Trajectory prediction method for high-speed train group tracking operation based on LSTM-KF model

FullText

0. introduction

As the brain and central nervous system of the high-speed railway system, the train operation control system (referred to as the train control system) plays a crucial role in ensuring the safe and efficient operation of trains^[1]At present, typical train control systems used in high-speed railways include the Chinese Train Control System (CTCS) Level 3 and CTCS Level 2, the European Train Control System (ETCS), and Japan's Automatic Train Control (ATC), all of which operate in a fixed block control mode^[2]To meet the growing demand for passenger transportation, the operating density of high-speed railway trains is constantly increasing, and the existing operating intervals are approaching the design limit of the system. Based on fixed block system^[3]The train control technology is difficult to meet the demand for further improving operational efficiency of high-speed railways. At present, achieving high-speed train group operation in existing networks is an important technical means to further shorten train operation intervals and improve system capacity, and is the core and key to the development of railways at home and abroad^[4].

In the operation control of high-speed train groups,nHeterogeneous trains will be interconnected through train to train communication, and operate in a coordinated and safe manner with shorter tracking distances under the constraints of the track. This is an important feature that distinguishes train group operation from traditional train operation^[5]Vehicle to vehicle communication technology plays a crucial role in building group operations. The basic principles of group operation control and its differences from traditional train operation control are as follows:Figure 1As shown, the tracking interval of trains under traditional train operation control and the inter group interval of group operation are bothDTrain tracking interval within the groupdClearly less thanDCompared with the traditional train operation control mode, multiple heterogeneous trains in a train group share real-time speed, position, acceleration, internal status and other information of train operation through train to train communication technology, thereby coupling to form an organic whole, which further reduces the tracking interval of the train group while ensuring safe and smooth operation^[6]However, the smooth operation of high-speed train groups is affected by the limited short distance signal transmission capacity and time-varying signal quality of train to train communication technology^[7]How to maximize the role of vehicle to vehicle communication technology and enhance the smoothness of formation operation is still an urgent problem to be solved. In order to reduce the impact of vehicle to vehicle communication delay, predicting the trajectory of the preceding vehicle within the group is an important means to further reduce the tracking distance of trains.

Figure  1.  Traditional and group operation modes of high-speed train

DownLoad: Full-Size Img PowerPoint

There is relatively little research on train trajectory prediction theory both domestically and internationally. He and others^[8]A subway train trajectory prediction method based on Long Short Term Memory (LSTM) network was proposed, and simulation experiments were conducted using actual train operation data. The results showed that the LSTM model had better long-term prediction accuracy than the Kalman Filter (KF) model; Yin et al^[9]The LSTM model combined with time delay information was used to predict the running trajectory of high-speed trains, and the effectiveness of the method was verified through simulation. Obviously, current research on train trajectories focuses on data-driven prediction methods.

It is worth noting that many scholars have conducted in-depth research on vehicle trajectory prediction and urban traffic flow prediction. The trajectory prediction methods used mainly include short-term physical model-based trajectory prediction and data-driven trajectory prediction. The trajectory prediction method based on physical models is to predict the changes in the vehicle's state over a period of time based on the relevant kinematic and dynamic laws of the vehicle. Lin et al^[10]Based on a two degree of freedom dynamic model, the future trajectory of the vehicle is predicted. In order to improve the accuracy of trajectory prediction, the author proposes a stable KF method to estimate the lateral velocity of the vehicle and external disturbances acting on the vehicle; De Miguel and others^[11]Using the recursive layer and variational autoencoder architecture model of deep learning for future local trajectories, it has been experimentally proven to achieve good results; Wang et al^[12]After comparing a large number of statistical models and deep learning models, it was concluded that the hybrid model of LSTM combined with random forest has better predictive performance; Lytrivis et al^[13]Considering the position, velocity, and heading angle of all surrounding vehicles, a kinematic model based on constant yaw rate and constant acceleration is used to collaboratively calculate the future position changes of the vehicles. At the same time, based on map information, the trajectory prediction effect is enhanced; Zhou Bing and others^[14]Predicting train trajectories based on motion models and considering the uncertainty of motion trajectories to make decisions on braking and collision avoidance for high-speed vehicles. This method can judge the probability of future collisions, but in high-speed situations, a single motion model prediction method can cause significant errors in the later stages of motion; Tong et al^[15]The Transformer model was used to predict and evaluate long-term trajectories, achieving good results; Reiter et al^[16]A quadratic programming algorithm was proposed for the most challenging motion planning problems in both long and short-term time ranges, and achieved good results in motion planning problems; Huang et al^[17]Based on differential positioning navigation system, different kinematic trajectory prediction models were studied, and it was pointed out that the main sources of error in kinematic trajectory prediction are changes in driving intention and behavior; Zhang Miao and others^[18]Propose intelligent train control methods based on driver experience; Sorstedt et al^[19]By considering the driver's control input, predict the driving trajectory for a period of time in the future. It can be seen that traditional trajectory prediction often uses a single motion model, and different motion models achieve different trajectory prediction results. However, the actual driving process of vehicles is very complex, which makes it difficult for any single motion model to accurately describe the motion state of the vehicle at any time. For feature extraction problems in complex environments, Xu et al^[20]A cascaded feature LSTM model was proposed, which can extract complex interaction feature information between pedestrians; Li et al^[21]A clustering convolutional LSTM algorithm was proposed for predicting vehicle trajectories, and experiments showed that the extraction of interactive information and selected features was accurate and met real-time requirements.

