Short-Term Online Forecasting for Passenger Origin–Destination (OD) Flows of Urban Rail Transit: A Graph–Temporal Fused Deep Learning Method

Abstract

Predicting short-term passenger flow accurately is of great significance for daily management and for a timely emergency response of rail transit networks. In this paper, we propose an attention-based Graph–Temporal Fused Neural Network (GTFNN) that can make online predictions of origin–destination (OD) flows in a large-scale urban transit network. In order to solve the key issue of the passenger hysteresis in online flow forecasting, the proposed GTFNN takes finished OD flow and a series of features, which are known or observable, as the input and performs multi-step prediction. The model is constructed from capturing both spatial and temporal characteristics. For learning spatial characteristics, a multi-layer graph neural network is proposed based on hidden relationships in the rail transit network. Then, we embedded the graph convolution into a Gated Recurrent Unit to learn spatial–temporal features. For learning temporal characteristics, a sequence-to-sequence structure embedded with the attention mechanism is proposed to enhance its ability to capture both local and global dependencies. Experiments based on real-world data collected from Chongqing’s rail transit system show that the metrics of GTFNN are better than other methods, e.g., the SMAPE (Symmetric Mean Absolute Percentage Error) score is about 14.16%, with a range from 5% to 20% higher compared to other methods.

1. Introduction

1.1. Background

Urbanization introduces growing populations in cities and leads to significant mobility and sustainability challenges. Predicting short-term passenger flow of urban rail transit systems, especially the origin–destination (OD) flow, is vital for providing full understanding of travel patterns and daily operation efficiency improvement. In case of emergencies or special incidents, an accurate online passenger flow forecasting can help to implement efficient response measures and enhance the service quality of public transport systems [1].

Nevertheless, traffic forecasting is challenging due to the complex temporal and spatial relationships among data. More specifically, in temporal dimension, the global features and local features both influence the evolution of passenger flow. Then, in the spatial dimension, the interrelationship between passenger flow and the complex rail network, both similarities and corrections (i.e., similarities or correlations among passenger flow series of different stations) [2], make the prediction of passenger flow much more complex.

In addition, passenger forecasting problems often contain a complex mix of inputs—including time-invariant, known future inputs, and other exogenous time series that are only observed historically—without any prior information on how they interact with the target [3].

In addition, OD prediction has its own characteristics. In the huge size of a rail transit network, online forecasting can only obtain data on the finished OD passenger flows, as passenger trips cannot finish immediately. The unknown passenger flow includes the current unfinished passenger flow as well as the future flow that will continuously enter the network. Meanwhile, the AFC (Auto Fare Collection) system has a delay time for updating OD flow information. Thus, the forecasting task of OD flow naturally takes temp finished OD flow as input and the actual OD flow as output.

To solve the above issues, we need a method that simultaneously considers spatial–temporal properties of OD passenger flow. The development of deep learning provides the possibility to find a reasonable solution. The temporal fusion transformer method [3] is explored, which can combine high-performance multi-horizon forecasting with interpretable insights into temporal dynamics. Furthermore, Graph Convolution Networks (GCN), which can automatically learn representations on non-Euclidean data (e.g., graphs), has been proposed and utilized in many scenarios [4,5,6,7].

Inspired by the above, a Graph–Temporal Fused Neural Network (GTFNN) is proposed in this paper. The most important feature of this model is the introduction of a graph neural network component in the attention-based sequence-to-sequence structure to fuse the hidden spatial relationships in a rail transit network into temporal relationships. This realizes the prediction of time-series OD passenger flow, which is highly dependent on the network structure.

1.2. Potential Contributions

This paper develops a method for forecasting OD flow of a network-level rail transit system, with considerations of the hysteresis characteristic and hidden complex spatial–tempol relationships. The potential contributions can be summarized as:

• Considering the hysteresis characteristic of OD flow, we proposed an online forecasting framework that maps the historical observable finished flow and features to the actual OD flow in the future.
• For better improving the prediction accuracy, we built a feature system including observable features and knowable features based on the temporal relationship. We also used specialized feature selection components for weighting the features.
• For capturing the hidden spatial relationships in the rail transit network, we introduced a multi-layer graph neural network as a key component in the method to depict the relationships among stations from different aspects.
• Based on sequence-to-sequence framework, a Graph–Temporal Fused Deep Learning model was built. In addition, an attention mechanism was attached to the model to fuse local and global temporal dependencies; then, we achieved the prediction of short-term online OD passenger flow.

For rail transit systems, obtaining accurate OD passenger flow data in real time is an important support for transportation organization. The train line planning, timetabling and station passenger flow planning all depend on accurate OD prediction results. Especially, the prediction of different OD passenger flow peaks can effectively guide the allocation of rail transportation resources. Therefore, the OD prediction model studied in this paper has strong practical significance.

2. Literature Reviews

We performed a literature review from three aspects: passenger flow prediction, graph convolution neural networks, and time series forecasting with attention-based deep neural networks. All three aspects are relevant to the study of this paper.

2.1. Passenger Flow Prediction

Passenger flow prediction is one of the most significant tasks in the field of intelligent transportation. There are two major categories of prediction methods covered in the flow prediction issue, which are statistical methods and machine learning methods [8]. Among the statistical algorithms, the Auto-Regressive Integrated Moving Average (ARIMA) model is representative. Williams et al. [9] and Lee and Fambro [10] applied the ARIMA model to realize urban freeway traffic flow prediction.

Recently, a large number of research studies in recent years have focused on applying machine learning or deep learning algorithms in transportation applications, which have generated considerable achievements. Huang et al. [11] claimed that their research is the earliest application of deep learning to traffic flow prediction research through the proposed Deep Belief Nets networks. Specific to subway passenger flow, Ni et al. [12] trained the Linear Regression model with event occurrences of information to tackle abnormal prediction. Sun et al. [13] proposed Wavelet-SVM and discussed the idea to predict different kinds of passenger flows. A combination of empirical mode decomposition and MLP was proposed by Wei and Chen [1] to forecast the short-term metro flow. Later on, a multi-scale radial basis function network [14] was proposed by the same authors to improve their forecasting accuracy. Furthermore, some researchers [15] focused on using the nonparametric regression model to predict flows of transfer stations instead of the whole subway system.

In the early days, OD predictions were more concerned with potential relationships in the time dimension. Some works [16,17,18] usually employed time series filtering models (e.g., Kalman filter) to estimate the OD flow. Recently, many studies have fused spatial relationships with temporal relationships. For instance, Liu et al. [19] proposed a Contextualized Spatial–temporal Network that incorporated local spatial context, temporal evolution context, and global correlation context to forecast taxi OD demand. Shi et al. [20] utilized long short-term memory (LSTM) units to extract temporal features for each OD pair and then learned the spatial dependency of origins and destinations by a two-dimensional graph convolutional network.

