Experimental Analysis and Machine Learning of Ground Vibrations Caused by an Elevated High-Speed Railway Based on Random Forest and Bayesian Optimization

Abstract

With the aim of predicting the environmental vibrations induced by an elevated high-speed railway, a machine learning method was developed by combining a random forest algorithm and Bayesian optimization, using a dataset from on-site experiments. When it comes to achieving a rapid and effective prediction of environmental vibrations, there is little research on comparisons between and verifications of different algorithms, and none on the parameter tuning and optimization of machine learning algorithms. In this paper, a field experiment is firstly carried out to measure the ground vibrations caused by high-speed trains running on a bridge, and then the environmental vibration characteristics are analyzed in view of ground accelerations and weighted vibration levels. Subsequently, three machine learning algorithms using linear regression, support vector machine, and random forest are developed using an experimental database, and their prediction performance is discussed. Finally, two optimization models for the hyperparameter set of the random forest algorithm are further compared. The results show that the integrated random forest algorithm has a higher accuracy in predicting environmental vibrations than linear regression and the support vector machine; the Bayesian optimization has an excellent performance and a high efficiency in achieving efficient and in-depth optimization of parameters and can be combined with the RF machine learning algorithm to effectively predict the environmental vibrations induced by the high-speed railway.

1. Introduction

The emergence of the high-speed railway (HSR) has stimulated economic development in Asia, Europe, and America. At the same time, it has also caused an increasing number of annoyances to the daily lives of residents along the railway line [1], affected the normal operation of nearby precision instruments [2], and impacted the integrity protection of ancient buildings and cultural relics [3,4]. In existing high-speed railway lines, a large proportion of the lines is composed of bridges [5], so it is of great significance to study the mechanism and propagation law of environmental vibrations around high-speed railway bridges with prediction models [6,7]. A half-vehicle roller rig (HVRR) [8] has been proposed to simulate the dynamic performance of railway vehicles at a lower cost. To study the instability of high-speed trains and the fatigue damage to the wheels and rails, a useful tool for engineers to estimate the wheel–rail contact forces from train motion signals has been presented [9]. All in all, the prediction and evaluation of environmental vibrations are of great significance for the planning of new HSR lines and environment management during the operating period of existing HSRs.

In 1971, Lang [10] provided the attenuation formula of the vibration level with distance based on field measurement results. ISO 14837-1 [11] focused on the emission–propagation–immission mechanisms of waves from the train-track system (source) to the building (receiver). Although the empirical formula is very convenient and practical [12,13,14], the prediction accuracy is not adequate, especially for some specific soil conditions [15,16]. Thus, analytical and numerical prediction methods have been more and more popular. To better understand the vibration source, an improved track vibration model based on the semi-analytical finite element (SAFE) allows multiple layers of support and the accurate shape of the rail cross-section to be considered [17]. Additionally, the influence of the vehicle’s speed, observation location, rail irregularity, subgrade bed stiffness, and vehicle type on the ground vibrations are investigated thoroughly with some semi-analytical vehicle–track–ground coupling models [18,19,20]. Since the vibration of the track structure and underlying soil induced by HSR trains differs notably from those caused by low-speed trains, numerical methods such as the 3D/2.5D finite element method (FEM) [21,22], the FEM-combining boundary element method (FEM-BEM) [23], TLM-perfectly matched layers (TLM-PML) [24], and some hybrid methods [25] have been proposed. Furthermore, nowadays, with the development of metamaterials, a general view related to low-frequency band gaps concentrated on metamaterials has been presented [26].

Due to the complex interactions involved in the generation and propagation of ground vibrations, most of the traditional prediction methods are very time-consuming. With the development of computer science, the feasibility of using machine learning techniques in predicting railway-induced environmental vibrations have been proven [27,28]. Among the algorithms of the machine learning method, the support vector machine [29], neural networks [30,31], and random forest [32,33] methods have been widely applied to efficiently predict soil vibration and analyze the influencing factors. Silka et al. established the RNN-LSTM neural network model, with which the prediction accuracy of ground vibrations is up to 99% on its validation set [34]. However, there are many hyperparameters in the machine learning method [35], and the diagnostic effectiveness depends significantly on using an appropriate set of hyperparameters [36]. Lyu et al. proposed a complete Bayesian optimization framework to handle multi-objective optimization problems [37]. The advantages of Bayesian optimization have also been validated with experiments [38,39,40,41].

