A New Hybrid Method for Establishing Landslide Displacement Point Forecasting, Interval Forecasting and Probabilistic Forecasting

: In addition to the inherent evolution trend, landslide displacement contains strong fluctuation 11 and randomness, the omni-directional landslide displacement prediction is more scientific than single 12 point prediction or interval prediction. In this work, a newly hybrid approach composed of double 13 exponential smoothing (DES), variational mode decomposition (VMD), long short-term memory 14 network (LSTM) and gaussian process regression (GPR), was proposed for point, interval and 15 probabilistic prediction of landslide displacement. The proposed model includes two parts: (i) predicting 16 the inherent evolution trend of landslide displacement by DES-VMD-LSTM; (ii) evaluating the 17 uncertainty in the first prediction based on the GPR model. In the first part, DES is used to predict the 18 trend displacement, VMD is used to extract the periodic and stochastic displacement from the residual 19 displacement, and then LSTM is used to predict them. The triggering factors of periodic and stochastic 20 displacement are screened by maximum information coefficient (MIC), and the screened factors are 21 decomposed into low-and high-frequency components by VMD, to predict periodic and stochastic 22 displacement respectively. The first cumulative displacement prediction results are achieved by adding 23 the predicted trend, periodic and stochastic displacement. By setting the first predicted displacement as 24 input and actual displacement as expected output, the point, interval and probability prediction of 25 displacement are realized in GPR model. The plausibility of this method was validated firstly with the 26 data from Bazimen (BZM) and Baishuihe (BSH) landslide in the Three Gorges Reservoir area. This 27 model has potential capacity to realize deterministic prediction of displacement and exhibit uncertainty 28 contained in displacement. A comparing study shows that this method has a high performance at point, 29 interval and probability prediction of displacement. 30