For ride-hailing applications, the origin and destination of a passenger are known once a taxi request is generated. However, in online metro systems, the destination of a passenger is unknown until it reaches the destination station; thus, the operators cannot obtain the complete OD distribution immediately to forecast the future OD demand. To address this issue, Gong et al. [21] used some indication matrices to mask and neglect the potential unfinished metro orders. Lingbo Liu et al. [22] handled this task by learning a mapping from the historical incomplete OD demands to the future complete OD demands.

2.2. Graph Convolution Neural Networks

To fit the required input format of Convolutional Neural Network and Recurrent Neural Network, some works [23,24] divided the studied network system into regular grid cells and transformed the raw traffic data to tensors. However, this preprocessing manner has limitations in handling the traffic systems with irregular topologies, such as rail transit systems and road networks.

Graph Convolutional Networks can be used to improve the generality of deep learning-based forecasting methods in networks. For instance, Li et al. [25] modeled the traffic flow as a diffusion process on a directed graph and captured the spatial dependency with bi-directional random walks. Song et al. [26] developed a Spatial–Temporal Synchronous Graph Convolutional Network (STSGCN), which captured the complex localized spatial–temporal correlations through a spatial–temporal synchronous modeling mechanism. In the article by Han et al. [27], graph convolution operations were applied to capture the irregular spatial–temporal dependencies along with the metro network. Geng X. et al. [28] developed a ST-MGCN, which incorporated a neighborhood graph (NGraph), a transportation connectivity graph (TGraph), and a functional similarity graph (FGraph) for ride-hailing demand prediction. Lingbo Liu et al. [22] designed a more flexible historical ridership data network based on this and fully explored the inter-station flow similarity and OD correlation for virtual graph construction.

2.3. Time Series Forecasting with Attention-Based Deep Neural Networks

Attention mechanisms are used in translation [29], image classification [30] or tabular learning [31] to identify salient portions of input for each instance using the magnitude of attention weights. Recently, they have been adapted for time series with interpretability motivations [32,33,34], using LSTM-based [35] and transformer-based [34] architectures.

Deep neural networks have increasingly been used in time series forecasting, which demonstrates stronger performance over traditional time-series models [32,36,37]. More recently, transformer-based architectures have been explored in Li et al. [34], which proposes the use of convolutional layers for local processing and a sparse attention mechanism to increase the size of the receptive field during forecasting. Despite their simplicity, iterative methods rely on the assumption that only the target needs to be recursively fed into future inputs.

The Multi-horizon Quantile Recurrent Forecaster (MQRNN) [38] uses LSTM or convolutional encoders to generate context vectors. In Fan et al. [39], a multi-modal attention mechanism was used with LSTM encoders to construct context vectors for a bi-directional LSTM decoder. Despite performing better than LSTM-based iterative methods, interpretability remains challenging for such standard direct methods.

In general, deep learning models involve a large number of input features. The significance of features is difficult to measure, which is not beneficial to the large-scale application of prediction models. Some methods are used for the significance analysis of different input features in time series forecasting. For example, Interpretable Multi-Variable LSTMs [40] partition the hidden state such that each variable contributes uniquely to its own memory segment and weights memory segments to determine variable contributions. Methods combining temporal importance and variable selection have also been considered [33], which compute a single contribution coefficient based on attention weights from each. However, in addition to the shortcoming of modeling only one-step-ahead forecasts, existing methods also focus on instance-specific (i.e., sample-specific) interpretations of attention weights—without providing insights into global temporal dynamics. The usage of an attention mechanism can provide an improvement for this issue. Temporal Fusion Transformer (TFT) [3] is able to analyze global temporal relationships and allows users to interpret global behaviors of the model on the whole dataset—specifically in the identification of any persistent patterns (e.g., seasonality or lag effects) and regimes present.

From the existing studies, it can be seen that the current deep learning techniques have started to take into account the temporal and spatial fusion features into the prediction models. To solve the prediction problem in this paper, a systematic consideration of the hysteresis of OD traffic, the hidden spatial relationships in complex networks, the different input features and their temporal characteristics, and how to ensure the capability to capture both local and global dependencies in the temporal dimension is needed.

3. Methodology

3.1. Short-Term Online Framework Considering Hysteresis of Passenger OD Flow

The online short-term online framework for forecasting passenger OD flow should consider hysteresis of the OD passenger, as well as the hidden temporal and spatial relationships.

3.1.1. Solution for Passenger Hysteresis

Passenger OD flow has a hysteresis characteristic that the destination of a passenger is unknown until it reaches the destination station, which provides a challenge to the short-term online forecasting. The passengers cannot finish their trips in a short forecasting time step (e.g., 15 min or even 5 min); furthermore, in each time step, we cannot obtain the actual or complete OD distribution for forecasting dynamically. This characteristic has attracted the attention of one existing research study. Gong et al. [21] used some indication matrices to mask and neglect the potential unfinished trip in urban rail networks.

In order to handle this issue, this paper proposes a framework that maps a time-dependent finished trip to the actual entered trips based on Equation (1).

$${{\displaystyle \sum }}_{t}ODflo{w}_t^{en}=ODflo{w}_t^{un}+{{\displaystyle \sum }}_{t}ODflo{w}_t^{fi}$$

(1)

where $ODflo{w}_t^{en}$ represents the actual or entered OD flow of time step $t$, which can be obtained in historical data but cannot be calculated instantaneously. $ODflo{w}_t^{en}$ is the object we wish to forecast and the guidance of system management. $ODflo{w}_t^{fi}$ represents the finished OD flow of time step $t$, which can be obtained from historical data and instantaneous observations. The gap between the two cumulative values of $ODflo{w}_t^{en}$ and $ODflo{w}_t^{fi}$ in the temporal dimension is caused by an unfinished journey. We denote this gap as the $ODflo{w}_t^{un}$ that represents the unfinished OD flow of time step $t$.

Based on this relationship, it is clear that one of the core aspects of the forecasting process is how to establish a mapping between the observable finished passenger flow and the actual passenger flow.