Although the Bayesian optimization method has been paid more and more attention, it is rarely used in the prediction of environmental vibrations induced by elevated high-speed railways. In addition, single machine learning algorithms are often tried, but few comparisons between and verifications of different algorithms are performed. In this paper, the field measurement of ground-borne vibrations from elevated high-speed railways is first carried out, with which the ground vibration characteristics are analyzed and the experimental database is obtained. Subsequently, three kinds of machine learning algorithms are utilized in developing machine learning prediction models of ground vibrations, and their efficiency and accuracy are compared. Finally, by comparing the effectiveness of random search optimization and Bayesian optimization, the integrated machine learning prediction method is proposed and further validated using the experimental database.

2. Materials and Methods

2.1. Overview

The emergence of the high-speed railway (HSR) has caused an increasing number of annoyances to the daily lives of residents along the railway line, so the prediction and evaluation of environmental vibrations are of great significance. A field experiment was first carried out to measure the ground vibrations caused by high-speed trains to obtain the dataset. Then, a data-driven model combined with BO and a random forest algorithm was proposed to analyze and predict the environmental vibrations of the high-speed railway. Figure 1 gives the analysis procedure, which can be divided in the following steps:

(1)

A field experiment was carried out to measure the ground vibrations caused by high-speed trains running on bridge. The layout of measurement points on ground is demonstrated in Figure 1. During the test, a total of 14 sets of valid data were collected, in which the high-speed train traveled at speeds of 200–335 km/h.

(2)

Then, the environmental vibration characteristics were analyzed in view of ground accelerations and weighted vibration levels. Some assessments of environmental vibrations were generated from experimental datasets.

(3)

Based on the assessments, factors with great effects were selected as input features for the machine learning prediction model, and the initial dataset was generated.

(4)

The data set was used to create prediction models with three different machine learning algorithms based on linear regression, support vector machine, and random forest. Those models were compared using 5-fold cross-validation (subgraph lying in top right corner of Figure 1), and the results showed that random forest has the best performance.

(5)

Two optimized vibration prediction models were created on the basis of the random forest algorithm, and the random forest hyperparameters were optimized with the mean value of 5-fold cross-validation as the objective function of BO and RS. The performance of the two optimized models was compared.

(6)

Results showed that Bayesian optimization has excellent performance and high efficiency in achieving efficient and in-depth optimization of parameters and can be combined with the RF machine learning algorithm to effectively predict the environmental vibrations induced by the high-speed railway, which can be seen in subgraph lying in lower right corner of Figure 1.

2.2. Experimental Analysis of Ground Vibration by Elevated HSR

The experimental field is an open farmland with flat terrain near Datong-Xi’an high-speed railway (Figure 2a), in which the HSR concrete bridge consists of a series of simply supported beams with a standard span of 32 m and double track lines.

The layout of seven measuring points on the ground surface is shown in Figure 2b. The intersection of the pier centerline and the ground horizontal line is referred to be the coordinate origin, the train running direction is defined as the horizontal x-axis, the transverse direction perpendicular to the track is the y-axis, and the vertical downward direction is the z-axis. Herein, the distance between the measurement points and the track centerline is denoted by D, so the measuring point of D = 1 m is located at the bottom of the pier since the pier diameter is 2 m.

The sensors used for measuring the ground vibrations were 941B-type acceleration sensors, and three sensors were simultaneously installed at each measuring point to measure the vibration accelerations in the x-, y- and z- directions as shown in Figure 2. In addition, an INV3020S 28-bit network distributed synchronous acquisition instrument was used for data acquisition in the experiment, and the sampling frequency was 512 Hz.

