A bank landslide step displacement prediction method

Abstract

A bank landslide step type displacement prediction method, comprising the following steps: collecting historical time series monitoring data of the target landslide area, and preprocessing the original monitoring data; adopting an optimized sliding t The test method identifies the mutation points in the displacement data, calculates the critical value, constructs the continuous mutation intensity according to the identification result, and further calculates the threshold value; the grey correlation degree analysis method is adopted, the monthly reservoir water level change W2, the monthly rainfall R1 and the mutation intensity characteristic M1 are screened out as key input variables; the accumulated displacement signal is decomposed into a trend component, a periodic component and a random component by using the variational mode decomposition; a long short-term memory network model is constructed to predict each component; the prediction results of each component are superimposed to obtain the final accumulated displacement prediction value, and the model precision is evaluated. The prediction method can improve the prediction precision of the prediction model.

Description

Technical Field

[0001] This invention belongs to the technical field of geological disaster prediction, and specifically relates to a method for predicting step displacement of reservoir bank landslides. Background Technology

[0002] Reservoir bank landslides are among the most common and destructive geological hazards during reservoir operation. Affected by the periodic fluctuations in reservoir water levels and long-term seasonal rainfall, the deformation of reservoir bank landslides typically undergoes two alternating stages: stable deformation and sudden acceleration, exhibiting typical step-type deformation characteristics. This dynamic switching of deformation states introduces significant uncertainty into displacement prediction, posing a major challenge to achieving accurate landslide early warning.

[0003] Currently, prediction models for step-type landslide displacement are mainly divided into two categories: one is based on physical mechanisms, which reflects the mechanical mechanism of landslide deformation by establishing a seepage-stress-displacement coupling equation. However, these models have numerous parameters, are computationally complex, and are difficult to update in real time. The other category is based on data-driven intelligent prediction models, such as artificial neural networks, support vector machines, and long short-term memory networks. Existing data-driven models typically combine signal decomposition methods with deep learning algorithms, first using wavelet transform, empirical mode decomposition, or variational mode decomposition to capture the details of the displacement curve before making predictions.

[0004] However, the aforementioned hybrid models only focus on the algorithm's fitting performance, neglecting the deformation formation mechanism of step-type landslides. Periodic fluctuations in reservoir water levels and seasonal rainfall are the fundamental causes of abrupt deformation changes. Existing models fail to accurately identify and extract information on abrupt deformation change points, resulting in insufficient prediction accuracy and poor generalization ability at critical moments of deformation state transition. Summary of the Invention

[0005] This invention provides a method for predicting step displacement in reservoir bank landslides, in order to solve the problems of poor ability to capture step deformation characteristics and low prediction accuracy of traditional prediction models.

[0006] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is as follows: A method for predicting step displacement in reservoir bank landslides includes the following steps: Step 1: Data Acquisition and Preprocessing: Collect historical time-series monitoring data of the target landslide area and preprocess the raw monitoring data; Step 2, Mutation Point Identification and Feature Construction: Utilizing an optimized sliding... t The test method identifies abrupt change points in displacement data, calculates critical values, constructs continuous abrupt change intensities based on the identification results, and then calculates the threshold. Step 3: Screening of Influencing Factors: The grey relational analysis method was used to quantify the correlation between multiple candidate influencing factors and landslide displacement, and the monthly reservoir water level change W2, monthly rainfall R1 and abrupt change intensity characteristics M1 were selected as key input variables. Step 4, Displacement Decomposition: The cumulative displacement signal is decomposed into trend component IMF1, periodic component IMF2, and random component IMF3 using variational mode decomposition. Step 5: Component Prediction: Construct a long short-term memory network model and predict each component; Step 6, Displacement Reconstruction and Prediction: The prediction results of each component in Step 5 are superimposed to obtain the final cumulative displacement prediction value, and the model accuracy is evaluated.

[0007] Furthermore, in step one, the preprocessing of the original monitoring data includes missing value imputation, moving average denoising, and max-min normalization, and finally the data is mapped to the interval [0, 1].

[0008] Furthermore, in step two, the optimized sliding... t The test method includes two contrasting subsequences of a given displacement-time series. and The optimized formula for calculating the sliding t-test is: ; in, and Representing time points i Displacement values ​​and weights; and The length of the subsequence; ; in, , They represent and The variance.

