algorithm evenly searches for the optimal solution among multiple objectives, thereby achieving a balance among multiple objectives.

Solving for the Pareto optimal solution for trend displacement. Using NSGA-II to optimize the variables α and β of the DES method. To enhance the model’s performance on new datasets, objective functions are established to minimize three evaluation metrics on the test set: f1(α,β)=RMSE, f2(α,β)=MAE, and f3(α,β)=−R2. The decision variables are the two parameters, α and β, in the DES model. The multi-objective problem is described as follows: (4)min{Objective1=f1(α,β)Objective2=f2(α,β)Objective3=f3(α,β) (5)s.t.{0<α<10<β<1where, “min” means minimizing the function, (6)f1(α,β)=RMSE=1n∑i=1n(Yt+i−y^t+i)2=1n∑i=1n(Yt+i−(Lt+Tt))2, (7)f2(α,β)=MAE=1n∑i=1n|Yt+i−y^t+i|=1n∑i=1n|Yt+i−(Lt+Tt)|, (8)f3(α,β)=−R2=−(1−∑i=1n(Yt+i−y^t+i)2∑i=1n(Yt+i−Y¯)2)=−(1−∑i=1n(Yt+i−(Lt+Tt))2∑i=1n(Yt+i−Y¯)2),where Yt+i is the actual value, y^t+i is the predicted value, Y¯ is the mean of the actual values, and n is the sample size, all of which belong to the test set. The hyperparameter settings for NSGA-II are shown in Table 3.

By manually selecting features for three baseline models (RBF neural network, BiLSTM neural network, GRU neural network), a feature dataset for periodic displacement is constructed to prepare for the model validation of the proposed network, which does not require manual feature selection.

(1)

Feature selection methods

In order to enhance the baseline model’s prediction accuracy, we evaluated the linear and dependency relationships between influencing factors and periodic displacements in the extended dataset using the MIC and Pearson correlation coefficient [37]. As the final dataset for periodic displacements, factors with Pearson’s correlation coefficient absolute values and MIC values of at least 0.12 and 0.4 were chosen. (2)

BiLSTM/GRU/RBF network architecture

BiLSTM/GRU neural network. BiLSTM networks, which are generated from enhancements to LSTM networks. In BiLSTM, the LSTM network is constructed as two oppositely oriented LSTM layers (Fig. 5a), which deal with the forward and reverse directions of the sequence data. LSTM is a model for optimizing Recurrent Neural Networks (RNN) [38], with a total of three computational mechanisms, namely, forgetting gates, input gates, and output gates (Fig. 5b). Each LSTM recurrent unit has an external state of the previous moment and the current moment’s input as the unit input. The internal structure of GRU is more concise than LSTM (Fig. 5c). It consists of an update gate that controls the retention of old information and the introduction of new information, and a reset gate that determines how much of the old information has been forgotten (Fig. 5d).

RBF neural network. The RBF neural network is a 3-layer feed-forward network with a single hidden layer and its action function is a Gaussian basis function [39]. The j-th neuron of the hidden layer is calculated in the following manner: (9)hj=exp⁡(−‖x−cj‖22bj2),j=1,2,…where, cj=[cj1,…,cji] and b=[b1,…,bi]T represent the center vector and the width vector of the j-th hidden layer neuron of the RBF network. The output of the RBF neural network is given by Eq. (10). (10)yj(t)=ω1h1+ω2h2+…+ωjhj

(1)

Attention mechanism

Global Attention Mechanism. The global attention technique is implemented by using the popular Keras core layer (Dense). The specific steps include: input transformation, weight calculation, normalization, and application of weights. The Dense layer takes all the transformed hidden states of BiLSTM as input, calculates a raw attention score for each time step, and then converts it into normalized attention weights through the softmax function. This approach dynamically generates independent attention weights Ghwi for the hidden states at each time step by attending to the entire input sequence data.

Local Attention Mechanism. As shown in the shaded part of Fig. 6a,b, the cumulative displacement and periodic displacement of monitoring stations ZG118 and ZG93 synchronously show an upward trend, and the change is obvious. While the cumulative displacement maintains a relatively stable stage, the periodic displacement shows a decreasing trend. From this analysis, the short-term rapid increase of landslide cumulative displacement primarily originates from the increase in periodic displacement, so we must draw the model’s attention to the increasing trend while predicting the periodic displacement. As per the principle of change, this paper proposes a two-way local attention mechanism applied to BiLSTM, designing local attention weights for the forward and reverse sequences of the periodic displacement.