During the test, the high-speed train was CRH380A with an 8-car train formation, and the axle load of the vehicle was about 140 kN. As shown in Figure 3, the vehicle wheelbase L₁ = 2.5 m, the center distance between adjacent bogies of adjacent vehicles L₂ = 7.625 m, the center distance between front and rear bogies of the same vehicle L₃ = 17.375 m, and the total length of a single car L₄ = 25 m. During the test, a total of 14 sets of valid data were collected. During this test, the high-speed railway line had not started official operation, so the train was tested with a trial run. In the trial run, the train speeds were 200–335 km/h.

2.3. Developing of Machine learning Prediction Method

In order to effectively predict the environmental vibrations induced by the HSR train in a variety of conditions, the machine learning technique is introduced in the following section. Machine learning can automatically predict ground vibrations based on large quantities of existing sampling data by means of computer training and learning. The pro-posed prediction procedure is shown in Figure 4.

The proposed prediction method includes two parts: (a) training and (b) predicting. First, training is required, in which the existing dataset is used to train the model in order to predict the ground vibration. As shown in Figure 4, the input dataset for training can be obtained from the experimental data, such as site soil parameters, deterministic parameters of the high-speed railway and bridge, and so on. After the training of the machine learning algorithm, the classifier model can be achieved, from which the amplitude, VAL, and VL of ground vibrations can be output. Secondly, other new parameters of the train, bridge, and soil are input into the feature extractor and classifier model, and then the expected prediction can be obtained with the machine learning model.

In the following, the reliability of linear regression (LR), support vector machine (SVM), and random forest (RF) is discussed and their advantages in solving environmental vibration problems are compared. The LR is a kind of quantitative statistical analysis method that describes the linear dependence between variables. Different from linear regression, the SVM algorithm adopts a non-linear mapping kernel function to map the data from the input space to a high-dimensional feature space. In the high-dimensional space, the samples are linearly separable and nonlinear treatment is possible with kernel functions.

Different from the above two algorithms, the RF algorithm constitutes multiple decision trees with random data, and the final classification results are obtained by voting. RF is an ensemble learning algorithm for solving classification and regression problems. The training data of each tree is random and the split fields of each node in the tree are random, that is, the feature dimensions of the selected samples are different when the tree is split. With the introduction of these two randomness, there are obvious differences among the decision trees of the random forest, which makes for better generalization performance.

In order to compare the accuracy of the above three algorithms, fifty sets of experimental data of ground vibrations were used as the training dataset. Also, the train speed V (200–335 km/h), the distance D from measuring point to pier center (1–40 m), the central frequency fc (1–100 Hz), and the vibration direction were taken as eigenvalues, and the VAL of ground vibration was taken as the prediction value.

2.4. Hyperparameter Tuning and Optimization Method

In machine learning prediction models, if the hyperparameters cannot be chosen and found correctly, underfitting or overfitting problems might be caused. Currently, two common optimization methods for hyperparameter sets are grid search [42] and random search [43]. Grid search determines the optimal value by finding all the points in the search range, while random search does not test all the values between the upper bound and the lower bound, but randomly selects sample points in the search range. When the number of searches is the same, random search will try more parameter values than grid search. However, grid search and random search will ignore the information of the previous point when testing a new point. Bayesian optimization algorithm uses a completely different method from grid search and random search when searching for the optimal parameters, and it uses Gaussian process as the probabilistic model and probability of improvement as the acquisition function that can make full use of the previous information, as shown in Figure 5.

In the hyperparameters of random forest, “n_estimators” is defined as the number of decision trees, and it is easy for the algorithm to be ill-fitted if “n_estimators” is too small. In general, the more decision trees there are, the better the algorithm is, but the computing cost will also increase. When the number of trees exceeds a critical value, the effect of the algorithm is not significantly improved. In addition, “max_features” depends on the maximum feature number to be considered when the optimal model of the decision tree is constructed, and also is the size of the random subset of features to be considered when dividing nodes. “min_samples_split” is the minimum number of samples divisible by nodes, and “max_depth” is the maximum depth of the decision tree. The above four hyperparameters were selected to be optimized and the range of every parameter can be determined via trial calculation and referring to the existing literature [44], as shown in

Table 1. Range of RF parameters.

Hyperparameter	Min	Max
n_estimators	10	300
max_depth	2	40
min_samples_split	2	25
max_features	1	4

.