decomposition, long short-term memory network; gaussian process regression 1. Introduction Landslide is a geological phenomenon of slope rock and soil sliding along the through shear failure surface, which is combining results of different environmental factors such as topography, lithology, hydrology and human activities, etc.The landslide not only lead to environmental deterioration, but damage the infrastructure and cause casualties.The Three Gorges Reservoir area is one of the areas prone to landslides, over 4200 landslides were recorded there (Yin et al., 2010).In this area, the extremely unstable hydrological environment, including the excessive rainfall and periodic reservoir water level, leads to frequently landslide disasters.For landslide, the displacement evolution is often used as an indicator to predict the behavior of failure process.High precision prediction can greatly reduce the loss brought by landslide disasters.
Accurate prediction of landslide displacement is very challenging, as the failure process of landslide is dynamic, nonlinear and stochastic (Cai et al., 2016).To date, there are three main approaches for landslide displacement prediction, involving physical model, statistical model and intelligent model (Ma et al., 2017).In the physical model, the displacement is predicted by using the model based upon creep theory and specific physical properties of geotechnical materials, for instance tertiary creep (Saito, 1965), Hayashi model (Setsuo et al., 1988) and Fukuzono model (Fukuzono, 1985).Statistical models use to predict landslide displacement with principle of time series analysis, like exponential smoothing model (Gould et al., 2008), autoregressive integrated moving average (Carlà et al., 2016), etc. Intelligent model developed on the basis of machine learning or deep learning algorithm is a promising method for landslide displacement prediction, such as back propagation neural network (BP) (Mayoraz and Vulliet, 2002), support vector regression (SVR) (Feng et al., 2004), extreme learning machine (ELM) (Lian et al., 2014).Recently, many hybrid models have been developed to improve the prediction accuracy of landslide displacement (Wen et al., 2017;Zhou et al., 2016).
The aforementioned methods mainly focus on how to obtain accurate point prediction, while ignore the uncertainty accumulated in landslide displacement prediction.Landslide displacement prediction contains various uncertainties, including measurement error, systematic error, inherent randomness, model uncertainty and so on (Ma et al., 2018).These uncertainties will weak the prediction accuracy in traditional point prediction model.Therefore, it is necessary to establish the probability prediction model for landslide displacement.The pre-existing probabilistic prediction models of landslide displacement are established dominantly based on the prediction interval (PI) technology.For example, Lian et al. (Lian et al., 2016a) designed a PI model on foundation of switched neural network for landslide displacement prediction, by using bootstrap, kernel based extreme learning (KELM) machine and artificial neural network (ANN).Their follow-up study replaced KELM with random vector functional link networks (RVLFNs) to improve the quality of PI (Lian et al., 2018).Furthermore, by combining the lower and upper bound estimation (LUBE) model with other methods, a PI model is established to explain the uncertainty in landslide displacement prediction (Lian et al., 2016b;Wang et al., 2019).
Although these models based on PI have been successfully applied to landslide displacement prediction, there are still some shortcomings.For example, the PI models based on bootstrap technology need massive resampling process, which significantly rise the computing complexity and the data volume.
Moreover, these models focus mainly on the uncertainty of landslide displacement prediction namely interval prediction, while ignores the point prediction, which could not give a complete probability prediction result.
Gaussian process regression (GPR) is a new machine learning method based on Bayesian theory and statistical theory, which is suitable for complex regression problems such as high dimension, small sample size and nonlinear (Anjishnu et al., 2013).GPR performs well in landslide displacement prediction in different fields (Liu et al., 2014;Liu et al., 2012).Unlike the traditional method, GPR can directly predict the distribution of target sequence, wherein the point prediction, interval prediction and probability prediction results can be achieved at the same time.Since GPR focus major on the error of prediction subject, it alone is difficult to obtain ideal prediction during landslide displacement prediction.
Combining with other methods could improve its unique probabilistic prediction characteristics.
The total landslide displacement curve in the Three Gorges Reservoir area contains trend, periodic and stochastic components (Huang et al., 2016;Li et al., 2012;Lian et al., 2015).To achieve highprecision prediction results, decomposing the total displacement into these three components and predicting individually is necessary.The combination of DES and VMD is able to decompose superior time series.DES can decompose landslide displacement into linear and nonlinear components.The linear component is trend displacement that can be predicted directly based on historical displacement, The nonlinear component including periodic and stochastic item are accurately separated with the help of VMD, and are predicted independently by using LSTMs.The LSTM is widely favored by researchers because of its high performance in dealing with time series problems.The LSTM model has a excellent performance in landslide displacement prediction (Xu and Niu, 2018).
In this paper a new landslide displacement prediction model DES-VMD-LSTM-GPR is proposed.
The main research contents are as follows: (1) establishing a more accurate point prediction model based on DES-VMD-LSTM; (2) introducing a GPR model to assess the uncertainty in the first prediction, as well as to achieve interval and probability prediction; (3) the proposed method is applied to study BZM and BSH landslide, and its effectiveness is validated through comparing with different prediction methods.

Double exponential smoothing
The DES is an important method of time series prediction, which can accurately extract the trend characteristics of time series (Holt, 2004).The basic idea of DES that all historical data have an impact on the forecast data, but the recent data has a greater impact than older data, and the influence changes geometrically with time (Wu et al., 2016).The three main formulas of DES are as follows: Where   and   represent stable and trend component of series at time  ;   is the is the observation; α and β are smoothing parameters within [0,1] ;  + is the prediction value at time  + , where m = 1.

Variational mode decomposition
The VMD is an adaptive, completely non recursive modal decomposition and signal processing method, which can decompose the time series with high complexity and strong nonlinearity into the specified number of intrinsic mode functions (IMFs) (Dragomiretskiy and Zosso, 2014).The core idea of VMD is to construct and solve variational problems.Firstly, the variational problem is constructed, i.e., assuming that the original sequence  is decomposed into  specified IMF components, each IMF sequence is guaranteed to have a central frequency and a finite bandwidth, and the sum of estimated bandwidth for IMF is the minimum, and the constraint condition is that the sum of all IMF is equal to the original sequence , then the corresponding constraint variational expression is as follows: Where  is the number of specified mode decomposition, {  } = { 1 ,  2 , ⋯ ,   } and {  } = { 1 ,  2 , ⋯ ,   } the  mode components IMF and their corresponding central frequencies after decomposition, () is the Dirac delta function.
To solve Eq. ( 2), the Lagrange multiplication operator λ is introduced to transform the constrained variational problem into the unconstrained variational problem, and the expression is as follows: Where α is the quadratic penalty factor, which can reduce the interference of gaussian noise.By using the alternating direction multiplier iterative algorithm combined with Parseval/Plancherel and Fourier equidistant transform, the   ,   and λ are obtained by alternating optimization iteration: (5) Where  the noise tolerance, which meets the fidelity requirements of signal decomposition,  ̂ +1 () ,  ̂() ,  () and λ () correspond to Fourier transform of   +1 () ,   () , () and λ() respectively.