3.1.2. Consideration for Temporal Relationships

Each OD pair $i,j$ associates with a set of features and targets at each time step $t\in \left[0,m_{max}\right]$. To facilitate the following description, we hereby make conventions on the notation, model, and framework of the study. The framework of the forecasting task can be illustrated as Equation (2).

$${\hat{Y}}_{pt,m}^A=GTFNN\left({\overrightarrow{Y}}_{pt,-h,-\pi }^A,{\overrightarrow{X}}_{pt,-h,-\pi }^F,O_{pt,-h,-\pi },K_{pt,-h,m},G\left({\overrightarrow{X}}_{pt,-h,-\pi }^F,O_{pt,-h,-\pi }^s,K_{pt,-h,m}^s\right)\right)$$

(2)

In Equation (2), ${\hat{Y}}_{pt,m}^A$ is the $m$-step-ahead ($m$ is the length of the predicted sequence) forecast result at prediction time $pt$. $GTFNN(·)$ refers to the proposed forecasting model. ${\overrightarrow{X}}_{pt,-h,-\pi }^F$ is the vector of historical finished OD flow of time range $\left[pt-h,pt-\pi \right]$. $h$ refers to the horizon of the input sequence, and $\pi$ refers to the blind spot for data updates caused by the information system update mechanism. ${\overrightarrow{Y}}_{pt,-h,-\pi }^A$ is the target vector of forecasting, a vector of historical actual or entered OD flow. $O_{pt,-h,-\pi }$ refers to the observed input features that can be obtained in historical data only, $K_{pt,-h,m}$ refers to the known input features that can be obtained in the whole range of time, and $G\left({\overrightarrow{X}}_{pt,-h,-\pi }^F,O_{pt,-h,-\pi }^s,K_{pt,-h,m}^s\right)$ represents the graphic structures called multi-layer networks that can help the framework to extract hidden localized features among stations. Within Figure 1, we show the relationship of the above concepts on the time axis to facilitate the construction of the following model.

Theoretically, the OD pairs among stations should be fully connected, i.e., if the number of stations is $N$, the number of OD pairs is $N\cdot \left(N-1\right)$. However, the OD matrix is relatively sparse. Thus, we only consider the OD flow from station $i$ to the top-$\kappa$ stations where its passengers are most likely to reach, as well as the total OD flow to the remaining stations. The details of the mentioned notations are shown in

Table 1. Key notations in forecasting task.

Index	Notation	Description
1	${\overrightarrow{X}}_{pt,-h,-\pi }^F$	A vector of input finished passenger flow sequence, where $pt$ is the time step of forecasting. $h$ refers to the length of the input sequence and $\pi$ refers to the blind spot for data updates caused by the information system update mechanism, where $h>0$ and $\pi <h$. ${\overrightarrow{X}}_{pt,-h,-\pi }^F$ represents the finished passenger flow of time steps $\left\{pt-h+1,pt-h+2,\cdots pt-\pi \right\}$. ${\overrightarrow{X}}_{pt,-h,-\pi }^F\triangleq \left\{X_{pt-n+1}^F,X_{pt-n+2}^F,\cdots X_{pt-\pi }^F\right\}$
2	$X_t^F$	The finished OD flows of time step $t$ with the origin station $i\in \left[1,N\right]$. $X_t^F\triangleq \left\{x_{1,t}^F,x_{2,t}^F,\cdots ,x_{i,t}^F,\cdots ,x_{N,t}^F\right\}\in {ℝ}^{\kappa \times N}$
3	$x_{i,t}^F$	The top $\kappa -1$ finished OD pairs that originate from station $i$ at time step $t$. $x_{i~\kappa ,t}^{IC}$ represents the rest of the finished OD flow of station $i$. $x_{i,t}^F\triangleq \left\{x_{i~1,t}^F,x_{i~2,t}^F,\cdots ,x_{i~j,t}^F,\cdots ,x_{i~\kappa -1,t}^F,x_{i~\kappa ,t}^F\right\}\in {ℝ}^{\kappa }$
4	$x_{i~j,t}^F$	The finished OD flow traveled from station $i$ to station $j$ at time step $t$.
5	${\hat{Y}}_{pt,m}^A$	A vector of predicted actual (or entered) passenger flow sequence, where $pt$ is the predicted or query time step. $m$ refers to the length of the predicted sequence where. $A$ indicates that the variable represents the actual (or entered) passenger flow of time steps $\left\{pt+1,pt+2,\cdots pt+m\right\}$. ${\hat{Y}}_{pt,m}^A\triangleq \left\{Y_{pt+1}^A,Y_{pt+2}^A,\cdots Y_{pt+m}^A\right\}$
6	${\overrightarrow{Y}}_{pt,-h,-\pi }^A$	A vector of historical actual (or entered) passenger flow sequence, where $pt$ is the time step of forecasting. $h$ refers to the length of the sequence, and $\pi$ refers to the blind spot for data updates caused by the information system update mechanism, where $h>0$ and $\pi <h$. ${\overrightarrow{Y}}_{pt,-h,-\pi }^A$ represents the actual passenger flow of time steps $\left\{pt-h+1,pt-h+2,\cdots pt-\pi \right\}$. ${\overrightarrow{Y}}_{pt,-h,-\pi }^A\triangleq \left\{Y_{pt-n+1}^A,Y_{pt-n+2}^A,\cdots Y_{pt-\pi }^A\right\}$
7	$Y_t^A$	The actual (or entered) OD flows of time step $t$ with the origin station $i\in \left[1,N\right]$. $Y_t^A\triangleq \left\{y_{1,t}^A,y_{2,t}^A,\cdots ,y_{i,t}^A,\cdots ,y_{N,t}^A\right\}\in {ℝ}^{\kappa \times N}$
8	$y_{i,t}^A$	The top $\kappa -1$ actual (or entered) OD pairs that originate from station $i$ at time step $t$. $x_{i~\kappa ,t}^C$ represents the rest of the actual (or entered) OD flow of station $i$. $y_{i,t}^A\triangleq \left\{y_{i~1,t}^A,y_{i~2,t}^A,\cdots ,y_{i~j,t}^A,\cdots ,y_{i~\kappa -1,t}^A,y_{i~\kappa ,t}^A\right\}\in {ℝ}^{\kappa }$
9	$y_{i~j,t}^A$	The actual (or entered) OD flow traveled from station $i$ to station $j$ at time step $t$.
10	$O_{pt,-h,-\pi }$	The observed input features that can only be obtained in historical data. We define $O_{pt,-h,-\pi }\triangleq \left\{O_{pt,-h,-\pi }^s,O_{pt,-h,-\pi }^g\right\}$
11	$K_{pt,-h,m}$	The known input features that can be obtained in the whole range of time. We define $K_{pt,-h,m}\triangleq \left\{K_{pt,-h,m}^s,K_{pt,-h,m}^g\right\}$

.

The details of $O_{pt,-h,-\pi }$ and $K_{pt,-h,m}$ are shown in

Table 2. Time-dependent features.