The vehicle trajectory prediction method based on data-driven methods mainly uses a large number of data sequences and data-driven models (such as deep learning network models) to predict the future driving trajectory of vehicles by learning and mining knowledge from the data. The data-driven vehicle trajectory prediction method is mainly based on the trajectory prediction model of deep learning network, which predicts the driving trajectory by considering the relationship and influence of time and spatial variables. Xing et al^[22]The Gaussian mixture model was used to recognize driving styles, and LSTM network and Recurrent Neural Network (RNN) were used to predict the trajectory of vehicles ahead, achieving good results; Kim and others^[23]Based on the long short-term memory model in recurrent neural networks, analyze the temporal behavior of vehicles in traffic environments and predict their future driving trajectories. Compare and analyze the predicted results with commonly used KF based methods. The trajectory prediction method based on deep neural network learning models can predict vehicle trajectories for a longer period of time, but it requires a higher database; Dequaire et al^[24]A recurrent neural network was used to describe the temporal and spatial changes in the traffic environment, and a unified motion tracking and prediction learning model for traffic participants such as vehicles and pedestrians was constructed; Xie et al^[25]After comparing multiple prediction models, a sequence prediction model combining LSTM and particle swarm optimization algorithm was proposed, and its accuracy was demonstrated through experiments; Deo et al^[26]The LSTM method was used to predict the trajectory of vehicles around highways; Zhao et al^[27]A traffic flow prediction model based on LSTM was proposed, with the motivation of utilizing a two-dimensional network composed of memory units in LSTM to simultaneously consider the spatiotemporal correlation of traffic sequence information, and the model was verified to have better predictive performance.

Unlike existing research, this paper proposes a novel hybrid model that integrates data-driven LSTM models and physics based KF models to predict train trajectories. In the proposed LSTM-KF hybrid model, the LSTM model is used to analyze time series data and explore the long-term dependencies of train trajectory data, which can be assumed to be the observation data of the KF model. The KF model combines the train dynamics mechanism to extract local features of train operation data, thereby smoothing the train trajectory predicted by the LSTM model. In addition, this article designs a real-time algorithm that can effectively implement the LSTM-KF hybrid model, relying on the high-speed railway control system simulation testing platform of Beijing Jiaotong University, and conducting simulation experiments using standard line data designed based on actual railway conditions in China.

1. The trajectory prediction problem of high-speed trains

In order to more accurately describe the problems in the process of high-speed train trajectory prediction, the following definitions are made.

Definition (Train Trajectory)^[8]During the one-way operation of high-speed trains, the changes in train position and speed along the track can be considered as longitudinal variables. The train trajectory within a certain period of time can be represented as

In the formula:X_{N, t}Starting from timet, including continuousNTrain trajectory at a sampling time;s_tFor the momenttCorresponding train position;v_tFor the momenttCorresponding train speed.

Generally, historical train operation data can be represented as

In the formula:O_{N, h, t}To terminate attTime, including continuityNTrain historical operation data at each sampling moment;u_tFor the momenttCorresponding train control commands;S_{N, h, t}To terminate at timet, including continuousNThe position sequence in the historical train operation data at each sampling time;V_{N, h, t}To terminate attTime, including continuityNThe speed sequence in the historical train operation data at each sampling time.

The train trajectory prediction model can be represented as a mapping function for historical operating data^[9]That is

In the formula:$\hat{\boldsymbol{X}}_{M, \mathrm{p}, t}$Starting from time$t+1$, including continuous$M$The final predicted train trajectory output at each sampling moment, where$M$Also known as historical time steps;$\boldsymbol{\varPhi}(\cdot)$Mapping function from historical train trajectories to predicted train trajectories;$\hat{s}_{t+1}$Based on the current situation$t$Time prediction$t+1$Train position at time;$\hat{v}_{t+1}$Based on the current situation$t$Time prediction$t+1$Train speed at the moment.

