A comparison between ARIMA, LSTM, ARIMA-LSTM and SSA for cross-border rail freight traffic forecasting: the case of Alpine-Western Balkan Rail Freight Corridor

ABSTRACT In this paper, we model the intensity of cross-border railway traffic on the Alpine-Western Balkan Rail Freight Corridor (AWB RFC). For each of the four border crossing points: Dimitrovgrad, Presevo, Sid, and Subotica, time series composed of 102 monthly export and import railway freight traffic observations are used for training and testing of alternative forecasting models. Traditional ARIMA, Long-Short-Term Memory (LSTM) neural network, hybrid ARIMA-LSTM and Singular Spectrum Analysis (SSA) models, are fitted to each of the time series. For all the considered time series, the best model was chosen based on the lowest values of commonly used metrics for measuring the performance of forecasting models. LSTM models outperformed all other models with the highest prediction accuracy while SSA models exhibited the lowest accuracy. By utilizing advanced forecasting models, this research contributes to finding effective solutions for addressing the issue of inadequate planning of border crossing procedures in railway traffic.


Introduction
Rail freight transport represents the most environmentally friendly mode of transport and plays an important role in the freight transport market (European Environment Agency 2020; Gholamizadeh, Zarei, and Yazdi 2022).Besides its environmental advantage, rail freight transport can provide more reliable, safer, cheaper, and faster transport service under a higher level of harmonization of transport and technological processes (European Union Agency for Railways 2018; Zunder and Islam 2018).Increasing requirements in terms of quality and availability of rail freight services in Europe have led to the need for the creation of a single European rail area by establishing international rail corridors (DG Move 2011).Creating dedicated rail freight corridors enhances the competitiveness of railways, leveraging their capacity and suitability for efficient and cost-effective long-distance freight transportation (Shi et al. 2014).
The Alpine-Western Balkan Rail Freight Corridor (AWB RFC) belongs to a recently extended network of TEN-T corridors.This corridor connects Central Europe and South-East Europe.In addition, together with the branches Xb (Nis-Sofia-Istanbul) and Xc (Belgrade-Novi Sad-Budapest) it brings significant improvements to railway transport in the direction from Central Europe to Turkey.The AWB RFC is the first RFC to include Serbia in the European rail network for competitive freight.However, to utilize the potential of AWB RFC it is necessary to improve services and infrastructure along the RFC.One of the main bottlenecks is the inefficient border-crossing process.The average stopping times of freight trains at the AWB RFC border crossing points (BCP) last for around several hours.These times can be much longer leading to disturbances in train traffic from both directions.The main reasons lie in the insufficient number of locomotives, the insufficient number of train crews and the low level of synchronization among neighboring railway infrastructure managers.Besides the technical means for eliminating these bottlenecks (such as equipping the BCPs with communication technologies), technological improvements may enable more efficient and proactive decision-making facilitators of technological improvements.These can be advanced forecasting of import and export train flows passing through a BCP.The forecasting of railway border crossing traffic may represent an essential component of planning and control of the border crossing process.Accurate prediction of import and export train traffic may contribute to reducing transit time on corridors, increasing reliability of rail service, reducing costs for both operators and shippers, and higher satisfaction of shippers, which would ultimately lead to a higher share of rail transport.
Several forecasting approaches are applicable to forecasting train traffic through BCPs.In the context of this paper, all forecasting approaches can be classified as linear, nonlinear, and hybrid.Traditional linear models include historical average (Smith and Demetsky 1997;Stephanedes, Michalopoulos, and Plum 1981), smoothing techniques (Williams, Durvasula, and Brown 1998), and Autoregressive Integrated Moving Average (ARIMA) models (Box and Jenkins 1976;Box, Jenkins, and Reinsel 2008;Milenkovic et al. 2016).Purely linear models have limited performance in real-world time series modeling which is commonly characterized by a mix of linear and non-linear temporal patterns.Artificial Neural Networks (ANN) represent the most popular non-linear forecasting models.ANNs can handle complex patterns and generate models which adequately reflect non-linear relationships (Milenkovic et al. 2021).As a non-parametric model, ANN is adaptively determined based on the characteristics of the time series.However, the adoption of a single ANN may not be sufficient for modeling both linear and non-linear patterns well due to problems with misspecification, underfitting, or over-fitting of the model (de O. Santos Junior, de Oliveira, and de Mattos Neto 2019).Since the real-world time series are almost always linear and non-linear in terms of their correlation structure, neither the ARIMA nor the ANN can be applied individually in order to adequately model the time series.In that case, hybrid approaches that combine the classical statistical models and the ANNs are more appropriate choices so as to achieve better accuracy.ARIMA-Long Short-term Memory (LSTM) neural network which hybridizes the ARIMA model and the LSTM model to obtain the linear tendency and the non-linear tendency represents one very competitive alternative in comparison to the other approaches based on many recent contributions (Abebe et al. 2022;Deng, Fan, and Wu 2020;Fan et al. 2021;Khozani et al. 2022;Manowska et al. 2021).
In this paper, we model the cross-border train traffic flow time series by using the traditional ARIMA approach, Long Short Term Memory (LSTM), hybrid ARIMA-LSTM, and Singular Spectrum Analysis (SSA) approach.The traditional Box-Jenkins method is used for fitting the ARIMA models.In comparison with ARIMA models, LSTM models are capable of looking for non-linear, non-stationary and intermittent or transient behavior in an observed time series.The third used approach is based on combining the ARIMA with the LSTM neural network to capture the linear and non-linear dynamics in the time series.SSA represents an additional non-parametric technique based on the concept of separability between signal and noise components (Golyandina, Nekrtutkin, and Zhigljavsky 2001).Comparisons of the proposed approaches are performed for Dimitrovgrad, Presevo, Sid, and Subotica border crossing points, all within the railway network of Serbia.Performances of the approaches are evaluated according to a set of criteria relevant for export and import train traffic flows independently.At the time of our analysis, the time series of monthly train traffic flows from January 2013 to June 2021 (102 monthly observations) were available.
The paper is organized as follows.In the next section, a review of relevant literature is given.Section 3 describes the methodology used in the modeling and forecasting of train traffic flows through selected border crossing points.In Section 4, the proposed models have been tested on eight different time series related to the export and import flows of Dimitrovgrad, Sid, Presevo, and Subotica border crossing points.The performances of the proposed models are compared in Section 4. Concluding remarks and future research directions are given in the last section.