3. Results and Discussion

3.1. Characteristics Analysis of Measured Ground Vibrations

Figure 6 shows the time histories and Fourier spectra of vertical ground accelerations at different distances Ds when the train speed is V = 335 km/h. From the time histories of acceleration from D = 1 m to D = 32 m on the ground, it can be seen that the peak value of acceleration decreases rapidly, while from D = 11 m to D = 32 m, the attenuation gradually slows down.

It can also be seen from Figure 6a that there is obvious cyclic loading phenomenon as a result of train axles for the vertical ground acceleration, and the loading time is about 2.84 s at the location of D = 1 m on ground, which is just equal to the theoretical calculation time of eight vehicles crossing the bridge $T=\frac{8\times 25+2\times 32}{335/3.6}=2.84\mathrm{s}$, indicating that the measured loading time is consistent with the theoretical loading time.

From the spectra of ground accelerations (Figure 6b), it can be seen that the vibration wave with high-frequency components of 30–100 Hz decays rapidly with the increase in distance, while that with low-frequency components of 0–30 Hz decays more slowly and propagates further.

Since the periodic excitation frequency from the running train can be calculated with f_i = V/L_i (i = 1, 2, 3, 4, and L_i can be seen in Figure 3) [45,46], the wheelbase loading frequency is f₁ = 37.2 Hz, the loading frequency from periodic center distance between adjacent trains is f₂ = 12.2 Hz, the loading frequency from periodic center distance between front and rear bogies of the same train is f₃ = 5.4 Hz, and the loading frequency from periodic carriage length is f₄ = 3.72 Hz. It can be interestingly observed from Figure 6b that several dominant frequencies in the spectra of ground vibration correspond to the periodic loading frequencies from the train, and the exciting frequency f₁ and f₂ are dominant in the four kinds of periodic frequencies excited by trains.

3.2. Assessment of Environmental Vibrations Generated from the HSR Bridge

Since the one-third octave spectrum can reflect the distribution of vibration energy within a different frequency bandwidth, it is usually used to evaluate the influence of environmental vibration on people’s living. According to international standard ISO 8041-1 [47], the evaluation indicators of vibration acceleration level (VAL, units: dB) in one-third octave is adopted herein, which can be calculated as

$$\mathrm{VL}=20\lg \left(\frac{a_{\mathrm{rms}}}{a_0}\right)$$

(1)

where a₀ represents the reference acceleration defined as 10⁻⁶ m/s², and a_rms represents the time-averaged root-mean-square (RMS) acceleration and can be calculated with

$$a_{\mathrm{rms}}=\sqrt{\frac{1}{T}{\displaystyle {\int }_0^T{a}_i^2\left(t\right)\mathrm{d}t}}$$

(2)

where a_i(t) denotes the instantaneous vibration acceleration in a specified axis as a function of the instantaneous time t, and T is the duration of the measurement.

We take a set of experimental data under the train speed V = 300 km/h as an example, and the VALs of all the measuring points on ground in the three directions are shown in Figure 7. It can be found that the variation of VALs with frequencies have similar tendencies, but the vertical vibrations are obviously greater than the horizontal (x-direction) and transverse (y-direction) vibrations.

Also, the characteristic frequencies corresponding to the vibration peaks in Figure 7, usually called predominant frequencies, are 2.5 Hz, 3.15 Hz, 6.3 Hz, 10 Hz, 31.5 Hz, and 50 Hz for the horizontal and transverse ground vibration, and 3.15 Hz, 6.3 Hz, 10 Hz, and 50 Hz for the vertical ground vibrations. Evidently, the environmental vibration has low-frequency properties, and the frequency band between 50–80 Hz is the dominant attenuation zones of environmental vibrations.