[0009] Furthermore, step two includes constructing a mutation intensity index. Slide soon t Test results t The statistic is input into the Sigmoid function, resulting in the mutation strength. The formula is as follows: ; in, t For sliding t Test statistic.

[0010] Furthermore, mutation intensity The value range is [0.5, 1), and... Defined as mutation intensity feature M1.

[0011] Furthermore, in step three, the grey relational analysis specifically includes: calculating the grey relational coefficient between each influencing factor and the cumulative displacement, the calculation formula of which is: ; in, The reference sequence is the cumulative displacement. To compare the series, i.e., the various influence factors, The resolution coefficient.

[0012] Furthermore, in step three, the top three influencing factors with the highest correlation values ​​are selected using the grey relational analysis method. These factors are, in order, the monthly reservoir water level change W2, the mutation intensity characteristic M1, and the monthly rainfall R1.

[0013] Furthermore, in step four, the original signal is decomposed using variational mode decomposition. f The corresponding constrained variational problem in the process is: ; in, These are the components obtained after decomposition. The actual center frequencies of each intrinsic mode function component; K The preset total number of modal components; k The index of the modal component; The Dirac function is used; * indicates convolution operation. The analytic signal representing each component; This represents the estimated center frequency of each analytical signal; This is the original cumulative displacement signal.

[0014] Furthermore, in step five, the Long Short-Term Memory (LSTM) network model includes: A univariate LSTM model is used for the trend component IMF1 and the stochastic component IMF3, with the input being the historical values ​​of each component over the previous two months. A multivariate LSTM model is used to predict the periodic component. Its inputs include the monthly reservoir water level change W2, the monthly rainfall R1, the abrupt change intensity feature M1, and the historical values ​​of the periodic component itself for the previous two months.

[0015] Furthermore, in step six, the displacement reconstruction formula is: ; in, For the predicted value of the trend component, For the predicted value of the periodic component, These are the predicted values ​​for the random components; Model accuracy is evaluated using root mean square error, mean absolute error, and mean absolute percentage error.

[0016] The present invention can achieve the following beneficial effects: 1. This invention introduces the optimized sliding t-test method into landslide displacement prediction for the first time. By identifying abrupt change points in the displacement sequence and constructing continuous abrupt change in intensity, it provides key deformation state switching information for deep learning models and improves the ability to capture step deformation.

[0017] 2. This invention organically integrates mutation point identification, variational mode decomposition, and long short-term memory networks to form a complete technical framework of "mechanism feature extraction - signal decomposition - component prediction - displacement reconstruction", which solves the problem that traditional models ignore deformation mechanisms and are difficult to predict step mutations.

[0018] 3. This invention uses grey relational analysis to screen key influencing factors and creatively introduces the mutation intensity feature M1 into the periodic component prediction, enabling the network to directly learn the causal relationship between external hydrological triggering and internal deformation state switching, resulting in high prediction accuracy and strong generalization ability. Attached Figure Description

[0019] The present invention will be further described below with reference to the accompanying drawings and embodiments: Figure 1 This is an overall flowchart of a method for predicting step displacement in reservoir bank landslides according to the present invention; Figure 2 The image shows the results of identifying abrupt change points using the optimized sliding t-test method of this invention. Figure 3 This is a comparison chart of the prediction errors of the model of this invention and the comparative model. Detailed Implementation

[0020] To facilitate understanding of this application, a more complete description will be provided below with reference to the accompanying drawings, which illustrate embodiments of the present application. However, the present application can be implemented in many different forms and is not limited to the embodiments described herein. Rather, these embodiments are provided so that the disclosure of this application will be thorough and complete.

[0021] like Figure 1 As shown, a method for predicting step displacement in reservoir bank landslides includes the following steps: Step 1: Data Acquisition and Preprocessing: Collect historical time-series monitoring data of the target landslide area, including reservoir water level elevation, rainfall and landslide monitoring point displacement data. Perform missing value imputation, moving average denoising and maximum-minimum normalization on the original monitoring data, and map the data to the interval [0, 1].

[0022] Step 2, Mutation Point Identification and Feature Construction: Utilizing an optimized sliding... tThe test method identifies abrupt changes in displacement data, sets significance levels and degrees of freedom to calculate critical values, and constructs continuous abrupt change intensities based on the identification results to calculate threshold values.

