# Prediction of Metro Train-Induced Tunnel Vibrations Using Machine Learning Method

Authors: Zhuosheng Xu; Meng Ma; Zikai Zhou; Xintong Xie; Haoxiang Xie; Bolong Jiang; Zhongshuai Zhang
Journal: Advances in Civil Engineering
 (2022)

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## Abstract

The tunnel vibration level is usually employed as a vibration source intensity of the empirical prediction method. Currently, the analogy test and data base are two main means to determine the vibration source intensity. To improve the accuracy efficiency, the machine learning (ML) method was introduced to predict the tunnel vibration responses. To acquire model training samples, the measurements were performed in 80 different running tunnel sections of Beijing metro lines. Two types of method, back propagation neural network (BPNN) and generalised regression neural network (GRNN) were employed, which can make full use of characteristics of measured samples and reduce the data noise. The results indicate that the prediction efficiency is high and the mean square errors of the two ML methods are acceptable. Accordingly, both of the ML methods can be used as the reference of vibration source intensity in metro train-induced environmental impact evaluation. GRNN has relatively better predicting ability than BPNN.


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## Body

## 1. Introduction

With the development of urban rail transit construction, the environmental vibration problem arising from metro operation is becoming more and more prominent [1–4]. A reasonable vibration mitigation design has put forward higher requirements to the environmental vibration prediction. Various types of prediction model can be employed in different construction stages of a metro project [5, 6]. In the feasibility study stage of developing a rail system, the scoping or preliminary prediction can be used to identify whether the environmental vibration is an issue for potential sensitive buildings along the rail transportation alignment. Empirical and semi-empirical models are widely used in this stage [7–9]. Recently, the machine learning (ML) method has been introduced in the scoping prediction, such as the researches by Paneiro et al. [10], Connolly et al. [11, 12], Chen et al. [13], Yao et al. [14], Paneiro et al. [15], Fang et al. [16] and Liang et al. [17]. In the scheme design stage, the determined prediction can be used, including various types of numerical [18–22], analytical/semi-analytical methods [23–26]. In the construction design stage, the detail prediction methods were developed, such as measurement-based transfer function method [27–29] and hybrid methods [30–34].A chain-type formula based on the assumption of uncoupled sub-systems is a classical empirical prediction method. Its idea was originally proposed by Kurzweil [35] and Melke [36], and developed in different standards and guidelines [9, 37–39]. In Chinese code HJ 453-2018 [38], the predicted environmental vibration level VLz can be calculated by the superposition of the vibration source level VLZ,0 and a series of vibration level correction terms, details can be found in reference [40]. The value of VLZ,0 is defined as the vertical weighted vibration acceleration level on the tunnel wall. Two main approaches can be used to determine VLZ,0. One is analogy test in a similar running tunnel section, which is regarded as the most accuracy method. Another is searching the database, in which the test has been performed in a similar tunnel section. However, there are disadvantages on both of the methods. The analogy test is time consuming, especially when the workload is heavy. Furthermore, the parameters of the test section should be consistent with the predicted section as far as possible. These parameters include train speed, route radius, tunnel shape and size, track type, soil parameters, etc. It is almost impossible to ensure that all parameters of the two sections are consistent, which introduces errors of the source intensity values. Owing to the large time and labor cost of the analogy test method, the database method is also recommended by standards. However, the database method has the same problem as the analogy test method, and the error is greater because the amount of data available for reference is not large enough.To solve the problem and improve the prediction efficiency ofVLZ,0, ML method was introduced in the present study. The in-situ measurements were performed in 80 different tunnel sections of Beijing metro lines and the model training samples of VLZ,0 were obtained. Finally, two types of ML method were analysed and the prediction results were validated.


## 2. Measurement in Metro Tunnel

### 2.1. Measurement Outline

To obtain the data ofVLZ,0 for training model, in-situ measurements were performed in 80 different running tunnel sections in Beijing metro. Various types of parameters were obtained and considered for each section, including track type, radius, tunnel shape, train speed and vehicle type, details as:(i)
Track type: regular slab track, steel spring floating slab track (FST), rubber isolator FST; ladder sleeper track, slab track with short sleeper, and slab track with elastic sleeper;(ii)
Radius: from 350 m to infinite (straight line);(iii)
Tunnel shape: horse-shoe tunnel and shield tunnel;(iv)
Train speed: between 15 and 92 km/h;(v)
Vehicle type: types A and B.Figure1 illustrates the measuring point location in a tunnel with FST. According to the specification HJ 453-2018 [38], the location of the vibration source intensity is defined at the tunnel wall, with 1.25 m height from the rail top. For the curved tunnel, the sensors were installed on the side of inner rails.Figure 1
Measuring point location on the tunnel wall (unit: mm).In these tests, the date acquisition equipment INV 3060S was employed with a maximum sampling frequency of 51.2 kHz. The accelerometer is Lance 0105T with a measurement range of 20 g and working frequency between 0.35 and 6000 Hz.