Class	Name	Notation	Time Range	Set
Observed input features ($O$)	Finished OD passenger flow in next time step	$o_t^1$	[pt – h + 1, pt − π]	$O_{pt,-h,-\pi }^s$
	Finished OD passenger flow in next two time steps	$o_t^2$
	Max. passenger flow	$o_t^3$		$O_{pt,-h,-\pi }^g$
	Min. passenger flow	$o_t^4$
Known input features ($K$)	Hour of the day	$k_t^1$	[pt – h + 1, pt + m]	$K_{pt,-h,m}^s$
	Day of the week	$k_t^2$
	Weather (Sunny, rainy, and cloudy)	$k_t^3$
	Passenger flow in the latest update time step	$k_t^4$		$K_{pt,-h,m}^g$
	Passenger flow in the two time steps before the latest update time
	Passenger flow in the three time steps before the latest update time	$k_t^6$
	Passenger flow in the four time steps before the latest update time	$k_t^7$
	Passenger flow in the same time step of the previous day	$k_t^8$
	Passenger flow in the same time step last week	$k_t^9$
	Passenger flow for the same time step two weeks ago	$k_t^{10}$

. These two sets mainly include sequenced (e.g., $O_{pt,-h,-\pi }^s$ and $K_{pt,-h,m}^s$) and graphic features (e.g., $O_{pt,-h,-\pi }^g$ and $K_{pt,-h,m}^g$) across time. These features are designed to help the model’s learning of the relationship among the different components of the historical data.

Unfold the features in Table 2 by spatial and temporal dimensions. Each element $k_t^j\in K_{pt,-h,m}^g$, $j\in \left\{4,5,\cdots 10\right\}$, can be denoted as:

$$o_t^j=\left\{o_{1,t}^j,o_{2,t}^j,\cdots ,o_{i,t}^j,\cdots ,o_{N,t}^j\right\}, j\in \left[1,4\right]$$

(3)

$$k_t^w=\left\{k_{1,t}^w,k_{2,t}^w,\cdots ,k_{i,t}^w,\cdots ,k_{N,t}^w\right\}, w\in \left[1,10\right]$$

(4)

Thus, each element $k_t^w$ can map to the station set of the network.

The input of the model is denoted as:

$$I=\left\{{\overrightarrow{X}}_{pt,-h,-\pi }^F,O_{pt,-h,-\pi },K_{pt,-h,m}\right\}$$

(5)

$$I_t=\left\{I_t^1,I_t^2,\cdots ,I_t^N\right\}$$

(6)

$$I_t^i=\left\{x_{i,t}^F,{\left\{o_{i,t}^j\right\}}_{j\in \left[1,4\right]},{\left\{k_{i,t}^w\right\}}_{w\in \left[1,10\right]}\right\}$$

(7)

Specifically, in Table 2, the features in set $K_{pt,-h,m}^s$ are dependent on the OD flow but are independent from the structure of networks. Thus, we define the input of the graph model in Equation (8).

$$I{G}_t=\left\{I{G}_t^1,I{G}_t^2,\cdots ,I{G}_t^N\right\}$$

(8)

$$I{G}_t^i=\left\{x_{i,t}^F,{\left\{o_{i,t}^j\right\}}_{j\in \left[1,4\right]},{\left\{k_{i,t}^w\right\}}_{w\in \left[4,10\right]}\right\}$$

(9)

3.1.3. Consideration for Spatial Relationships

The spatial relationships studied in this paper refer to the relationships among stations, which can influence the prediction of OD flow. We summarized four classes of spatial relationships with six derived graphs that exist in the urban transit system. A summarization of these relationships is shown in

Table 3. Multi-layer networks.

Class	Graph	Illustration	Notation	Weight of Edge
Station–line–network relationship	Station network	(a)	$G_{sn}=\left(N, E_{sn}, W_{sn}\right)$	$W_{sn}\left(i,j\right)=\frac{Se\left(i,j\right)}{{{\displaystyle \sum }}_{k=1}^{N}Se\left(i,k\right)}$	(10)
	Transfer network	(b)	$G_{tn}=\left(N, E_{tn}, W_{tn}\right)$	$W_{tn}\left(i,j\right)=\frac{Tr\left(i,j\right)}{{{\displaystyle \sum }}_{k=1}^{N}Tr\left(i,k\right)}$	(11)
Passenger flow characteristics relationship	Time series similarity	(c)	$G_{ts}=\left(N, E_{ts}, W_{ts}\right)$	$W_{ts}\left(i,j\right)=\frac{Ts\left(i,j\right)}{{{\displaystyle \sum }}_{k\in c}Ts\left(i,k\right)}$	(12)
	Peak hour factor similarity	(d)	$G_{ps}=\left(N, E_{ps}, W_{ps}\right)$	$W_{ps}\left(i,j\right)=\frac{Ps\left(i,j\right)}{{{\displaystyle \sum }}_{k\in c}Ps\left(i,k\right)}$	(13)
Line planning relationship	Line planning network	(e)	$G_{lp}=\left(N, E_{lp}, W_{lp}\right)$	$$\begin{gathered} W_{lp}\left(i,j\right) \\ =\frac{Fre\left(l\right)Lp\left(i,j\right)}{{{\displaystyle \sum }}_{k\in SL\left(l\right)}Lp\left(i,k\right)Fre\left(l\right)} \end{gathered}$$	(14)
Correlation relationship	Correlation of flow evolution	(f)	$G_{cf}=\left(N, E_{cf}, W_{cf}\right)$	$W_{cf}\left(i,j\right)=\frac{D\left(i,j\right)}{{{\displaystyle \sum }}_{k=1}^{N}D\left(i,k\right)}$	(15)
			$G=\left(N, E, W\right)$

.

• Station–line–network relationship

This is the basic tomography relationship of the rail transit network that determines the connections between each pair of stations. We focused on two networks of this relationship:

a. Station network

The station network $G_{sn}=\left(N, E_{sn}, W_{sn}\right)$ is directly constructed according to the connections of sections and stations of the studied rail transit network. An edge is formed to connect node $i$ and $j$ in $E_{sn}$ if the corresponding station $i$ and $j$ are connected in the real network.

b. Transfer network

The transfer network $G_{tn}=\left(N, E_{tn}, W_{tn}\right)$ is constructed by the connections of a station with its nearby transfer stations. An edge is formed to connect node $i$ and $j$ in $E_{tn}$ if the corresponding station $i$ and transfer station $j$ are connected by a station path along with the station network, and the path cannot contain other transfer station.

2.