Maximum information coefficient
Maximum information coefficient (MIC) is an important unsupervised feature extraction method, which can be used to reveal the degree of association between two random variables, including linear and non-linear relationship, which has the characteristics of breadth and fairness (Reshef et al., 2011).
Assuming that there are two random variables A = [  ,  = 1,2, ⋯ , ] and B = [  ,  = 1,2, ⋯ , ] , where  is the sample number.Then, the MIC between A and B is calculated as follows: (1) The data of A and B are taken out to form data set D = [(  ,   ),  = 1,2, ⋯ , ], and the data set D is sorted in a certain order; (2) The data set D is mapped to a two-dimensional plane.(4) Find out the maximum value max (|  ) of MI in all cases of partition, let  ′ (, , ) = max (|  ), and standardize it as follows: According to ()  , the MIC of random variables A and B can be obtained as follows: When MIC() = 0, it means that variable A and B do not depend on each other.On the contrary, when MIC() value is close to 1, there is a strong linear or nonlinear correlation between these two variables.

Long short-term memory network
The LSTM is a variant of recurrent neural network (RNN) proposed by Hochreiter and Schmiduber (Hochreiter and Schmidhuber, 1997).By improving the traditional RNN, LSTM solved the problem of vanishing and explosion gradient of RNN and realized the long-term memory of information.Due to its excellent performance in time series, LSTM has been successfully applied in many fields, such as speech recognition, natural language processing, wind power prediction and so on (Devi and Thongam, 2020;Yu et al., 2019;Zhang et al., 2019).
where   ,   ,   represents the weights;   ,   ,   are their corresponding bias values; ℎ −1 are output value of the memory cell at the previous moment;   are input value at the current time;   ,   ,   are the values of the input, forget, and output gates, respectively.

Gaussian process regression
GPR is an uncertainty model, which describes the regression problem from the perspective of probability (Carl Edward and Williams, 2005).Different from the traditional deterministic regression model, GPR directly models the function () and obtains the distribution of function ().Therefore, GPR can explore more possible regions of data than traditional models, so as to solve the problem of interval and probabilistic prediction.The general model of regression and prediction problem is shown as follows: where  is the observation with noise and  is the input eigenvector,  is the observation noise with gaussian distribution,   2 is the noise variance and   is the unit matrix.
GPR can accurately describe () by mean function () and kernel function (covariance function) (, ) after learning observation data.The expression is as follows: Where: () is a random variable composed of multi-dimensional gaussian distribution.According to the properties of multi-dimensional gaussian distribution, the prior distribution of observation  and the joint distribution of observed value  and predicted value  can be obtained by Eq. ( 13), ( 14) and ( 15), as follows: In general, the mean is subtracted when GPR data is preprocessed, that is, () = 0.In Eq. ( 16) and ( 17 Where: In Eq. ( 19),  ̅ is the mean value of predicted value y corresponding to test set  * ,   2 is the variance of y.The interval prediction results correspond to 95% confidence level, i.e.
The expression of probability prediction result for i-th prediction value   is as follows: According to the above process, it can be found that the kernel function plays a decisive role in the prediction performance of GPR.Common kernel functions include Radial-basis function kernel (RBF), Matern kernel, Rational quadratic kernel, Exp-sine-squared kernel, etc.The expression of RBF kernel is as follows,  1 and  2 are the hyperparameters to be optimized.in the sliding area of BZM.In the past few years, a large number of displacement monitoring data have been collected through these GPS stations.These data were collected and stored in the National Cryosphere Desert Data Center (Wu, 2016;Wu and Haifeng, 2016).Fig. 3 showed the displacement, rainfall and reservoir level data for BZM from January 2007 to December 2012.According to the displacement curve, it can be found that BZM landslide are still in unstable state and the deformation is increasing continuously during the entire monitoring period.The deformation of BZM landslide increases in stepwise, which is relate to the periodic change reservoir water level and rainfall.The displacement of the four monitoring points increased rapidly from May to July, which was physically consistent with the annual flood period and reservoir water level falling time.On the contrary, the displacement of BZM changes little in non-flood and reservoir impoundment period, and their deformation tends to be stable.variables (I1-I12) including the maximum continuous precipitation, cumulative precipitation and reservoir level average elevation of the current month, the previous month and the first two months are selected as candidate factors, as shown in Table 1.To screen the trigger factors from these candidate factors, the MIC value between candidate factors and the target displacement is calculated.For periodic displacement, MIC value greater than 0.4 is used as trigger factor, and for stochastic displacement, MIC value greater than 0.3 is regarded as trigger factor.
Finally, I1, I2, I4, I5, I9, I10, I11, I12 are chosen for the periodic displacement prediction task, and I1, I2, I4, I6, I7, I9, I10, I11, I12 are chosen for the stochastic displacement prediction task.(1) Collecting landslide displacement monitoring data (Y); (2) Prediction the trend displacement (T) based on DES; (3) Decomposing periodic displacement ( P ) and stochastic displacement ( S ) from residual displacement based on VMD; (4) Screening triggering factors based on MIC, and decomposition these factors by VMD; (5) Establishing periodic and stochastic displacement prediction models based on LSTM; (6) The first cumulative displacement prediction result is obtained by summing the prediction results of trend, periodic and stochastic displacement; Finally, the GPR model is established to realize the point, interval and probability prediction of landslide displacement by taking the predicted results of the sixth step as the input feature and the actual monitoring displacement as the expected output.

