Railway subgrade settlement prediction method based on IGM-PSVR model

Abstract

The present invention discloses a railway subgrade settlement prediction method based on the IGM-PSVR model. First, the subgrade settlement data is collected and preprocessed. Then, the data is subjected to a level ratio check, a corresponding differential equation is established, and the time response function is solved. The background value of the gray model is further optimized, and an improved gray model IGM(1,1) is constructed to obtain the prediction results of the model. Then, the particle swarm optimization algorithm is used to optimize the hyperparameters of the support vector regression (SVR) model, and a particle swarm optimized support vector machine model is constructed to obtain the prediction results of the model. Finally, the inverse sum of squared errors method is used to assign different weights to the two-step prediction results, and finally the combined prediction results are obtained. Experiments show that the combined model has higher prediction accuracy than the single model, can better reflect the trend of subgrade settlement changes, and plays an important role in ensuring the safety of railway transportation.

Description

Technical Field

[0001] The invention belongs to the technical field of railway roadbed safety, and in particular relates to a railway roadbed settlement prediction method based on a GM-SVR combined model. Background Art

[0002] During the railway operation stage, the roadbed is affected by various factors, resulting in uneven settlement, which seriously threatens the safety of passengers and drivers, and also affects the normal operation of the railway transportation system. Therefore, accurate prediction of roadbed settlement is particularly important. It can not only ensure the safety of passengers and drivers, but also provide maintenance plans for the railway transportation department in a timely manner, reduce the waste of manpower, material and financial resources, and play an important role in the smooth operation of trains.

[0003] Currently, there are two main approaches to railway subgrade settlement prediction: the first involves building a complex soil model based on subgrade soil parameter information to predict settlement; the second involves using a system formula method to establish a time-series prediction model based on measured settlement data. In my country, system analysis methods are often used for subgrade settlement prediction. Common system analysis methods include the exponential curve method, parabola method, hyperbola method, and gray system theory. Gray system methods require a small sample size, are computationally simple, and are easy to use, but they offer low medium- and long-term prediction accuracy and are sensitive to outliers. A literature review of existing technologies reveals that single models have limitations when addressing settlement prediction and are unable to provide accurate, real-time predictions. Summary of the Invention

[0004] In response to the above-mentioned problems, the present invention proposes a railway roadbed settlement prediction method based on the IGM-PSVR model, which can overcome the shortcomings of a single model, improve the settlement prediction accuracy, provide the railway transportation department with a roadbed design and maintenance plan, reduce the waste of manpower, material and financial resources, and improve the sustainable utilization of railway line equipment.

[0005] The above purpose is achieved through the following technical solutions:

[0006] The present invention provides a railway subgrade settlement prediction method based on the IGM-PSVR model, the method comprising the following steps:

[0007] S1. Setting observation points, collecting roadbed settlement data, and preprocessing the collected roadbed settlement data;

[0008] S2. First, perform a level ratio check. If the roadbed settlement data pre-processed in step S1 meets the requirements, establish the corresponding differential equation, solve it to obtain the time response function, further optimize the background value of the gray model, construct the improved gray model IGM(1,1) and obtain the IGM model prediction result; if the roadbed settlement data pre-processed in step S1 does not meet the requirements, continue to perform pre-processing.

[0009] S3. Comprehensively evaluate the accuracy of the grey model using the two indicators of posterior difference ratio C1 and small error probability p; if the accuracy of the prediction model meets the requirements, the prediction result is output, recorded as y1;

[0010] S4, normalizing the roadbed settlement data pre-processed in step S1 and constructing an SVR model;

[0011] S5. Use the particle swarm optimization algorithm (PSO) to optimize the hyperparameters of the support vector regression (SVR) model, build a particle swarm optimized support vector machine (PSVR) model and obtain the PSVR model prediction result, which is recorded as y2;

[0012] S6. Calculate the sum of squares of the errors between the predicted result y1 of the IGM model and the predicted result y2 of the PSVR model and the true value respectively, and determine the weights w1 and w2 of the grey model IGM model and PSVR model according to the inverse sum of squares error method; finally, obtain the predicted result of the IGM-PSVR model, which is recorded as y;

[0013] Furthermore, the preprocessing in step S1 refers to filling or removing outliers by interpolation. Specifically, for missing values, the average value of the previous and next items is taken to fill them; for outliers, the 3σ criterion is used to remove them; then the original data is smoothed and denoised by wavelet packet transform (db6 wavelet basis function, rigrsure soft threshold and 5-layer wavelet decomposition).