2. LSTM-KF trajectory prediction hybrid model

2.1 LSTM model

LSTM network is a variant of RNN network with a gate mechanism for capturing long-term dependencies^[28]LikeFigure 2As shown, The basic component of an LSTM network is a memory block consisting of a forget gate, an input gate, and an output gate. During the continuous data reception process, the memory block maintains a unit state to store historical information$\boldsymbol{C}_{t}$express. The two inputs of the memory block are the current state$\boldsymbol{x}_{t}$Output from the previous sequence$\boldsymbol{h}_{t-1}$The memory block will replace the previous one$\boldsymbol{h}_{t-1}$Compared to the current state$\boldsymbol{x}_{t}$Combined together, as$t$Actual input vector at the moment$\left(\boldsymbol{h}_{t-1}, \boldsymbol{x}_{t}\right)$Then, the combined input vector is used to calculate the output through three gates$\boldsymbol{h}_{t}$And update the cell state to$\boldsymbol{C}_{t}$.

Figure  2.  LSTM model memory block

DownLoad: Full-Size Img PowerPoint

The basic formula is

In the formula:f_tTo forget the doortOutput of time;σTo activate the function;i_tFor the input gate intOutput of time;ο_tFor the output gatetTime output;W_f、W_i、W_C、W_oThe weight matrix in the memory block corresponding to each gate;b_f、b_i、bC、b_oThe deviation matrices in the memory blocks corresponding to each gate* For convolution operation.

2.2 KF model

KF is a prediction model based on linear regression, which uses the state space of linear stochastic systems to describe it^[29]The KF model can be divided into two parts: prediction and update, where the prediction equation is

In the formula:$\hat{\boldsymbol{x}}_{t+1 \mid t}$Based on the current situation$t$Time prediction$t+1$The state of time;$\boldsymbol{B}_{t+1}$To control the matrix and reflect$t+1$Time system control input$u_{t}$The impact;$\boldsymbol{F}_{t+1}$do$t+1$Time state transition matrix;$\boldsymbol{P}_{t+1 \mid t}$Based on the current situation$t$Obtained at all times$t+1$Time prediction error covariance matrix;$Q_{t+1}$do$t+1$Covariance matrix of time system noise.

The update equation in the KF model can be expressed as

In the formula:$\boldsymbol{K}_{t+1}$do$t+1$Kalman gain at time;$\boldsymbol{H}_{t+1}$do$t+1$The time mapping matrix maps the state vector to the observed values;$\boldsymbol{R}_{t+1}$do$t+1$Observing noise at all times is the variance of the standard normal distribution;$\boldsymbol{z}_{t+1}$do$t+1$Observation values of time;$\boldsymbol{P}_{t+1}$do$t+1$Covariance matrix of prediction error at time.

2.3 LSTM-KF hybrid model

in compliance withFigure 3As shown, taking Train 1 (front car) and Train 2 (rear car) as examples, the principle of train trajectory prediction based on LSTM-KF hybrid model is explained as follows: current time$t$Train 2 combines the historical spatiotemporal trajectory data of Train 1 (historical time steps)$N$）Using the LSTM model to predict the next state of Train 1 in the long time domain (i.e. the assumed observation data of the KF model)$z_{t+1}$）, and capture corresponding reference control action information$u_{t}$Then filter the state information predicted by the LSTM model through the KF model to generate more accurate estimates$\hat{x}_{t+1 \mid t+1}$Next, we will$\hat{x}_{t+1 \mid t+1}$At the end of the input sequence of the LSTM model, update the input sequence and repeat the aforementioned rolling prediction steps until the final length of the prediction time step is obtained$M$The predicted trajectory; At this point, Train 2 has completed the current task$t$Predict the trajectory of train 1 ahead at the given time.

Figure  3.  Train track prediction principle based on LSTM-KF model

DownLoad: Full-Size Img PowerPoint

Define the train operation status as

A specific LSTM model has been designed for predicting train trajectoriesL(·), its structure is as followsFigure 4As shown, the corresponding function can be expressed as the following function

Figure  4.  LSTM-KF model framework

DownLoad: Full-Size Img PowerPoint

In the formula:$\boldsymbol{X}_{1, \mathrm{p}, t}$Based on the model$L(\bullet)$Predicted$t+1$The current train status corresponds to the predicted time step$1 ; \boldsymbol{X}_{N, \mathrm{~h}, t}$To terminate at time$t$, including continuous$N$Historical state data at each sampling moment;$\boldsymbol{U}_{N, \mathrm{~h}, t}$To terminate at time$t$, including continuous$N$Historical acceleration data at each sampling moment.