According to the sensitivity degree of the human body to environmental vibration with different frequencies in different directions, it is necessary to consider the frequency-weighted acceleration in evaluating the impact of environmental vibration on the human body. Therefore, the overall vibration effect of rail traffic on the nearby environment is evaluated with the frequency-weighted vibration level (VL), which is further put forward as:

$$\mathrm{VL}=20\lg \left(\frac{a_{\mathrm{rms}}^{\prime }}{a_0}\right)$$

(3)

where $a_{\mathrm{rms}}^{\prime }$ represents the frequency-weighted RMS acceleration and can be calculated by

$$a_{\mathrm{rms}}^{\prime }=\sqrt{{\displaystyle \sum _{i=1}^n\left(a_{\mathrm{rms}i}^2{k}_i^2\right)}}$$

(4)

in which k_i represents the weighting factor for the i-th one-third-octave band [47], a_rmsi is the RMS value of vibration acceleration in the i-th one-third-octave band, and n is the one-third-octave band numbers.

Figure 8 shows the VL contours of ground responses with the train speed and the distance. It can be observed from density distribution of the contours that the ground vibrations along the x-direction attenuate rapidly within D = 6 m and those along the y-direction and z-direction attenuate rapidly within D = 11 m for all the train speeds. Moreover, when the distance D is greater than 26 m, most of the environmental vibrations are less than 70 dB and attenuate slowly with the distance. In addition, the VL increases with train speed within D = 6 m, and the increased tendency becomes more and more gentle when D is greater than 11 m.

3.3. Determination of the Optimal Machine Learning Algorithm

In evaluating the performance of the machine learning algorithm, the following indexes are often used: mean square error (MSE), root mean square error (RMSE), absolute mean error (MAE), and determination coefficient (R²). They can be calculated with the following formulas:

$$\mathrm{RMSE}=\sqrt{\frac{1}{m}{\displaystyle \sum _{i=1}^m{(y_i-{\widehat{y}}_i)}^2}}$$

(5)

$$\mathrm{MSE}=\frac{1}{m}{\displaystyle \sum _{i=1}^m{(y_i-{\widehat{y}}_i)}^2}$$

(6)

$$\mathrm{MAE}=\frac{1}{m}{\displaystyle \sum _{i=1}^m\left|(y_i-{\widehat{y}}_i)\right|}$$

(7)

$${\mathrm{R}}^2=1-\frac{{\displaystyle \sum _{i=1}^m{({\widehat{y}}_i-y_i)}^2}}{{\displaystyle \sum _{i=1}^m{({\overline{y}}_i-y_i)}^2}}$$

(8)

where $y_i-{\widehat{y}}_i$ is the difference between real value in the test set and the predicted value, and m is the number of measurements. Obviously, the smaller the indicators MSE, RMSE, and MAE are, the better the fitting effect of the prediction model is; the value range of R² is between 0 and 1, and generally speaking, the larger R² is, the better the fitting effect of the model is.

When the regression fitting effect of the machine learning algorithm is evaluated, the same data set not only carries on the training, but also carries on the error estimation of the model, which is very inaccurate. To overcome this problem, a K-fold cross-validation approach is proposed herein. In the method, the original data is divided into K groups (generally equally divided), and each divided subset of data is considered to be a validation set separately, while the remaining K-1 groups of subset data are used as training sets. Thus, the K models can be obtained, and the average classification accuracy of the final validation set of these K models is used as the performance index, which improves the generalization performance of the regression model in the machine learning algorithm.

In the following, the data set was randomly divided into a training set and a test set in a ratio of 7:3, with 2822 training set samples and 1210 test set samples. The training data sets were trained using LR, SVM (including linear kernel, polynomial kernel, and gaussian radial basis function), decision tree regression, and RF, respectively. In addition, a 5-fold cross-validation was used to assess the performance of each regression diagnostic model: the training set was divided into 5 aliquots, and the model was trained with 80% data at a time, using 20% data to test the accuracy of fault diagnosis model. The comparison of indicators for several machine learning algorithms is shown in

Table 2. Comparison of indicators for several machine learning algorithms.

Model	MSE	RMSE	MAE	R2
LR	373.60	19.33	16.19	0.066
SVM-linear	381.60	19.53	16.16	0.046
SVM-polynomial	212.64	14.58	12.17	0.468
SVM-gaussian radial	239.57	15.48	12.93	0.401
Decision tree	129.99	11.40	9.06	0.675
RF	11.07	3.33	2.36	0.972

.