[0014] Furthermore, the specific method of step S2 is as follows:

[0015] S2.1. Perform level comparison check on the roadbed settlement data sequence pre-processed in step S1. Specifically, let the sequence composed of n roadbed settlement data pre-processed in step S1 be X. (0) ={x1 (0) ,x2 (0) ,…,x n (0)}, by calculating X (0) Level ratio: Check X (0) Are the level ratios in the interval If X (0) The level ratios are all in the range If X (0) The level ratio is not in the interval If the roadbed settlement data pre-processed in step S1 is smoothed or transformed until the level ratio verification meets the requirements;

[0016] S2.2, compare the X that has passed the level comparison test (0) The sequence is accumulated to obtain the new sequence X (1) ={x1 (1) ,x2 (1) ,…,x n (1)};

[0017] S2.3. Fit the new sequence obtained in step S2.2 with the first-order linear constant coefficient differential equation of formula (1) as follows:

[0018]

[0019] In the formula, X (1) Indicates the cumulative value of each observation point, a=(a ij ) n×n ,b=[b1,b2,…,b i ,…,b n ] T , both represent coefficient matrices; b i and a ij represents the i-th order coefficient, i = 1, 2, ..., n;

[0020] S2.4. Discretize equation (1) and obtain equation

[0021]

[0022] Where x i (0) (k) represents the grey differential equation; Represents the model base value, and is calculated as shown in formula (3):

[0023] z i (1) (k) = 0.5*[x i (1) (k)+x i (1) (k-1)] (3);

[0024] S2.5. Based on formulas (4)-(6), solve for parameters a and b:

[0025]

[0026]

[0027] In the formula, L and Y are intermediate variables introduced to simplify the calculation, and the superscript T represents the transpose of the matrix;

[0028] S2.6. According to the parameters a and b, the time response function of the grey model is:

[0029]

[0030] Finally, the grey model prediction value is obtained by performing cumulative reduction on formula (7).

[0031] Furthermore, according to step S2.6, the prediction accuracy of the model is highly dependent on the solved parameters a and b, and the calculation process of the coefficient matrices a and b depends on the model base value constructed by the original sequence in formula (3) The solution of , that is, the cardinality value The more accurate, the better the prediction effect, and the base value The calculation is actually solved by using the trapezoidal formula of numerical integration. In order to further enhance the accuracy of the basic numerical calculation, the integral formula is used for direct calculation:

[0032]

[0033] Each prediction process of the model adopts an iterative and progressive approach, that is, a and b are calculated based on the original data, and further predictions are made based on the above parameters. Remove the first value of the prediction sequence, use the remaining sequence to rebuild the parameters and update the base value Complete the next prediction task.

[0034] Furthermore, the accuracy of the prediction model is comprehensively evaluated in step S3, specifically, when the posterior difference ratio C of the grey model IGM(1,1) prediction is ≤ 0.65 and the small error probability p is ≥ 0.70%, the prediction result is output and recorded as y1.

[0035] Furthermore, in step S4, the roadbed settlement data pre-processed in step S1 is normalized and an SVR model is constructed; the normalized roadbed settlement data is expressed as:

[0036]

[0037] In the formula, x i represents the i-th observation value in the sedimentation sequence, max(x) and min(x) represent the maximum and minimum values ​​of the sedimentation sequence respectively;

[0038] The support vector regression function is expressed as:

[0039] f(x)=wω(x)+B (10)

[0040] Where w is the weight; B is the bias; ω(x) is the mapping function from low-dimensional space to high-dimensional space;

[0041] According to the principle of structured risk minimization, slack variables are introduced and the objective function of the SVR model is:

[0042]

[0043] The corresponding constraints are:

[0044]

[0045] Where, l ε is the insensitive loss function; C is the penalty coefficient; ξ i 、 are the upper and lower bounds of the training error under the width ε error constraint of the insensitive loss function;

[0046] Construct the Lagrangian function and transform the objective function optimization problem into solving the following equation:

[0047]

[0048] Finally, we simplify to:

[0049]

[0050] Where, and μ i Both represent Lagrange multiplier variables, K(x i ,x) is called the kernel function, where n represents the number of training samples, i.e., the n roadbed settlement data preprocessed in step S1; m represents the number of Lagrange multipliers introduced during the optimization process, typically 2n, corresponding to the Lagrange multipliers for the positive and negative errors of each sample. Common kernel functions include the linear kernel function, Gaussian radial basis kernel function, polynomial kernel function, and nonlinear kernel function of neurons. During parameter optimization, the kernel function coefficient parameter is only used when using Gaussian radial basis kernel function, polynomial kernel function, and nonlinear kernel function of neurons.

[0051] Furthermore, step S5 uses the particle swarm optimization algorithm (PSO) to optimize the hyperparameters of the support vector regression (SVR) model, constructs a particle swarm optimized support vector machine (PSVR) model and obtains the PSVR model prediction results. The specific process is as follows:

[0052] The performance of the SVR model directly depends on the selection of model hyperparameters. This patent uses the particle swarm optimization algorithm to optimize the kernel function, kernel function parameter (gamma), width ε of the insensitive loss function, and penalty factor C of the SVR model. The fitness of the optimization process is calculated using RMSE (Root Mean Squared Error). The optimal parameters are selected to construct the PSVR model. The steps for constructing the PSVR model are as follows:

[0053] Step 1: Parameter and population initialization. Parameter initialization includes the maximum number of iterations, inertia weight, learning factor, and the speed and position ranges of particle updates. The population size is initialized to M, which is the number of particles in the population. The particle position vector contains the SVR model parameters that need to be optimized.