### 2.2. Measurement Result

The vibration source intensityVLZ,0 is expressed as the maximum Z-vibration level, defined as:(1)VLz,0=maxtVLZt=maxt20lgaw,τta0,aw,τt=1τ∫0taw2ξdξ,where VLz(t) is the frequency-weighed vertical vibration acceleration level as a function of time t, aw,τt is the running root-mean-square weighted acceleration, awξ is the frequency-weighed instantaneous vibration acceleration at time ξ, τ is the integration time of the measurement, and t is the instantaneous time. The weighting factor suggested by ISO 2631/1 was employed in this study.An illustration of the calculation method for maximum Z-vibration level was shown in Figure2.Figure 2
Illustration of calculation method for maximum Z-vibration level.All the values ofVLZ,0 were averaged by five recording pass-by trains. Finally the averaged VLZ,0 of the 80 test sections were listed in Table 1. Figure 3 illustrates the VLZ,0 of different track type varies with train speeds. Generally, VLZ,0 increases with train speed, especially below 40 km/h. Generally speaking, the vibration reduction effect of steel spring FST is better than that of rubber isolator FST. However, sections 10 and 12 were measured in a deep buried horse-shoe tunnel, where the surrounding rock condition and tunnel shape affect the test results of the tunnel responses.Table 1
Average source intensity and parameter of 80 sections.
Test section nr.Track typeRadius (m)Tunnel typeSpeed (km/h)Vehicle typeAverageVLz,0 (dB)1Regular slab track650Shield tunnel70B77.82Steel spring FST650Shield tunnel70B60.533Steel spring FSTStraight lineShield tunnel70B59.54Regular slab trackStraight lineShield tunnel70B805Steel spring FSTStraight lineShield tunnel70B64.56Steel spring FSTStraight lineShield tunnel70B667Regular slab trackStraight lineShield tunnel75B64.78Steel spring FST450Horse-shoe tunnel59.8B45.19Regular slab track450Horse-shoe tunnel64.7B66.210Rubber isolator FST490Horse-shoe tunnel75.4B51.911Regular slab track505Horse-shoe tunnel66.2B64.912Rubber isolator FST505Horse-shoe tunnel77.5B52.413Regular slab track505Horse-shoe tunnel66.2B64.914Ladder sleeper trackStraight lineShield tunnel57.8B68.915Regular slab trackStraight lineShield tunnel60.4B82.516Steel spring FST650Shield tunnel88B73.6317Steel spring FST650Shield tunnel92B73.5618Ladder sleeper track650Shield tunnel90B88.7519Ladder sleeper trackStraight lineShield tunnel90B79.9620Ladder sleeper track650Shield tunnel75B82.9121Ladder sleeper track1156Shield tunnel87B75.9322Slab track with short sleeper350Shield tunnel15B6823Slab track with short sleeperStraight lineShield tunnel15B65.224Slab track with short sleeperStraight lineShield tunnel15B61.225Steel spring FSTStraight lineShield tunnel15B4726Slab track with elastic long sleeper350Shield tunnel15B63.227Steel spring FSTStraight lineShield tunnel15B53.128Slab track with short sleeper350Shield tunnel25B7429Slab track with short sleeperStraight lineShield tunnel25B67.230Slab track with short sleeperStraight lineShield tunnel25B59.231Steel spring FSTStraight lineShield tunnel25B50.532Slab track with elastic long sleeper350Shield tunnel25B69.133Slab track with elastic long sleeper500Shield tunnel25B68.134Slab track with elastic long sleeperStraight lineShield tunnel25B60.935Steel spring FSTStraight lineShield tunnel25B58.836Slab track with short sleeper350Shield tunnel35B76.337Slab track with short sleeperStraight lineShield tunnel35B77.638Slab track with short sleeperStraight lineShield tunnel35B69.539Steel spring FSTStraight lineShield tunnel35B54.540Slab track with elastic long sleeper350Shield tunnel35B74.741Slab track with elastic long sleeper500Shield tunnel35B80.742Slab track with elastic long sleeperStraight lineShield tunnel35B70.543Steel spring FSTStraight lineShield tunnel35B55.644Slab track with short sleeper350Shield tunnel45B78.745Slab track with short sleeperStraight lineShield tunnel45B81.746Slab track with short sleeperStraight lineShield tunnel45B71.847Steel spring FSTStraight lineShield tunnel45B56.348Slab track with elastic long sleeper350Shield tunnel45B74.849Slab track with elastic long sleeper500Shield tunnel45B78.250Slab track with elastic long sleeperStraight lineShield tunnel45B72.851Steel spring FSTStraight lineShield tunnel45B57.452Slab track with short sleeper350Shield tunnel57B81.453Slab track with short sleeperStraight lineShield tunnel57B82.454Slab track with short sleeperStraight lineShield tunnel57B72.655Steel spring FSTStraight lineShield tunnel57B57.256Slab track with elastic long sleeper350Shield tunnel57B74.657Slab track with elastic long sleeper500Shield tunnel57B77.258Slab track with elastic long sleeperStraight lineShield tunnel57B71.359Steel spring FSTStraight lineShield tunnel57B57.460Ladder sleeper trackStraight lineHorse-shoe tunnel64B68.79561Ladder sleeper trackStraight lineHorse-shoe tunnel64B68.6362Regular slab trackStraight lineShield tunnel67.7B8363Steel spring FST350Shield tunnel45B61.564Ladder sleeper trackStraight lineShield tunnel60.6B71.565Steel spring FSTStraight lineShield tunnel75B64.566Steel spring FSTStraight lineHorse-shoe tunnel68.6B61.567Steel spring FSTStraight lineHorse-shoe tunnel67.6B6868Regular slab trackStraight lineHorse-shoe tunnel68.3B7869Regular slab trackStraight lineHorse-shoe tunnel69.3B6870Regular slab trackStraight lineShield tunnel69.4B7871Regular slab trackStraight lineShield tunnel36B81.272Regular slab trackStraight lineShield tunnel40B84.973Regular slab trackStraight lineShield tunnel69B85.974Regular slab trackStraight lineShield tunnel76B8275Regular slab trackStraight lineShield tunnel78B81.676Regular slab trackStraight lineShield tunnel69B85.877Regular slab trackStraight lineShield tunnel69B78.878Regular slab trackStraight lineShield tunnel72B79.779Regular slab trackStraight lineShield tunnel68A84.580Regular slab trackStraight lineShield tunnel68A87.5Figure 3
Tunnel vibration responses vary with train speed.


