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Short-Term Passenger Flow Forecasting for Rail Transit considering Chaos Theory and Improved EMD-PSO-LSTM-Combined Optimization

Short-Term Passenger Flow Forecasting for Rail Transit considering Chaos Theory and Improved... Hindawi Journal of Advanced Transportation Volume 2023, Article ID 9524966, 16 pages https://doi.org/10.1155/2023/9524966 Research Article Short-Term Passenger Flow Forecasting for Rail Transit considering Chaos Theory and Improved EMD-PSO- LSTM-Combined Optimization 1 1 1 2 Lixin Zhao , Hui Jin , Xintong Zou , and Xiao Liu Liaoning University of Technology of China, Jinzhou, Liaoning 121000, China North China University of Technology, Beijing 100000, China Correspondence should be addressed to Hui Jin; qcjinhui@lnut.edu.cn Received 4 August 2022; Revised 20 January 2023; Accepted 5 April 2023; Published 3 May 2023 Academic Editor: Ren-Yong Guo Copyright © 2023 Lixin Zhao et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Tis paper proposes a prediction method based on chaos theory and an improved empirical-modal-decomposition particle- swarm-optimization long short-term-memory (EMD-PSO-LSTM)-combined optimization process for passenger fow data with high nonlinearity and dynamic space-time dependence, using EMD to process the original passenger fow data and generate several eigenmodal functions (IMFs) and residuals with diferent characteristic scales. Based on the chaos theory, each component of the PSO algorithm was improved by introducing an inertia factor to facilitate the adjustment of its search capability to improve optimization. Each subsequence of the phase-space reconstruction was built into an improved PSO-LSTM prediction model, and the output of each prediction model was summed to determine the fnal output. Experimental studies were performed using data from the North Railway Station of Chengdu Rail Transit, and the results showed that the proposed model can generate better prediction results. Te proposed model obtained root mean square error (RMSE) and mean absolute error (MAE) of 16.0908 and 11.3704, respectively. Compared with the LSTM, the improved PSO-LSTM, the improved EMD-PSO-LSTM, and the model proposed in this paper improved the RMSE values by 25.53%, 29.97%, and 58.76%, respectively, and the MAE values by 30.41%, 40.13%, and 63.08%, respectively, of the prediction results. However, short-time passenger fow prediction can be 1. Introduction a particularly challenging problem. Raw trafc-fow data are At present, urban rail transport continues to develop spatiotemporal data that simultaneously exhibit heteroge- well, as its speed, capacity, comfort, and safety help it neity and correlation, as well as strong nonlinearity and become the main mode of transport for the urban public. chaos. In addition, most existing research captures relatively Along with the gradually increasing intensity of pas- few trafc data attributes, resulting in unsatisfactory pre- senger fow, short-time passenger fow prediction has diction results. Consequently, real-time accurate short-time become particularly important, as accurate trafc pas- passenger fow prediction is critical. senger fow prediction helps urban trafc managers Short-term trafc forecasting models can be broadly better plan and manage their resources. Moreover, trafc classifed into four categories—traditional statistical learning short-time passenger fow prediction has become of algorithms, machine learning models, deep learning algo- strategic importance for the construction of urbanized rithms, and combinatorial models. Te tasks of statistical intelligent transport systems to relieve trafc pressure, learning and machine learning algorithms are similar, which adjust operating times, and plan future construction. It is involve inferring model parameters and ftting and pre- also the foundation of smart city ambitions and dicting data. However, the focus of the two difers—that is, construction. statistical learning algorithms are more concerned with the 2 Journal of Advanced Transportation nonsmooth signal processing methods—was examined for confdence of predictions, whereas machine learning algo- rithms are more concerned with the predictive efects of the rail trafc short-time passenger fow predictions. Te im- proved complete ensemble EMD with adaptive noise model. Statistical learning algorithms include autoregressive-integrated moving average (ARIMA) models (CEEMDAN) method was proposed and used for the de- [1], seasonal ARIMA models [2], and Kalman flter models composition of highway data by decomposing the time series [3]. Te advantage of these models is that they are simple to into diferent features, which could dramatically reduce the operate; however, owing to the complexity of changes in prediction error of the mainstream model. Te improved actual passenger fow data, there can be a certain subjective CEEMDAN-fuzzy entropy (FE)-temporal convolutional factor in their establishment, which can be easily infuenced network model was shown to exhibit high predictive accuracy and strong robustness when using the US101-S highway in by a priori assumptions that can be difcult to satisfy in practice, limiting their predictive performance. California as the research object. Wang Xiao Quan et al. [17] proposed an SVM model for short-time trafc fow pre- Machine learning models include support vector ma- chines (SVMs) [4], artifcial neural networks [5, 6], and diction, incorporating the principles of chaos theory to map trafc fows into a hyper-dimensional structure by per- Bayesian networks [7], which can capture the nonlinear data features of short-time passenger fow using their own forming phase-space reconstruction based on its nonlinear learning abilities. Leng et al. [8] established an improved characteristics, and to calculate the embedding dimension neural network prediction model that was optimized using parameter by using the maximum conditional entropy a genetic optimization algorithm that not only improved the method with a time delay parameter obtained using mutual convergence of its search capability but also its prediction information techniques. Finally, the reconstructed subseries accuracy. However, traditional machine learning models were used as inputs and predicted using genetic-algorithm optimized support vector regression. Numerical experiments cannot efectively process high-dimensional data, and the complex variability of nonlinearities in time series trafc showed that the proposed method exhibited excellent pre- dictive accuracy. Lingling Wu et al. [18] proposed an em- data can be difcult to capture. Moreover, their predictive performance depends on expert experience, and their pirical model decomposition and diferential evolution algorithm to optimize the back propagation neural network generalization ability is weak. Consequently, many scholars have researched deep learning models to handle high- for a short-time trafc fow prediction model. Tey used the dimensional spatiotemporal trafc data. EMD algorithm to decompose diferent modal components in Based on deep learning algorithms such as recurrent the trafc timing data step-by-step, generating a series of neural networks (RNNs) [9, 10] and long short-term eigenmodal functions (IMFs) and residuals at diferent scales memory (LSTM) neural networks [11], Huang et al. [12] to remove certain noise efects, thereby improving the ac- processed information from trafc sequence data using curacy of results. Te main contributions of this paper are as follows: long-and short-time neural networks and gated recurrent units in RNNs, and performed noise reduction of raw (1) Given the nonlinear characteristics of trafc fow and passenger fow data using wavelet transforms. Zhang et al. the fact that particle swarm optimization (PSO) al- [13] used multigraph convolutional neural networks to gorithms usually fall into local optimality, we explore the spatial features of trafc data. However, deep combined the chaos theory and an improved EMD- learning models are prone to overftting or underftting [14]. PSO-LSTM model to design a short-time passenger Tere is also a class of combinatorial models. Zhai fow forecasting method for urban rail transport, and et al.[15] proposed a hybrid trafc fow prediction method by applied it to the feld of rail transit passenger fow combining the k-nearest neighbor and LSTM algorithms forecasting for the frst time. based on the spatiotemporal features of transportation data. (2) We used the EMD algorithm to decompose the Teir experimental results showed that their proposed original time series data and perform phase-space model improved by 12.59% on average compared with the reconstruction using chaos theory principles to re- comparison model. Gao et al. [16] proposed a new hierar- construct the useful aspects of the EMD to further chical hybrid model to forecast short-term passenger fows, explore the internal characteristics of trafc fow and with an average absolute error of approximately 10% in the improve prediction accuracy. An improved forecasting results. Moreover, experiments showed that the PSO-LSTM prediction model was then developed for prediction results of this combined model exhibited greater each reconstructed subsequence. Owing to the ten- accuracy. dency of the original model to fall into a local op- However, the collected trafc fow data can be disturbed timum, we improved it to increase its PSO-seeking by noise factors, reducing the predictive performance of the search capability, and the predictions of each com- models. To minimize the impact of external factors on ponent were summed to produce the fnal output. forecast accuracy, a prediction model that employs chaos (3) We conducted the experiments using a dataset theory can directly analyze the intrinsic regularity of trafc comprising data from the North Railway Station of fow data through a priori cognition without establishing the Chengdu Municipal Railway to validate the ef- a subjective model. Consequently, considering the charac- fectiveness of the proposed model. Te results teristics of nonlinearity and nonsmoothness in urban rail showed that the proposed model performed better trafc time series