[0023] Step two aims to quantitatively extract key information on the transition of landslide deformation states from the original displacement sequence. First, an optimized sliding... t The test method identifies abrupt changes in a displacement sequence. This method detects statistically significant changes by calculating the difference between the means of two adjacent subsequences.

[0024] Optimized sliding t The testing method includes: two comparative subsequences of a given displacement time series. and The optimized formula for calculating the sliding t-test is: ; ; ; ; in, and Representing time points i Displacement values ​​and weights; and The length of the subsequence; This represents the mean of the displacement time series; Z The normalization factor is used to make... ; , and , They represent and The mean and variance; t The statistic satisfies the following degrees of freedom: of t Distribution pattern.

[0025] To ensure high confidence in identifying true deformation mutations and minimize false alarms caused by monitoring fluctuations, a significance level is set. Degrees of freedom By consulting t Obtaining critical values ​​from distribution tables This value serves as the threshold for mutation identification; if... This indicates that there are mutation points in these two subsequences. The higher the value, the higher the significance level.

[0026] Furthermore, in order to transform discrete mutation point information into continuous features that can be effectively utilized by deep learning models, the prediction method in this application constructs a "mutation intensity" index. .

[0027] Specifically, the sliding t Test results t The statistic is input into the Sigmoid function, and its range is mapped to the interval [0.5, 1). The mutation strength... The formula is as follows: ; in, t For sliding t Test statistic, mutation strength The value range is [0.5, 1).

[0028] critical value Substituting this into the equation, we obtain the corresponding mutation strength threshold as 0.9663. When... When this occurs, it indicates that the landslide is in a significant stage of accelerated step deformation; when A value close to 0.5 indicates a stable deformation state. This continuous abrupt change in intensity... It accurately depicts the gradual transition of the landslide between "stable" and "accelerating" states. For ease of subsequent description, this... Defined as mutation intensity feature M1.

[0029] Step 3: Screening of Influencing Factors: Using grey relational analysis, the correlation between multiple candidate influencing factors and landslide displacement was quantified, and monthly reservoir water level change W2, monthly rainfall R1, and abrupt change intensity characteristic M1 were selected as key input variables.

[0030] To determine the optimal input variables for the displacement prediction model, this invention employs grey relational analysis to quantify the correlation between multiple candidate influencing factors and the cumulative landslide displacement. The candidate factors include: monthly rainfall R1, two-month cumulative rainfall R2, three-month cumulative rainfall R3, monthly reservoir water level W1, monthly reservoir water level change W2, two-month reservoir water level change W3, and the abrupt change intensity feature M1 constructed in step two. The calculation results show that the correlation degree of monthly reservoir water level change W2 is the highest at 0.871, followed by the abrupt change intensity feature M1 at 0.856, and monthly rainfall R1 at 0.777. Therefore, W2, R1, and M1 are selected as the key driving factors for displacement prediction.

[0031] Grey relational analysis specifically includes: calculating the grey relational coefficient between each influencing factor and the cumulative displacement, and the calculation formula is as follows: ; in, The reference sequence is the cumulative displacement. To compare the series, i.e., the various influence factors, To determine the resolvability coefficient, the top three influencing factors with the highest correlation values ​​were selected: monthly reservoir water level change W2, abrupt change intensity characteristic M1, and monthly rainfall R1.

[0032] Step 4, Displacement Decomposition: Use variational mode decomposition to decompose the cumulative displacement signal into trend components, periodic components, and random components.

[0033] Cumulative displacement signals are non-stationary time-series data, making direct prediction difficult. This invention employs variational mode decomposition to decompose them into three physically meaningful intrinsic mode functions: the trend component IMF1, the periodic component IMF2, and the random component IMF3. Specifically, IMF1 reflects the long-term creep trend of landslides controlled by geological conditions, IMF2 characterizes periodic fluctuations influenced by seasonal variations in reservoir water levels and rainfall, and IMF3 represents random uncertainties such as monitoring noise.