Literature review
There are different classifications of quantitative forecasting approaches.All quantitative approaches can be decomposed into univariate, or projective, and multivariate, or causal (Milenkovic et al. 2021).Both categories can be decomposed into parametric and nonparametric (Milenkovic et al. 2016), whereas within these two groups, there are linear, non-linear, and hybrid approaches (Petropoulos et al. 2022).Excellent recent reviews of the contributions related to the traffic flow forecasting can be found in Medina-Salgado et al. (2022), Kashyap et al. (2022), andLiu et al. (2021).
Relevant transport flow forecasting contributions are reviewed in this section and are classified based on the forecasting method, the mode of transportation, and the application area.Table 1 summarizes the applications of the traditional linear or parametric forecasting techniques for the forecasting of transport flows.
In the domain of passenger traffic, Grubb and Mason (2001) used the Holt-Winters method for a lengthy time series about air passenger traffic.Zhi- Peng et al. (2008) proposed an improved adaptive exponential smoothing model for short-term travel time forecasting of the urban arterial street.Chen, Chang, and Chang (2009) proposed the Holt-Winters method, the SARIMA model, and the GM(1,1) gray forecasting model to replicate the monthly inbound air travel arrivals to Taiwan and to compare the models' forecasting performance.Ding et al. (2010) propose a space-time ARIMA (STARIMA) model to predict the traffic volume in urban areas.Kim et al. (2011) forecasted the future Korea's entire container handling volume by using the ARIMA model, time series analysis, and system dynamics.Ge, Zheng, and Hou (2013) compared the exponential smoothing with the trend-moving average method for bus passenger traffic prediction.Li (2013) compared the exponential smoothing and the brown exponential smoothing for freight turnover forecasting.Kumar and Vanajakshi (2015) proposed a prediction scheme using the Seasonal ARIMA (SARIMA) model for shortterm prediction of road traffic flow using only limited input data.Milenkovic et al. (2016) developed Seasonal ARIMA (SARIMA) model for forecasting railway passenger demand.Tang and Deng (2016) applied ARIMA(1,1,8) model to adequately fit the civil aviation passenger turnover.Dantas, Oliveira, and Repolho (2017) combined the Bootstrap aggregating (Bagging) method with the exponential smoothing Holt-Winters to predict the future demand for passenger air transportation.Miller (2019) applied the ARIMA methodology to develop forecasts for three time series of monthly archival trucking prices.Zhao, Cai, and Zheng (2018) modeled the railway freight volume by establishing an autoregressive integrated moving average model (ARIMA model), using the original data of railway freight volume in the Ningxia Hui Autonomous Region to do an empirical analysis.Alhindawi et al. (2020) applied the double exponential smoothing for the projection of GHG emissions from the road transport sector.Jighjigh et al. (2021) formulated a multiplicative Holt-Winters method to forecast the volume of passenger traffic in Nigerian airports in the future.Moiseev (2021) applied an exponential smoothing model in the oil tanker shipping market forecasting.Sitzimis (2022) compared Winters' multiplicative method, simple seasonal model, decomposition The literature has also recognized the significance of employing non-linear dynamics forecasting models in addressing transportation-related problems (Table 2).Mostafa (2004) investigated the Suez Canal traffic forecasting and compared the