data, the empirical mode decomposition than existing methods. (EMD) algorithm—which can be applied to nonlinear and Journal of Advanced Transportation 3 n− 1 2. Methods X(t) � 􏽘 f (t) + RES (t). (4) i n 2.1. EMD Algorithm. EMD can be used for analyzing both i�1 linear and smooth, and nonlinear and nonsmooth signals. Te core of this approach is to gradually smoothen the signal and decompose the vibration modes in the signal into a fnite 2.2. Chaos Teory. Te study of the chaos theory began in number of components that tend to be smoothed based on 1980 with the phase-space reconstruction theory proposed diferent characteristic scales or trends. Moreover, in short- by Packard et al. Te theory states that the evolution of each component in a chaotic system is jointly dictated by the period passenger fow forecasting for rail transport, the transformation of nonlinear and nonsmooth passenger fow other individual components of these interactions, and that the variable contains information about the long-term signals into linear and smooth signals better refects their intrinsic physical meaning [18]. Compared with other evolution of all the variables in the system. Te basic signal-processing methods, EMD methods are more in- principle of phase-space reconstruction is the delayed em- direct, intuitive, and adaptive. bedding theorem proposed by Takens [19]. For chaotic time EMD divides the raw trafc fow signal into several series, chaotic models can be built and predicted in the so- empirical mode components (IMFs) based on its adaptive called phase space, wherein phase-space reconstruction timing analysis, each of which contains local features of based on the chaos theory is an essential component in the diferent feature scales in the previous trafc fow signal, and processing of a chaotic time series. Tere are two key index residuals (RES), which represent the mean or trend in the values in the phase-space reconstruction algorithm—the embedding dimension (d) and time delay (τ)—and in [19], original trafc fow signal. Each IMF must satisfy two conditions at the same time—that is, the diference between the parameters of both the embedding dimension and time the number of extremes and zeros in the domain of def- delay are only proved via theoretical studies, and no specifc nition should not exceed 1, and the mean value of the upper formula is given. In practical applications, the time delay and and lower envelope functions should be 0 [18]. Te process is embedding dimension parameters should be calculated as follows: considering the actual situation because raw trafc fow time-series data are infuenced by external variables. Step 1: In the original trafc signal, all extreme points In a chaotic system, a set of observations that vary with are calculated, and the upper and lower envelopes are time can be obtained by examining them—noted as chaotic ftted m (t)m (t) with the cubic spline interpolation 1 2 time series Y , i � 1, 2, . . . , n—and a set of m-dimensional function to calculate the mean value m(t) of the upper vectors can be constructed using the observations, as follows: and lower envelopes. Te mean of the original signal envelope is calculated as follows: Y � y , y(i + τ), · · ·, y(i + (m − 1)τ) , (5) i i m (t) + m (t) 1 2 where τ � k∆t, k � 1, 2, . . ., ∆t denotes the time interval, Y (1) m(t) � . i denotes a sample point in the constructed phase space with m components, and m � n − (m − 1)k. If the parameters of Step 2: Subtracting the original sequence m(t) from m are chosen appropriately, then Y can represent the state X(t) to obtain a new sequence, as follows: in the original system and dynamic characteristics of pri- mary passenger fow data in the multidimensional h (t) � X(t) − m(t), (2) phase space. where h (t) satisfes the IMF condition, so that the frst f (t) IMF component is obtained. If h (t) is still 1 1 2.3. LSTM-Based Short-Time Passenger Flow Forecasting for unstable, the abovementioned process is repeated once Rail Transit. LSTM is particularly suitable for processing with h (t) instead of X(t) until the resulting average trafc data sequences with certain time intervals. In the envelope tends to 0, defning the f component as h . 1 1k LSTM neural network structure, each neuron comprises three gating units as a solution to the drawback of dis- f � h . (3) i ik appearing gradients owing to a long time series. Te LSTM model comprises four main parts—memory cells, forgetting Step 3: Te original sequence X(t) is subtracted from gate, input gate, and output gate [20]—as shown in Figure 1. the frst IMF component to obtain the frst f (t) Te LSTM network structure is shown in Figure 2. diference sequence with the high-frequency compo- Here, the memory cell is the core component of the nent and removed RES (t). Te above processing of entire LSTM model used for storing the cell states of past RES (t) is used to obtain a second empirical modal information, and the output of the memory cell at moment t component until it is no longer possible to disaggregate can be expressed as follows: it, with the last one obtained being a residual RES (t). After decomposition, which represents the actual av- C � tan h W · 􏼂h , Z 􏼃 + b 􏼁, t c t− 1 t c erage trend of the primary series X(t), the original (6) C � f · C + i · C , sequence X(t) can be expressed as follows: t t t− 1 t t 4 Journal of Advanced Transportation c t t-1 × + tanh σσ tanh σ h h t-1 Figure 1: LSTM structure. h h t+1 t-1 t c c t-1 t × + tanh σσ σ tanh h h t-1 t Z Z t t+1 t-1 Figure 2: LSTM network structure. where Z denotes the input at the present time, i denotes the Te output gate result can be determined via three main t t input gate, which refreshes the stored information in the cell components—the previous moment’s input information, state, and C is the update to C . the information stored after the cell state is updated, and the t− 1 t Te forgetting gate is used to determine which part of the output information at the last moment. Tus, the output at information in the cell state needs to be removed by fusing moment t can be expressed as follows: information from the preceding point in time and that on O � σ W · 􏼂h , Z 􏼃 + b 􏼁, t O t− 1 t O the time at hand. Te output of the forgetting gate at mo- (9) ment t is obtained as follows: h � O · tan h C􏼁 , t t t f � σ􏼐W · 􏼂h − 1, Z 􏼃 + b 􏼑, (7) where W denotes the weight value, b denotes the bias term, t f t t f and tan h denotes the hyperbolic slice employment factor. where f denotes the output value of the forgetting gate, σ denotes the activation function, W denotes the weighting 2.4. Improved PSO-LSTM Algorithm for Rail Transit Short- matrix representing the forgetfulness gate, and b denotes Time Passenger Flow Prediction. PSO is an exploratory the bias term for the forgotten door. method and a classical swarm intelligence algorithm used for Unlike the forgetting gate, the input gate decides which solving the optimal search problem. Te principle un- information can enter the unit state based on its threshold, derpinning this optimization method originated from the and the candidate value vector is created in the tanh layer to search learning of particle foraging behavioral approach, generate candidate memories, wherein new information wherein each bird is abstractly viewed as a particle and used passing through the screening is added to the unit state to to represent a feasible solution [21]. By evaluating the de- replenish the lost attribute information. In addition, the grees of superiority and inferiority of each particle through input gate updates the information stored in the cell state, a ftness value, a series of random searches are performed, and its output at time t is calculated as follows: and the current optimal solution search is dynamically i � σ W · 􏼂h − 1, Z 􏼃 + b 􏼁. (8) t t t t i tracked by exchanging information with other particles, discovering information, and adaptively changing the Journal of Advanced Transportation 5 Start Initialize particle swarm and parameter setting T=1 To calculate the objective function Update individual best pbest and group best gbest Determining whether the NO convergence basis is satisfied Update the position vector and YES velocity vector of each particle Output optimal results and parameters Evaluate the fitness value of the function for each particle End Update the historical optimal position of each particle Update global optimal position T=T+1 Figure 3: Improved PSO solution process. direction of the next search by collective information sharing function of the PSO algorithm [22]. When the particles such that the group can determine the optimal destination update their velocity and position vectors with each re- location. peated motion, the best value of these two results can be To overcome the shortcomings of traditional PSO obtained by tracking the positions through which the algorithms, which can easily fall into optimal solutions, particles and swarm pass. Te specifc method can be the particle swarm algorithm was improved by in- expressed as follows: troducing an inertia factor, which can reasonably and v � v + c pbest − x􏼁 + c gbest − x􏼁 , i i 1 i i 2 i i efectively regulate the global search and partial search (10) x � x + v , capabilities of the algorithm, such that it can change i i i during the PSO search process based on the search 6 Journal of Advanced Transportation where i � (1, 2, . . . , N), N denotes the total number of PSO-LSTM passenger fow forecasting algorithm based on particles in the swarm, x denotes the current position of the the chaos theory is shown in Figure 6. particle, v denotes the velocity of the particle, c and c i 1 2 Step 1: By performing EMD on the rail trafc short- denote the learning factors, pbest denotes the most opti- time passenger fow data, several IMFs and RES terms mum point for a mass to pass through, and gbest denotes can be obtained. the optimal position experienced by the swarm as a whole Step 2: Te decomposed components are screened, [20]. which are then reconstructed in the phase space. v &9; � w × v + c pbest − x􏼁 + c gbest − x􏼁 , i i 1 i i 2 i i Step 3: An improved PSO-LSTM prediction model is (11) developed for each component after phase-space re- w&9; � w + w − w · 1 − , 􏼁 􏼒 􏼓 