[0054] Step 2: Based on the initialization value of each particle in the population, calculate the fitness value of each particle and record the historical optimal solution p of each particle best , obtain the global optimal position g of the particle swarm best ;

[0055] Step 3: Update the particle velocity based on formula (15) and update the particle position based on formula (16);

[0056]

[0057] in, and are the velocity and position of particle q at iteration t+1, w t is the inertia weight of the t-th iteration, represents the speed of particle q in t iterations, c1 and c2 represent learning factors, which are set to 2, r1 and r2 represent random numbers in the range of [0,1], and p best,q represents the best historical position of particle q, represents the position of particle q at t iterations, g best In order to adapt to the nonlinear requirements of roadbed settlement data and further increase the global exploration and local optimization capabilities of the optimization algorithm, this patent uses the nonlinear inertia weight calculation method given by formula (17) to improve the PSO algorithm.

[0058]

[0059] Among them, w begin and w end They represent the inertia weight of the first iteration and the inertia weight of the last iteration respectively, Iter represents the current iteration, and Iter_max represents the maximum iteration;

[0060] Step 4: Calculate the fitness value of each particle, update the particle's historical optimal position and the global optimal position of the particle swarm, and determine whether the maximum number of iterations has been reached. If so, return g best , if otherwise continue to Step3;

[0061] Step 5: The optimal hyperparameter g returned in step 4 best Substitute the SVR model, build the particle swarm optimized support vector machine model (PSVR) and obtain the PSVR model prediction results.

[0062] Furthermore, the specific method of step S6 is: first calculate the sum of squares of the prediction errors of a single model e γ :

[0063]

[0064] Where: x τγ is the τth original data of the γth model; is the τth predicted value corresponding to the γth model; is the prediction error;

[0065] The weight coefficient of a single model is:

[0066]

[0067] Where, z represents the total number of models.

[0068] The beneficial effects of the present invention compared to the prior art are:

[0069] Aiming at the changing characteristics and laws of roadbed settlement data, the present invention improves the classic grey model to construct an IGM model based on roadbed settlement changes, introduces an improved PSO algorithm to optimize the SVR model to construct a PSVR model, and further completes the evaluation. The proposed IGM-PSVR prediction model can overcome the shortcomings of a single model, improve the settlement prediction accuracy, provide roadbed design and maintenance solutions for railway transportation departments, reduce the waste of manpower, material and financial resources, and improve the sustainable utilization of railway line equipment. BRIEF DESCRIPTION OF THE DRAWINGS

[0070] Figure 1 Flow chart of the method of the present invention;

[0071] Figure 2 The roadbed settlement diagram shown in the embodiment of the present invention;

[0072] Figure 3 is the prediction result of a single prediction model based on the grey model GM(1,1);

[0073] Figure 4It is the prediction result of a single prediction model based on the SVR support vector regression prediction method;

[0074] Figure 5 is the prediction result based on the IGM-PSVR model prediction method;

[0075] Figure 6 It is a relative error comparison diagram of the prediction values ​​of the grey model GM (1, 1), the SVR support vector regression prediction model and the IGM-PSVR model of the present invention. DETAILED DESCRIPTION

[0076] The present invention proposes a railway roadbed settlement prediction method based on the IGM-PSVR model. In order to make the purpose, technical solutions and advantages of the present invention clearer, the embodiment of the present invention is further described in detail with reference to the accompanying drawings. The system flow chart is as follows: Figure 1 shown.

[0077] The present invention provides a railway subgrade settlement prediction method based on the GM-SVR combined model, the method comprising the following steps:

[0078] S1. Set up observation points, collect roadbed settlement data, and preprocess the collected roadbed settlement data; the preprocessing refers to using interpolation method to fill or remove outliers. Specifically, for missing values, mean processing is used to take the average of the previous and next items; for outliers, use the 3σ criterion to remove them, and then use wavelet packet transform (db6 wavelet basis function, rigrsure soft threshold and 5-layer wavelet decomposition) to smooth and reduce noise on the original data.