Point prediction metrics
The root mean square error (RMSE) and coefficient of determination ( 2 ) are used to assess the performance of point predict accuracy, which can be expressed by the following: Where Te is the sample number of test set,   is predicted value,   and  ̅  are actual monitoring value and its mean value, respectively.

Interval prediction metrics
Two common indexes are used to evaluate the interval prediction quality of GPR model, including coverage probability (PICP) and mean prediction interval width (MPIW).The expression is as follows: In Eq. ( 26), ( 27) and ( 28),    and    represent the upper and lower limit of the prediction interval at  time step under the confidence level of ;   is a Boolean variable, which represents the number of observations falling into the prediction interval [   ,    ].
In addition to PICP and MPIW, another comprehensive index average interval score (AIS) is used to measure the quality of the forecast interval (Winkler, 1972).The interval score (IS) of prediction interval at time  is defined as follows: Where    is the width of the prediction interval at time : Thus, the AIS is calculated as follows:

Probabilistic prediction metrics
Continuous ranked probability score (CRPS ) is an effective index to evaluate the accuracy of probabilistic prediction (Alessandrini et al., 2015).CRPS can be defined as follows: Where (  ) and (  ) are the probability density function and cumulative distribution function

Periodic and stochastic displacement prediction
Due to the results of each run are slightly different, the averaged value of run 5 times is taken as the final results.Fig. 8 shows the prediction results of periodic and stochastic displacement based on LSTM, GRU, SVR and BP.Overall, the prediction effect is considerable.Table 3 listed the prediction accuracy and running time of the four methods.The RMSE and R 2 values of these predictors are very close, which demonstrate that the four methods are highly competitive.Considering all evaluation criteria, the dynamic approaches (LSTM and GRU) is higher accuracy than that of static approaches (BP and SVR), as the memory function make LSTM and GRU more professional in dealing with time series problem.
Besides, the prediction accuracy of GRU is slightly higher than LSTM, because GRU is extended from LSTM through combining forget gate and input gate, and the model parameters are reduced by about 1/3.
Although the prediction accuracy obtained by the dynamic algorithm is high, it is at the expense of time.
Table 3 shows that the running time of LSTM for periodic displacement prediction is about 29 times that of SVR and 8 times that of BP.Another phenomenon can be found from Table 3, that is, the prediction accuracy of periodic displacement is higher than that of stochastic displacement, and the prediction stability of stochastic displacement is poor.The possible reasons could be: (1) Stochastic displacement is affected by many factors, such as human activities, observation error, temporary reservoir water level regulation, nonseasonal rainfall, etc. which are not entirely considered in this work; (2) Stochastic displacement contains many strong randomness and uncertainty, the existing static or dynamic algorithms could not capture the potential laws accurately.