construction, and the improved PSO algorithm is then e s e used to fnd the optimal parameters and train the where w denotes the weight at the beginning of the inertia LSTM model. factor, w denotes the weight at the end of the inertia factor, t Step 4: Te predicted values of each component are denotes the maximum number of iterations after all itera- superimposed and ftted to obtain the fnal prediction tions are completed, and t denotes the number of iterations results of the model. at the current moment [23]. Step 5: Te fnal prediction results are output [26]. Te improved PSO solution can be obtained by con- structing the LSTM short-time passenger fow prediction 4. Results: Case Studies model and using a modifed PSO optimization algorithm to discover the optimal parameters for the LSTM prediction 4.1. AFC Data and Processing. Te operational data of the model [24], as shown in Figure 3. Te pseudocode of the North Railway Station of Chengdu Metro Line 1—that is, its improved PSO algorithm is shown in Figure 4. incoming passenger fow data from January 4–13, Te specifc operational steps are as follows: 2020—were selected for this study. In the Chengdu rail transit system, the AFC platform recorded the entrance and Step 1: Initialize the particle swarm and set the relevant exit information of each passenger using smart card data parameters, including the population size, random from the automatic ticketing system at each metro station. position, and velocity. Inbound trafc data at the metro stations were obtained Step 2: Determine the ftness value of each particle, as between 5 : 00 and 00 : 55 the following day, with a data well as the optimal positions for the particle and collection interval of 5 min. Te data contained a total of particle population to pass. 2,260 time series, each of which included the start time, end Step 3: Determine whether the particles satisfy the time, input fow, and output fow. Te Chengdu Metro Rail convergence condition, and if they do, output the re- Transit North Station road network map is shown in sult. If they do not satisfy the convergence condition, Figure 7. continue with the following steps. A total of 2260 time series of data were input in the Step 4: Te velocity vector is updated with the opti- experiments of this paper. Te input data are inbound passenger fow in person/5 min. Te output data are the mum positions passed by the particles and particle swarm, and the position vector of the particles is inbound passenger fow in person/5 min. Te simulation updated with the updated velocity vector, after which environment used to test the predictive performance of all optimal particles are updated. the model in this study was MATLAB 2019a. Generally, the larger the training set, the more accurate the pre- Step 5: Return to Step 3 until the convergence con- diction results. Terefore, to take full advantage of the dition is met, before outputting the optimal result and data, the frst 90% of the original trafc data (that of eight number of iterations. days from January 4–11) were used in the training set, and the remaining 10% (January 12) were used in the 3. Design of Short-Time Passenger Flow test set. Forecasting Algorithm for Rail Transit First, anomalies and missing data from the original data Based on Chaos Theory and Improved EMD- were processed, wherein the anomalous data were consid- PSO-LSTM ered as missing data. Lagrangian interpolation methods were used to process the missing data, wherein four neighboring Te EMD algorithm can decompose the time series of rail data before and after the missing datum are selected for transit short-time passenger fow into IMF components of interpolation to ensure the reliability of the diferent frequencies based on their intrinsic characteristics, interpolated data. which describe the local characteristics of the original series Te data was then normalized using the min-max more clearly. An improved PSO-LSTM prediction model method as follows: can then be separately built for each subsequence, before y − y ∗ min adding the predicted values of each subsequence to obtain y � , (12) y − y max min the fnal output [25]. Te model construction process is shown in Figure 5. Te pseudocode of the improved EMD- Journal of Advanced Transportation 7 Input: Population size M, random position P, velocity V Output: Outputs optimal results and related parameters 1 Initialize the particle swarm 2 Set population size M, random position P, velocity V 3 Set the parameter T←1 4 Computes the objective function value 5 Update individual optimals Update (pbest) , population optimal Update (gbest) 6 If (convergence condition satisfied) 7 Output optimal results and related parameters 8 quit ( ) 9 Else (does not meet the convergence condition) 10 Updates the particle's position vector P and velocity vector V 11 Calculate the adaptation value W for each particle 12 Update the historical optimal position of each particle Update (pbest) 13 Update the global optimal location of the population Update (gbest) 14 Set the parameter T← T+1 15 Recalculate the objective function value 16 End if Figure 4: Pseudocode of the improved PSO algorithm. where y and y are the minimum and maximum values series can be divided into nine empirical modal components min max of the trafc fow, respectively, and y and y are the trafc and one residual component. Te EMD results are shown in fow data before and after normalization, respectively. Figure 10. It can be observed from the fgure that the IMF1, IMF2, and IMF3 empirical modal components have higher fre- 4.2. Prediction Results of the LSTM Model. Te LSTM rail quencies and are high-frequency components of the original trafc short-time passenger fow prediction model was rail trafc passenger fow data. IMF4, IMF5, and IMF6 established, the results of which are shown in Figure 8. empirical modal components have more obvious periodicity and are low-frequency components of the original rail trafc passenger fow data. Te residuals are the overall trend of the 4.3. Prediction Results of the Improved PSO-LSTM Model. time series data and are the trend components of the original Te prediction results show that the prediction performance rail trafc passenger fow data. Te EMD of the rail trafc of the LSTM model is poor. Consequently, an improved short-time passenger fow time series provides a clearer PSO-LSTM model was introduced, the prediction results of understanding of the passenger trafc fow data fuctuation which are shown in Figure 9. By running the model through and overall trend. repeated iterations, its prediction results were found to be optimal with an optimal number of hidden nodes of 167, optimal learning rate of 0.0310, and optimal number of 4.4.2. Phase Space Reconstruction Based on Chaos Teory. iterations of 30. Te EMD of the rail trafc fow change sequence can determine the trafc fow fuctuations more accurately. However, because the three components—that is, IMF7, 4.4. Prediction Results Based on Chaos Teory and Improved IMF8, and IMF9—do not show the intrinsic properties of EMD-PSO-LSTM Model the data, the model only selects the remaining seven 4.4.1. EMD of Trafc Flow Change Series. Because the components, reconstructing them in phase space by original rail trafc fow data have characteristics of non- fnding the time delay using the mutual information linearity and nonsmoothness [27], the noise in it will have method [28] and the embedding dimension using the Cao some infuence on the prediction results, resulting in in- method. If phase-space reconstruction parameters are accurate prediction results. Te noise in the time series data carefully selected, the reconstructed phase space can can be mitigated through the EMD algorithm, thus im- describe the states in the original system, and the mul- proving the predictive power of the model. Based on the tidimensional phase space can show the dynamic char- EMD algorithm, the EMD of the rail trafc fow variation acteristics of the trafc fow change sequence [29]. Te 8 Journal of Advanced Transportation Original time series of traffic flow Empirical modal decomposition IMF1 IMF2 IMFk RESn Phase space Phase space Phase space Phase space reconstruction reconstruction reconstruction reconstruction based on chaos based on chaos based on chaos based on chaos theory theory theory theory Improved PSO-LSTM Improved PSO-LSTM Improved PSO-LSTM Improved PSO-LSTM prediction prediction prediction prediction model model model model Superimpose the prediction results of each subsequence Final Forecast Results Figure 5: Improved EMD-PSO-LSTM model construction process. remaining seven component phase-space reconstruction Figures 11 more specifcally, the superimposed predictions parameters are listed in Table 1. are shown in Figures 11(a)–11(g), which show the plots of After phase-space reconstruction, the improved PSO the PSO search results for each component, whereas algorithm for the components was used to determine the Figures 11(h)–11(n) show the predicted results for each optimal number of hidden nodes, learning rate, and number component. Figure 12 shows the predictions for each component overlay. of iterations of the LSTM prediction model. Te optimal parameters of each component are listed in Table 2. 