Passenger flow characteristics relationship

When two stations are located in different areas but have the same function (e.g., office, education, business districts), it makes sense that the evolution of the passenger flow will be similar. We chose two kinds of feature to measure similarities.

a. Time series similarity

The daily passenger flow data along the time axis will form a time series. The similarity among the time series belonging to different stations can construct the edges and weights among stations. By these, the time series similarity $G_{ts}=\left(N, E_{ts}, W_{ts}\right)$ is built by a similarity measurement with threshold. We illustrate the details about how to construct $G_{ts}$ in the following part of measurement weights.

b. Peak hour factor similarity

Likewise, the peak hour factor was also chosen as a feature to measure similarity between any pair of stations. The peak hour factor is calculated by the formula:

$$p_i=\frac{\max \left({\overrightarrow{Y}}_{pt,-h_1,-h_2}^A\right)}{\mathrm{avg}\left({\overrightarrow{Y}}_{pt,-h_1,-h_2}^A\right)}$$

(16)

where $h_1$ and $h_2$ are the operational beginning and end time points of a day. The function $\max (·)$ can find the maximum OD flow in vector ${\overrightarrow{Y}}_{pt,-h_1,-h_2}^A$, and the function $\mathrm{avg}(·)$ can calculate the average flow of off-peak hours.

By these, the peak hour factor similarity $G_{ps}=\left(N, E_{ps}, W_{ps}\right)$ is built by a similarity measurement based on the second-norm. We illustrate the details about how to construct $G_{ps}$ in the following part of measurement weights.

3.

Line planning relationship

Although urban rail transit has a natural physical topology, passengers can only move with the trains. Thus, the line planning has a huge impact on passenger OD, especially in terms of short-term resolution. A line plan is one of the most basic papers for rail transit operations. A line is often taken to be a route in a high-level infrastructure graph ignoring precise details of platforms, junctions, etc. In addition, a line is a route in the network together with a stopping pattern for the stations along that route, as a line may either stop at or bypass a station on its route [41]. We define a line plan as a set of such routes, each with a series of way stations, a stopping pattern and frequency, which together must meet certain targets such as providing minimal service at every station.

a. Line planning network

The line planning network $G_{lp}=\left(N, E_{lp}, W_{lp}\right)$ describes the connected relationships formed by the line planning. This correlation has a huge impact on passenger travel. $E_{lp}$ and $W_{lp}$ are determined by the shopping pattern and running frequency.

4.

Correlation relationship

For representing potential large OD pairs or potential travel demand hidden in the urban rail transit, we built a network to represent the correlation relationships.

a. Correlation of flow evolution

OD flow between every two stations is not uniform, and the direction of passenger flow implicitly represents the correlation of two stations. For instance, if: (I) the majority of inflow of station $a$ stream to station $b$, or (II) the outflow of station $a$ primarily comes from station $b$, we believe that the stations $a$ and $b$ are highly correlated.

According to the above discussions, we defined the graphs by nodes $N$, edges $E$ and the weights $W$. In this context, we set node $n\in N$, where $n\in N$ represents a real station. The graphs share the same nodes but have their own edges and weights, that is, $E\triangleq \left\{E_{sn},E_{tn},E_{ts},E_{ps},E_{cd},E_{cf}\right\}$ and $W\triangleq \left\{W_{sn},W_{tn},W_{ts},W_{ps},W_{cd},W_{cf}\right\}$. Specifically, we denote $W\in {ℝ}^{6\times N\times N}$ as the weights of all edges, and for each $W_z\in W$, $z\in \left\{sn,tn,ts,ps,lp,cf\right\}$. An overall design of the graphs is shown in Table 3.

We denote $W_z\left(i,j\right)$ as the weight of edge $\left(i,j\right)$. In Table 3, the fifth column summarizes the calculation methods of different weights.

For calculating $W_{sn}\left(i,j\right)$ and $W_{tn}\left(i,j\right)$, $Se\left(i,j\right)$ represents the connection function of nodes (i.e., station) $i$ and $j$. $Se\left(i,j\right)=1$ if there exists a section between node $i$ and $j$, else $Se\left(i,j\right)=0$, and we set $Se\left(i,i\right)=0$. Likewise, $Tr\left(i,j\right)$ represents the connection function of a station with the nearby transfer stations. $Tr\left(i,j\right)=1$ if there exists a path without another transfer station between station $i$ and transfer station $j$, or else $Tr\left(i,j\right)=0$.

Theoretically, similarity exists between any two stations. However, in large-scale networks, the number of station pairs is large, and the similarity of most of them is small, which leads to a complex similarity graph. For this reason, we used a combination of clustering method and threshold selection to first find the potential groups to which the stations belong and then excluded the station pairs with small similarity. Thus, for calculating $Ts\left(i,j\right)$ in $W_{ts}\left(i,j\right)$, using the time series as inputs, we first obtained the similarity relationships of different stations based on the clustering method [42] and obtained clusters of stations, denoted as $C$. Then, for each $c\in C$, a predefined similarity threshold was set to control the number of similarities. Based on the finite similarity relationships, we built the edge set $E_{ts}$ and used Equations (17)–(20) to calculate $Ts\left(i,j\right)$ in category $c\in C$.

$$Ts\left(i,j\right)=\exp \left(-SBD\left({\left\{sum\left(y_{i,t}^A\right)\right\}}_{t\in T_h},{\left\{sum\left(y_{j,t}^A\right)\right\}}_{t\in T_h}\right)\right)$$

(17)

$$sum\left(y_{i,t}^A\right)={{\displaystyle \sum }}_{j\in \left[1,\kappa \right]}{y}_{i~j,t}^A$$

(18)

$$SBD\left({\overrightarrow{r}}_i,{\overrightarrow{r}}_j\right)=1-\underset{w}{max}\left(\frac{C{C}_w\left({\overrightarrow{r}}_i,{\overrightarrow{r}}_j\right)}{||{\overrightarrow{r}}_i||\cdot ||{\overrightarrow{r}}_j||}\right)$$

(19)

$$C{C}_w\left({\overrightarrow{r}}_i,{\overrightarrow{r}}_j\right)=\left\{\begin{array}{c}{\displaystyle {\displaystyle \sum }_{l=1}^{2m-w}}{r}_{i,l+w-m}\cdot r_{j,l},w\ge m\\ C{C}_{-w}\left({\overrightarrow{r}}_j,{\overrightarrow{r}}_i\right), w<m\end{array}\right.$$

(20)

In Equation (17), the function $SBD$ [43] (shape-based distance) is set for measuring the distance between two temporal sequences with equal length. Specifically, the $SBD$ is calculated by Equation (11), where ${\overrightarrow{r}}_i,{\overrightarrow{r}}_j\in R_I$ is the flow time series of OD $i$ and $j$ and $||\cdot ||$ refers to the second norm operator. $C{C}_w\left({\overrightarrow{r}}_i,{\overrightarrow{r}}_j\right)$ represents the cross-correlation between ${\overrightarrow{r}}_i$ and ${\overrightarrow{r}}_j$. In addition, we set $Ts\left(i,i\right)$ to 0.