Point prediction
The cumulated displacement point prediction include two parts, the first is the sum of trend, periodic and stochastic component prediction, and the second is GPR modeling prediction.As illustrated in Fig. 9(a) and (b), these two prediction results are basically comparable.The evaluation criterions in Fig. 9 (c) displays that the accuracy of GPR second prediction is slightly improved, but it is not apparent, which means that GPR will slightly improve the point prediction performance of the cumulative displacement.The point prediction and interval prediction results are compared with the calculation results in previous literature to verify the effectiveness of the proposed method.For point prediction, the proposed method is compared with GA-SVM (Zhou et al., 2016), ABC-ELM (Yan et al., 2019), LSTM (Yang et al., 2019) and PSO-SVR (Zhang et al., 2015).The prediction metrics are listed in Table 7.The LSTM based predictor performs better than other methods, and the prediction accuracy of the proposed method and LSTM in Yan et al. ( 2019) is almost the same, which are mainly attributed to the unique time processing advantages of LSTM.In the architecture of LSTM, the forget gate can control how much information at the previous time is used as the input of the current time, and it can make full use of historical information to learn rules.For static models like SVM and BP, there is no connections in different time steps, and rules can only be learned from one time point, and the response between output and input cannot be fully learned.Another reason why the proposed method has high prediction accuracy is the reasonable trigger factor selection.Reservoir level and rainfall are the dominant triggering factors of reservoir bank slope deformation.In traditional predictors, a pseudo periodic displacement combining the periodic and random displacement is predicted according to rainfall and reservoir level factors.These method neglects the periodic and random characteristics of trigger factors.In this work, the periodic and stochastic displacement are accurately separated from the residual displacement by using VMD model, and the trigger factor is also decomposed into low-and high-frequency components, which have similar trend respectively with periodic and stochastic displacement (Fig. 6), which is the key to improve the prediction performance of the model.
The interval prediction feasibility of the proposed approach is assessed by comparison with other four methods: DES-PSO-ELM (Wang et al., 2019), SBS-RVFLN (Lian et al., 2018), SB-KELM (Lian et al., 2016a) and Bootstrap-KELM-BPNN (Li et al., 2019a).Three aspects need to be discussed: (1) all the methods show satisfactory coverage probabilities.The PICP values of most methods are 100%, except for DES-PSO-ELM and Bootstrap-KELM-BPNN (approaching or exceeding 95% confidence).These results indicated that all models are reliable and have good extrapolation prediction ability in landslide displacement interval prediction.(2) Regarding the interval prediction, the proposed method performs better or similarly to the other methods in the two applications.It is known from Table 5, the smaller the absolute value of AIS indicate the narrower width of prediction interval.The AIS absolute value of the proposed method is higher than DES-PSO-ELM and SBS-RVFLN but lower than SB-KELM and Bootstrap-KELM-BPNN, which shows that the proposed method tends to establish a narrower prediction interval, namely better interval prediction quality.
(3) Compared with other methods, the proposed method is simple and can obtain more comprehensive prediction information.As the SBS-RVFLN, SB-KELM and Bootstrap-KELM-BPNN is generated based on bootstrap technology, it requires at least B times of repeated sampling calculation (B = 50 in SB-KELM ).The prediction accuracy of SBS-RVFLN and SB-KELM also depends strongly on the accuracy of K-means classifier.These models are too complex and time-consuming.Although DES-PSO-ELM is simple, in which only single interval prediction can be obtained.only the deterministic prediction of landslide displacement can be realized, but also the variability and uncertainty related to landslide displacement prediction can be estimated.
Although the DES-VMD-LSTM-GPR model achieved a good performance in landslide displacement prediction, it has some drawbacks.As a deep learning algorithm, LSTM needs to adjust more model parameters than the classical machine learning algorithm, and has a strong dependence on the amount of data.If the amount of training data is insufficient, the prediction accuracy will be affected.
Landslide displacement monitoring is based on month.It takes years or even decades to obtain a large amount of data.Therefore, the method of developing larger datasets in a limited monitoring cycle should be considered in the later stage.Another drawback is the gaussian distribution assumption of GPR model.
The landslide displacement has strong volatility and randomness.The gaussian distribution assumption may not be the optimal distribution.More distributions need to be tested in the future, such as Weibull, Rayleigh and Beta distribution.