4.4.4. Prediction Results of the Improved EMD-PSO-LSTM 4.4.3. Prediction Results Based on Chaos Teory and Im- Model. To further validate the prediction efect of the combined optimization model based on the chaos theory proved EMD-PSO-LSTM Model. Owing to the nonlinear and nonsmooth characteristics of the original rail trafc and the improved EMD-PSO-LSTM model, a set of com- parison models were added—that is, phase-space re- passenger fow data, the noise in its time series data has a certain infuence on the prediction results, resulting in construction of the components without considering the inaccurate predictions. Consequently, using the improved chaos theory. Te components obtained after EMD were PSO-LSTM model, the trafc fow variation sequence could screened, and the remaining seven components were se- be empirically decomposed. Based on the chaos theory, the lected to build the improved PSO-LSTM prediction model. decomposed subsequence could be reconstructed in the Te optimal number of hidden nodes, learning rate, and phase space and an improved EMD-PSO-LSTM-combined number of iterations of the LSTM prediction model were optimization model was constructed. Te PSO search and determined using the improved PSO algorithm. Te optimal prediction results for each component are shown in parameters of each component are listed in Table 3. Te Journal of Advanced Transportation 9 Input: Rail transit short-term passenger flow data Output: Outputs the final prediction and performs an error analysis 1 Enter the short-term passenger flow data of rail transit 2 EMD empirical modal decomposition of the data 3Fori←0 to n do 4 Modal function INF[i]←Data[i] 5 Filter the individual components after decomposition Choose () 6 Performing Phase Space Reconstruction Recon () 7 Establishing an Improved Predictive Model for Components Predict () 8 Look for the optimal parameter M 9 Train the LSTM model 10 End for 11 For i←0 to n do 12 Overlay the prediction results for each subseries 13 Sum+=Pre[i] 14 End for 15 Outputs the final prediction Figure 6: Pseudocode of the improved EMD-PSO-LSTM passenger fow forecasting algorithm. SubwayStation 0 0.5 1 km Subway Figure 7: Chengdu Metro Rail Transit North Railway Station road network map. 10 Journal of Advanced Transportation 0 50 100 150 200 250 Time (min) Original LSTM Figure 8: LSTM model prediction results. 0 50 100 150 200 250 Time (min) Original Improved PSO–LSTM Figure 9: Improved PSO-LSTM model prediction results. prediction results shown in Figures 13(a)–13(g) show the where x denotes the actual inbound trafc at time i, x 􏽢 i i plots of the PSO search results for each component. denotes the forecast inbound trafc at time i, x indicates the Figures 13(h)–13(n) show the predicted results for each average value of trafc volume, and N denotes the total component. Figure 14 shows the predictions for each volume of inbound trafc in the trafc sequence. component overlay. To visually evaluate the prediction results of the four prediction models, the aforementioned errors were used to compare and analyze the strengths and weaknesses of 4.5. Evaluation Indicators. To better compare the pre- the model predictions [17]. Te evaluation indicator dictions between the LSTM model, improved PSO-LSTM values of the prediction results are listed in Table 4, and model, improved EMD-PSO-LSTM model, and improved the percentage improvements in the prediction results are EMD-PSO-LSTM model are based on the chaos theory. Te listed in Table 5. mean absolute error (MAE) and root mean square error Te comparative analysis of the error metrics shown in (RMSE) and the coefcient of determination (R ) metrics Tables 4 and 5 indicates that the improved EMD-PSO-LSTM were chosen to compare their errors [30]. Teir formulas can prediction model based on the chaos theory exhibits higher be expressed as follows: prediction accuracy. In addition, the improved PSO-LSTM 􏽶������������� and improved EMD-PSO-LSTM models and the improved EMD-PSO-LSTM model based on the chaos theory improve RMSE � x − x 􏽢 , 􏽘 􏼁 i i the passenger fow prediction results by using the LSTM i�1 model prediction results as a benchmark. Te percentage improvement of RMSE values is 25.53, 29.97, and 58.76%, 􏼌 􏼌 (13) 􏼌 􏼌 􏼌 􏼌 respectively, and the percentage improvement of MAE MAE � 􏽘􏼌x − x 􏽢 􏼌, i i values is 30.41, 40.13, and 63.08%, and the percentage im- i�1 provement of R values is 13.36, 16.31, and 32.30%. Tese 􏽐 x − x􏼁 results indicate that the proposed model has great potential 2 i�1 i R � , n 2 for short-time passenger fow forecasting applications in rail 􏽐 x − x􏼁 i�1 transit. Flow (person/5 min) Flow (person/5 min) Journal of Advanced Transportation 11 –50 0 500 1000 1500 2000 2500 3000 –50 0 500 1000 1500 2000 2500 3000 –50 0 500 1000 1500 2000 2500 3000 –50 0 500 1000 1500 2000 2500 3000 –50 0 500 1000 1500 2000 2500 3000 –50 0 500 1000 1500 2000 2500 3000 –50 0 500 1000 1500 2000 2500 3000 –50 0 500 1000 1500 2000 2500 3000 –50 0 500 1000 1500 2000 2500 3000 –50 0 500 1000 1500 2000 2500 3000 Figure 10: EMD results. Table 1: Phase space reconstruction parameters. Portion size IMF1 IMF2 IMF3 IMF4 IMF5 IMF6 RESn Time delay τ 99 99 4 8 23 36 99 Embedded dimensions m 6 6 6 5 5 4 6 Table 2: Optimal parameters of each subsequence after phase-space reconstruction. Optimal parameters IMF1 IMF2 IMF3 IMF4 IMF5 IMF6 RESn Hidden nodes 172 184 126 150 200 139 100 Learning rate 0.0051 0.0010 0.0010 0.0010 0.0088 0.0010 0.0010 Number of iterations 30 30 30 30 30 30 30 To further validate the experimental results, the prediction, including the model radial basis function neural EMD-PSO optimization algorithm based on the chaos network (RBF) and multilayer perceptron (MLP), were also theory was compared with combinations of deep learning- compared. Te experimental results and error results are based deep belief networks (DBN) and gated recurrent unit presented in Figure 15 and Table 6, respectively. (GRU) neural networks. Moreover, combinations of neural Te experimental results showed that the proposed network prediction commonly used in the feld of trafc fow EMD-PSO-LSTM model of rail transit short-time RESn IMF9 IMF8 IMF7 IMF6 IMF5 IMF4 IMF3 IMF2 IMF1 12 Journal of Advanced Transportation 18 9 2.8 3.5 16 3 2.6 8.5 14 2.5 2.4 12 2 10 1.5 2.2 7.5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 Evolutionary number Evolutionary number Evolutionary number Evolutionary number Best Best Best Best Average Average Average Average (a) (b) (c) (d) 9 7 300 8 6 6 4 100 5 4 3 -100 1 2 3 4 5 1 2 3 4 5 0 50 100 150 200 250 1 2 3 4 5 Evolutionary number Evolutionary number Time (min) Evolutionary number Best Best Original Best Average Average Predicted Average (e) (f) (g) (h) 300 300 300 300 200 200 100 100 100 0 0 0 -100 0 50 100 150 200 250 0 50 100 150 200 250 0 50 100 150 200 250 0 50 100 150 200 250 Time (min) Time (min) Time (min) Time (min) Original Original Original Original Predicted Predicted Predicted Predicted (i) (j) (k) (l) 300 300 0 50 100 150 200 250 0 50 100 150 200 250 Time (min) Time (min) Original Original Predicted Predicted (m) (n) Figure 11: Optimization search results and predictions for each component based on chaos theory : (a) IMF1’s search results, (b) IMF2’s search results, (c) IMF3’s search results, (d) IMF4’s search results, (e) IMF5’s search results, (f ) IMF6’s search results, (g) RESn’s search results, (h) IMF1’s predicted results, (i) IMF2’s predicted results, (j) IMF3’s predicted results, (k) IMF4’s predicted results, (l) IMF5’s predicted results, (m) IMF6’s predicted results, and (n) RESn’s predicted results. 50 100 150 200 250 Time (min) Original improved EMD-PSO-LSTM model based on chaos theory. Figure 12: Prediction results of the improved EMD-PSO-LSTM model based on chaos theory. Adaptation Flow (person/5 min) Adaptation Adaptation Flow (person/5 min) Flow (person/5 min) Adaptation Flow (person/5 min) Adaptation Flow (person/5 min) Adaptation Flow (person/5 min) Adaptation Flow (person/5 min) Flow (person/5 min) Journal of Advanced Transportation 13 Table 3: Optimal parameters of each subsequence after phase-space reconstruction. Optimal parameters IMF1 IMF2 IMF3 IMF4 IMF5 IMF6 RESn Hidden nodes 126 116 139 100 182 100 200 Learning rate 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 0.0013 Number of iterations 30 30 30 30 30 30 30 7 5 15.6 15.4 15.2 14.8 14.6 4 1 1 2345 1 2345 1 2345 1.5 234 2.5 3.5 4.5 5 Evolutionary number Evolutionary number Evolutionary number Evolutionary number Best Best Best Best Average Average Average Average (a) (b) (c) (d) 8 300 7 9 8.5 0 50 100 150 200 250 1 2345 1 2345 1 2345 Evolutionary number Evolutionary number Evolutionary number Time (min) Best Original Best Best Average Average Average Predicted (e) (f) (g) (h) 300 300 300 300 200 200 200 200 100 100 100 0 0 0 0 0 50 100 150 200 250 050 100 150 200 250 0 50 100 150 200 250 0 50 100 150 200 250 Time (min) Time (min) Time (min) Time (min) Original Original Original Original Predicted Predicted Predicted Predicted (i) (j) (k) (l) 300 300 050 100 150 200 250 050 100 150 200 250 Time (min) Time (min) Original Original Predicted Predicted (m) (n) Figure 13: Optimization search results and predictions for each component: (a) IMF1’s search results, (b) IMF2’s search results, (c) IMF3’s search results, (d) IMF4’s search results, (e) IMF5’s search results, (f) IMF6’s search results, (g) RESn’s search results, (h) IMF1’s predicted results, (i) IMF2’s predicted results, (j) IMF3’s predicted results, (k) IMF4’s predicted results, (l) IMF5’s predicted results, (m) IMF6’s predicted results, and (n) RESn’s predicted results. Adaptation Adaptation Flow (person/5 min) Adaptation Flow (person/5 min) Adaptation Flow (person/5 min) Adaptation Adaptation Flow (person/5 min) Flow (person/5 min) Flow (person/5 min) Flow (person/5 min) Adaptation 14 Journal of Advanced Transportation -50 0 50 100 150 200 250 Time (min) Original Improved EMD-PSO-LSTM Figure 14: Improved EMD-PSO-LSTM model prediction results. Table 4: Results of various models. Evaluation indicators Predictive models RMSE MAE R (%) LSTM 39.0150 30.7984 71.02 Improved PSO-LSTM 29.0557 21.4331 80.51 Improved EMD-PSO-LSTM 27.3210 18.4379 82.60 Improved EMD-PSO-LSTM based on chaos theory 16.0908 11.3704 93.96 Table 5: Percentage improvements in the prediction results. Evaluation indicators (%) Predictive models RMSE MAE R (%) Improved PSO-LSTM 25.53 30.41 13.36 Improved EMD-PSO-LSTM 29.97 40.13 16.31 Improved EMD-PSO-LSTM based on chaos theory 58.76 63.08 32.30 0 50 100 150 200 250 Time (min) True value GRU MLP LSTM RBF DBN Figure 15: Experimental results of the fve models. Flow (person/5 min) Passenger Flow (person/5 min) Journal of Advanced Transportation 15 Table 6: Results of various models. Evaluation indicators Algorithm run time Predictive models Training RMSE MAE R (%) and testing (s) Improved EMD-PSO-MLP 25.4751 17.5730 84.38 984.65 Improved EMD-PSO-RBF 22.2670 17.6294 86.70 1092.47 Improved EMD-PSO-DBN 17.1435 13.4770 92.15 1168.54 Improved EMD-PSO-GRU 18.5085 14.3520 91.32 945.28 Improved EMD-PSO-LSTM 16.0908 11.3704 93.96 967.53 [3] J. Guo, W. Huang, and B. M. 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Short-Term Passenger Flow Forecasting for Rail Transit considering Chaos Theory and Improved EMD-PSO-LSTM-Combined Optimization