[0079] S2. First, perform a level ratio check. If the roadbed settlement data pre-processed in step S1 meets the requirements, establish the corresponding differential equation, solve it to obtain the time response function, further optimize the background value of the grey model, construct the improved grey model IGM (1,1) and obtain the IGM model prediction result. If the roadbed settlement data pre-processed in step S1 does not meet the requirements, continue to perform pre-processing. The specific method is as follows:

[0080] S2.1. In order to ensure that the grey GM (1, 1) model can be established, the original data needs to be checked. Therefore, firstly, the sequence of roadbed settlement data pre-processed in step S1 is subjected to level comparison verification. Specifically, let the sequence composed of n roadbed settlement data pre-processed in step S1 be X. (0) ={x1 (0) ,x2 (0) ,…,x n (0)}, by calculating X (0) Level ratio: Check X (0)Are the level ratios in the interval If X (0) The level ratios are all in the range If X (0) The level ratio is not in the interval If the roadbed settlement data pre-processed in step S1 is smoothed or transformed until the level ratio verification meets the requirements;

[0081] X that passed the level comparison test (0) The sequence is accumulated to obtain the new sequence X (1) ={x1 (1) ,x2 (1) ,…,x n (1)};

[0082] S2.3, the new sequence is monotonically increasing data, which is approximately consistent with exponential growth. Therefore, the new sequence obtained in step S2.2 is fitted with the first-order linear constant coefficient differential equation of formula (1) as follows:

[0083]

[0084] In the formula, X (1) Indicates the cumulative value of each observation point, a=(a ij ) n×n ,b=[b1,b2,…,b i ,…,b n ] T , both represent coefficient matrices; b i and a ij represents the i-th order coefficient, i = 1, 2, ..., n;

[0085] S2.4. Discretize equation (1) and obtain equation

[0086]

[0087] Where x i (0) (j) represents the grey differential equation; Represents the model base value, and is calculated as shown in formula (3):

[0088] z i (1) (k) = 0.5*[x i (1) (k)+x i (1) (k-1)] (3);

[0089] S2.5. Based on formulas (4)-(6), solve for parameters a and b:

[0090]

[0091] In the formula, L and Y are intermediate variables introduced to simplify the calculation, and the superscript T represents the transpose of the matrix;

[0092] S2.6. According to the parameters a and b, the time response function of the grey model is:

[0093]

[0094] Finally, the grey model prediction value is obtained by performing cumulative reduction on formula (7).

[0095] According to step S2.6, the prediction accuracy of the model is highly dependent on the solved parameters a and b, and the calculation process of the coefficient matrices a and b depends on the model base value constructed by the original sequence in formula (3) The solution of , that is, the cardinality value The more accurate, the better the prediction effect, and the base value The calculation is actually solved by using the trapezoidal formula of numerical integration. In order to further enhance the accuracy of the basic numerical calculation, the integral formula is used for direct calculation:

[0096]

[0097] Each prediction process of the model adopts an iterative and progressive approach, that is, a and b are calculated based on the original data, and further predictions are made based on the above parameters. Remove the first value of the prediction sequence, use the remaining sequence to rebuild the parameters and update the base value Complete the next prediction task.

[0098] S3. The prediction accuracy of the gray model IGM(1,1) is comprehensively evaluated using two metrics: the posterior difference ratio C1 and the probability of small error p. See Table 1 for details. When the C value is small, the original data has a larger dispersion, while the residual dispersion is smaller, indicating higher prediction accuracy. Furthermore, an increase in the p value indicates that there are more points where the difference between the residual and the mean residual is less than 0.6745. Therefore, a larger p value indicates a better prediction model. If the prediction model meets the accuracy requirements, the prediction result is output, denoted as y1.

[0099] S4, normalizing the roadbed settlement data pre-processed in step S1 and constructing an SVR model; the normalized roadbed settlement data is expressed as:

[0100]

[0101] In the formula, x irepresents the i-th observation value in the sedimentation sequence, max(x) and min(x) represent the maximum and minimum values ​​of the sedimentation sequence respectively;

[0102] The support vector regression function is expressed as:

[0103] f(x)=wω(x)+B (10)

[0104] Where w is the weight; B is the bias; ω(x) is the mapping function from low-dimensional space to high-dimensional space;

[0105] According to the principle of structured risk minimization, slack variables are introduced and the objective function of the SVR model is:

[0106]

[0107] The corresponding constraints are:

[0108]

[0109] Where k ε is the insensitive loss function; C is the penalty coefficient; ξ i 、 are the upper and lower bounds of the training error under the width ε error constraint of the insensitive loss function;

[0110] Construct the Lagrangian function and transform the objective function optimization problem into solving the following equation:

[0111]

[0112] Finally, we simplify to:

[0113]

[0114] Where, and μ i Both represent Lagrange multiplier variables, K(x i ,x) is called the kernel function, where n represents the number of training samples, i.e., the n roadbed settlement data preprocessed in step S1; m represents the number of Lagrange multipliers introduced during the optimization process, typically 2n, corresponding to the Lagrange multipliers for the positive and negative errors of each sample. Common kernel functions include the linear kernel function, Gaussian radial basis kernel function, polynomial kernel function, and nonlinear kernel function of neurons. During parameter optimization, the kernel function coefficient parameter is only used when using Gaussian radial basis kernel function, polynomial kernel function, and nonlinear kernel function of neurons.