performance of the ANNs with that of the ARIMA models on an example of a large monthly dataset.Murat and Ceylan (2006) developed an ANN model for transport energy demand forecasting.Blinova (2007) proposed a neural network model to forecast the air intraregional and interregional passenger traffic flows.Zhang and Liu (2009) proposed the least squares support vector machines (LS-SVMs) to forecast the travel time index.Yang, Jin, and Wang (2011) proposed a wavelet transform-SVM combined model to forecast the freight index of Panamax bulk carriers.Bao, Xiong, and Hu (2012) proposed an ensemble empirical mode decomposition (EEMD) based support vector machines (SVMs) modeling framework, incorporating a slope-based method (EEMD-Slope-SVMs) to model the monthly air passenger traffic series, including the six selected airlines in USA and UK.Wei and Chen (2012) combined the empirical mode decomposition (EMD), and the back-propagation neural networks (BPNN) are developed to predict the short-term passenger flow in metro systems.Wang and Shi (2013) proposed a short-term traffic speed forecasting hybrid model (Chaos-Wavelet Analysis-Support Vector Machine) to model the real traffic speed data.Jiang, Zhang, and Chen (2014) developed a short-term high-speed railway demand forecasting approach by combining the ensemble empirical mode decomposition (EEMD) and the gray support vector machine (GSVM) models.Cong, Wang, and Li (2016) combined the least squares support vector machine (LSSVM) with the fruit fly optimization algorithm (FFOA), in order to study the potential of traffic flow forecasting.Glišović et al.Non-linear models such as ANN are usually difficult to interpret and test for the statistical significance of the parameters (Medeiros and Veiga 2000).Over the recent years, new formulations appeared that combine the traditional linear time series models and non-linear models to handle both linear and non-linear structures in time series equally well (Table 3).
For the prediction of the number of goods subject to inspection at the European Border Inspections Post, Ruiz-Aguilar, Turias, and Jimenez-Come (2014) applied a hybrid twostep procedure based on integrating the data obtained from the SARIMA model in the artificial neural network model (ANN).Xie, Wang, and Lai (2014) proposed two hybrid approaches based on seasonal decomposition and the least squares support vector regression (LSSVR) model for short-term forecasting of air passengers.A hybrid model combining symbolic regression and ARIMA was proposed by Li et al. (2018) for metro passenger flow forecasting.Xu, Chan, and Zhang (2019)   accuracy of traffic count on four main arterial roads in Sydney.Ge et al. (2021) proposed a hybrid of ARIMA and fuzzy support vector regression machine (FSVR) to predict the passenger flow at the Shanghai-Guangzhou high-speed railway.
Based on this review of relevant literature, we may draw the following conclusions: . Forecasting models were developed mainly for non-rail and/or passenger transportation; .To our knowledge, there are no past attempts that paid particular attention to forecasting traffic flow on railway border crossings.
To eliminate these research gaps, we model the cross-border train traffic flows by applying ARIMA, LSTM, ARIMA-LSTM, and SSA approaches.Comprehensiveness as an additional feature of this approach is based on separate analysis, assessment, and forecasting of the import and export train flow for each of the four border crossing points, which produces a more detailed input to the corridor managers as support for solving a diversity of corridor-related decision-making problems.