(a) Window size for the local attention mechanism

The window size of the local attention mechanism is fixed to 2-time steps (forward sequence: focus only on the current month displacement and the preceding month displacement in the forward periodic displacement sequence; reverse sequence: focus only on the current month displacement and the next month displacement in the reverse periodic displacement sequence), and the window sliding step size is one month.

(b) Weights of the local attention mechanism

Since the periodic displacement shows a notable increasing trend in the short term and to prevent data leakage, the historical rate of change of the periodic displacement is adopted as the local attention weights. To convert the final attention scores to the range between 0 and 1 and focus the local attention mechanism on periodic displacement sequences with an increasing trend, ending with the application of the sigmoid function. The attention weights for the forward and reverse periodic displacement sequences are denoted as wFi and wRi, as shown in Eqs. (11) and (12): (11)wFi={0,i=0sigmoid((PFi−PF(i−1))/1),i≠0(i=0,1,2,…,n) (12)wRi={0,i=n−1sigmoid(−(PR(i+1)−PRi)/1),i≠n−1(i=0,1,2,…,n)where, PFi and PRi represent the forward and reverse periodic displacements at the time step i.

Apply the forward attention weights to the hidden states of the forward LSTM layer and the reverse attention weights to the hidden states of the reverse LSTM layer (Eqs. (13) and (14)). Then, concatenate the weighted hidden states of the forward and reverse directions at corresponding time steps to obtain the hidden states of the BiLSTM after the application of bidirectional local attention weights, as shown in Eq. (15), Fig. 7:(13)hwFi=hFi⋅wFi (14)hwRi=hRi⋅wRi (15)Lhwi=hwFi⊕hwR0

In summary, the global-local hybrid attention mechanism is simultaneously applied to the weighted concatenated hidden states following the BiLSTM hidden states, as shown in Eq. (16): (16)bhi=Ghwi+Lhwi

(2)

Loss function

The PI is an important concept in statistics, which provides a range to estimate the possible values of future observations. To construct a PI, the mean of the predicted values is used as the center of the interval, and the standard deviation of the residuals is considered to reflect the variability of the data. In the study, the PI is built on the foundation of the normal distribution, with a confidence level set at 95%. The mathematical formula for the PI is Eq. (17): (17)PI=y^±Z1−α⋅Spred,where, y^ is the model’s predicted value, Z1−α is the critical value of the normal distribution, and Spred is the standard error of the prediction. From Eq. (17), the width of the PI WPI′ is derived as: (18)WPI′=(y^+Z1−α⋅Spred)−(y^−Z1−α⋅Spred)=2⋅Z1−α⋅Spred

Construct a loss function that uses the width of the PI as a penalty term, to penalize the width when the model’s predicted value deviates from the observed value. Therefore, this paper replaces the model’s predicted standard error Spred with the batch Mean Absolute Error (MAEbatch), and the improved width of the PI is: (19)WPI=2⋅Z1−α⋅MAEbatch

The final loss function is as in Eq. (20):(20)Loss=MSE+λ⋅WPI,where, λ is a hyperparameter that balances the weights of the MSE and WPI. (3)

Periodic displacement prediction based on BiLSTM-RBF-Attention

Fig. 8 shows the structure of the proposed model, which directly takes the original time series data as its input (ZG118: reservoir level and historical displacement; ZG93: monthly cumulative rainfall, reservoir level, and historical displacement). All of the weighted hidden states bhi of BiLSTM are used as inputs to the RBF neural network. The “expand_dims” operation in TensorFlow is used to add the necessary dimensions to the input of the RBF to integrate the weighted hidden states of the BiLSTM. The specific process includes: First, obtain the weighted hidden states of the BiLSTM as the input tensor “inputs”, whose shape is (batch_size, time_steps, 2LSTM_units). Second, add a single dimension after the feature dimension through “tf.expand_dims (inputs, -1)” to convert it to “(batch_size, time_steps, 2LSTM_units, 1)”. Then, broadcast the RBF center tensor “centers” with the shape of “(2LSTM_units, RBF_neurons)” to match the shape of the input tensor, broadcasting it to (1, 1, 2LSTM_units, RBF_neurons). After broadcasting, the shapes of the two are compatible, enabling element-wise difference calculation. By using the method of taking the output of BiLSTM as the input of RBF, the effective integration of the two models is achieved.