[0115] S5. Use the particle swarm optimization algorithm (PSO) to optimize the hyperparameters of the support vector regression (SVR) model, build a particle swarm optimized support vector machine (PSVR) model, and obtain the PSVR model prediction results. The specific process of S5 is as follows: The performance of the SVR model directly depends on the selection of model hyperparameters. This patent uses the particle swarm optimization algorithm to optimize the kernel function, kernel function parameter (gamma), insensitive loss function width (ε), and penalty factor (C) of the SVR model. The fitness calculation of the optimization process uses RMSE (Root Mean Squared Error) to select the optimal parameters to build the PSVR model. The steps for building the PSVR model are as follows:

[0116] Step 1: Parameter and population initialization. Parameter initialization includes the maximum number of iterations, inertia weight, learning factor, and the speed and position ranges of particle updates. The population size is initialized to M, which is the number of particles in the population. The particle position vector contains the SVR model parameters that need to be optimized.

[0117] Step 2: Based on the initialization value of each particle in the population, calculate the fitness value of each particle and record the historical optimal solution p of each particle best , obtain the global optimal position g of the particle swarm best ;

[0118] Step 3: Update the particle velocity based on formula (15) and update the particle position based on formula (16);

[0119]

[0120] in, and are the velocity and position of particle q at iteration t+1, w t is the inertia weight of the t-th iteration, represents the speed of particle q in t iterations, c1 and c2 represent learning factors, which are set to 2, r1 and r2 represent random numbers in the range of [0,1], and p best,q represents the best historical position of particle q, represents the position of particle q at t iterations, g best In order to adapt to the nonlinear requirements of roadbed settlement data and further increase the global exploration and local optimization capabilities of the optimization algorithm, this patent uses the nonlinear inertia weight calculation method given by formula (17) to improve the PSO algorithm.

[0121]

[0122] Among them, w begin and w endThey represent the inertia weight of the first iteration and the inertia weight of the last iteration respectively, Iter represents the current iteration, and Iter_max represents the maximum iteration;

[0123] Step 4: Calculate the fitness value of each particle, update the particle's historical optimal position and the global optimal position of the particle swarm, and determine whether the maximum number of iterations has been reached. If so, return g best , if otherwise continue to Step3;

[0124] Step 5: The optimal hyperparameter g returned in step 4 best Substitute the SVR model, build the particle swarm optimized support vector machine model (PSVR) and obtain the PSVR model prediction result, which is recorded as y2.

[0125] S6. This embodiment adopts the IGM-PSVR parallel model to predict railway subgrade settlement. The gray IGM model has better linear prediction ability, and the PSVR has better nonlinear data prediction ability. The parallel combination of these two models makes up for the shortcomings of a single model, improves the prediction accuracy of subgrade settlement, and provides technical support for the safe and smooth operation of trains.

[0126] This embodiment adopts the GM-SVR parallel model to predict railway subgrade settlement. The grey GM (1, 1) model has better linear prediction capability, while the SVR has better nonlinear data prediction capability. Combining these two models in parallel makes up for the shortcomings of a single model, improves the prediction accuracy of subgrade settlement, and provides technical support for the safe and smooth operation of trains.

[0127] In a combined prediction model, it is crucial to determine the weight coefficients. The arithmetic mean method, the optimal weighting method, or the inverse sum of squared errors method are generally used to determine the weight coefficients. The arithmetic mean method is usually used when the importance of the models is not well understood, but because it assigns the same weight to all models and does not consider priority, it may lead to poor prediction results. The optimal weighting method is computationally complex and has significant limitations in practical applications. The inverse sum of squared errors method, on the other hand, reflects the importance of each model in the combined prediction by calculating the sum of squared errors of the prediction models. The weights assigned are more reasonable and effective, and are easy to operate and can achieve good prediction results. Therefore, this embodiment uses the inverse sum of squared errors method to determine the optimal combined weights of the prediction models.

[0128] Calculate the sum of squares of the errors between the prediction result y1 of the grey model GM(1,1) and the prediction result y2 of the SVR model and the true value respectively. According to the inverse method of the sum of squares of errors, determine the weights w1 and w2 of the grey model GM(1,1) and the SVR model. The specific method is:

[0129] First calculate the sum of squared prediction errors e for a single model γ :

[0130]

[0131] Where: x τγ is the τth original data of the γth model; is the τth predicted value corresponding to the γth model; is the prediction error;

[0132] The weight coefficient of a single model is:

[0133]

[0134] Where, z represents the total number of models.