## 2.1. Measurement Outline

To obtain the data ofVLZ,0 for training model, in-situ measurements were performed in 80 different running tunnel sections in Beijing metro. Various types of parameters were obtained and considered for each section, including track type, radius, tunnel shape, train speed and vehicle type, details as:(i)
Track type: regular slab track, steel spring floating slab track (FST), rubber isolator FST; ladder sleeper track, slab track with short sleeper, and slab track with elastic sleeper;(ii)
Radius: from 350 m to infinite (straight line);(iii)
Tunnel shape: horse-shoe tunnel and shield tunnel;(iv)
Train speed: between 15 and 92 km/h;(v)
Vehicle type: types A and B.Figure1 illustrates the measuring point location in a tunnel with FST. According to the specification HJ 453-2018 [38], the location of the vibration source intensity is defined at the tunnel wall, with 1.25 m height from the rail top. For the curved tunnel, the sensors were installed on the side of inner rails.Figure 1
Measuring point location on the tunnel wall (unit: mm).In these tests, the date acquisition equipment INV 3060S was employed with a maximum sampling frequency of 51.2 kHz. The accelerometer is Lance 0105T with a measurement range of 20 g and working frequency between 0.35 and 6000 Hz.


## 2.2. Measurement Result

The vibration source intensityVLZ,0 is expressed as the maximum Z-vibration level, defined as:(1)VLz,0=maxtVLZt=maxt20lgaw,τta0,aw,τt=1τ∫0taw2ξdξ,where VLz(t) is the frequency-weighed vertical vibration acceleration level as a function of time t, aw,τt is the running root-mean-square weighted acceleration, awξ is the frequency-weighed instantaneous vibration acceleration at time ξ, τ is the integration time of the measurement, and t is the instantaneous time. The weighting factor suggested by ISO 2631/1 was employed in this study.An illustration of the calculation method for maximum Z-vibration level was shown in Figure2.Figure 2
Illustration of calculation method for maximum Z-vibration level.All the values ofVLZ,0 were averaged by five recording pass-by trains. Finally the averaged VLZ,0 of the 80 test sections were listed in Table 1. Figure 3 illustrates the VLZ,0 of different track type varies with train speeds. Generally, VLZ,0 increases with train speed, especially below 40 km/h. Generally speaking, the vibration reduction effect of steel spring FST is better than that of rubber isolator FST. However, sections 10 and 12 were measured in a deep buried horse-shoe tunnel, where the surrounding rock condition and tunnel shape affect the test results of the tunnel responses.Table 1
Average source intensity and parameter of 80 sections.
Test section nr.Track typeRadius (m)Tunnel typeSpeed (km/h)Vehicle typeAverageVLz,0 (dB)1Regular slab track650Shield tunnel70B77.82Steel spring FST650Shield tunnel70B60.533Steel spring FSTStraight lineShield tunnel70B59.54Regular slab trackStraight lineShield tunnel70B805Steel spring FSTStraight lineShield tunnel70B64.56Steel spring FSTStraight lineShield tunnel70B667Regular slab trackStraight lineShield tunnel75B64.78Steel spring FST450Horse-shoe tunnel59.8B45.19Regular slab track450Horse-shoe tunnel64.7B66.210Rubber isolator FST490Horse-shoe tunnel75.4B51.911Regular slab track505Horse-shoe tunnel66.2B64.912Rubber isolator FST505Horse-shoe tunnel77.5B52.413Regular slab track505Horse-shoe tunnel66.2B64.914Ladder sleeper trackStraight lineShield tunnel57.8B68.915Regular slab trackStraight lineShield tunnel60.4B82.516Steel spring FST650Shield tunnel88B73.6317Steel spring FST650Shield tunnel92B73.5618Ladder sleeper track650Shield tunnel90B88.7519Ladder sleeper trackStraight lineShield tunnel90B79.9620Ladder sleeper track650Shield tunnel75B82.9121Ladder sleeper track1156Shield tunnel87B75.9322Slab track with short sleeper350Shield tunnel15B6823Slab track with short sleeperStraight lineShield tunnel15B65.224Slab track with short sleeperStraight lineShield tunnel15B61.225Steel spring FSTStraight lineShield tunnel15B4726Slab track with elastic long sleeper350Shield tunnel15B63.227Steel spring FSTStraight lineShield tunnel15B53.128Slab track with short sleeper350Shield tunnel25B7429Slab track with short sleeperStraight lineShield tunnel25B67.230Slab track with short sleeperStraight lineShield tunnel25B59.231Steel spring FSTStraight lineShield tunnel25B50.532Slab track with elastic long sleeper350Shield tunnel25B69.133Slab track with elastic long sleeper500Shield tunnel25B68.134Slab track with elastic long sleeperStraight lineShield tunnel25B60.935Steel spring FSTStraight lineShield tunnel25B58.836Slab track with short sleeper350Shield tunnel35B76.337Slab track with short sleeperStraight lineShield tunnel35B77.638Slab track with short sleeperStraight lineShield tunnel35B69.539Steel spring FSTStraight lineShield tunnel35B54.540Slab track with elastic long sleeper350Shield tunnel35B74.741Slab track with elastic long sleeper500Shield tunnel35B80.742Slab track with elastic long sleeperStraight lineShield tunnel35B70.543Steel spring FSTStraight lineShield tunnel35B55.644Slab track with short sleeper350Shield tunnel45B78.745Slab track with short sleeperStraight lineShield tunnel45B81.746Slab track with short sleeperStraight lineShield tunnel45B71.847Steel spring FSTStraight lineShield tunnel45B56.348Slab track with elastic long sleeper350Shield tunnel45B74.849Slab track with elastic long sleeper500Shield tunnel45B78.250Slab track with elastic long sleeperStraight lineShield tunnel45B72.851Steel spring FSTStraight lineShield tunnel45B57.452Slab track with short sleeper350Shield tunnel57B81.453Slab track with short sleeperStraight lineShield tunnel57B82.454Slab track with short sleeperStraight lineShield tunnel57B72.655Steel spring FSTStraight lineShield tunnel57B57.256Slab track with elastic long sleeper350Shield tunnel57B74.657Slab track with elastic long sleeper500Shield tunnel57B77.258Slab track with elastic long sleeperStraight lineShield tunnel57B71.359Steel spring FSTStraight lineShield tunnel57B57.460Ladder sleeper trackStraight lineHorse-shoe tunnel64B68.79561Ladder sleeper trackStraight lineHorse-shoe tunnel64B68.6362Regular slab trackStraight lineShield tunnel67.7B8363Steel spring FST350Shield tunnel45B61.564Ladder sleeper trackStraight lineShield tunnel60.6B71.565Steel spring FSTStraight lineShield tunnel75B64.566Steel spring FSTStraight lineHorse-shoe tunnel68.6B61.567Steel spring FSTStraight lineHorse-shoe tunnel67.6B6868Regular slab trackStraight lineHorse-shoe tunnel68.3B7869Regular slab trackStraight lineHorse-shoe tunnel69.3B6870Regular slab trackStraight lineShield tunnel69.4B7871Regular slab trackStraight lineShield tunnel36B81.272Regular slab trackStraight lineShield tunnel40B84.973Regular slab trackStraight lineShield tunnel69B85.974Regular slab trackStraight lineShield tunnel76B8275Regular slab trackStraight lineShield tunnel78B81.676Regular slab trackStraight lineShield tunnel69B85.877Regular slab trackStraight lineShield tunnel69B78.878Regular slab trackStraight lineShield tunnel72B79.779Regular slab trackStraight lineShield tunnel68A84.580Regular slab trackStraight lineShield tunnel68A87.5Figure 3
Tunnel vibration responses vary with train speed.