It is worth noting that the "historical" data may not necessarily be the actual operation data of the train, and may also be the predicted values output by the LSTM-KF hybrid model. For example, the current time ist， The LSTM-KF hybrid model has already rolled out the prediction outputt+The train status value at time 1 is relative tot+In terms of the state predicted by the model after 1 moment, it is' historical 'data.

Furthermore, future control behavior data of the preceding vehicle is extracted through differential calculation, i.e

In the formula:$\boldsymbol{U}_{M, \mathrm{p}, t}$Starting from time$t$, including continuous$M$The future reference control sequence of the preceding vehicle at each sampling moment;$u_{t+i \mid t}$Based on$t$Time prediction$t+i$Future reference control behavior of the vehicle before the time;$v_{t+i \mid t}$Predicting trajectories for LSTM models$\boldsymbol{U}_{M, \mathrm{p}, t}$pass the civil examinations$i$Train speed at one time step;$\Delta \tau$For the sampling interval.

For high-speed trains, the approximate expression of the dynamic equation is^[29]

Equation (20) is used as the prediction equation of the KF model for train trajectory prediction, where$\hat{x}_{t+1 \mid t}$and$\hat{x}_{t \mid t}$Can be defined as$\left(\hat{s}_{t+1 \mid t}, \hat{v}_{t+1 \mid t}\right)^{\mathrm{T}}$and$\left(\hat{s}_{t \mid t}, \hat{v}_{t \mid t}\right)^{\mathrm{T}}$(or$\left(\hat{s}_{t}, \hat{v}_{t}\right)^{\mathrm{T}}$）Afterwards, a complete KF model can be constructed according to the steps in section 2.2. The structure of the KF model is as followsFigure 4As shown, the corresponding parameters are as follows

Based on the above ideas, this article proposes a train trajectory prediction algorithm based on the LSTM-KF hybrid model, which can predict the current train trajectory at a given timetAccurately predict and output the futureMThe train trajectory at a certain moment. The train trajectory prediction algorithm has two advantages: (1) the algorithm uses the LSTM model to learn the rules of the train's historical trajectory, and uses the train operation mechanism included in the KF model to correct the trajectory prediction value, which enhances the interpretability of the model while achieving high-precision train trajectory prediction; (2) Complex LSTM models can be trained offline, while KF models themselves are lightweight, and the combination of the two can make the algorithm easy to deploy in real-time online.

2.4 Train trajectory prediction algorithm based on LSTM-KF model

The process of train trajectory prediction algorithm based on LSTM-KF model is as follows:Figure 5As shown.

Figure  5.  Algorithm flow of LSTM-KF model

DownLoad: Full-Size Img PowerPoint

Firstly, fromtStart predicting the timing and initialize the stepsjTo 1, organize the collected data of train operation, select data from longer operating stages, and calculate estimated data to supplement the discontinuous collected data during the selected stages. Set the training dataset and validation dataset in a certain proportion, and normalize the data to prevent large-scale data from having a significant impact on small-scale data. Secondly, select the latest set of data from the training dataset as historical data. Next, build an LSTM model and train it using training data to extract the motion patterns during train operation. Fix the model parameters after convergence.

Use LSTM model to predict historical data sets. In the prediction result correction section, the KF model is used to correct the prediction results of the test model. Normalize the output of LSTM, linearly calculate the train state data according to the train dynamics equation, and use the covariance matrix to correct the linearly calculated train state. Then, linearly combine the predicted data of the LSTM model with the KF calculation results. Finally, normalize the fused data, include it in the latest data position of the historical dataset, and remove the oldest data to complete the rolling update.

3. case study

3.1 Data preprocessing

This article relies on the laboratory high-speed railway control system simulation testing platform, and the line is tested using standard line data designed based on the actual line conditions of Chinese railways, such asFigure 6As shown, it consists of three standard station areas: A, B, and C. The case study of trajectory prediction is carried out by taking the three detection characteristics (position, speed and control command) directly collected by the train auto drive system in the simulation test platform as the input of the prediction model.

Figure  6.  High-speed railway train group tracking operation scenario

DownLoad: Full-Size Img PowerPoint

The data is recorded by the Juridical Recording Unit (JRU) in the vehicle, with a sampling interval of ΔτFor 0.5 seconds, perform the following 5 preprocessing operations on the data.

(1) Missing data points are filled with a combination of historical mean and pre - and post values.

(2) The distance traveled by the train within the unit sampling interval is calculated by subtracting the train positions at the previous and next time points.

(3) After checking for missing and invalid data, standardize each dimension feature.

(4) Acceleration is estimated using velocity and sampling interval.

(5) Historical data is processed into a data with historical time stepsNA prediction dataset with a scrolling window step size of 1. For the convenience of comparing and verifying the training effect of the model, 80% of the data is selected for training, and the remaining 20% is used for testing.