[0034] Variational mode decomposition transforms the original signal f Decomposed into K One intrinsic mode function In this application, variational mode decomposition decomposes the cumulative displacement signal into three intrinsic mode functions (IMFs), corresponding to the trend component IMF1, the periodic component IMF2, and the random component IMF3, respectively. The corresponding constrained variational problem is: ; in, These are the components obtained after decomposition. The actual center frequency of each IMF component; K is the preset total number of modal components, which is 3 in this invention; k The index of the modal component; The Dirac function is used; * indicates convolution operation. The analytic signal representing each component; This represents the estimated center frequency of each analytical signal; ( t () represents the original cumulative displacement signal.

[0035] To transform the constrained optimization problem into an unconstrained problem, a penalty factor is introduced. With Lagrange multipliers Construct the augmented Lagrangian function: ; in, The penalty factor is set to 2500 in this invention; For Lagrange multipliers; ( t () is the Lagrange multiplier; It is the time domain numberk There are several intrinsic mode components. This problem is solved iteratively using an alternating direction multiplier iterative algorithm, the core formula of which is as follows: ; ; ; Among them, superscript n , n +1 represents the calculation results of the nth and (n+1)th iterations, respectively; Indicates the first k Modal components Frequency domain representation after Fourier transform; Represents the original signal Frequency domain representation after Fourier transform; Represents the frequency domain spectrum of the i-th modal component; Represents the Lagrange multiplier Frequency domain representation after Fourier transform; As a penalty factor; Represents the angular frequency variable; The actual center frequency of each IMF component; This represents the integral operation in the positive frequency domain, used to calculate the energy-weighted center frequency of a mode; This represents the dual update step size, controlling the iterative update rate of the Lagrange multiplier.

[0036] Step 5: Component Prediction: Construct a long short-term memory network model to predict each component. The trend component and the random component use a univariate LSTM model, while the periodic component uses a multivariate LSTM model with W2, R1, and M1 as inputs.

[0037] Long Short-Term Memory (LSTM) network models include univariate LSTM models and multivariate LSTM models. Univariate LSTM models are used to predict the trend component IMF1 and the random component IMF3, with their inputs being the historical values ​​of each component over the previous two months. Multivariate LSTM models are used to predict the periodic component, with their inputs including the monthly reservoir water level change W2, monthly rainfall R1, abrupt change intensity feature M1, and the historical values ​​of the periodic component itself over the previous two months.

[0038] The LSTM network structure includes an input layer, hidden layers, and an output layer. The hidden layer uses LSTM units to construct a single-layer recurrent neural network. The forward propagation equations for the forget gate, input gate, output gate, and cell state are as follows: ; in, These are the outputs of the forget gate, input gate, and output gate, respectively. In cellular state, This is the hidden layer state. This is the Sigmoid activation function.

[0039] Step 6, Displacement Reconstruction and Prediction: The prediction results of the three components in Step 5 are superimposed to obtain the final cumulative displacement prediction value, and the model accuracy is evaluated.

[0040] The displacement reconstruction formula is: ; in, For the predicted value of the trend component, For the predicted value of the periodic component, These are the predicted values ​​for the random components.

[0041] Model accuracy is evaluated using root mean square error, mean absolute error, and mean absolute percentage error, calculated using the following formulas: ; ; ; in, This represents the actual displacement value. This is the average actual displacement. To predict displacement values, N This represents the number of samples.

[0042] In one specific embodiment, the Baishuihe landslide, a typical step-type landslide in the Three Gorges Reservoir area, is used as the research object to verify the effectiveness of the step-type displacement prediction method for reservoir bank landslides based on mutation point identification and deep learning proposed in this invention.

[0043] The Baishuihe landslide is located in Zigui County, Hubei Province, approximately 56 km from the Three Gorges Dam. The landslide body is about 500 m long, 430 m wide, with a volume of approximately 6.45 million cubic meters and an average thickness of 30 m. Since the Three Gorges Reservoir began impounding water in July 2003, 11 GPS displacement monitoring points have been deployed on the landslide. Among them, monitoring point ZG118, located in the central area of ​​the second-level platform of the landslide, is the most severely deformed part. Daily displacement data has been continuously collected from this point since July 2003, making it the most representative monitoring point for analyzing the deformation characteristics of the Baishuihe landslide. This embodiment uses the monthly average displacement data from monitoring point ZG118 from April 2004 to July 2014, along with monthly rainfall and reservoir water level data during the same period, for model construction and verification.