## 3. PredictingVLz,0 Using ML Method

To provide a fast and accuracy prediction ofVLz,0 based on the measured samples, two types of ML methods were used: back propagation neural network (BPNN) and generalised regression neural network (GRNN).


### 3.1. BPNN Based Prediction

#### 3.1.1. Method Introduction

BPNN is a type of multilayer feedforward neural network which can acquire output vectors by processing input vectors through hidden layers (Figure4). The output error can be evaluated by the error function. The error back propagation can be carried out by the gradient descent method based on the output error. Then, the connection weight wi and threshold bi between neurons can be modified. Finally, the error of the neural network can be decreased to the minimum. The weight adjustment can be regarded as a prior probability distribution of the weight and threshold. Then, the posterior probability distribution of the weight and threshold are adjusted based on different input data. Finally, the network parameters can be modified and the generalisation ability of network is improved.Figure 4
Topology of BPNN.The BPNN can be optimised by introducing the Bayes’ principle, by which a modification function for the performance function is introduced:(2)Fw=αEw+Edβ,where, α and β are hyper-parameters; Ew is the network coefficient related to the weight; Ed is the conventional error term. Ew and Ed can be expressed as:(3)Ew=1m∑i=1mwi2,ED=1N∑i=1Nyi−ti2,where m and N are the neurons number of output and hidden layers, wi is the initial weight, and yi is the output vector.When predicting with BPNN, the initialised weightwi and hyper-parameters α and β need to be randomised firstly. Subsequently, the training set P is input as training samples. After training, the weight wMP is calculated at which the grad of equation (2) is the minimum. Finally, the hyper-parameters can be calculated:(4)aMP=γ2EwwMP,βMP=N−γ2EDwMP,where, γ can be calculated based on wMP and the renewed values of α and β can be re-determined. In equation (5), αMP and βMP are the α and β when w values wMP. The steps above are repeated until the network converge [41].According to the number of data eigenvalues, six nodes were set in the input layer, and the network layer number was set as 3 to lower the complexity of network and pretend over-fitting. One node was set in the output layer, and output value was the predicted vibration source intensity, e.g. maximum Z-vibration level.The mean squared error (MSE) was used to analyse the prediction performance of the network. MSE is defined as:(5)MSE=1n∑i=1Ny^i−yi2,where yi and y^i are the true and predicted values of the test set, respectively; N is the number of output layer.In this study, the value ofVLZ,0 is expressed in dB, so MSE is measured in dB2. Generally, the VLZ,0 is between 60 and 80 dB. If the predicted average percentage error is 10%, the absolute error is approximately between 6 and 8 dB. That is, if MSE is below 36 dB2, it can be regarded as an acceptable result.The relationship between node number of hidden layers, eigenvalue number and input/output node number can be expressed as [42].(6)s=m+n+a,where s and m are node numbers of the hidden layer and input layer.