[0135] From this we can see that the larger the sum of squared errors of the prediction model, the larger the prediction error value of the prediction model, and the smaller the weight coefficient in the combined model. The smaller the sum of squared errors of the prediction model, the smaller the prediction error value of the prediction model, and the larger its weight coefficient in the combined model. Finally, the prediction result y is obtained:

[0136] y=w1*y1+w2*y2 (20)

[0137] Experimental verification: This experiment takes the subgrade settlement of the Xinjiang Kuitun Letuyi section of the railway as the research object. The poor geological bodies distributed in the Kuitun Letuyi section of the railway are mainly loess, and the soil quality has a great impact on the stability of the subgrade. Figure 2 shown.

[0138] In areas with severe settlement and deformation, static levels are set up to collect roadbed settlement data with an accuracy of up to 0.01mm, and there are 10 observation points in total. Due to space limitations, this paper selects the most representative set of settlement observations, namely the K2047+750 observation pile data for experimental verification. The time span is from March 2022 to August 2022, totaling 160 days. Due to the long time, missing values ​​and outliers are inevitable in the collected data. For missing values, mean processing is adopted, and the average of the previous and next items is taken; for outliers, the 3σ criterion is used to eliminate singular values ​​in the monitoring data to prevent erroneous conclusions.

[0139] Since the grey model requires that the input data must be equidistant in time, after preliminary data screening, a total of 56 periods of settlement data were obtained, with each period of settlement data having an interval of 3 days. The first 80% of the data set was used as the training set, and the last 20% of the data set was used as the test set.

[0140] When predicting railway subgrade settlement, the prediction accuracy and reliability of the evaluation model play a vital role. The smaller the following evaluation index, the higher the prediction accuracy of the model. This paper mainly uses the following evaluation indexes for evaluation:

[0141] (1) Mean Absolute Error

[0142]

[0143] (2) Mean Absolute Percentage Error

[0144]

[0145] (3) Mean Squared Error

[0146]

[0147] (4) Root Mean Squared Error

[0148]

[0149] Among the above evaluation indicators, is the predicted value; i is the true value.

[0150] Depend on Figure 2 It can be seen that the roadbed settlement is a time series with a trend. For the improved grey model IGM (1,1), the original data sequence is checked for level ratios, and it is determined that all level ratios fall within the acceptable coverage interval. A model was established and used to predict railway subgrade settlement data for the next 11 periods. The improved grey model IGM(1,1) was rated for accuracy, with a posterior error ratio of 0.17 and a small error probability of 0.89. According to the grading in Table 1, the grey model's accuracy is considered good.

[0151] For the PSVR model, the input data is normalized, the model is further trained, the model parameters are determined, the PSVR model is established, and the output results of the model are denormalized to obtain the settlement prediction value.

[0152] Table 1 Improved grey model IGM (1,1) accuracy comparison table

[0153] Model accuracy level Posterior difference ratio C1 Small error probability p / % Level 1 (Excellent) C1≤0.35 p≥0.95 Level 2 (good) 0.35＜C1≤0.50 0.80≤p≤0.95 Level 3 (pass) 0.50＜C1≤0.65 0.70≤p≤0.80 Level 4 (Unqualified) 0.65＜C1 p＜0.70

[0154] In order to verify the prediction performance of the combined prediction method proposed in this invention, the grey model GM (1, 1), the SVR support vector regression model and the IGM-PSVR model prediction method of this invention were used to predict and analyze the roadbed settlement data series. The result curves of the three prediction methods are shown in Figure 2. Figures 3 to 5 shown.

[0155] Figure 3 is the prediction result of a single prediction model based on the grey model GM(1,1). Figure 3 It can be seen that the changing trend of the roadbed settlement fitting value based on the grey model GM (1, 1) is consistent with the changing trend of the actual value as a whole, but the predicted value deviates from the true value and the error is large.

[0156] Figure 4 is the prediction result of a single prediction model based on the SVR support vector regression prediction method. Figure 4 It can be seen that the predicted values ​​of the first few periods based on SVR support vector regression are very close to the actual values ​​of settlement, but the errors between the data of the last few periods and the actual values ​​are large.

[0157] Figure 5 This is the prediction result of the combined IGM-PSVR model prediction method of the present invention. Compared with the single GM (1,1) grey model prediction method and the SVR support vector regression prediction method, the prediction results of the IGM-PSVR model are closer to the actual value of railway subgrade settlement, with improved prediction accuracy and precision. Only in the seventh period did the prediction accuracy decline, but the error was only 0.1mm, which does not interfere with actual engineering tasks and can meet the requirements of real-world applications.