Conclusions
Landslide displacement prediction is the key of landslide early warning ability.To overcome the

Fig 1 .
Fig 1.(a) LSTM network; (b) LSTM cell structure The reason why LSTM can deal with the long-term dependencies problem is that it is equipped with several gates to control the magnitude based on the standard RNN, and these gates can control the flow of information.As shown in Fig. 1(b), the input gate controls how much input information can enter the memory cell at the current time, and the output gate controls the output information entering other cells or as the final result.The forget gate controls the retention degree of data information from the previous moment.Three gates interaction controls memory cell output status.The functions of the three gates are defined as follows:

2. 2
Materials2.2.1 Bazimen landslideBazimen landslide (BZM) is located at the estuary of the right bank of Xiangxi River, a tributary of the north bank of the Yangtze River in Guizhou town, Zigui County, 31 km away from the Three Gorges dam (Fig.2(a)).The BZM landslide mass is dustpan-shaped with the west higher than the east, and the main sliding direction is about N110°.The plane shape of the landslide is irregular fan-shaped, with homologous gullies developed on both sides of the landslide, and the back edge is in the shape of an armchair.SeeYang et al. (2019) for more information.

Fig 2 .
Fig 2. Geographical location and topographic map of Bazimen landslide (BZM) In order to monitor the displacement of bank landslide in real time in Three Gorges Reservoir area,

Fig 3 .
Fig 3. Displacement monitoring curve of Bazimen (BZM) landslide2.2.2 Landslide displacement decompositionThe landslide displacement curve is a non-stationary time series that increases obviously with time.Many researches have shown that the landslide displacement curve can be divided into three components, namely trend component T(t), periodic component P(t) and stochastic component S(t), as shown in Eq. (17).Among them, the T(t) increases monotonically with time and is controlled by lithology, topography and other factors; the P(t) is approximately a periodic function with time, which is affected by seasonal rainfall and periodic regulation of reservoir water level; the S(t) is a near white noise sequence, which is affected by wind load, non-seasonal rainfall and temporary regulation of reservoir

Fig 4 .
Fig 4. The decomposition results of trend displacement and residual displacement based on DES 2.2.3 Triggering factor selection and decomposition Selecting appropriate triggering factors is the precondition for high prediction accuracy.First, the candidate factors are determined, and then the triggering factors are filtered from the candidate factors according to the MIC value.As described in section 2.2.2, the deformation of landslide is mainly related to historical deformation, rainfall and reservoir level fluctuation.Furthermore, Fig 2 indicated that the rainfall and reservoir level factors exert a hysteresis effect on landslide displacement.Therefore, 12

Fig 5 .
Fig 5. Factor I1-I12 VMD decomposition.(a): low frequency component, (b): high frequency component Corresponding to the periodic and stochastic displacement, the time series of trigger factors can also be decomposed into two subsequences by VMD method, as shown in Fig 5.The high-frequency component has the characteristics of periodic variation, while the low-frequency component fluctuates with time.Fig. 6 shows the relationship between low and high frequency components of I1 and periodic and stochastic displacement.The variation characteristics of low-frequency component and periodic displacement are generally the same, and the evolution trend of high-frequency component and stochastic displacement is similar.Therefore, this paper will use the high-frequency component of the trigger factor to predict the periodic displacement and its low-frequency component to predict the stochastic displacement.

Fig 6 .
Fig 6.(a) Relationship between low frequency component of factor I1 and periodic displacement; (b) relationship between high frequency component of factor I1 and stochastic displacement.2.3 Prediction process Fig. 7 describes the analytical process of the predict model in the form of a flowchart.The main steps are as follows:

Fig 7 .
Fig 7. Flow chart of landslide displacement point, interval and probabilistic prediction 2.3.1 Method parameter setting DES, VMD, LSTM and GPR are the main components of the hybrid method proposed.To test the effectiveness, BP, SVR and GRU are selected to replace LSTM for comparative analysis.In these of  time step prediction value, respectively; (  −   ) is a Boolean variable.The relatively low values of CRPS indicate high-performance of probabilistic prediction.3. Results and discussion 72 displacement data points were collected from 2007 to 2012, which are divided into two parts: 60 displacement data in 2007-2011 as training set and 12 data in 2012 as testing set.Since the prediction results of trend displacement have been analyzed in section 2.2.2, the following part mainly discusses the prediction results of periodic, stochastic and cumulative displacement.To maximally preserve the distribution of the data, the original data is normalized to [−1,1] using linear normalization for training and prediction.