Analogously, we calculated $Ps\left(i,j\right)$ for $W_{ps}\left(i,j\right)$ by the cluster–threshold–calculation framework. The classical k-means method is used for clustering, and the similarity is measured by the second norm operator, as shown in Equation (21).

$$Ps\left(i,j\right)=||P_i,P_j||$$

(21)

where $P_i$ and $P_j$ are the peak hour factor of station $i$ and $j$.

In Equation (15), the weight $W_{lp}\left(i,j\right)$ takes line plans of urban transit into consideration. We set $l\in L$ as a line plan and $SL\left(l\right)$ as the stations in the route of $l$. $Lp\left(i,j\right)=0$ represents that there exists an edge between station $i$ and $j$ in terms of a train running line $l$, and $Fre\left(l\right)$ is the running frequency of the train running line $l$.

In Equation (16), the $D\left(i,j\right)$ is the total number of passengers that traveled from station $j$ to station $i$ in the whole dataset.

3.2. Multi-Layer Graph Neural Networks Model (MGNN) for Structured Forecasting

3.2.1. Structure of the Multi-Layer Graph Neural Networks Model (MGNN)

In previous works [3,16,44], sequenced features and graphic features have both been proven to be useful for traffic state prediction. One key issue in designing MGNN is how to fuse the temporal features and the spatial features during the training of the model. In the context of network-level passenger flow prediction, we specifically call the features that can be represented by the graphs the spatial features, or otherwise, the temporal features. For fusing the temporal features and the local features, we adopted the designs of the Graph Convolution Gated Recurrent Unit (GC-GRU) and Fully Connected Gates Recurrent Unit (FC-GRU) proposed by [3].

3.2.2. Convolution Operation for MGNN by GC-GRU

A GC-GRU [22] was introduced for spatial–temporal feature learning of MGNN. By using this GC-GRU, we could effectively learn spatial temporal features from the OD flow data. The convolution operation was designed as follows:

The parameters of this graph convolution are denoted as $\mathrm{\Theta }$. Following the definition in Equation (9), the input of graph convolution is set as $I{G}_t=\left\{I{G}_t^1,I{G}_t^2,\cdots ,I{G}_t^N\right\}$. By definition of convolution, the output feature $f\left(I{G}_t^i\right)\in {ℝ}^d$ of $I{G}_t^i$ is computed by:

$$\begin{array}{c}f\left(I{G}_t^i\right)={\mathrm{\Theta }}_{l}I{G}_t^i+{\displaystyle \sum }_{j\in {\mathcal{N}}_{sn}\left(i\right)}{W}_{sn}\left(i,j\right)\odot {\mathrm{\Theta }}_{sn}I{G}_t^i+{\displaystyle \sum }_{j\in {\mathcal{N}}_{tn}\left(i\right)}{W}_{tn}\left(i,j\right)\odot {\mathrm{\Theta }}_{tn}I{G}_t^i\\ +{\displaystyle \sum }_{j\in {\mathcal{N}}_{ts}\left(i\right)}{W}_{ts}\left(i,j\right)\odot {\mathrm{\Theta }}_{ts}I{G}_t^i+{\displaystyle \sum }_{j\in {\mathcal{N}}_{ps}\left(i\right)}{W}_{ps}\left(i,j\right)\odot {\mathrm{\Theta }}_{ps}I{G}_t^i\\ +{\displaystyle \sum }_{j\in {\mathcal{N}}_{lp}\left(i\right)}{W}_{lp}\left(i,j\right)\odot {\mathrm{\Theta }}_{lp}I{G}_t^i+{\displaystyle \sum }_{j\in {\mathcal{N}}_{cf}\left(i\right)}{W}_{cf}\left(i,j\right)\odot {\mathrm{\Theta }}_{cf}I{G}_t^i\end{array}$$

(22)

where $\odot$ is the Hadamard product and $\mathrm{\Theta }\triangleq \left\{{\mathrm{\Theta }}_{sn},{\mathrm{\Theta }}_{tn},{\mathrm{\Theta }}_{ts},{\mathrm{\Theta }}_{ps},{\mathrm{\Theta }}_{lp},{\mathrm{\Theta }}_{cf}\right\}$. Specifically, ${\mathrm{\Theta }}_{l}I{G}_t^i$ is the self-loop for all graphs, and ${\mathrm{\Theta }}_l$ is the learnable parameters. ${\mathrm{\Theta }}_{sn}$ denotes the parameters of the station network graph $G_{sn}$, and ${\mathcal{N}}_{sn}\left(i\right)$ represents the neighbor set of node $i$ in $G_{sn}$. Other notations ${\mathrm{\Theta }}_{tn},{\mathrm{\Theta }}_{ts},{\mathrm{\Theta }}_{ps},{\mathrm{\Theta }}_{lp},{\mathrm{\Theta }}_{cf}$, ${\mathcal{N}}_{sn}\left(i\right),{\mathcal{N}}_{tn}\left(i\right),{\mathcal{N}}_{ts}\left(i\right),{\mathcal{N}}_{ps}\left(i\right),{\mathcal{N}}_{lp}\left(i\right)$ and ${\mathcal{N}}_{cf}\left(i\right)$ have similar semantic meanings. $d$ is the dimensionality of feature $f\left(I{G}_t^i\right)$. In this manner, a node can dynamically receive information from some highly corrected neighbor nodes. For convenience, we denoted the graph convolution in Equation (22) as $I{G}_t\ast \mathrm{\Theta }$ in the following.