It can be found from Table 2 that the MSE, RMSE, and MAE of the RF algorithm are the smallest among the six kinds of machine learning algorithms while the R² is the greatest, which indicates that RF is the optimal algorithm in predicting the environmental vibrations induced by high-speed railways. In order to further investigate the prediction performance of LR, SVM, and RF, the prediction values are compared with the actual data, as shown in Figure 9.

Figure 9a reflects the fit goodness of the three kinds of machine learning algorithms with the experiment actual data, and Figure 9b illustrates the proximity of the predicted values to the true data (straight line y = x). It can be observed that the random forest (RF) algorithm has the best performance and best prediction effect among the above mentioned four kinds of algorithms. Therefore, the random forest is chosen for the follow-up work.

3.4. Determination of the Optimal Optimization Method

In this paper, two kinds of optimization methods for RF prediction model were respectively applied: Bayesian optimization (BO) and random search (RS). The RMSE of 5-fold cross validation was taken as the objective function, in which 50 iterations were performed in the BO method, while 300 iterations were performed in the RS method. Firstly, the optimal combination of RF parameters was obtained by means of random search and Bayesian optimization. Then, the prediction performance of RF model was evaluated under the optimal RF parameters, as shown in Figure 10. It can be seen that objective function MSE can reach 9.75 using Bayesian optimization and 10.19 using random search optimization; additionally, the determination coefficient R² can reach up to 0.976 using Bayesian optimization, higher than that using RF and RF-RS. Therefore, the Bayesian optimization can search the optimal parameter set and can make the cross-validation of RF in the training set so accurate that the prediction accuracy can be higher on the test data.

The relationship between the iteration number and the objective function is shown in Figure 11 for the BO method and the RS method, respectively. In Figure 11a, the objective function shows a falling trend in the iteration of the Bayesian optimization algorithm. In Figure 11b, the random search parameters are randomly distributed due to random sampling. In fact, the parameter set obtained by Bayesian optimization after 29 iterations has an obvious convergence process with the optimal parameter set, while the random search has no obvious convergence process with the optimal point search.

All in all, because of the active optimization strategy, the Bayesian optimization algorithm can avoid the evaluation of many useless sampling points, accurately describe the distribution of the objective function, and efficiently find the optimal parameter combination. Random search parameters depend on the number of sampling times; the random sampling point does not fall easily on the optimal combination, so the hit ratio of the optimal parameters is not as good as that of Bayesian optimization. Therefore, the RF machine learning algorithm based on Bayesian optimization is strongly suggested in the prediction of environmental vibrations induced by high-speed railway.

4. Conclusions

In this paper, an on-site vibration experiment of a high-speed railway was carried out, and the characteristics and attenuation law of ground vibration were analyzed. A machine learning technique was constructed using the random forest algorithm based on Bayesian optimization by comparing several different algorithms, and thus a prediction method based on experimental dataset and machine learning is proposed for environmental vibrations caused by HSR trains, and the following conclusions can be drawn:

(1)

The proposed prediction model constructed with the random forest algorithm based on Bayesian optimization is proven to be effective;

(2)

The cyclic loading of vehicles can be observed obviously in the time history of ground acceleration when the observation point is close to the center line of the bridge pier. Also, the periodic exciting frequencies produced by the characteristic parameters of the HSR train can be obviously found as the predominant frequencies in the frequency spectra of ground vibrations;

(3)

For the ground vibrations generated by the elevated HSR railway, the medium-high frequency component within 30–200 Hz attenuates rapidly with the increase in distance, while the low frequency component within 0–30 Hz attenuates slowly and travels far;

(4)

Compared with the linear regression and support vector regression, the random forest algorithm in machine learning has a higher prediction accuracy and is the preferred method in the intellectual prediction of environmental vibration;

(5)

The Bayesian optimization has excellent performance and high efficiency in achieving efficient and in-depth optimization of parameters, and can be combined with the RF machine learning algorithm to effectively predict the environmental vibrations induced by a high-speed railway;

(6)

The size of the high-speed railway vibration database is an important factor that restricts the effectiveness of machine learning. In this paper, the amount of experimental data is small, and the expanding of database scope from a large number of on-site experiments is the focus in the future research.