This article usesZ-Score normalization is used to scale features with typical normal distribution attributes, with a mean of 0 and variance of 1, to standardize the data.

3.2 Model configuration

The specific configuration of constructing a LSTM-KF hybrid model for train trajectory prediction includes loss function, activation function, optimization algorithm, and parameter settings to ensure the accuracy and convergence of the model.

(1) Loss function

Measure the degree of deviation between the predicted and actual values of the model using a loss function. The better the loss function, the better the performance of the model. In order to measure the Euclidean distance error of each trajectory while considering regression problems, this paper uses Mean Square Error (MSE), i.e

In the formula:$E$Mean square error;$y_{i}$For the th$k$Step true value;$\hat{y}_{i}$For the th$k$Step prediction value;$n$For the total time step.

(2) Activation function

This article uses the Rectified Linear Unit (ReLU) function as the activation function for train operation status. This function solves the gradient vanishing problem, and compared to Sigmoid and tanh, ReLU has faster computation and convergence speed.

(3) Optimization algorithm

The convergence of deep learning models is highly dependent on the selected optimizer. The Adaptive Moment Estimation (ADAM) algorithm is suitable for problems with large amounts of data or parameters because it has high computational efficiency, requires minimal memory, and is invariant to rescaling the diagonals of gradients. Therefore, this paper uses an ADAM optimizer to construct the model on each train trajectory dataset.

(4) Parameter settings

The parameters of the prediction model are as followsTable 1As shown, the historical time step N refers to a single sample in the historical data that contains the pastNThe state and control variables at each moment, predicting the time step sizeMIt refers to predicting trajectories that include the futureMThe state of a moment. In particular, LSTM networks with different structures are tested, such as the number of LSTM layers and the number of hidden units in each layer, and KF observation covariance matrix is adjusted in the experimentQCovariance matrix of process errorR.

Table  1.  Experimental comparison parameter settings

参数	取值
LSTM网络结构	100×100
预测时窗/s	6
历史时间步长	12
未来时间步长	30
状态迁移矩阵	$\left[\begin{array}{cc}0 & 0.5 \\ 0 & 1\end{array}\right]$
观测矩阵	$\left[\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right]$
观测协方差矩阵	$\left[\begin{array}{cc}10 & 0 \\ 0 & 10\end{array}\right]$
过程误差协方差矩阵	$\left[\begin{array}{cc}0.1 & 0 \\ 0 & 0.1\end{array}\right]$

 | Show Table

DownLoad: CSV

3.3 performance index

Compare the trajectory prediction results of various models using three metrics: Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Mean Absolute Percentage Error (MAPE)E₁、E₂、E₃show

3.4 Result analysis

3.4.1   R、QThe impact on LSTM-KF hybrid model

After fixing the parameters of the LSTM model, explore$\boldsymbol{R} 、\boldsymbol{Q}$The impact of two covariance matrices on the prediction results is adjusted by adjusting the covariance of the KF model observation matrix$R$Covariance of Process Error Matrix$Q$realization. According to equation (10), the Kalman gain and$\boldsymbol{R} 、\boldsymbol{Q}$The ratio of matrix coefficients is related. For the convenience of evaluation$\boldsymbol{R} 、\boldsymbol{Q}$The function of a matrix is to$\boldsymbol{R} 、\boldsymbol{Q}$Equal to multiplying the identity matrix by constant coefficients$R 、Q$Afterwards, design experiments$R / Q$respectively$0.01 、0.1 、1 、10$Four different situations to explore$\boldsymbol{R} 、\boldsymbol{Q}$The impact of both on the prediction results.

Figure 7The results of the LSTM-KF model for predicting train speed indicate that:R=Q=At 10 o'clock, the RMSE and MAE evaluation index values gradually decreased from around 48 on the vertical axis, indicating a large initial prediction error. As the rolling prediction progressed, the predicted values gradually approached the actual values; existR=1 andQ=At 10 o'clock, the speed prediction value of the LSTM-KF model is more biased towards the prediction value of the LSTM model. In addition, the three indicators RMSE, MAE, and MAPE decrease with increasing time steps, especially after the 18th time step when they tend to stabilizeR/QThe LSTM-KF model performs well in prediction; equalR=10 andQ=At 1 o'clock, the predicted values of the LSTM-KF model tend to be more biased towards the predicted values of the KF model prediction equation, resulting in poorer prediction performance; equalR=0.1 andQ=At 10 o'clock, compared to othersR/QThe RMSE, MAE, and MAPE indicators corresponding to the LSTM-KF model are relatively small and stable.