Fig 8 .
Fig 8. Prediction results of periodic and stochastic displacement based on LSTM, GRU, SVR, and BP

Fig 9 .
Fig 9. (a) First cumulative displacement prediction result; (b) Second cumulative displacement prediction result; (c) Point prediction accuracy metrics of two predictions before and after GPR modeling.3.3Interval predictionThe interval prediction results based on DES-VMD-LSTM-GPR, DES-VMD-SVR-GPR, DES-VMD-BP-GPR and DES-VMD-GRU-GPR method are shown in Fig.10.At the 95% confidence level, it can be found that the prediction interval obtained by the four methods can cover the true displacement well, whether in the training set or in the test set.It proves the feasibility of these methods for landslide displacement interval prediction.

Fig 11 .
Fig 11.The probabilistic prediction results by DES-VMD-LSTM-GPR method.Table 6.The probabilistic prediction metric of four method Metric DES-VMD-LSTM-GPR DES-VMD-SVR-GPR DES-VMD-BP-GPR DES-VMD-GRU-GPR CRPS 7.809 6.431 11.128 9.3713.5Application and comparative analysisTo further test the method, the proposed method is used to analyze the Baishuihe landslide (BSH) from Three Gorges Reservoir area.GPS monitoring point ZG118 in BSH landslide is used to test the model, and the prediction results are shown in Fig.12(testing set: 2012.01-2012.12).Firstly, the prediction quality by the proposed method is very good, whether point prediction (RMSE：7.161；R 2 ： Then, the A is divided into  portion and the B is divided into  portion to form the grid G .For a fixed D , different partition methods can obtain G with different grid number  * , the grid number should satisfy  *  <  0.6 .Based on this, different distribution |  of dataset D can be obtained under For different |  , the corresponding MI, i.e. (|  ), is calculated by Eq. (7), where   (, ) is the joint probability distribution of A and B ;   () and   () are the marginal 

Table 1 .
Candidate factors and its MIC value with periodic and stochastic displacement methods, the parameter values of DES and VMD methods have been determined in Section 2.2.2 and 325 2.2.3, and the main parameters and their values of the remaining five methods are listed in Table2.

Table 3 .
Prediction accuracy of periodic and stochastic displacement

Table 5 .
Interval prediction performance of all studied methodsThe hybrid methods based on GPR can not only provide accurate point prediction and reliable interval prediction, but provide probability prediction results.The Fig.11visualized the probability distribution curve of BZM landslide at different time points.The six probability density curves are very complete, without unordinary value, which indicates that the probability density curves obtained by the proposed method is appropriate.In January, March, May and November, the monitored displacement is close to the center of the curve, which indicates a high prediction accuracy of these points.In July and September, the monitored displacement line is deviated from the center of the curve, caused by increasing prediction error.For the probability prediction of test set, the monitored displacements distribute evenly on both sides of the curve center, which proves the reliability of the probability prediction result.If all monitored displacements concentrate at the center or locate far from the center, it may reduce the reliability of probability prediction.The evaluating indicator of the probabilistic prediction results i.e.CRPS index are summarized in

Table 6 .
The CRPS is used to evaluate the entire probability density distribution curve, including the quality of point and interval prediction, as well as the performance of probability prediction.The CRPS of BZM landslide based on the four methods are similar and ranges 6.431 ~ 11.128.And the CRPS rank are consistent with the results of point prediction and interval prediction.

Table 7 .
Comparisons of point prediction constructed using the proposed method and these methods in

Table 8 .
Comparisons of interval prediction at the 95% confidence level constructed using the proposed method and the method in the previous references Temporal deformation of landslide is a noisy and non-stationary process, which is impacted by many internal and external factors, such as lithology, geological structure, rainfall, reservoir level, monitoring error and so on.Due to the complex nonlinear relationship between various influencing factors and landslide deformation, accurate landslide displacement prediction is very difficult.Different from the previous blindly pursuing high prediction accuracy, this paper combines the respective advantages of LSTM and GPR, and propose a new idea of landslide displacement prediction.Based on this model, not