Methodology
The flow chart in Figure 1 illustrates the methodology applied in this paper.Each forecasting approach requires a set of steps to be conducted.The ARIMA modeling is based on the Box and Jenkins methodology that includes the identification of a suitable model, estimation of parameters and diagnostic checking.The LSTM neural network includes the transformation of a sequence of observations and the definition of the model.The ARIMA-LSTM is composed of filtering the linear tendencies in the data and passing on the residual values to the LSTM model.Decomposition and reconstruction represent the main steps of the SSA approach.The included forecasting models are fitted on a training data sample and their optimal configurations are selected and compared on a test data sample.The model which produces the best performances can be used for forecasting freight train flows at a specific border-crossing.The following subsections contain a brief description of the included approaches.

ARIMA
ARIMA models are composed of the Autoregressive (AR) model, the Moving Average (MA) model, and the ARMA as a combination of AR and MA models (Box, Jenkins, and Reinsel 2008).The AR model includes a linear combination of past values of the variable.AR model of order p (AR(p)) can be expressed as: where c is a constant and 1 t is a white noise sequence assumed to be a normal random variable with zero mean and variance s 2 .The MA model uses past forecast errors as forecast variables.The MA model of order q (MA(q)) can be expressed as: To apply the ARIMA models the time series needs to be stationary.The letter 'I' (integrated) means that the first-order difference is applied to transform a considered time series into stationary.The full ARIMA model can be written as follows: Y ′ t is differenced time series.Equivalent integrated form of any ARIMA model looks as follows: B represents the backshift operator, whose effect on a time series Y t can be summarized as: Seasonal ARIMA models (SARIMA) represent an extension to cover seasonal variations in the time series.Equation ( 6) represents a SARIMA(p, d, q) × (P, D, Q) model where F and Q are the seasonal ARMA coefficients, and seasonal differencing operator (1 − B S ) D of order D is applied to eliminate seasonal patterns.

Long short term memory (LSTM) models
LSTM provides a solution for long-term dependency problems thanks to improved recursive neural network architecture with feedback, so it can process not only the individual data points but the entire sequences as well (Manowska et al. 2021;Sepp and Schmidhuber 1997).An LSTM neural network is composed of one input layer, one recurrent hidden layer, and one output layer.Improvement in architecture relates to replacing the hidden layer of RNN cells with LSTM cells (hereafter memory blocks) to achieve long-term memory.The self-connected LSTM memory blocks enable the model to learn the long-term dependencies while handling sequential data (Somu, Raman, and Ramamritham 2020).In contrast to RNNs, which have only one hidden state, in the LSTM neural network to each cell two states are transferred, the cell state and the hidden state.The cell state enables long-term memory capability, whereas the hidden state enables a working memory capability that contains only the near-past information and uncontrollably overwrites at every step.Memory blocks that are responsible for memorizing, and manipulations between blocks are done by special multiplicative units called gates.The gates control the flow of information (Hrnjica and Mehr 2020;Ma et al. 2015).The input gate controls the flow of input activations into the memory cell.The output gate controls the output flow of the cell activation.Besides these two gates, there is also a forget gate that filters the information from the input and previous output and decides which one should be remembered, forgotten, and dropped (Hrnjica and Mehr 2020).Besides the gates, the core of the memory cell is a recurrently self-connected linear unit-Constant Error Carousel (CEC), whose activation represents the cell state.Due to the presence of CEC, the problem of vanishing or exploding gradient is solved since multiplicative gates can learn to open and close enabling the LSTM cell state to enforce the constant error flow.Figure 2 provides an insight into the internal architecture of LSTM.Symbols i, f , C and o represent the input gate, forget gate the cell state vector and the output gate, respectively.s and tanh are sigmoid and hyperbolic tangent activation functions, respectively.The element-wise multiplication of two vectors is denoted by ⊗.