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10.1155/2023/9524966
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Hindawi Journal of Advanced Transportation Volume 2023, Article ID 9524966, 16 pages https://doi.org/10.1155/2023/9524966 Research Article Short-Term Passenger Flow Forecasting for Rail Transit considering Chaos Theory and Improved EMD-PSO- LSTM-Combined Optimization 1 1 1 2 Lixin Zhao , Hui Jin , Xintong Zou , and Xiao Liu Liaoning University of Technology of China, Jinzhou, Liaoning 121000, China North China University of Technology, Beijing 100000, China Correspondence should be addressed to Hui Jin; qcjinhui@lnut.edu.cn Received 4 August 2022; Revised 20 January 2023; Accepted 5 April 2023; Published 3 May 2023 Academic Editor: Ren-Yong Guo Copyright © 2023 Lixin Zhao et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Tis paper proposes a prediction method based on chaos theory and an improved empirical-modal-decomposition particle- swarm-optimization long short-term-memory (EMD-PSO-LSTM)-combined optimization process for passenger fow data with high nonlinearity and dynamic space-time dependence, using EMD to process the original passenger fow data and generate several eigenmodal functions (IMFs) and residuals with diferent characteristic scales. Based on the chaos theory, each component of the PSO algorithm was improved by introducing an inertia factor to facilitate the adjustment of its search capability to improve optimization. Each subsequence of the phase-space reconstruction was built into an improved PSO-LSTM prediction model, and the output of each prediction model was summed to determine the fnal output. Experimental studies were performed using data from the North Railway Station of Chengdu Rail Transit, and the results showed that the proposed model can generate better prediction results. Te proposed model obtained root mean square error (RMSE) and mean absolute error (MAE) of 16.0908 and 11.3704, respectively. Compared with the LSTM, the improved PSO-LSTM, the improved EMD-PSO-LSTM, and the model proposed in this paper improved the RMSE values by 25.53%, 29.97%, and 58.76%, respectively, and the MAE values by 30.41%, 40.13%, and 63.08%, respectively, of the prediction results. However, short-time passenger fow prediction can be 1. Introduction a particularly challenging problem. Raw trafc-fow data are At present, urban rail transport continues to develop spatiotemporal data that simultaneously exhibit heteroge- well, as its speed, capacity, comfort, and safety help it neity and correlation, as well as strong nonlinearity and become the main mode of transport for the urban public. chaos. In addition, most existing research captures relatively Along with the gradually increasing intensity of pas- few trafc data attributes, resulting in unsatisfactory pre- senger fow, short-time passenger fow prediction has diction results. Consequently, real-time accurate short-time become particularly important, as accurate trafc pas- passenger fow prediction is critical. senger fow prediction helps urban trafc managers Short-term trafc forecasting models can be broadly better plan and manage their resources. Moreover, trafc classifed into four categories—traditional statistical learning short-time passenger fow prediction has become of algorithms, machine learning models, deep learning algo- strategic importance for the construction of urbanized rithms, and combinatorial models. Te tasks of statistical intelligent transport systems to relieve trafc pressure, learning and machine learning algorithms are similar, which adjust operating times, and plan future construction. It is involve inferring model parameters and ftting and pre- also the foundation of smart city ambitions and dicting data. However, the focus of the two difers—that is, construction. statistical learning algorithms are more concerned with the 2 Journal of Advanced Transportation nonsmooth signal processing methods—was examined for confdence of predictions, whereas machine learning algo- rithms are more concerned with the predictive efects of the rail trafc short-time passenger fow predictions. Te im- proved complete ensemble EMD with adaptive noise model. Statistical learning algorithms include autoregressive-integrated moving average (ARIMA) models (CEEMDAN) method was proposed and used for the de- [1], seasonal ARIMA models [2], and Kalman flter models composition of highway data by decomposing the time series [3]. Te advantage of these models is that they are simple to into diferent features, which could dramatically reduce the operate; however, owing to the complexity of changes in prediction error of the mainstream model. Te improved actual passenger fow data, there can be a certain subjective CEEMDAN-fuzzy entropy (FE)-temporal convolutional factor in their establishment, which can be easily infuenced network model was shown to exhibit high predictive accuracy and strong robustness when using the US101-S highway in by a priori assumptions that can be difcult to satisfy in practice, limiting their predictive performance. California as the research object. Wang Xiao Quan et al. [17] proposed an SVM model for short-time trafc fow pre- Machine learning models include support vector ma- chines (SVMs) [4], artifcial neural networks [5, 6], and diction, incorporating the principles of chaos theory to map trafc fows into a hyper-dimensional structure by per- Bayesian networks [7], which can capture the nonlinear data features of short-time passenger fow using their own forming phase-space reconstruction based on its nonlinear learning abilities. Leng et al. [8] established an improved characteristics, and to calculate the embedding dimension neural network prediction model that was optimized using parameter by using the maximum conditional entropy a genetic optimization algorithm that not only improved the method with a time delay parameter obtained using mutual convergence of its search capability but also its prediction information techniques. Finally, the reconstructed subseries accuracy. However, traditional machine learning models were used as inputs and predicted using genetic-algorithm optimized support vector regression. Numerical experiments cannot efectively process high-dimensional data, and the complex variability of nonlinearities in time series trafc showed that the proposed method exhibited excellent pre- dictive accuracy. Lingling Wu et al. [18] proposed an em- data can be difcult to capture. Moreover, their predictive performance depends on expert experience, and their pirical model decomposition and diferential evolution algorithm to optimize the back propagation neural network generalization ability is weak. Consequently, many scholars have researched deep learning models to handle high- for a short-time trafc fow prediction model. Tey used the dimensional spatiotemporal trafc data. EMD algorithm to decompose diferent modal components in Based on deep learning algorithms such as recurrent the trafc timing data step-by-step, generating a series of neural networks (RNNs) [9, 10] and long short-term eigenmodal functions (IMFs) and residuals at diferent scales memory (LSTM) neural networks [11], Huang et al. [12] to remove certain noise efects, thereby improving the ac- processed information from trafc sequence data using curacy of results. Te main contributions of this paper are as follows: long-and short-time neural networks and gated recurrent units in RNNs, and performed noise reduction of raw (1) Given the nonlinear characteristics of trafc fow and passenger fow data using wavelet transforms. Zhang et al. the fact that particle swarm optimization (PSO) al- [13] used multigraph convolutional neural networks to gorithms usually fall into local optimality, we explore the spatial features of trafc data. However, deep combined the chaos theory and an improved EMD- learning models are prone to overftting or underftting [14]. PSO-LSTM model to design a short-time passenger Tere is also a class of combinatorial models. Zhai fow forecasting method for urban rail transport, and et al.[15] proposed a hybrid trafc fow prediction method by applied it to the feld of rail transit passenger fow combining the k-nearest neighbor and LSTM algorithms forecasting for the frst time. based on the spatiotemporal features of transportation data. (2) We used the EMD algorithm to decompose the Teir experimental results showed that their proposed original time series data and perform phase-space model improved by 12.59% on average compared with the reconstruction using chaos theory principles to re- comparison model. Gao et al. [16] proposed a new hierar- construct the useful aspects of the EMD to further chical hybrid model to forecast short-term passenger fows, explore the internal characteristics of trafc fow and with an average absolute error of approximately 10% in the improve prediction accuracy. An improved forecasting results. Moreover, experiments showed that the PSO-LSTM prediction model was then developed for prediction results of this combined model exhibited greater each reconstructed subsequence. Owing to the ten- accuracy. dency of the original model to fall into a local op- However, the collected trafc fow data can be disturbed timum, we improved it to increase its PSO-seeking by noise factors, reducing the predictive performance of the search capability, and the predictions of each com- models. To minimize the impact of external factors on ponent were summed to produce the fnal output. forecast accuracy, a prediction model that employs chaos (3) We conducted the experiments using a dataset theory can directly analyze the intrinsic regularity of trafc comprising data from the North Railway Station of fow data through a priori cognition without establishing the Chengdu Municipal Railway to validate the ef- a subjective model. Consequently, considering the charac- fectiveness of the proposed model. Te results teristics of nonlinearity and nonsmoothness in urban rail showed that the proposed model performed better trafc time series data, the empirical mode decomposition than existing