Since the above-mentioned operation is conducted on a spatial dimension, we embedded the graph convolution in a Gated Recurrent Unit (GRU) to learn spatial–temporal features. Specifically, the reset gate $R_t=\left\{R_t^1,R_t^2,\cdots ,R_t^N\right\}$, update gate $Z_t=\left\{Z_t^1,Z_t^2,\cdots ,Z_t^N\right\}$ new information $N_t=\left\{N_t^1,N_t^2,\cdots ,N_t^N\right\}$ and hidden state $H_t=\left\{H_t^1,H_t^2,\cdots ,H_t^N\right\}$ are computed by:

$$H_t=GC\text{-}GRU\left(I{G}_t,H_{t-1}\right):$$

(23)

$$R_t=\sigma \left({\mathrm{\Theta }}_{rx}\times I{G}_t+{\mathrm{\Theta }}_{rh}\times H_{t-1}+b_r\right)$$

(24)

$$Z_t=\sigma \left({\mathrm{\Theta }}_{zx}\times I{G}_t+{\mathrm{\Theta }}_{zh}\times H_{t-1}+b_z\right)$$

(25)

$$N_t=\tanh \left({\mathrm{\Theta }}_{nx}\times I{G}_t+R_t\odot \left({\mathrm{\Theta }}_{nh}\times H_{t-1}+b_n\right)\right)$$

(26)

$$H_t=\left(1-Z_t\right)\odot N_t+Z_t\odot H_{t-1}$$

(27)

where $\sigma$ is the sigmoid function and the $H_{t-1}$ is the hidden state at last $t-1$ iteration. ${\mathrm{\Theta }}_{rx}$, ${\mathrm{\Theta }}_{zx}$, ${\mathrm{\Theta }}_{zh}$, ${\mathrm{\Theta }}_{nx}$ and ${\mathrm{\Theta }}_{nh}$ denote the graph convolution parameters. $b_r$, $b_z$ and $b_n$ are bias terms. The feature dimensions of $R_t^i$, $Z_t^i$, $N_t^i$ and $H_t^i$ are set to $d$.

3.2.3. The Combination of GC-GRU and FC-GRU

The proposed MGNN can conduct convolution on graphic space, and FC-GRU can learn the inputs from a sequenced view. We combined the GC-GRU and FC-GRU and denoted the combination as GFGRU. Many GFGRUs are organized according to the Encoder–Decoder framework, and we built a Sequence-to-Sequence structure, as shown in Figure 2. Specifically, the inputs of GFGRU are $I_t$ and ${\tilde{H}}_{t-1}$, where ${\tilde{H}}_{t-1}$ is the output hidden state of the last iteration.

In GFGRU, GC-GRU utilizes the accumulated information in ${\tilde{H}}_{t-1}$ to update the hidden state, rather than take the original $H_t$ as input. Thus, Equation (23) becomes Equation (28).

$$H_t=GC\text{-}GRU\left(I{G}_t,{\tilde{H}}_{t-1}\right)$$

(28)

For FC-GRU, we first transformed $I_t$ and ${\tilde{H}}_{t-1}$ to an embedded $I_t^e\in {ℝ}^d$ and $H_{t-1}^e\in {ℝ}^d$ with two fully connect (FC) layers. Then, we fed $I_t^e$ and $H_{t-1}^e$ into a common GRU [45] implemented with a full connection to generate a global hidden state $H_t^f\in {ℝ}^d$, which can be express as:

$$I_t^e=FC\left(I_t\right)$$

(29)

$$H_{t-1}^e=FC\left({\tilde{H}}_{t-1}\right)$$

(30)

$$H_t^f=FC-GRU\left(I_t^e,H_{t-1}^e\right)$$

(31)

Finally, we incorporated $H_t$ and $H_t^g$ to generate a combined hidden state ${\tilde{H}}_t=\left\{{\tilde{H}}_t^1,{\tilde{H}}_t^2,\cdots ,{\tilde{H}}_t^N\right\}$ with a fully connected layer:

$${\tilde{H}}_t^i=FC\left(H_t^i\oplus H_t^g\right)$$

(32)

where $\oplus$ denotes an operator of feature concatenation.

3.3. Graph–Temporal Fused Neural Network (GTFNN)

3.3.1. Overall for the Proposed GTFNN

When building the GTFNN framework, we need to address two main issues: firstly, the model has a comprehensive but complex input. The relationship (e.g., linear or nonlinear) between the complex inputs and the forecasting model is difficult to determine; second, time series data naturally have local features (e.g., change-points, etc.) and global features (e.g., series trends and attention at different time positions), and the framework we designed needs to take both types of features into account to ensure better forecast performance. Thus, three important designs are considered in GTFNN:

• Gating structure GRN

A gating structure GRN can decide if non-linear learning is required. It can skip over any unused components of the architecture, which provide adaptive depth and network complexity to accommodate a wide range of datasets and scenarios.

2.

Feature selection layer

While the designed features may be available, their relevance and specific contribution to the output are typically unknown. The GTFNN also uses specialized components for the judicious selection of relevant features and a series of gating layers to suppress unnecessary components, enabling high performance in a wide range of regimes. We adopted the feature selection network proposed by Lim et al. [3] to tune the weights of input features at each time step dynamically. This feature selection network was designed based on the gating mechanism.

3.

Sequence-to-sequence layer with attention mechanisms

For learning both local and global temporal relationships from time-varying inputs, a sequence-to-sequence layer is employed for local processing, whereas long-term dependencies are captured using an interpretable multi-head attention block. The GTFNN employs a self-attention mechanism to learn long-term relationships across different time steps [3], which is modified from multi-head attention in transformer-based architectures [29,34] to enhance explainability.

Figure 3 shows the high-level architecture of GTFNN, where individual components are described in detail in the subsequent sections.

3.3.2. Gating Structure GRN

With the motivation of giving the model the flexibility to apply non-linear processing only where needed, the Gated Residual Network (GRN) was chosen as a building block of GTFNN. The GRN takes in a primary input $I_t$ designed in Equation (6) and obtains:

$$GR{N}_{\omega }\left(I_t\right)=LayerNorm\left(I_t+GL{U}_{\omega }\left({\eta }_1\right)\right)$$

(33)

$${\eta }_1=W_{1,\omega }{\eta }_2+b_{1,\omega }$$

(34)

$${\eta }_2=ELU\left(W_{2,\omega }{I}_t+b_{2,\omega }\right)$$

(35)

where $ELU$ is the Exponential Linear Unit activation function [46], ${\eta }_1 \in R^{d_{model}}$, ${\eta }_2\in R^{d_{model}}$ are intermediate layers, $LayerNorm(·)$ is standard layer normalization [47], and $\omega$ is an index to denote weight sharing. When $W_{2,\omega }a+b_{2,\omega }\gg 0$, the ELU activation would act as an identity function and when $W_{2,\omega }a+b_{2,\omega }\ll 0$, the ELU activation would generate a constant output, resulting in linear layer behavior. The structure of GRN is shown in Figure 4.