[0158] Figure 6 The relative error comparison chart of the prediction values ​​of the three models is shown in Figure 2. Figure 6 It can be seen that the grey model prediction method has the largest prediction error and the least ideal prediction results. Specifically, the error is small in the first few periods, but the error gradually increases in the later periods. This also shows that the grey model is only suitable for short-term predictions, not for medium- and long-term predictions. Compared with the grey prediction model, the SVR model has a smaller error and its prediction results are significantly better than the grey prediction model. The prediction method based on the IGM-PSVR model has the smallest error, with the relative error always fluctuating within a range of 0.1mm. This shows that this method can well predict the changes in railway subgrade settlement and meet the requirements of practical engineering applications.

[0159] Table 2 shows the analysis of prediction results of different prediction methods

[0160] Evaluation index model GM Model SVR model IGM-PSVR combined model MAE 0.178 0.093 0.060 MAPE 1.932 0.687 0.501 MSE 0.039 0.011 0.005 RMSE 0.198 0.099 0.071

[0161] Table 2 shows the prediction statistics of the three models. The MAE, MAPE, MSE, and RMSE of the IGM-PSVR combined model are 0.06, 0.501, 0.005, and 0.071, respectively. Compared with the GM(1,1) grey model, the MAE, MAPE, MSE, and RMSE of the GM-SVR combined model are reduced by 66.3%, 74.1%, 87.1%, and 64.1%, respectively. Compared with the SVR model, the MAE, MAPE, MSE, and RMSE are reduced by 35.5%, 27.1%, 54.6%, and 28.3%, respectively.

[0162] The prediction results of the three prediction models were compared through statistical data: in the field of roadbed settlement prediction, the prediction method based on the IGM-PSVR combined model is better than the SVR support vector regression model and the GM (1, 1) grey model.

Claims

1. The railway subgrade settlement prediction method based on the IGM-PSVR model is characterized by: The method comprises the following steps: S1. Setting observation points, collecting roadbed settlement data, and preprocessing the collected roadbed settlement data; S2, first perform level ratio verification. If the roadbed settlement data after S1 preprocessing does not meet the requirements, continue to perform preprocessing; If the roadbed settlement data pre-processed in step S1 meets the requirements, the corresponding differential equation is established, the time response function is obtained by solving it, the background value of the grey model is further optimized, and the improved grey model IGM(1,1) is constructed; S3. The improved grey model IGM(1,1) is comprehensively evaluated for its accuracy through the two indicators of posterior difference ratio C1 and small error probability p; and the prediction result is output, recorded as y1; S4, normalizing the roadbed settlement data pre-processed in step S1 and constructing an SVR model; S5. Use the particle swarm optimization algorithm (PSO) to optimize the hyperparameters of the support vector regression model (SVR), build a particle swarm optimized support vector machine model (PSVR model), and obtain the prediction result of the PSVR model, which is recorded as y2; S6. Calculate the sum of squares of the errors between the prediction result y1 of the IGM(1,1) model and the prediction result y2 of the PSVR model and the true value respectively. Determine the weights w1 and w2 of the grey model IGM(1,1) model and PSVR model according to the inverse sum of squares error method. Finally, obtain the prediction result y of the IGM-PSVR model. In step S4, the roadbed settlement data pre-processed in step S1 is normalized and an SVR model is constructed. The normalized roadbed settlement data is expressed as: In the formula, x i represents the i-th observation value in the sedimentation sequence, max(x) and min(x) represent the maximum and minimum values ​​of the sedimentation sequence respectively; The support vector regression function is expressed as: f(x)=wω(x)+B (10) Where w is the weight; B is the bias; ω(x) is the mapping function from low-dimensional space to high-dimensional space; According to the principle of structured risk minimization, slack variables are introduced and the objective function of the SVR model is: The corresponding constraints are: Where, l ε is the insensitive loss function; C is the penalty coefficient; ξ i 、 are the upper and lower bounds of the training error under the width ε error constraint of the insensitive loss function; Construct the Lagrangian function and transform the objective function optimization problem into solving the following equation: Finally, we simplify to: Where, and μ i Both represent Lagrange multiplier variables, K(x i ,x) is called the kernel function, n represents the number of training samples, that is, the n roadbed settlement data after preprocessing in step S1; m represents the number of Lagrange multipliers introduced in the optimization process, which is 2n, corresponding to the Lagrange multipliers of the positive error and negative error of each sample; Step S5 uses the particle swarm optimization algorithm PSO to optimize the hyperparameters of the support vector regression SVR model, constructs the particle swarm optimized support vector machine model PSVR model and obtains the PSVR model prediction results. The specific process is as follows: The performance of the SVR model directly depends on the selection of model hyperparameters. The particle swarm optimization algorithm is used to optimize the kernel function, kernel function parameter gamma, insensitive loss function width ε, and penalty factor C of the SVR model. The fitness of the optimization process is calculated using RMSE (Root Mean Squared Error). The optimal parameters are selected to build the PSVR model. The steps for building the PSVR model are as follows: Step 1: Parameter and population initialization. Parameter initialization includes the maximum number of iterations, inertia weight, learning factor, and the speed and position ranges of particle updates. The population size is initialized to M, which is the number of particles in the population. The particle position vector contains the SVR model parameters that need to be optimized. Step 2: Based on the initialization value of each particle in the population, calculate the fitness value of each particle and record the historical optimal solution p of each particle best , obtain the global optimal position g of the particle swarm best ; Step 3: Update the particle velocity based on formula (15) and update the particle position based on formula (16); in, and are the velocity and position of particle q at iteration t+1, w t is the inertia weight of the t-th iteration, represents the speed of particle q in t iterations, c1 and c2 represent learning factors, which are set to 2, r1 and r2 represent random numbers in the range of [0,1], and p best,q represents the best historical position of particle q, represents the position of particle q at t iterations, g best In order to adapt to the nonlinear requirements of roadbed settlement data and further increase the global exploration and local optimization capabilities of the optimization algorithm, the nonlinear inertia weight calculation method given by formula (17) is used to improve the PSO algorithm; Among them, w begin and w end They represent the inertia weight of the first iteration and the inertia weight of the last iteration respectively, Iter represents the current iteration, and Iter_max represents the maximum iteration; Step 4: Calculate the fitness value of each particle, update the particle's historical optimal position and the global optimal position of the particle swarm, and determine whether the maximum number of iterations has been reached. If so, return g best , if otherwise continue to Step3; Step 5: The optimal hyperparameter g returned in step 4 best Substitute the SVR model, build the particle swarm optimized support vector machine model PSVR and obtain the PSVR model prediction results; The specific method of step S2 is as follows: S2.