#### 3.1.2. Sample Training and Results

According to equation (6), the test begins with three nodes of the hidden layers. With the same training set and test set, the node number of the hidden layer can be increased gradually. The training is repeated three times for each hidden layer, and the averaged value can be obtained. The training results are illustrated in Figure 5 and the final optimised node number was six.Figure 5
MSE of different node number of hidden layer.The model was building using the neural network toolbox in MATLAB. The activation function for the hidden layer was Sigmod function and for the output layer was purelin function. The maximum convergence times was set as 1000, and the maximum training accuracy was 0. The Bayesian regularisation method (trainbr) was selected as the training method. A total of eight groups of data were randomly selected for test and the rest 72 groups of data were used as the train set. After training the neural network with the train set, the test set was predicted.When training, a weight matrix was randomly generated and then weight was modified with the transmission error.Figure6 illustrates the training results by comparing measured and predicted values. Based on equation (5), MSE of this test set can be calculated as 29.98, which proves the accuracy of training model is good enough to perform the prediction. According to detailed information in Figure 6, the absolute error of MSE is controlled within 10%, ranging from −7.57% to 9.86%, which demonstrates an acceptable prediction ability of this model.Figure 6
Predicted result of back propagation neural network.


#### 3.1.3. Test and Verification

To ensure the generalisation ability of the network, a cross validation was performed. After arranging data randomly, the data were divided into ten subsets. Every subset was selected as a test set, and the remaining were training sets. The cross validation were repeated ten times following the above steps. Figure7 demonstrates MSE of ten cross validation results and the averaged MSE is 32.28, which demonstrates a good generalisation ability of this method.Figure 7
Cross validation of source intensity prediction.Furthermore, the method of leave-one-out-cross-validation (LOOCV) was used to calculate and analyse the errors between measured and predicted values. Figure8 illustrates the error normal distribution of BPNN using LOOCV.Figure 8
Error distribution of vibration source intensity predicted using BPNN.Based on the verification of LOOCV and cross validation, the prediction error is generally below 10% and the average MSE is less than 35. Accordingly, BPNN can be used to undertake a preliminary prediction of the source intensity vibration of a running metro train.Moreover, coefficient of determinationR2 can be used to evaluate linear correlation degree of network fitting result. The coefficient of determination is defined as(7)R2=SSRSST,SSR=∑iy^i−y¯i2,SST=∑iyi−y¯i2,where, yi is the true value, y^i is the predicted value, and y¯i is the average of true value.Coefficient of determinationR2 ranges from 0 to 1. The closer R2 approaches to 1, the better the linear correlation degree of network fitting result is. After calculation, R2 of BPNN is 0.9464, which reflects the BPNN model can also be used to predict the vibration source intensity.


### 3.2. GRNN Based Prediction

#### 3.2.1. Method Introduction

GRNN is a type of radial basis function (RBF) neural network, which has strong non-linear mapping capability, fault tolerance and robustness. Besides, it has the advantage in the approaching ability and learning speed. The model of radial basis neural network includes two main independent variable and basis function. The independent variable is the Euclidean distance between the points to be measured and sampling. The basis function is a radial function. GRNN can transform a multi-dimensional problem into a one-dimensional problem. After transformation, the independent variable of one-dimensional issue turns into the Euclidean distance mentioned above. Any function can be obtained by making weighted combination of basis functions. Figure9 demonstrates the topology of GRNN.Figure 9
Topology of GRNN.In the calculation and analyse by GRNN, the training setP is firstly input as a learning sample, and then SD and SNi are calculated and output to the summation layer. SD and SNi are neurons in the hidden layer, calculated by two different methods. One is the summation of denominator neurons, i.e. straightly summing up all the neurons in hidden layers. The other is the summation of molecular neurons, i.e. making a weighted summation of neurons in mode layers. SD and SNi can be calculated by(8)SD=∑i=1nPi=∑i=1nexpx−xiTx−xi2σi2,i=1,2,3…n,SNi=∑i=1nYiPi=∑i=1nyijexpX−XiTX−Xi2σi2,j=1,2,3…n,where σi is a network expansion constant, yij is the connecting weight of the i-th neuron in the summation layer and the j-th neuron in the mode layer. Finally, the output network predicted value can be obtained: yj = SD/SNi [43].According to the characteristic of RBF neural network, the data should be expressed in the form of scientific counting before normalisation. Then, the route radius was multiplied by 0.01 and the train speed was multiplied by 0.1. Based on the fast learning speed of GRNN, building a large network and performing mutual authentications are possible. Then, the network was built for all 80 groups of data one by one. Thei-th group of data was selected as the test set and the remaining were the train sets. Thus, a total of 80 networks can be established and the data can be made full used.


#### 3.2.2. Results, Test and Verification

After training, 80 groups of predicted results can be obtained using LOOCV. Figure10 demonstrates the comparison between predicted and measured values.Figure 10
GRNN based predicted result.Figure11 demonstrates MSE values under different values of σ. According to Figure 11, the value of σ have little influence on the result of this experiment. When σ = 1 the MSE value is the minimum. Accordingly, this value was determined in this experiment.Figure 11
MSE of GRNN under different values ofσ.Both of the MSE and coefficient of determinationR2 were used to evaluate linear correlation degree of network fitting result. After calculation, MSE is 17.9205, and R2 is 0.8153. Figure 12 demonstrates the error distribution. For its normal fitting curve, average value is 0.004664 and standard deviation is 0.07571.Figure 12
Source intensity prediction error distribution predicted by GRNN.The above results and verification proves that GRNN can be used to predict the vibration source intensity with considering different parameters. Compared with BPNN, both of the calculation efficiency and the accuracy of GRNN are higher. Accordingly, GRNN is more recommended to predict the vibration source intensity.