methods. (EMD) algorithm—which can be applied to nonlinear and Journal of Advanced Transportation 3 n− 1 2. Methods X(t) � 􏽘 f (t) + RES (t). (4) i n 2.1. EMD Algorithm. EMD can be used for analyzing both i�1 linear and smooth, and nonlinear and nonsmooth signals. Te core of this approach is to gradually smoothen the signal and decompose the vibration modes in the signal into a fnite 2.2. Chaos Teory. Te study of the chaos theory began in number of components that tend to be smoothed based on 1980 with the phase-space reconstruction theory proposed diferent characteristic scales or trends. Moreover, in short- by Packard et al. Te theory states that the evolution of each component in a chaotic system is jointly dictated by the period passenger fow forecasting for rail transport, the transformation of nonlinear and nonsmooth passenger fow other individual components of these interactions, and that the variable contains information about the long-term signals into linear and smooth signals better refects their intrinsic physical meaning [18]. Compared with other evolution of all the variables in the system. Te basic signal-processing methods, EMD methods are more in- principle of phase-space reconstruction is the delayed em- direct, intuitive, and adaptive. bedding theorem proposed by Takens [19]. For chaotic time EMD divides the raw trafc fow signal into several series, chaotic models can be built and predicted in the so- empirical mode components (IMFs) based on its adaptive called phase space, wherein phase-space reconstruction timing analysis, each of which contains local features of based on the chaos theory is an essential component in the diferent feature scales in the previous trafc fow signal, and processing of a chaotic time series. Tere are two key index residuals (RES), which represent the mean or trend in the values in the phase-space reconstruction algorithm—the embedding dimension (d) and time delay (τ)—and in [19], original trafc fow signal. Each IMF must satisfy two conditions at the same time—that is, the diference between the parameters of both the embedding dimension and time the number of extremes and zeros in the domain of def- delay are only proved via theoretical studies, and no specifc nition should not exceed 1, and the mean value of the upper formula is given. In practical applications, the time delay and and lower envelope functions should be 0 [18]. Te process is embedding dimension parameters should be calculated as follows: considering the actual situation because raw trafc fow time-series data are infuenced by external variables. Step 1: In the original trafc signal, all extreme points In a chaotic system, a set of observations that vary with are calculated, and the upper and lower envelopes are time can be obtained by examining them—noted as chaotic ftted m (t)m (t) with the cubic spline interpolation 1 2 time series Y , i � 1, 2, . . . , n—and a set of m-dimensional function to calculate the mean value m(t) of the upper vectors can be constructed using the observations, as follows: and lower envelopes. Te mean of the original signal envelope is calculated as follows: Y � y , y(i + τ), · · ·, y(i + (m − 1)τ) , (5) i i m (t) + m (t) 1 2 where τ � k∆t, k � 1, 2, . . ., ∆t denotes the time interval, Y (1) m(t) � . i denotes a sample point in the constructed phase space with m components, and m � n − (m − 1)k. If the parameters of Step 2: Subtracting the original sequence m(t) from m are chosen appropriately, then Y can represent the state X(t) to obtain a new sequence, as follows: in the original system and dynamic characteristics of pri- mary passenger fow data in the multidimensional h (t) � X(t) − m(t), (2) phase space. where h (t) satisfes the IMF condition, so that the frst f (t) IMF component is obtained. If h (t) is still 1 1 2.3. LSTM-Based Short-Time Passenger Flow Forecasting for unstable, the abovementioned process is repeated once Rail Transit. LSTM is particularly suitable for processing with h (t) instead of X(t) until the resulting average trafc data sequences with certain time intervals. In the envelope tends to 0, defning the f component as h . 1 1k LSTM neural network structure, each neuron comprises three gating units as a solution to the drawback of dis- f � h . (3) i ik appearing gradients owing to a long time series. Te LSTM model comprises four main parts—memory cells, forgetting Step 3: Te original sequence X(t) is subtracted from gate, input gate, and output gate [20]—as shown in Figure 1. the frst IMF component to obtain the frst f (t) Te LSTM network structure is shown in Figure 2. diference sequence with the high-frequency compo- Here, the memory cell is the core component of the nent and removed RES (t). Te above processing of entire LSTM model used for storing the cell states of past RES (t) is used to obtain a second empirical modal information, and the output of the memory cell at moment t component until it is no longer possible to disaggregate can be expressed as follows: it, with the last one obtained being a residual RES (t). After decomposition, which represents the actual av- C � tan h W · 􏼂h , Z 􏼃 + b 􏼁, t c t− 1 t c erage trend of the primary series X(t), the original (6) C � f · C + i · C , sequence X(t) can be expressed as follows: t t t− 1 t t 4 Journal of Advanced Transportation c t t-1 × + tanh σσ tanh σ h h t-1 Figure 1: LSTM structure. h h t+1 t-1 t c c t-1 t × + tanh σσ σ tanh h h t-1 t Z Z t t+1 t-1 Figure 2: LSTM network structure. where Z denotes the input at the present time, i denotes the Te output gate result can be determined via three main t t input gate, which refreshes the stored information in the cell components—the previous moment’s input information, state, and C is the update to C . the information stored after the cell state is updated, and the t− 1 t Te forgetting gate is used to determine which part of the output information at the last moment. Tus, the output at information in the cell state needs to be removed by fusing moment t can be expressed as follows: information from the preceding point in time and that on O � σ W · 􏼂h , Z 􏼃 + b 􏼁, t O t− 1 t O the time at hand. Te output of the forgetting gate at mo- (9) ment t is obtained as follows: h � O · tan h C􏼁 , t t t f � σ􏼐W · 􏼂h − 1, Z 􏼃 + b 􏼑, (7) where W denotes the weight value, b denotes the bias term, t f t t f and tan h denotes the hyperbolic slice employment factor. where f denotes the output value of the forgetting gate, σ denotes the activation function, W denotes the weighting 2.4. Improved PSO-LSTM Algorithm for Rail Transit Short- matrix representing the forgetfulness gate, and b denotes Time Passenger Flow Prediction. PSO is an exploratory the bias term for the forgotten door. method and a classical swarm intelligence algorithm used for Unlike the forgetting gate, the input gate decides which solving the optimal search problem. Te principle un- information can enter the unit state based on its threshold, derpinning this optimization method originated from the and the candidate value vector is created in the tanh layer to search learning of particle foraging behavioral approach, generate candidate memories, wherein new information wherein each bird is abstractly viewed as a particle and used passing through the screening is added to the unit state to to represent a feasible solution [21]. By evaluating the de- replenish the lost attribute information. In addition, the grees of superiority and inferiority of each particle through input gate updates the information stored in the cell state, a ftness value, a series of random searches are performed, and its output at time t is calculated as follows: and the current optimal solution search is dynamically i � σ W · 􏼂h − 1, Z 􏼃 + b 􏼁. (8) t t t t i tracked by exchanging information with other particles, discovering information, and adaptively changing the Journal of Advanced Transportation 5 Start Initialize particle swarm and parameter setting T=1 To calculate the objective function Update individual best pbest and group best gbest Determining whether the NO convergence basis is satisfied Update the position vector and YES velocity vector of each particle Output optimal results and parameters Evaluate the fitness value of the function for each particle End Update the historical optimal position of each particle Update global optimal position T=T+1 Figure 3: Improved PSO solution process. direction of the next search by collective information sharing function of the PSO algorithm [22]. When the particles such that the group can determine the optimal destination update their velocity and position vectors with each re- location. peated motion, the best value of these two results can be To overcome the shortcomings of traditional PSO obtained by tracking the positions through which the algorithms, which can easily fall into optimal solutions, particles and swarm pass. Te specifc method can be the particle swarm algorithm was improved by in- expressed as follows: troducing an inertia factor, which can reasonably and v � v + c pbest − x􏼁 + c gbest − x􏼁 , i i 1 i i 2 i i efectively regulate the global search and partial search (10) x � x + v , capabilities of the algorithm, such that it can change i i i during the PSO search process based on the search 6 Journal of Advanced Transportation where i � (1, 2, . . . , N), N denotes the total number of PSO-LSTM passenger fow forecasting algorithm based on particles in the swarm, x denotes the current position of the the chaos theory is shown in Figure 6. particle, v denotes the velocity of the particle, c and c i 1 2 Step 1: By performing EMD on the rail trafc short- denote the learning factors, pbest denotes the most opti- time passenger fow data, several IMFs and RES terms mum point for a mass to pass through, and gbest denotes can be obtained. the optimal position experienced by the swarm as a whole Step 2: Te decomposed components are screened, [20]. which are then reconstructed in the phase space. v &9; � w × v + c pbest − x􏼁 + c gbest − x􏼁 , i i 1 i i 2 i i Step 3: An improved PSO-LSTM prediction model is (11) developed for each component after phase-space re- w&9; � w + w − w · 1 − , 􏼁 􏼒 􏼓 construction, and the improved PSO algorithm is