ARIMA-LSTM models
ARIMA models can only recognize linear patterns of the time series whereas LSTM models are capable of mining the non-linear relationships.Since it is proven that some of the time series related to rail cross-border traffic have some remaining non- The modeling flow chart of ARIMA-LSTM approach is shown in Figure 3.In essence, the residuals of ARIMA models are regarded as the input of the LSTM model, and the LSTM model is utilized to train the non-linear tendency by modeling the residual series (Deng, Fan, and Wu 2020).The linear part (L t ) and non-linear part (N t ) were combined to obtain the prediction results (Y t ) of the hybrid ARIMA-LSTM model: 3.4.Singular spectrum analysis (SSA) algorithm Singular Spectrum Analysis (SSA) represents a nonparametric spectral estimation method based on the combination of time series analysis, dynamical systems, and signal processing (Harris and Yuan 2010;Hassani 2007;Hassani and Mahmoudvand 2013;Hassani and Zhigljavsky 2009;Stratigakos et al. 2021).The algorithm is composed of two main stages: decomposition and reconstruction.In the first stage, the time series is represented as a spectrum of independent components such as trend, periodic oscillatory, and noise.At the embedding step, the univariate time series is transformed into a trajectory matrix, which has the properties of a Henkel matrix with equal elements on the antidiagonal.Consider that there are N realizations X(t) = {x(1), x(2), . . ., x(N)} of a stochastic process {x(t); t = 1, 2, . . ., N}.The trajectory matrix is obtained as follows: where L represents the length of the selected window and K = N + 1 − Lare the selected lagged vectors.
The main objective of the singular value decomposition is to express the Hankel matrix as a sum of weighted orthogonal matrices as follows: where U 1 , U 2 , . . ., U L are the corresponding eigenvectors and r is the rank of H.
The second stage, called reconstruction, is accomplished in two steps: grouping and diagonal averaging.In the grouping step, the set of matrices After that, all matrices within each subset are summed.For m = 2, there are only two subsets, subset E 1 = d i=1 H i that represents the signal component, and E 2 = r i=d+1 H i associated with the noise component.Finally, the diagonal averaging represents the transformation of each reconstructed trajectory matrix (16) into new time series of length N.
Elements of the new time series are extracted from H by following calculations: Finally, the original time series X N is expressed as a sum of d principal vectors:

ARIMA results
Table 4 summarizes the process of the ARIMA model selection.In the preliminary step, the time series is visually examined, the procedure for detecting outliers is applied and In the first step, the ADF test is applied for detecting non-stationarity.The test provides a p-value, which is used to assess the stationarity of the series.A p-value below a certain threshold (commonly 0.05) indicates that the series is stationary, while a p-value above the threshold suggests non-stationarity.In cases where a time series is non-stationary (such as the export and import flow in the border crossings of Sid and Subotica), it needs to be transformed or adjusted to achieve stationarity using techniques like the first order differencing.In the second step, based on a visual plot of ACF and PACF, a general structure of the ARIMA model is proposed.Significant values of correlograms refer to the lag values in the Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) that fall outside the confidence bounds (at the 5% level), and indicate significant correlations.These significant values in the correlograms suggest the presence of a pattern or relationship between the observations at different time lags and provide initial guidance to determine the most appropriate AR(I)MA model for a given time series.The third step includes determining the best model based on the criteria of minimum AICc.The values of MAPE and Adjusted R-squared are calculated for selected models.In the last step, Ljung-Box statistics is applied to check if the selected model is correctly specified.The p-value greater than the significance level (0.05) indicates that the model is considered adequately specified in terms of residual autocorrelation.
The BDS test is applied to check if there is remaining non-linearity in time series.According to the BDS test, non-linearity exists if the residual datasets, after fitting a selected ARIMA model, still contain non-linear components (the null hypothesis of i.i.d. for the residuals cannot be rejected at the 5% level of confidence).In that case, there are p-values which are lower than 0.05.for any of embedding dimensions (m = 2,3), whereas epsilon (0.5-1.5).m represents the embedding dimension.ε is equal to 0.5, 1.0 and 1.5 times the standard deviation.The critical value for the confidence level of 5% is 1.96.Table 2 contains the outputs of the BDS test for all considered border crossing points.From the results of the BDS test applied to the residuals of chosen ARIMA models, it can be noticed that in five time series there is a remaining non-linearity in the residuals (bold p-values in Table 5).In the cases of Presevo Import, Sid Import, and Subotica Import time series, according to the BDS test, the hypothesis of the nonlinearity in the residuals of the ARIMA models can be rejected.

LSTM results
The LSTM network was built based on the Keras framework of the Python 3.10 platform.Before modeling, each training data that belonged to a time series for each border crossing was normalized or rescaled from the original range so that all values were within the range of 0 and 1.Then the methods and parameters of the LSTM model needed to be configured.Depending on the time series, the hidden layer was built from 100 to 200 LSTM cells, the number of iterations varied from 100 to 300 and the batch size spanned from 2 to 12.The activation function was set to the rectified linear activation function (ReLu), the loss function was MSE,

ARIMA-LSTM results
The best ARIMA-LSTM model configurations for residuals of each of the time series with associated MAPE values for training and testing samples are given in Table 7.For each follows:

Concluding remarks
In this paper, we analyzed import and export freight traffic flows on the four main border crossings on Serbian railway network.The four prediction models (ARIMA, LSTM, ARIMA-LSTM, SSA) are applied for the modeling of the considered eight time series.Three evaluation indices were selected to evaluate the prediction accuracy of considered proposed the SARIMA-SVR model to forecast the statistical indicators in the aviation industry.Shahriari et al. (2020) combined the bootstrap with the conventional parametric ARIMA model to improve the prediction

Figure 1 .
Figure 1.Comparative analysis of ARIMA, ARIMA-LSTM and SSA methods for modeling of freight wagon flows at border crossing points.