## 3.1. BPNN Based Prediction

### 3.1.1. Method Introduction

BPNN is a type of multilayer feedforward neural network which can acquire output vectors by processing input vectors through hidden layers (Figure4). The output error can be evaluated by the error function. The error back propagation can be carried out by the gradient descent method based on the output error. Then, the connection weight wi and threshold bi between neurons can be modified. Finally, the error of the neural network can be decreased to the minimum. The weight adjustment can be regarded as a prior probability distribution of the weight and threshold. Then, the posterior probability distribution of the weight and threshold are adjusted based on different input data. Finally, the network parameters can be modified and the generalisation ability of network is improved.Figure 4
Topology of BPNN.The BPNN can be optimised by introducing the Bayes’ principle, by which a modification function for the performance function is introduced:(2)Fw=αEw+Edβ,where, α and β are hyper-parameters; Ew is the network coefficient related to the weight; Ed is the conventional error term. Ew and Ed can be expressed as:(3)Ew=1m∑i=1mwi2,ED=1N∑i=1Nyi−ti2,where m and N are the neurons number of output and hidden layers, wi is the initial weight, and yi is the output vector.When predicting with BPNN, the initialised weightwi and hyper-parameters α and β need to be randomised firstly. Subsequently, the training set P is input as training samples. After training, the weight wMP is calculated at which the grad of equation (2) is the minimum. Finally, the hyper-parameters can be calculated:(4)aMP=γ2EwwMP,βMP=N−γ2EDwMP,where, γ can be calculated based on wMP and the renewed values of α and β can be re-determined. In equation (5), αMP and βMP are the α and β when w values wMP. The steps above are repeated until the network converge [41].According to the number of data eigenvalues, six nodes were set in the input layer, and the network layer number was set as 3 to lower the complexity of network and pretend over-fitting. One node was set in the output layer, and output value was the predicted vibration source intensity, e.g. maximum Z-vibration level.The mean squared error (MSE) was used to analyse the prediction performance of the network. MSE is defined as:(5)MSE=1n∑i=1Ny^i−yi2,where yi and y^i are the true and predicted values of the test set, respectively; N is the number of output layer.In this study, the value ofVLZ,0 is expressed in dB, so MSE is measured in dB2. Generally, the VLZ,0 is between 60 and 80 dB. If the predicted average percentage error is 10%, the absolute error is approximately between 6 and 8 dB. That is, if MSE is below 36 dB2, it can be regarded as an acceptable result.The relationship between node number of hidden layers, eigenvalue number and input/output node number can be expressed as [42].(6)s=m+n+a,where s and m are node numbers of the hidden layer and input layer.


### 3.1.2. Sample Training and Results

According to equation (6), the test begins with three nodes of the hidden layers. With the same training set and test set, the node number of the hidden layer can be increased gradually. The training is repeated three times for each hidden layer, and the averaged value can be obtained. The training results are illustrated in Figure 5 and the final optimised node number was six.Figure 5
MSE of different node number of hidden layer.The model was building using the neural network toolbox in MATLAB. The activation function for the hidden layer was Sigmod function and for the output layer was purelin function. The maximum convergence times was set as 1000, and the maximum training accuracy was 0. The Bayesian regularisation method (trainbr) was selected as the training method. A total of eight groups of data were randomly selected for test and the rest 72 groups of data were used as the train set. After training the neural network with the train set, the test set was predicted.When training, a weight matrix was randomly generated and then weight was modified with the transmission error.Figure6 illustrates the training results by comparing measured and predicted values. Based on equation (5), MSE of this test set can be calculated as 29.98, which proves the accuracy of training model is good enough to perform the prediction. According to detailed information in Figure 6, the absolute error of MSE is controlled within 10%, ranging from −7.57% to 9.86%, which demonstrates an acceptable prediction ability of this model.Figure 6
Predicted result of back propagation neural network.


### 3.1.3. Test and Verification

To ensure the generalisation ability of the network, a cross validation was performed. After arranging data randomly, the data were divided into ten subsets. Every subset was selected as a test set, and the remaining were training sets. The cross validation were repeated ten times following the above steps. Figure7 demonstrates MSE of ten cross validation results and the averaged MSE is 32.28, which demonstrates a good generalisation ability of this method.Figure 7
Cross validation of source intensity prediction.Furthermore, the method of leave-one-out-cross-validation (LOOCV) was used to calculate and analyse the errors between measured and predicted values. Figure8 illustrates the error normal distribution of BPNN using LOOCV.Figure 8
Error distribution of vibration source intensity predicted using BPNN.Based on the verification of LOOCV and cross validation, the prediction error is generally below 10% and the average MSE is less than 35. Accordingly, BPNN can be used to undertake a preliminary prediction of the source intensity vibration of a running metro train.Moreover, coefficient of determinationR2 can be used to evaluate linear correlation degree of network fitting result. The coefficient of determination is defined as(7)R2=SSRSST,SSR=∑iy^i−y¯i2,SST=∑iyi−y¯i2,where, yi is the true value, y^i is the predicted value, and y¯i is the average of true value.Coefficient of determinationR2 ranges from 0 to 1. The closer R2 approaches to 1, the better the linear correlation degree of network fitting result is. After calculation, R2 of BPNN is 0.9464, which reflects the BPNN model can also be used to predict the vibration source intensity.