then e s e used to fnd the optimal parameters and train the where w denotes the weight at the beginning of the inertia LSTM model. factor, w denotes the weight at the end of the inertia factor, t Step 4: Te predicted values of each component are denotes the maximum number of iterations after all itera- superimposed and ftted to obtain the fnal prediction tions are completed, and t denotes the number of iterations results of the model. at the current moment [23]. Step 5: Te fnal prediction results are output [26]. Te improved PSO solution can be obtained by con- structing the LSTM short-time passenger fow prediction 4. Results: Case Studies model and using a modifed PSO optimization algorithm to discover the optimal parameters for the LSTM prediction 4.1. AFC Data and Processing. Te operational data of the model [24], as shown in Figure 3. Te pseudocode of the North Railway Station of Chengdu Metro Line 1—that is, its improved PSO algorithm is shown in Figure 4. incoming passenger fow data from January 4–13, Te specifc operational steps are as follows: 2020—were selected for this study. In the Chengdu rail transit system, the AFC platform recorded the entrance and Step 1: Initialize the particle swarm and set the relevant exit information of each passenger using smart card data parameters, including the population size, random from the automatic ticketing system at each metro station. position, and velocity. Inbound trafc data at the metro stations were obtained Step 2: Determine the ftness value of each particle, as between 5 : 00 and 00 : 55 the following day, with a data well as the optimal positions for the particle and collection interval of 5 min. Te data contained a total of particle population to pass. 2,260 time series, each of which included the start time, end Step 3: Determine whether the particles satisfy the time, input fow, and output fow. Te Chengdu Metro Rail convergence condition, and if they do, output the re- Transit North Station road network map is shown in sult. If they do not satisfy the convergence condition, Figure 7. continue with the following steps. A total of 2260 time series of data were input in the Step 4: Te velocity vector is updated with the opti- experiments of this paper. Te input data are inbound passenger fow in person/5 min. Te output data are the mum positions passed by the particles and particle swarm, and the position vector of the particles is inbound passenger fow in person/5 min. Te simulation updated with the updated velocity vector, after which environment used to test the predictive performance of all optimal particles are updated. the model in this study was MATLAB 2019a. Generally, the larger the training set, the more accurate the pre- Step 5: Return to Step 3 until the convergence con- diction results. Terefore, to take full advantage of the dition is met, before outputting the optimal result and data, the frst 90% of the original trafc data (that of eight number of iterations. days from January 4–11) were used in the training set, and the remaining 10% (January 12) were used in the 3. Design of Short-Time Passenger Flow test set. Forecasting Algorithm for Rail Transit First, anomalies and missing data from the original data Based on Chaos Theory and Improved EMD- were processed, wherein the anomalous data were consid- PSO-LSTM ered as missing data. Lagrangian interpolation methods were used to process the missing data, wherein four neighboring Te EMD algorithm can decompose the time series of rail data before and after the missing datum are selected for transit short-time passenger fow into IMF components of interpolation to ensure the reliability of the diferent frequencies based on their intrinsic characteristics, interpolated data. which describe the local characteristics of the original series Te data was then normalized using the min-max more clearly. An improved PSO-LSTM prediction model method as follows: can then be separately built for each subsequence, before y − y ∗ min adding the predicted values of each subsequence to obtain y � , (12) y − y max min the fnal output [25]. Te model construction process is shown in Figure 5. Te pseudocode of the improved EMD- Journal of Advanced Transportation 7 Input: Population size M, random position P, velocity V Output: Outputs optimal results and related parameters 1 Initialize the particle swarm 2 Set population size M, random position P, velocity V 3 Set the parameter T←1 4 Computes the objective function value 5 Update individual optimals Update (pbest) , population optimal Update (gbest) 6 If (convergence condition satisfied) 7 Output optimal results and related parameters 8 quit ( ) 9 Else (does not meet the convergence condition) 10 Updates the particle's position vector P and velocity vector V 11 Calculate the adaptation value W for each particle 12 Update the historical optimal position of each particle Update (pbest) 13 Update the global optimal location of the population Update (gbest) 14 Set the parameter T← T+1 15 Recalculate the objective function value 16 End if Figure 4: Pseudocode of the improved PSO algorithm. where y and y are the minimum and maximum values series can be divided into nine empirical modal components min max of the trafc fow, respectively, and y and y are the trafc and one residual component. Te EMD results are shown in fow data before and after normalization, respectively. Figure 10. It can be observed from the fgure that the IMF1, IMF2, and IMF3 empirical modal components have higher fre- 4.2. Prediction Results of the LSTM Model. Te LSTM rail quencies and are high-frequency components of the original trafc short-time passenger fow prediction model was rail trafc passenger fow data. IMF4, IMF5, and IMF6 established, the results of which are shown in Figure 8. empirical modal components have more obvious periodicity and are low-frequency components of the original rail trafc passenger fow data. Te residuals are the overall trend of the 4.3. Prediction Results of the Improved PSO-LSTM Model. time series data and are the trend components of the original Te prediction results show that the prediction performance rail trafc passenger fow data. Te EMD of the rail trafc of the LSTM model is poor. Consequently, an improved short-time passenger fow time series provides a clearer PSO-LSTM model was introduced, the prediction results of understanding of the passenger trafc fow data fuctuation which are shown in Figure 9. By running the model through and overall trend. repeated iterations, its prediction results were found to be optimal with an optimal number of hidden nodes of 167, optimal learning rate of 0.0310, and optimal number of 4.4.2. Phase Space Reconstruction Based on Chaos Teory. iterations of 30. Te EMD of the rail trafc fow change sequence can determine the trafc fow fuctuations more accurately. However, because the three components—that is, IMF7, 4.4. Prediction Results Based on Chaos Teory and Improved IMF8, and IMF9—do not show the intrinsic properties of EMD-PSO-LSTM Model the data, the model only selects the remaining seven 4.4.1. EMD of Trafc Flow Change Series. Because the components, reconstructing them in phase space by original rail trafc fow data have characteristics of non- fnding the time delay using the mutual information linearity and nonsmoothness [27], the noise in it will have method [28] and the embedding dimension using the Cao some infuence on the prediction results, resulting in in- method. If phase-space reconstruction parameters are accurate prediction results. Te noise in the time series data carefully selected, the reconstructed phase space can can be mitigated through the EMD algorithm, thus im- describe the states in the original system, and the mul- proving the predictive power of the model. Based on the tidimensional phase space can show the dynamic char- EMD algorithm, the EMD of the rail trafc fow variation acteristics of the trafc fow change sequence [29]. Te 8 Journal of Advanced Transportation Original time series of traffic flow Empirical modal decomposition IMF1 IMF2 IMFk RESn Phase space Phase space Phase space Phase space reconstruction reconstruction reconstruction reconstruction based on chaos based on chaos based on chaos based on chaos theory theory theory theory Improved PSO-LSTM Improved PSO-LSTM Improved PSO-LSTM Improved PSO-LSTM prediction prediction prediction prediction model model model model Superimpose the prediction results of each subsequence Final Forecast Results Figure 5: Improved EMD-PSO-LSTM model construction process. remaining seven component phase-space reconstruction Figures 11 more specifcally, the superimposed predictions parameters are listed in Table 1. are shown in Figures 11(a)–11(g), which show the plots of After phase-space reconstruction, the improved PSO the PSO search results for each component, whereas algorithm for the components was used to determine the Figures 11(h)–11(n) show the predicted results for each optimal number of hidden nodes, learning rate, and number component. Figure 12 shows the predictions for each component overlay. of iterations of the LSTM prediction model. Te optimal parameters of each component are listed in Table 2. 