[0044] Step 1: Data Acquisition and Preprocessing Cumulative displacement, monthly average reservoir water level elevation, and monthly rainfall data were collected from the ZG118 monitoring point of the Baishuihe landslide, spanning from April 2004 to July 2014. The original monitored displacements showed a clear step-like upward trend and exhibited a strong temporal correlation with monthly rainfall and reservoir water level fluctuations. Missing value imputation, moving average denoising, and max-min normalization were applied to the data to map them to the [0,1] interval. The dataset was divided chronologically, with data from April 2004 to December 2011 serving as the training set and data from January 2012 to July 2014 serving as the test set.

[0045] Step 2: Mutation Point Identification and Feature Construction Optimized sliding t The mutation point is identified using the test method, and a significance level is set. Degrees of freedom The critical value was calculated. The mutation strength is constructed using the Sigmoid function. The threshold is thus calculated. .like Figure 2 As shown, when The region with a value exceeding 0.9663 precisely corresponds to the step segment of the cumulative displacement, verifying the effectiveness of the method in identifying step deformation.

[0046] Step 3: Impact Factor Screening Grey relational analysis was used to calculate the grey relational coefficients between each influencing factor and the cumulative displacement. The selected influencing factors included: monthly rainfall R1, cumulative rainfall over two months R2, cumulative rainfall over three months R3, monthly reservoir water level W1, monthly reservoir water level change W2, two-month reservoir water level change W3, and the abrupt change intensity feature M1 constructed in this invention.

[0047] The calculation results show that the correlation coefficients of all factors are greater than 0.6. Among them, the correlation coefficient of monthly reservoir water level change W2 is the highest (0.871), followed by the abrupt change intensity feature M1 (0.856), and the correlation coefficient of monthly rainfall R1 is 0.777. Therefore, W2, R1, and M1 were selected as key input variables.

[0048] Step 4: Displacement Decomposition Variational mode decomposition is used to decompose the cumulative displacement signal. The number of modes is set. Punishment factor The cumulative displacement was ultimately decomposed into three components: IMF1, IMF2, and IMF3. Among them, IMF1 is the trend component, which shows a monotonically increasing trend, reflecting the long-term creep characteristics of the landslide; IMF2 is the periodic component, which shows periodic fluctuations, corresponding to the seasonal changes in reservoir water level and rainfall; and IMF3 is the random component, which shows irregular fluctuations, representing uncertain factors such as measurement noise.

[0049] Step 5: Component Prediction Construct an LSTM model to predict each component: Trend component and random component: Univariate LSTM models are constructed for prediction, with the input being the historical values ​​of the previous two months.

[0050] Periodic component: A multivariate LSTM model is constructed for prediction. The input features of the model include the historical values ​​of the periodic component for the previous two months, the monthly reservoir water level change W2, the monthly rainfall R1, and the abrupt change intensity feature M1.

[0051] This model aims to learn the evolution law of periodic components under the combined effect of external environmental driving force and internal deformation state switching.

[0052] The LSTM network structure consists of an input layer (corresponding to the number of input features), a hidden layer (128 LSTM units), and an output layer (1 neuron). The activation function is tanh, the optimizer is Adam, the learning rate is 0.001, and the number of iterations is 500.

[0053] Step Six: Displacement Reconstruction and Prediction The prediction results of the three components are superimposed to obtain the cumulative displacement prediction value, which is then compared with the measured value. The IMP-VMD-LSTM model of this invention has an RMSE of 17.58 mm, a MAE of 14.72 mm, and a MAPE of 0.64% on the test set.

[0054] To verify the superiority of this invention, three comparative models were set up: (1) VMD-LSTM (without mutation intensity feature M1); (2) LSTM (without decomposition, without M1); and (3) ANN (Artificial Neural Network). Predictions were performed on the same training and test sets, and the results are shown in Table 1 and... Figure 3 As shown.

[0055] Table 1 Comparison of prediction performance of different models

[0056] Therefore, the IMP-VMD-LSTM model proposed in this invention significantly outperforms the comparative models in all metrics. Particularly during the periods of step deformation (June to August 2012, May to July 2013, and April to June 2014), the proposed model accurately predicts the abrupt increase in displacement, while the comparative models exhibit significant prediction lag. This fully demonstrates that by introducing the abrupt change intensity feature M1, the proposed model successfully captures the deformation state switching patterns that traditional models cannot learn, validating its effectiveness and advancement.