## 3.1.1. Method Introduction

BPNN is a type of multilayer feedforward neural network which can acquire output vectors by processing input vectors through hidden layers (Figure4). The output error can be evaluated by the error function. The error back propagation can be carried out by the gradient descent method based on the output error. Then, the connection weight wi and threshold bi between neurons can be modified. Finally, the error of the neural network can be decreased to the minimum. The weight adjustment can be regarded as a prior probability distribution of the weight and threshold. Then, the posterior probability distribution of the weight and threshold are adjusted based on different input data. Finally, the network parameters can be modified and the generalisation ability of network is improved.Figure 4
Topology of BPNN.The BPNN can be optimised by introducing the Bayes’ principle, by which a modification function for the performance function is introduced:(2)Fw=αEw+Edβ,where, α and β are hyper-parameters; Ew is the network coefficient related to the weight; Ed is the conventional error term. Ew and Ed can be expressed as:(3)Ew=1m∑i=1mwi2,ED=1N∑i=1Nyi−ti2,where m and N are the neurons number of output and hidden layers, wi is the initial weight, and yi is the output vector.When predicting with BPNN, the initialised weightwi and hyper-parameters α and β need to be randomised firstly. Subsequently, the training set P is input as training samples. After training, the weight wMP is calculated at which the grad of equation (2) is the minimum. Finally, the hyper-parameters can be calculated:(4)aMP=γ2EwwMP,βMP=N−γ2EDwMP,where, γ can be calculated based on wMP and the renewed values of α and β can be re-determined. In equation (5), αMP and βMP are the α and β when w values wMP. The steps above are repeated until the network converge [41].According to the number of data eigenvalues, six nodes were set in the input layer, and the network layer number was set as 3 to lower the complexity of network and pretend over-fitting. One node was set in the output layer, and output value was the predicted vibration source intensity, e.g. maximum Z-vibration level.The mean squared error (MSE) was used to analyse the prediction performance of the network. MSE is defined as:(5)MSE=1n∑i=1Ny^i−yi2,where yi and y^i are the true and predicted values of the test set, respectively; N is the number of output layer.In this study, the value ofVLZ,0 is expressed in dB, so MSE is measured in dB2. Generally, the VLZ,0 is between 60 and 80 dB. If the predicted average percentage error is 10%, the absolute error is approximately between 6 and 8 dB. That is, if MSE is below 36 dB2, it can be regarded as an acceptable result.The relationship between node number of hidden layers, eigenvalue number and input/output node number can be expressed as [42].(6)s=m+n+a,where s and m are node numbers of the hidden layer and input layer.


## 3.1.2. Sample Training and Results

According to equation (6), the test begins with three nodes of the hidden layers. With the same training set and test set, the node number of the hidden layer can be increased gradually. The training is repeated three times for each hidden layer, and the averaged value can be obtained. The training results are illustrated in Figure 5 and the final optimised node number was six.Figure 5
MSE of different node number of hidden layer.The model was building using the neural network toolbox in MATLAB. The activation function for the hidden layer was Sigmod function and for the output layer was purelin function. The maximum convergence times was set as 1000, and the maximum training accuracy was 0. The Bayesian regularisation method (trainbr) was selected as the training method. A total of eight groups of data were randomly selected for test and the rest 72 groups of data were used as the train set. After training the neural network with the train set, the test set was predicted.When training, a weight matrix was randomly generated and then weight was modified with the transmission error.Figure6 illustrates the training results by comparing measured and predicted values. Based on equation (5), MSE of this test set can be calculated as 29.98, which proves the accuracy of training model is good enough to perform the prediction. According to detailed information in Figure 6, the absolute error of MSE is controlled within 10%, ranging from −7.57% to 9.86%, which demonstrates an acceptable prediction ability of this model.Figure 6
Predicted result of back propagation neural network.


## 3.1.3. Test and Verification

To ensure the generalisation ability of the network, a cross validation was performed. After arranging data randomly, the data were divided into ten subsets. Every subset was selected as a test set, and the remaining were training sets. The cross validation were repeated ten times following the above steps. Figure7 demonstrates MSE of ten cross validation results and the averaged MSE is 32.28, which demonstrates a good generalisation ability of this method.Figure 7
Cross validation of source intensity prediction.Furthermore, the method of leave-one-out-cross-validation (LOOCV) was used to calculate and analyse the errors between measured and predicted values. Figure8 illustrates the error normal distribution of BPNN using LOOCV.Figure 8
Error distribution of vibration source intensity predicted using BPNN.Based on the verification of LOOCV and cross validation, the prediction error is generally below 10% and the average MSE is less than 35. Accordingly, BPNN can be used to undertake a preliminary prediction of the source intensity vibration of a running metro train.Moreover, coefficient of determinationR2 can be used to evaluate linear correlation degree of network fitting result. The coefficient of determination is defined as(7)R2=SSRSST,SSR=∑iy^i−y¯i2,SST=∑iyi−y¯i2,where, yi is the true value, y^i is the predicted value, and y¯i is the average of true value.Coefficient of determinationR2 ranges from 0 to 1. The closer R2 approaches to 1, the better the linear correlation degree of network fitting result is. After calculation, R2 of BPNN is 0.9464, which reflects the BPNN model can also be used to predict the vibration source intensity.


## 3.2. GRNN Based Prediction

### 3.2.1. Method Introduction

GRNN is a type of radial basis function (RBF) neural network, which has strong non-linear mapping capability, fault tolerance and robustness. Besides, it has the advantage in the approaching ability and learning speed. The model of radial basis neural network includes two main independent variable and basis function. The independent variable is the Euclidean distance between the points to be measured and sampling. The basis function is a radial function. GRNN can transform a multi-dimensional problem into a one-dimensional problem. After transformation, the independent variable of one-dimensional issue turns into the Euclidean distance mentioned above. Any function can be obtained by making weighted combination of basis functions. Figure9 demonstrates the topology of GRNN.Figure 9
Topology of GRNN.In the calculation and analyse by GRNN, the training setP is firstly input as a learning sample, and then SD and SNi are calculated and output to the summation layer. SD and SNi are neurons in the hidden layer, calculated by two different methods. One is the summation of denominator neurons, i.e. straightly summing up all the neurons in hidden layers. The other is the summation of molecular neurons, i.e. making a weighted summation of neurons in mode layers. SD and SNi can be calculated by(8)SD=∑i=1nPi=∑i=1nexpx−xiTx−xi2σi2,i=1,2,3…n,SNi=∑i=1nYiPi=∑i=1nyijexpX−XiTX−Xi2σi2,j=1,2,3…n,where σi is a network expansion constant, yij is the connecting weight of the i-th neuron in the summation layer and the j-th neuron in the mode layer. Finally, the output network predicted value can be obtained: yj = SD/SNi [43].According to the characteristic of RBF neural network, the data should be expressed in the form of scientific counting before normalisation. Then, the route radius was multiplied by 0.01 and the train speed was multiplied by 0.1. Based on the fast learning speed of GRNN, building a large network and performing mutual authentications are possible. Then, the network was built for all 80 groups of data one by one. Thei-th group of data was selected as the test set and the remaining were the train sets. Thus, a total of 80 networks can be established and the data can be made full used.