Model parameter settings: the model was trained in CPU mode on Intel (R) Iris (R) Xe Graphics integrated graphics card, implemented using TensorFlow 2.15.0 and keras framework, and the programming language was Python (Python 3.9). After experimental testing, the training parameters of BiLSTM-RBF-Attention were finally set as follows: epochs = 200, learning rate (Adam optimizer) = 0.001, batch = 2, LSTM_units (ZG118) = 64, LSTM_units (ZG93) = 32, RBF_units (ZG118) = 96, RBF_units (ZG93) = 64, and the loss function penalty coefficients λ are 1×10−5 and 1×10−4. Setting independent random seeds for the data characteristics of different landslide monitoring points, at ZG118: (NumPy: 0, TensorFlow: 1); at ZG93: (NumPy: 6, TensorFlow: 8).

Fig. 12a,b shows the decomposition results of the reservoir level and monthly cumulative rainfall. The extended dataset for the baseline model is obtained from these results (see Table 5).

The feature selection processes using Pearson and MIC are presented in Figs. 13a,b and 14a,b for the extended datasets for monitoring points ZG118 and ZG93. In summary, the final dataset index for monitoring point ZG118 were determined as Pearson-selected features: 9, 15, 12, 6, 18, 13 and MIC-selected features: 3, 2, 23, 17, 1, 10, 13. while for ZG93, the corresponding index were Pearson: 5, 4, 6, 2, 3, 9, 21, 22, 7, 18, 20, 13 and MIC: 17, 13. Consequently, the surface displacement and deformation of the ZG118/ZG93 landslide monitoring site are influenced to varying degrees by the fluctuation components of the characterisation variables. For the ZG118/ZG93, feature factor 13 (Reservoir level IMF3) is chosen in both feature selection techniques, demonstrating both a complex nonlinear connection and a high linear correlation with the periodic displacement. It has more stability as a baseline model feature factor.

The results of the periodic displacement prediction for the ZG118 and ZG93 (Fig. 15a,b) demonstrate that the BiLSTM-RBF model proposed in this study, which integrates a global-local hybrid attention mechanism, significantly outperforms baseline models requiring manual feature selection. As shown in Fig. 16a,b, the BiLSTM-RBF model achieved the lowest RMSE (ZG118: 10.45 mm; ZG93: 10.16 mm) and MAE (ZG118: 7.46 mm; ZG93: 9.15 mm) values and the highest R² (ZG118: 0.91; ZG93: 0.90) scores at both monitoring points (the “None” option in feature selection methods indicates the direct use of monthly cumulative rainfall, reservoir water levels, and historical displacement as inputs). Notably, after feature selection based on the Pearson and MIC, the RBF neural network exhibited superior predictive performance compared to predictions using raw data inputs.

Among baseline models, the BiLSTM and RBF models showed better prediction accuracy for ZG118 than for ZG93 after feature decomposition. Furthermore, analysis of the ZG118 prediction results revealed that although RBF-MIC and BiLSTM-Pearson demonstrated higher accuracy, their evaluation metrics remained inferior to those of the proposed model. For ZG93 with a smaller sample size, baseline models exhibited suboptimal predictive performance even after feature selection, while the proposed model consistently achieved the best and most stable results. Moreover, the PIs of ZG118 and ZG93 both meet the expected coverage requirements for the actual data.

As shown in Fig. 17a,b, at the ZG118 and ZG93 monitoring points, compared with the RBF, BiLSTM, and GRU series models, the median error of BiLSTM-RBF is significantly lower and the dispersion is smaller (in the box-plot, the box of BiLSTM-RBF is the shortest and the whiskers are the narrowest), which proves that its performance advantage is not due to random fluctuations but a stable effect brought by the model structure (hybrid attention, bidirectional time-series modeling, and RBF fusion).

In order to verify the effectiveness and role of the introduced components, as well as the impact of different activation functions on local attention, ablation experiments were conducted on the Baishuihe ZG118 and ZG93 datasets. To ensure that the comparison between ablation experiments is only caused by the difference of the target variable, and all experiments use the same random seed as the main model. In Table 6, “✓” and “❌” respectively indicate whether the component is enabled, and each row gives the corresponding indicator.

The results in Table 6 indicate that: (1)

Whether attention is introduced or not, the fusion of BiLSTM and RBF is superior to single BiLSTM or RBF, proving the effectiveness of this structure itself.

In summary, it shows that the global attention mechanism effectively captures the long-term dependency relationship of landslide displacement, while the bidirectional local attention mechanism accurately extracts short-term local features.