Figure 2 .
Figure 2. Memory cell and gate units of LSTM memory block.
Within the territory of the Republic of Serbia, there are four main BCPs on the AWP RFC with different operational conditions and volumes of freight traffic.These are Dimitrovgrad, Presevo, Sid, and Subotica.This study focuses on the observation and prediction of import and export cross-border freight train flows for these BCPs.The historical data for import and export flows are presented in Figures4 and 5respectively.The time series data have been obtained from the Public Enterprise 'Serbian Railways'.The sample data are monthly observations of freight train flows on four BCPs covering the period from January 2013 to June 2021.The first 96 monthly observations are used as a training dataset, whereas the remaining 6 observations serve for the verification of selected models.Both types of traffic (import and export) for each BCP are independently investigated and the appropriate models are estimated.ARIMA, LSTM, and ARIMA-LSTM are implemented by the use of R and Python software packages, and SSA is developed in Python.

Figure 4 .
Figure 4. Border-crossing points: railway import flows expressed in the number of freight trains (January 2012-June 2021).

Figure 5 .
Figure 5. Border-crossing points: railway export flows expressed in the number of freight trains (January 2012-June 2021).
15) where Y i and Y i represent the actual and predicted values of the time series in period i, respectively.MAE represents the mean of absolute errors.MAPE is one of the most commonly used criteria to measure forecast accuracy.It represents the sum of the individual absolute errors divided by the actual observation.RMSE represents a square root of the average squared error.Figure 8 graphically illustrates the comparison of forecasting accuracy of the proposed methods for import freight train flows on all border crossing points.In terms of performances on a training sample, the SSA shows the lowest values of

Table 1 .
Saxena and Yadav (2022)ature: application of linear forecasting techniques.multiplicativetrend, and seasonal model with the Box-Jenkins ARIMA approach for forecasting passenger traffic in Greek coastal shipping.Saxena and Yadav (2022)conducted a scenario-based to estimate the effect of COVID-19 on railway freight transport in India and developed an ARIMA forecasting model to analyze rail freight volume and the corresponding revenue loss.

Table 2 .
Jiang and Luo (2022)021)ture: application of non-linear forecasting techniques.presented a hybrid model based on the integration of the genetic algorithm (GA) and the artificial neural networks (ANN) for forecasting the monthly volume of passengers within the Serbian railways.Yang et al. (2016)developed the prediction model of bus arrival time based on a Support Vector Machine with a genetic algorithm (GA-SVM).Wang et al. (2018)proposed a novel hybrid model combining the support vector machine overall online (SVMOOL) model and the support vector machine partial online model (SVMPOL) for short-term metro ridership forecasting.Gallo et al. (2019)proposed the Artificial Neural Networks (ANNs) approach for forecasting metro on-board passenger flows as a function of passenger counts at station turnstiles.Based on the Baltic Supermax Index and historical decision data of different companies,Guan et al. (2019)used the support vector machine model to predict the dry bulk carrier route selection.Milenkovic et al. (2021)proposed a fuzzy neural network prediction approach based on metaheuristics for container flow forecasting.Peng et al. (2020)proposed a spatio-temporal incidence dynamic graph convolution neural networks (Dynamic-GRCNN) framework for urban traffic passenger flow prediction.Huang et al. (2021)proposed the Gray model GM (1,1) and Back Propagation (BP) neural network model for the simulation and forecasting the logistics demand of Guangdong province from 2000 to 2019.Jiang and Luo (2022)made a comprehensive review of the application of Graphical Neural Networks for road traffic forecasting.

Table 3 .
Classification of literature: application of hybrid forecasting techniques.

Table 4 .
ARIMA model selection for railway border crossing points.

Table 5 .
Non-linearity testing for ARMA residuals of time series.

Table 8 .
SSA parameters and the accuracy of proposed SSA models for considered time series.