Figure  7.  Evaluation indexes of speed prediction

DownLoad: Full-Size Img PowerPoint

Figure 8The results of the LSTM-KF model for predicting train position indicate that:R=10 andQ=The curves of the three evaluation indicators at 1 hour slowly increase over time and tend to stabilize; From the speed and position prediction results,R=10 andQ=The parameter setting of 1 can enable the LSTM-KF model to achieve better prediction performance. Therefore, chooseR=0.1 andQ=The covariance matrix of 10 is used as a parameter for the LSTM-KF model in subsequent experiments.

Figure  8.  Evaluation indexes of position prediction

DownLoad: Full-Size Img PowerPoint

3.4.2   The effectiveness of multiple trajectory prediction models under different operating conditions

In order to verify the performance of the proposed model, data was collected on the high-speed rail control system simulation testing platform of Beijing Jiaotong University. The data was divided into a training set and a testing set, and the model was trained on the training set and validated on the testing set.

Figure 930 time step train speed curves and actual train speed values predicted by different models under cruising conditions. This experiment compared the performance of LSTM, RNN, and LSTM-KF in 30 step prediction. During the experiment, the last 12 sets of training data (historical data) were taken to predict the first set of data (future data), and then the last 11 sets of training data (historical data) and the predicted data (the first set of future data) were merged into 12 data sets to predict the second set of data in the future. Further realizing the rolling prediction process.

Figure  9.  Speed prediction results of different models under cruising conditions

DownLoad: Full-Size Img PowerPoint

causeFigure 9It can be seen that a single LSTM model has low accuracy in speed prediction, manifested as a true speed of 97 m · s^-1The predicted value is 119 m · s^-1, with a difference of 22 m · s^-1The RNN model is relatively accurate, with predicted values near the true value, which is 97 m · s^-1The predicted value is 98 m · s^-1, with a difference of 1 m · s^-1The LSTM-KF model predicts that the distance between the velocity trajectory and the true value is relatively small, with a true value of 97 m · s^-1The predicted value is 98 m · s^-1, with a difference of 1 m · s^-1The predicted trajectory shows that the model has good predictive performance.

Figure 10For the prediction of train position and the actual value of train position by different models, it can be seen that a single LSTM model has a large prediction deviation for position, with a predicted value of 12171 m and an actual value of 11373 m, a difference of 798 m; The RNN model's prediction of location is not ideal, with a predicted value of 10462m and a true value of 11373m, a difference of 911m; The LSTM-KF model is relatively accurate in predicting location, with a predicted value of 17121 m and a true value of 17043 m, a difference of 78 m.

Figure  10.  Position prediction results of different models under cruising conditions

DownLoad: Full-Size Img PowerPoint

Figure 11The reason why the predicted results of the LSTM-KF model are different from those of the LSTM model with the same parameters for the acceleration predicted by different models and the actual value of train acceleration is that the acceleration is calculated by dividing the difference between the speed values predicted by the two LSTM models before and after by the time interval between the two predictions. Similarly, due to the addition of a fully connected network model, there are certain connections between various features. Therefore, it is normal for the corrected data values of the KF model to have an impact on the prediction results of some other features. causeFigure 11It can be seen that the LSTM model is more accurate in predicting acceleration; The RNN model and LSTM-KF model are inaccurate in fitting acceleration.

Figure  11.  Acceleration prediction results of different models under cruising conditions

DownLoad: Full-Size Img PowerPoint

Figure 12It can be concluded that the LSTM-KF model performs well in predicting the distance between two vehicles and their true values for different models, and is close to the true values; The RNN model has a good fitting effect on the indicator of distance from the preceding vehicle, with a large starting point error, and the error gradually decreases with time; The LSTM model does not have ideal fitting effect on the indicator of distance from the preceding vehicle, and the prediction deviation is relatively large.

Figure  12.  Prediction results of distances between two trains under cruising conditions using different models

DownLoad: Full-Size Img PowerPoint

Figure 13It can be seen that the RNN model and LSTM-KF model are more accurate in predicting the speed difference between two vehicles and their true values for different models, but the prediction results of a single LSTM model are not very ideal. For the problem of initial value deviation in other models without KF filtering algorithm for correction, after analysis, it is believed that the reason is that predicting multiple dimensions of data at the same time causes interference in the calculation process of the loss function of key data in different dimensions (such as non critical data such as the speed of the preceding vehicle), such as position prediction. The range of position is 0-12 km, which is normalized to 0-1 by conventional methods, while the range of speed difference is -100~100 m · s^-1For the loss function, a speed change of 1 m · s^-1The weight is 1/200, and the same weight is required for a position of 60 meters. Therefore, in multidimensional prediction models, a combination of data models and mechanistic models can be used to correct the predicted values of key data (such as velocity and position) through mechanistic models, reducing the errors of key data within an acceptable range.