4.4.4. Prediction Results of the Improved EMD-PSO-LSTM 4.4.3. Prediction Results Based on Chaos Teory and Im- Model. To further validate the prediction efect of the combined optimization model based on the chaos theory proved EMD-PSO-LSTM Model. Owing to the nonlinear and nonsmooth characteristics of the original rail trafc and the improved EMD-PSO-LSTM model, a set of com- parison models were added—that is, phase-space re- passenger fow data, the noise in its time series data has a certain infuence on the prediction results, resulting in construction of the components without considering the inaccurate predictions. Consequently, using the improved chaos theory. Te components obtained after EMD were PSO-LSTM model, the trafc fow variation sequence could screened, and the remaining seven components were se- be empirically decomposed. Based on the chaos theory, the lected to build the improved PSO-LSTM prediction model. decomposed subsequence could be reconstructed in the Te optimal number of hidden nodes, learning rate, and phase space and an improved EMD-PSO-LSTM-combined number of iterations of the LSTM prediction model were optimization model was constructed. Te PSO search and determined using the improved PSO algorithm. Te optimal prediction results for each component are shown in parameters of each component are listed in Table 3. Te Journal of Advanced Transportation 9 Input: Rail transit short-term passenger flow data Output: Outputs the final prediction and performs an error analysis 1 Enter the short-term passenger flow data of rail transit 2 EMD empirical modal decomposition of the data 3Fori←0 to n do 4 Modal function INF[i]←Data[i] 5 Filter the individual components after decomposition Choose () 6 Performing Phase Space Reconstruction Recon () 7 Establishing an Improved Predictive Model for Components Predict () 8 Look for the optimal parameter M 9 Train the LSTM model 10 End for 11 For i←0 to n do 12 Overlay the prediction results for each subseries 13 Sum+=Pre[i] 14 End for 15 Outputs the final prediction Figure 6: Pseudocode of the improved EMD-PSO-LSTM passenger fow forecasting algorithm. SubwayStation 0 0.5 1 km Subway Figure 7: Chengdu Metro Rail Transit North Railway Station road network map. 10 Journal of Advanced Transportation 0 50 100 150 200 250 Time (min) Original LSTM Figure 8: LSTM model prediction results. 0 50 100 150 200 250 Time (min) Original Improved PSO–LSTM Figure 9: Improved PSO-LSTM model prediction results. prediction results shown in Figures 13(a)–13(g) show the where x denotes the actual inbound trafc at time i, x 􏽢 i i plots of the PSO search results for each component. denotes the forecast inbound trafc at time i, x indicates the Figures 13(h)–13(n) show the predicted results for each average value of trafc volume, and N denotes the total component. Figure 14 shows the predictions for each volume of inbound trafc in the trafc sequence. component overlay. To visually evaluate the prediction results of the four prediction models, the aforementioned errors were used to compare and analyze the strengths and weaknesses of 4.5. Evaluation Indicators. To better compare the pre- the model predictions [17]. Te evaluation indicator dictions between the LSTM model, improved PSO-LSTM values of the prediction results are listed in Table 4, and model, improved EMD-PSO-LSTM model, and improved the percentage improvements in the prediction results are EMD-PSO-LSTM model are based on the chaos theory. Te listed in Table 5. mean absolute error (MAE) and root mean square error Te comparative analysis of the error metrics shown in (RMSE) and the coefcient of determination (R ) metrics Tables 4 and 5 indicates that the improved EMD-PSO-LSTM were chosen to compare their errors [30]. Teir formulas can prediction model based on the chaos theory exhibits higher be expressed as follows: prediction accuracy. In addition, the improved PSO-LSTM 􏽶������������� and improved EMD-PSO-LSTM models and the improved EMD-PSO-LSTM model based on the chaos theory improve RMSE � x − x 􏽢 , 􏽘 􏼁 i i the passenger fow prediction results by using the LSTM i�1 model prediction results as a benchmark. Te percentage improvement of RMSE values is 25.53, 29.97, and 58.76%, 􏼌 􏼌 (13) 􏼌 􏼌 􏼌 􏼌 respectively, and the percentage improvement of MAE MAE � 􏽘􏼌x − x 􏽢 􏼌, i i values is 30.41, 40.13, and 63.08%, and the percentage im- i�1 provement of R values is 13.36, 16.31, and 32.30%. Tese 􏽐 x − x􏼁 results indicate that the proposed model has great potential 2 i�1 i R � , n 2 for short-time passenger fow forecasting applications in rail 􏽐 x − x􏼁 i�1 transit. Flow (person/5 min) Flow (person/5 min) Journal of Advanced Transportation 11 –50 0 500 1000 1500 2000 2500 3000 –50 0 500 1000 1500 2000 2500 3000 –50 0 500 1000 1500 2000 2500 3000 –50 0 500 1000 1500 2000 2500 3000 –50 0 500 1000 1500 2000 2500 3000 –50 0 500 1000 1500 2000 2500 3000 –50 0 500 1000 1500 2000 2500 3000 –50 0 500 1000 1500 2000 2500 3000 –50 0 500 1000 1500 2000 2500 3000 –50 0 500 1000 1500 2000 2500 3000 Figure 10: EMD results. Table 1: Phase space reconstruction parameters. Portion size IMF1 IMF2 IMF3 IMF4 IMF5 IMF6 RESn Time delay τ 99 99 4 8 23 36 99 Embedded dimensions m 6 6 6 5 5 4 6 Table 2: Optimal parameters of each subsequence after phase-space reconstruction. Optimal parameters IMF1 IMF2 IMF3 IMF4 IMF5 IMF6 RESn Hidden nodes 172 184 126 150 200 139 100 Learning rate 0.0051 0.0010 0.0010 0.0010 0.0088 0.0010 0.0010 Number of iterations 30 30 30 30 30 30 30 To further validate the experimental results, the prediction, including the model radial basis function neural EMD-PSO optimization algorithm based on the chaos network (RBF) and multilayer perceptron (MLP), were also theory was compared with combinations of deep learning- compared. Te experimental results and error results are based deep belief networks (DBN) and gated recurrent unit presented in Figure 15 and Table 6, respectively. (GRU) neural networks. Moreover, combinations of neural Te experimental results showed that the proposed network prediction commonly used in the feld of trafc fow EMD-PSO-LSTM model of rail transit short-time RESn IMF9 IMF8 IMF7 IMF6 IMF5 IMF4 IMF3 IMF2 IMF1 12 Journal of Advanced Transportation 18 9 2.8 3.5 16 3 2.6 8.5 14 2.5 2.4 12 2 10 1.5 2.2 7.5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 Evolutionary number Evolutionary number Evolutionary number Evolutionary number Best Best Best Best Average Average Average Average (a) (b) (c) (d) 9 7 300 8 6 6 4 100 5 4 3 -100 1 2 3 4 5 1 2 3 4 5 0 50 100 150 200 250 1 2 3 4 5 Evolutionary number Evolutionary number Time (min) Evolutionary number Best Best Original Best Average Average Predicted Average (e) (f) (g) (h) 300 300 300 300 200 200 100 100 100 0 0 0 -100 0 50 100 150 200 250 0 50 100 150 200 250 0 50 100 150 200 250 0 50 100 150 200 250 Time (min) Time (min) Time (min) Time (min) Original Original Original Original Predicted Predicted Predicted Predicted (i) (j) (k) (l) 300 300 0 50 100 150 200 250 0 50 100 150 200 250 Time (min) Time (min) Original Original Predicted Predicted (m) (n) Figure 11: Optimization search results and predictions for each component based on chaos theory : (a) IMF1’s search results, (b) IMF2’s search results, (c) IMF3’s search results, (d) IMF4’s search results, (e) IMF5’s search results, (f ) IMF6’s search results, (g) RESn’s search results, (h) IMF1’s predicted results, (i) IMF2’s predicted results, (j) IMF3’s predicted results, (k) IMF4’s predicted results, (l) IMF5’s predicted results, (m) IMF6’s predicted results, and (n) RESn’s predicted results. 50 100 150 200 250 Time (min) Original improved EMD-PSO-LSTM model based on chaos theory. Figure 12: Prediction results of the improved EMD-PSO-LSTM model based on chaos theory. Adaptation Flow (person/5 min) Adaptation Adaptation Flow (person/5 min) Flow (person/5 min) Adaptation Flow (person/5 min) Adaptation Flow (person/5 min) Adaptation Flow (person/5 min) Adaptation Flow (person/5 min) Flow (person/5 min) Journal of Advanced Transportation 13 Table 3: Optimal parameters of each subsequence after phase-space reconstruction. Optimal parameters IMF1 IMF2 IMF3 IMF4 IMF5 IMF6 RESn Hidden nodes 126 116 139 100 182 100 200 Learning rate 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 0.0013 Number of iterations 30 30 30 30 30 30 30 7 5 15.6 15.4 15.2 14.8 14.6 4 1 1 2345 1 2345 1 2345 1.5 234 2.5 3.5 4.5 5 Evolutionary number Evolutionary number Evolutionary number Evolutionary number Best Best Best Best Average Average Average Average (a) (b) (c) (d) 8 300 7 9 8.5 0 50 100 150 200 250 1 2345 1 2345 1 2345 Evolutionary number Evolutionary number Evolutionary number Time (min) Best Original Best Best Average Average Average Predicted (e) (f) (g) (h) 300 300 300 300 200 200 200 200 100 100 100 0 0 0 0 0 50 100 150 200 250 050 100 150 200 250 0 50 100 150 200 250 0 50 100 150 200 250 Time (min) Time (min) Time (min) Time (min) Original Original Original Original Predicted Predicted Predicted Predicted (i) (j) (k) (l) 300 300 050 100 150 200 250 050 100 150 200 250 Time (min) Time (min) Original Original Predicted Predicted (m) (n) Figure 13: Optimization search results and predictions for each component: (a) IMF1’s search results, (b) IMF2’s search results, (c) IMF3’s search results, (d) IMF4’s search results, (e) IMF5’s search results, (f) IMF6’s search results, (g) RESn’s search results, (h) IMF1’s predicted results, (i) IMF2’s predicted results, (j) IMF3’s predicted results, (k) IMF4’s predicted results, (l) IMF5’s predicted results, (m) IMF6’s predicted results, and (n) RESn’s predicted results. Adaptation Adaptation Flow (person/5 min) Adaptation Flow (person/5 min) Adaptation Flow (person/5 min) Adaptation Adaptation Flow (person/5 min) Flow (person/5 min) Flow (person/5 min) Flow (person/5 min) Adaptation 14 Journal of Advanced Transportation -50 0 50 100 150 200 250 Time (min) Original Improved EMD-PSO-LSTM Figure 14: Improved EMD-PSO-LSTM model prediction results. Table 4: Results of various models. Evaluation indicators Predictive models RMSE MAE R (%) LSTM 39.0150 30.7984 71.02 Improved PSO-LSTM 29.0557 21.4331 80.51 Improved EMD-PSO-LSTM 27.3210 18.4379 82.60 Improved EMD-PSO-LSTM based on chaos theory 16.0908 11.3704 93.96 Table 5: Percentage improvements in the prediction results. Evaluation indicators (%) Predictive models RMSE MAE R (%) Improved PSO-LSTM 25.53 30.41 13.36 Improved EMD-PSO-LSTM 29.97 40.13 16.31 Improved EMD-PSO-LSTM based on chaos theory 58.76 63.08 32.30 0 50 100 150 200 250 Time (min) True value GRU MLP LSTM RBF DBN Figure 15: Experimental results of the fve models. Flow (person/5 min) Passenger Flow (person/5 min) Journal of Advanced Transportation 15 Table 6: Results of various models. Evaluation indicators Algorithm run time Predictive models Training RMSE MAE R (%) and testing (s) Improved EMD-PSO-MLP 25.4751 17.5730 84.38 984.65 Improved EMD-PSO-RBF 22.2670 17.6294 86.70 1092.47 Improved EMD-PSO-DBN 17.1435 13.4770 92.15 1168.54 Improved EMD-PSO-GRU 18.5085 14.3520 91.32 945.28 Improved EMD-PSO-LSTM 16.0908 11.3704 93.96 967.53 [3] J. Guo, W. Huang, and B. M. 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Published: May 3, 2023

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