### 3.2.2. Results, Test and Verification

After training, 80 groups of predicted results can be obtained using LOOCV. Figure10 demonstrates the comparison between predicted and measured values.Figure 10
GRNN based predicted result.Figure11 demonstrates MSE values under different values of σ. According to Figure 11, the value of σ have little influence on the result of this experiment. When σ = 1 the MSE value is the minimum. Accordingly, this value was determined in this experiment.Figure 11
MSE of GRNN under different values ofσ.Both of the MSE and coefficient of determinationR2 were used to evaluate linear correlation degree of network fitting result. After calculation, MSE is 17.9205, and R2 is 0.8153. Figure 12 demonstrates the error distribution. For its normal fitting curve, average value is 0.004664 and standard deviation is 0.07571.Figure 12
Source intensity prediction error distribution predicted by GRNN.The above results and verification proves that GRNN can be used to predict the vibration source intensity with considering different parameters. Compared with BPNN, both of the calculation efficiency and the accuracy of GRNN are higher. Accordingly, GRNN is more recommended to predict the vibration source intensity.


## 3.2.1. Method Introduction

GRNN is a type of radial basis function (RBF) neural network, which has strong non-linear mapping capability, fault tolerance and robustness. Besides, it has the advantage in the approaching ability and learning speed. The model of radial basis neural network includes two main independent variable and basis function. The independent variable is the Euclidean distance between the points to be measured and sampling. The basis function is a radial function. GRNN can transform a multi-dimensional problem into a one-dimensional problem. After transformation, the independent variable of one-dimensional issue turns into the Euclidean distance mentioned above. Any function can be obtained by making weighted combination of basis functions. Figure9 demonstrates the topology of GRNN.Figure 9
Topology of GRNN.In the calculation and analyse by GRNN, the training setP is firstly input as a learning sample, and then SD and SNi are calculated and output to the summation layer. SD and SNi are neurons in the hidden layer, calculated by two different methods. One is the summation of denominator neurons, i.e. straightly summing up all the neurons in hidden layers. The other is the summation of molecular neurons, i.e. making a weighted summation of neurons in mode layers. SD and SNi can be calculated by(8)SD=∑i=1nPi=∑i=1nexpx−xiTx−xi2σi2,i=1,2,3…n,SNi=∑i=1nYiPi=∑i=1nyijexpX−XiTX−Xi2σi2,j=1,2,3…n,where σi is a network expansion constant, yij is the connecting weight of the i-th neuron in the summation layer and the j-th neuron in the mode layer. Finally, the output network predicted value can be obtained: yj = SD/SNi [43].According to the characteristic of RBF neural network, the data should be expressed in the form of scientific counting before normalisation. Then, the route radius was multiplied by 0.01 and the train speed was multiplied by 0.1. Based on the fast learning speed of GRNN, building a large network and performing mutual authentications are possible. Then, the network was built for all 80 groups of data one by one. Thei-th group of data was selected as the test set and the remaining were the train sets. Thus, a total of 80 networks can be established and the data can be made full used.


## 3.2.2. Results, Test and Verification

After training, 80 groups of predicted results can be obtained using LOOCV. Figure10 demonstrates the comparison between predicted and measured values.Figure 10
GRNN based predicted result.Figure11 demonstrates MSE values under different values of σ. According to Figure 11, the value of σ have little influence on the result of this experiment. When σ = 1 the MSE value is the minimum. Accordingly, this value was determined in this experiment.Figure 11
MSE of GRNN under different values ofσ.Both of the MSE and coefficient of determinationR2 were used to evaluate linear correlation degree of network fitting result. After calculation, MSE is 17.9205, and R2 is 0.8153. Figure 12 demonstrates the error distribution. For its normal fitting curve, average value is 0.004664 and standard deviation is 0.07571.Figure 12
Source intensity prediction error distribution predicted by GRNN.The above results and verification proves that GRNN can be used to predict the vibration source intensity with considering different parameters. Compared with BPNN, both of the calculation efficiency and the accuracy of GRNN are higher. Accordingly, GRNN is more recommended to predict the vibration source intensity.


## 4. Conclusion

To improve the prediction efficiency ofVLZ,0 in the empirical prediction formula, ML method was introduced in the present study. In-situ measurements were performed in 80 different running tunnel sections of Beijing metro and the model training samples of VLZ,0 were obtained. Two types of ML method were employed and compared for the prediction results. The results indicate that:(1)
Both of BPNN and GRNN can be used to predict the tunnel vibration responsesVLZ,0. Proved by LOOCV, predicting by neural network has good extensionality. In the preliminary prediction phase, the neural network based prediction results can be used as reference values in the preliminary prediction.(2)
GRNN has relatively better predicting ability than BPNN.This study only explores the application of ML for predictingVLZ,0. As the number and quality of test samples used for training determine the accuracy of prediction results, more test work is suggested to be carried out in future to enrich training samples.



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*Source: 4031050-2022-06-19.xml*