Figure  13.  Prediction results of speed differences between two trains under cruising conditions using different models

DownLoad: Full-Size Img PowerPoint

Table 2For several common evaluation metrics for position prediction using different models, it can be seen that, taking RMSE as an example, in the comparison of multidimensional output prediction models, the RMSE metric of the LSTM-KF model is 7% of that of the traditional LSTM model and 15% of that of the traditional RNN model.

Table  2.  Evaluation of position prediction results of different models under cruising conditions

时间步	RMSE/m			MAE/m			MAPE/%
	RNN	LSTM-KF	LSTM	RNN	LSTM-KF	LSTM	RNN	LSTM-KF	LSTM
1	1 456.41	78.95	2 799.74	1 456.41	78.95	2 799.74	0.08	0.00	0.14
5	1 591.03	97.88	2 464.24	1 586.20	96.49	2 453.65	0.08	0.01	0.12
10	1 904.61	132.90	2 234.40	1 870.70	127.31	2 211.47	0.10	0.01	0.11
15	2 431.12	163.92	2 083.96	2 311.75	155.26	2 053.49	0.12	0.01	0.11
20	3 345.82	185.87	1 961.03	3 011.15	176.17	1 921.79	0.14	0.01	0.10
25	4 966.02	196.82	1 850.34	4 144.52	187.91	1 799.06	0.18	0.01	0.09

 | Show Table

DownLoad: CSV

Table 3For several common evaluation metrics for speed prediction in multiple models, it can be seen that taking RMSE as an example, the speed prediction results under cruising conditions show that the RMSE evaluation metric of the LSTM-KF model is 14% of that of the traditional LSTM model and 30% of that of the traditional RNN model.

Table  3.  Evaluation of speed prediction results of different models under cruising conditions

时间步	RMSE/(m·s-1)			MAE/(m·s-1)			MAPE/%
	RNN	LSTM-KF	LSTM	RNN	LSTM-KF	LSTM	RNN	LSTM-KF	LSTM
1	1.02	1.58	22.07	1.02	1.58	22.07	0.01	0.02	0.19
5	1.16	0.85	19.28	1.15	0.77	19.17	0.01	0.01	0.16
10	1.40	0.65	16.40	1.37	0.53	15.87	0.01	0.01	0.14
15	1.76	0.55	14.08	1.68	0.40	12.93	0.02	0.00	0.12
20	2.34	0.51	12.40	2.13	0.39	10.64	0.02	0.00	0.10
25	3.31	0.51	11.16	2.83	0.42	8.87	0.03	0.00	0.08

 | Show Table

DownLoad: CSV

Figure  14.  Position prediction results of different models under acceleration conditions

DownLoad: Full-Size Img PowerPoint

Figure 15To accelerate the prediction results and their true values of different model speeds under different operating conditions, it can be seen that the LSTM model has a large deviation, while the RNN model's prediction results are more accurate. This is related to the different data weights used in multidimensional data prediction. A decrease in the error of one dimension may lead to an increase in the error of other dimensions, while the LSTM-KF hybrid model has better prediction performance.

Figure  15.  Speed prediction results of different models under acceleration conditions

DownLoad: Full-Size Img PowerPoint

Figure  16.  Position prediction results of different models under deceleration conditions

DownLoad: Full-Size Img PowerPoint

Figure 17To compare the speed prediction results of different models under deceleration conditions with the actual values, it can be concluded that the LSTM model and LSTM-KF model have more accurate prediction results for the data, and the predicted values of the starting and ending points of the LSTM-KF model are relatively close. Therefore, overall, the KF model has a good corrective effect on the prediction results of the LSTM model, while the RNN model is more prone to gradient explosion and requires the addition of more optimization methods such as gradient pruning.

Figure  17.  Speed prediction results of different models under deceleration conditions

DownLoad: Full-Size Img PowerPoint

Table 4Based on the actual values of acceleration and deceleration conditions and the predicted values of the LSTM-KF model, the following conclusions can be drawn.