[0057] The above description is only a preferred embodiment of this application and is not intended to limit this application. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this application should be included within the protection scope of this application.

Claims

1. A method for predicting step displacement in reservoir bank landslides, characterized in that, Includes the following steps: Step 1: Data Acquisition and Preprocessing: Collect historical time-series monitoring data of the target landslide area and preprocess the raw monitoring data; Step 2, Mutation Point Identification and Feature Construction: Employing an optimized sliding... t The test method identifies abrupt change points in displacement data, calculates critical values, constructs continuous abrupt change intensities based on the identification results, and then calculates the threshold. Step 3: Screening of Influencing Factors: The grey relational analysis method was used to quantify the correlation between multiple candidate influencing factors and landslide displacement, and the monthly reservoir water level change W2, monthly rainfall R1 and abrupt change intensity characteristics M1 were selected as key input variables. Step 4, Displacement Decomposition: The cumulative displacement signal is decomposed into trend component IMF1, periodic component IMF2, and random component IMF3 using variational mode decomposition. Step 5: Component Prediction: Construct a long short-term memory network model and predict each component; Step 6, Displacement Reconstruction and Prediction: The prediction results of each component in Step 5 are superimposed to obtain the final cumulative displacement prediction value, and the model accuracy is evaluated.

2. The method for predicting step displacement of reservoir bank landslides according to claim 1, characterized in that: In step one, the preprocessing of the raw monitoring data includes missing value imputation, moving average denoising and max-min normalization, and finally the data is mapped to the interval [0, 1].

3. The method for predicting step displacement of reservoir bank landslides according to claim 1, characterized in that: In step two, the optimized sliding t The test method includes two contrasting subsequences of a given displacement-time series. and The optimized formula for calculating the sliding t-test is: ； in, and Representing time points i Displacement values ​​and weights; and The length of the subsequence; ； in, , They represent and The variance.

4. The method for predicting step displacement of reservoir bank landslides according to claim 3, characterized in that: Step two includes constructing a mutation strength index. Slide soon t Test results t The statistic is input into the Sigmoid function, resulting in the mutation strength. The formula is as follows: ； in, t For sliding t Test statistic.

5. The method for predicting step displacement of reservoir bank landslides according to claim 4, characterized in that: Mutation strength The value range is [0.5, 1), and... Defined as mutation intensity feature M1.

6. The method for predicting step displacement of reservoir bank landslides according to claim 1, characterized in that: In step three, the grey relational analysis specifically includes: calculating the grey relational coefficient between each influencing factor and the cumulative displacement, and the calculation formula is as follows: ； in, The reference sequence is the cumulative displacement. To compare the series, i.e., the various influencing factors, The resolution coefficient.

7. The method for predicting step displacement of reservoir bank landslides according to claim 6, characterized in that: In step three, the top three influencing factors with the highest correlation values ​​were selected using the grey relational analysis method. These factors were, in order, monthly reservoir water level change W2, mutation intensity characteristic M1, and monthly rainfall R1.

8. The method for predicting step displacement of reservoir bank landslides according to claim 1, characterized in that: In step four, the original signal is decomposed using variational mode decomposition. f The corresponding constrained variational problem in the process is: ； in, These are the components obtained after decomposition. The actual center frequencies of each intrinsic mode function component; K The preset total number of modal components; k The index of the modal component; The Dirac function is used; * indicates convolution operation. The analytic signal representing each component; This represents the estimated center frequency of each analytical signal; This is the original cumulative displacement signal.

9. The method for predicting step displacement of reservoir bank landslides according to claim 1, characterized in that: In step five, the long short-term memory network model includes: A univariate LSTM model is used for the trend component IMF1 and the stochastic component IMF3, with the input being the historical values ​​of each component over the previous two months. A multivariate LSTM model is used to predict the periodic component. Its inputs include the monthly reservoir water level change W2, the monthly rainfall R1, the abrupt change intensity feature M1, and the historical values ​​of the periodic component itself for the previous two months.

10. The method for predicting step displacement of reservoir bank landslides according to claim 1, characterized in that: In step six, the displacement reconstruction formula is: ； in, For the predicted value of the trend component, For the predicted value of the periodic component, These are the predicted values ​​for the random components; Model accuracy is evaluated using root mean square error, mean absolute error, and mean absolute percentage error.

Images