1. Perform level comparison check on the roadbed settlement data sequence pre-processed in step S1. Specifically, let the sequence composed of n roadbed settlement data pre-processed in step S1 be X. (0) ={x1 (0) ,x2 (0) ,…,x n (0) }, by calculating X (0) Level ratio: Check X (0) Are the level ratios in the interval If X (0) The level ratios are all in the range If X (0) The level ratio is not in the interval If the roadbed settlement data pre-processed in step S1 is smoothed or transformed until the level ratio verification meets the requirements; S2.2, compare the X that has passed the level comparison test (0) The sequence is accumulated to obtain the new sequence X (1) ={x1 (1) ,x2 (1) ,…,x n (1) }; S2.

3. Fit the new sequence obtained in step S2.2 with the first-order linear constant coefficient differential equation of formula (1) as follows: In the formula, X (1) Indicates the cumulative value of each observation point, a=(a ij ) n×n ,b=[b1,b2,…,b i ,…,b n ] T , both represent coefficient matrices; b i and a ij represents the i-th order coefficient, i = 1, 2, ..., n; S2.

4. Discretize equation (1) and obtain equation Where x i (0) (k) represents the grey differential equation; Represents the model base value, and is calculated as shown in formula (3): z i (1) (k)=0.5*[x i (1) (k)+x i (1) (k-1)] (3); S2.

5. Based on formulas (4)-(6), solve for parameters a and b: In the formula, L and Y are intermediate variables introduced to simplify the calculation, and the superscript T represents the transpose of the matrix; S2.

6. According to the parameters a and b, the time response function of the grey model is: Finally, the grey model prediction value is obtained by cumulative reduction of formula (7); The calculation process of coefficient matrices a and b depends on the model base value constructed by the original sequence in formula (3) The solution of , that is, the cardinality value The more accurate, the better the prediction effect, and the base value The calculation is actually solved by numerical integration. In order to further enhance the accuracy of the basic numerical calculation, the integration interval is further refined into H, and [k-1, k] = [c, d] is recorded. The interval [c, d] is divided into H equal parts, and the node x k =c+kh, where k=0,1,2…,H, Solve using the complex trapezoidal formula: Each prediction process of the model adopts an iterative and progressive approach, that is, a and b are calculated based on the original data, and further predictions are made based on the above parameters. Remove the first value of the prediction sequence, use the remaining sequence to rebuild the parameters and update the base value Complete the next prediction task; The accuracy of the prediction model is comprehensively evaluated in step S3. Specifically, when the posterior difference ratio C of the prediction of the grey model IGM(1,1) is less than or equal to 0.65 and the small error probability p is greater than or equal to 0.70%, the prediction result is output and recorded as y1. The specific method of step S6 is: first calculate the sum of squares of the prediction errors of a single model e γ : Where: x τγ is the τth original data of the γth model; is the τth predicted value corresponding to the γth model; is the prediction error; The weight coefficient of a single model is: Where, z represents the total number of models.

2. The railway subgrade settlement prediction method based on the IGM-PSVR model according to claim 1 is characterized in that: The preprocessing in step S1 refers to filling or removing outliers by interpolation. Specifically, for missing values, the average value of the previous and next items is used to fill them; for outliers, the 3σ criterion is used to remove them; and then the original data is smoothed and denoised by wavelet packet transform.

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