Table  4.  Real values and predicted values of LSTM-KF model under different conditions

时间步	加速工况-位置/m		加速工况-速度/(m·s-1)		减速工况-位置/m		减速工况-速度/(m·s-1)
	真实值	LSTM-KF模型预测值	真实值	LSTM-KF模型预测值	真实值	LSTM-KF模型预测值	真实值	LSTM-KF模型预测值
1	4 806.00	4 892.41	70.56	74.69	18 499.00	18 894.95	96.67	96.88
2	4 806.00	4 892.41	71.11	74.89	18 519.00	18 983.17	96.11	96.03
3	4 849.00	4 936.37	71.39	74.89	18 499.00	19 082.20	95.83	95.25
4	4 877.00	4 964.85	71.67	74.87	18 519.00	19 189.04	95.00	94.16
5	4 949.00	5 037.90	71.94	74.82	18 499.00	19 300.63	94.72	93.36
6	4 992.00	5 081.46	72.50	74.97	18 517.00	19 414.51	94.44	92.60
7	5 021.00	5 111.03	72.78	74.90	18 499.00	19 529.68	93.89	91.75
8	5 065.00	5 155.61	73.06	74.84	18 517.00	19 644.79	93.33	90.96
9	5 094.00	5 185.20	73.61	74.95	18 498.00	19 759.37	93.06	90.37
10	5 138.00	5 229.78	73.89	74.89	18 517.00	19 872.41	92.50	89.69
11	5 168.00	5 260.35	74.17	74.83	18 497.00	19 983.59	92.22	89.22
12	5 212.00	5 304.92	74.72	74.98	18 517.00	20 092.02	91.39	88.53
13	5 242.00	5 335.48	75.00	74.95	18 497.00	20 197.32	91.11	88.16
14	5 287.00	5 381.00	75.28	74.99	18 515.00	20 294.13	90.56	87.74
15	5 317.00	5 411.49	75.56	75.04	18 498.00	20 388.78	90.28	87.49
16	5 363.00	5 457.98	76.11	75.30	18 515.00	20 480.60	89.72	87.10
17	5 393.00	5 488.45	76.39	75.39	18 496.00	20 569.20	89.17	86.74
18	5 439.00	5 534.91	76.94	75.69	18 515.00	20 654.33	88.89	86.53
19	5 470.00	5 566.35	77.22	75.82	18 496.00	20 735.88	88.61	86.33
20	5 516.00	5 612.77	77.50	75.98	18 514.00	20 813.79	87.78	85.86
21	5 547.00	5 644.17	77.78	76.15	18 496.00	20 888.06	87.50	85.68
22	5 594.00	5 691.55	78.33	76.54	18 513.00	20 958.73	86.94	85.37
23	5 625.00	5 722.92	78.61	76.77	18 496.00	21 025.86	86.67	85.20
24	5 672.00	5 770.26	78.89	77.03	18 512.00	21 089.52	86.11	84.90
25	5 704.00	5 802.59	79.44	77.50	18 496.00	21 149.84	85.56	84.60
26	5 751.00	5 849.89	79.72	77.81	18 512.00	21 206.93	85.00	84.31
27	5 783.00	5 882.19	80.00	78.16	18 495.00	21 260.98	84.72	84.15
28	5 832.00	5 931.46	80.56	78.74	18 511.00	21 312.05	84.17	83.86
29	5 864.00	5 963.72	80.83	79.16	18 495.00	21 360.27	83.89	83.71
30	5 912.00	6 011.96	81.11	79.64	18 511.00	21 405.73	83.33	83.42

 | Show Table

DownLoad: CSV

The position prediction results under acceleration conditions show that the error between the predicted value and the true value of the LSTM-KF model is about 100 m, and the error between the predicted value and the true value of the speed is about 1.5 m · s^-1Under deceleration conditions, the error between the predicted position value and the true value gradually increases, and the error between the predicted speed value and the true value is about 2 m · s^-1To compare and analyze the overall prediction performance, taking the position prediction results under acceleration conditions as an example, the average is calculated. The predicted mean of the LSTM-KF model is 5440 m, and the true mean is 5346 m; The prediction mean of the traditional LSTM model under acceleration conditions is 5640 m, while the prediction result of the traditional RNN model is 8037 m; The error between the mean and true mean of the LSTM-KF model is 94 m, the error between the mean and true mean of the traditional LSTM model is 294 m, and the error between the mean and true mean of the traditional RNN model is 2691 m.

The position prediction results under deceleration conditions show that the predicted mean of the LSTM-KF model is 20 317 m, and the true mean is 18 506 m; The prediction mean of the traditional LSTM model is 22641 m, while the prediction result of the traditional RNN model is 22585 m; The error between the mean and true mean of LSTM-KF is 1181m, the error between the mean and true mean of traditional LSTM model is 4135m, and the error between the mean and true mean of traditional RNN model is 4079m. In comparison, the predicted mean of LSTM-KF model is closer.

4. 结语

(2) In order to make the data more accurate, the input data of the model not only uses velocity and position, but also adds acceleration dimension. By increasing the richness of the data, the deep learning model can fit the data features through multiple dimensions, thus achieving more accurate results.

(4) The research conducted in this article is based on the comparison of traditional models, and the correction effect of KF on new models has not been explored yet. Further research can be attempted.