In Equation (33), Gated Linear Units (GLUs) [48] are selected as the gating components to provide the flexibility to suppress any part of the architecture that can be skipped in the scenario. Denoting $\gamma \in R^{d_{model}}$ as the input of the $GLU$, we obtain Equation (36).

$$GL{U}_{\omega }\left(\gamma \right)=\sigma \left(W_{3,\omega }\gamma +b_{3,\omega }\right)\odot \left(W_{4,\omega }\gamma +b_{4,\omega }\right)$$

(36)

where $\sigma (.)$ is the sigmoid activation function, $W(.) \in R^{d_{model}\times d_{model}}$, $b(.)\in R^{d_{model}}$ are the weights and biases, $\odot$ is the element-wise Hadamard product, and $d_{model}$ is the hidden state size. GLU allows GTFNN to control the extent to which the GRN contributes to the original input $I_t$, potentially skipping over the layer entirely if necessary, as the GLU outputs could be all close to 0 in order to suppress the nonlinear contribution [3].

3.3.3. Instance-Wise Feature Selection Layer

Instance-wise feature selection is provided by the feature selection networks applied to all input features. Entity embeddings [49] is used for categorical variables as feature representations, and linear transformations for continuous variables—transforming each input variable into a $d_{model}$-dimensional vector. All inputs make use of separate feature selection networks with distinct weights.

Let ${\xi }_t^{\left(j\right)}\in {ℝ}^{d_{model}}$ denote the transformed input of the $j$th feature at time $t$, with ${\Xi }_t={\left[{\xi }_t^{{\left(j\right)}^T},\dots , {\xi }_t^{{\left(m_{\chi }\right)}^T}\right]}^T$ being the flattened vector of all past inputs at time $t$. Feature selection weights are generated by feeding ${\Xi }_t$ through a GRN, followed by a Softmax layer:

$$v_{{\chi }_t}=\mathrm{Softmax}\left({\mathrm{GRN}}_{v_{\chi }}\left({\Xi }_t\right)\right)$$

(37)

where $v_{{\chi }_t}\in {ℝ}^{m_{\chi }}$ is a vector of feature selection weights.

At each time step, an additional layer of non-linear processing is employed by feeding each ${\xi }_t^{\left(j\right)}$ through its own GRN:

$${\tilde{\xi }}_t^{\left(j\right)}={\mathrm{GRN}}_{\tilde{\xi }\left(j\right)}\left({\xi }_t^{\left(j\right)}\right)$$

(38)

where ${\tilde{\xi }}_t^{\left(j\right)}$ is the processed feature vector for variable $j$. We note that each variable has its own $GR{N}_{\xi \left(j\right)}$, with weights shared across all time steps $t$. Processed features are then weighted by their variable selection weights and are combined:

$${\tilde{\xi }}_t={\displaystyle \sum }_{j=1}^{m_{\chi }}{v}_{{\chi }_t}^{\left(j\right)}{\tilde{\xi }}_t^{\left(j\right)}$$

(39)

where $v_{{\chi }_t}^{\left(j\right)}$ is the $j$th element of vector $v_{{\chi }_t}$.

3.3.4. Sequence-to-Sequence Layer with Attention Mechanisms

The local features of the time series that are identified in relation to their surrounding values, such as anomalies, change-points etc., are significant. Thus, we applied a sequence-to-sequence layer to build an Encoder–Decoder structure with feeding ${\tilde{\xi }}_{t-h:t}$ into the encoder and ${\tilde{\xi }}_{t+1:t+m_{max}}$ into the decoder. This then generates a set of uniform temporal features that serve as inputs into the decoder itself, denoted by $\varphi \left(t, n\right)\in \left\{\varphi \left(t,-h\right), . . . ,\varphi \left(t,m_{max}\right)\right\}$ with $n$ being a position index. We also employed a gated skip connection over this layer:

$$\tilde{\varphi }\left(t,n\right)=LayerNorm({\tilde{\xi }}_{t+n}+GL{U}_{\overline{\varphi }}\left(\varphi \left(t,n\right)\right)$$

(40)

where $n\in \left[-h,m_{max}\right]$ is the position index.

(1)

Temporal self-attention layer

Following the sequence-to-sequence layer and the gate layer, we applied a modified self-attention layer [3]. All temporal features are grouped into a matrix, i.e., $\Theta \left(t\right)={\left[\theta \left(t,-h\right),\dots ,\theta \left(t, m\right)\right]}^T$, and interpretable multi-head attention [3] is applied at each forecast time step, with the number of time steps feeding into the attention layer $N=m_{max}+h+1$:

$$\begin{gathered} B\left(t\right)=InterpretableMultiHead\left(\Theta \left(t\right), \Theta \left(t\right), \Theta \left(t\right)\right) \\ =\left[\beta \left(t,-h\right),\dots \beta \left(t,m_{max}\right)\right] \end{gathered}$$

(41)

Decoder masking [29,34] is applied to the multi-head attention layer to ensure that each temporal dimension can only attend to features preceding it.

$$InterpretableMultiHead\left(Q, K,V\right)=\tilde{H}{W}_H$$

(42)

$$\tilde{H}=\tilde{A}\left(Q, K\right)V{W}_V=\left\{\frac{1}{H}{\displaystyle \sum }_{h=1}^{n_H}A\left(Q{W}_Q^{\left(h\right)},K{W}_K^{\left(h\right)}\right)\right\}V{W}_V$$

(43)

$$=\frac{1}{H}{\displaystyle \sum }_{h=1}^{n_H}Attention\left(Q{W}_Q^{\left(h\right)},K{W}_K^{\left(h\right)},V{W}_V\right)$$

(44)

$$A\left(Q,K\right)=\mathrm{Softmax}\left(Q{K}^{\mathrm{T}}/\sqrt{d_{attn}}\right)$$

(45)

$$Attention\left(Q,K,V\right)=A\left(Q,K\right)V$$

(46)

where $V\in R^{N\times d_V}$, $K\in R^{N\times d_{attn}}$ and $Q\in R^{N\times d_{attn}}$ are attention mechanism scale values, keys and queries. $d_V=d_{attn}=d_{mpdel}/n_H$, and $n_H$ is the number of heads. $A(·)$ is a normalization function. $W_K^{\left(h\right)}\in R^{d_{model}\times d_{attn}}$, $W_Q^{\left(h\right)}\in R^{d_{model}\times d_{attn}}$ are head-specific weights for keys, queries. $W_V\in R^{d_{model}\times d_V}$ are value weights shared across all heads, and $W_H\in R^{d_{attn}\times d_{model}}$ is used for final linear mapping.

The self-attention layer allows GTFNN to pick up long-range dependencies that may be challenging for RNN-based architectures to learn. Following the self-attention layer, an additional gating layer is also applied to facilitate training:

$$\delta \left(t,n\right)=LayerNorm\left(\theta \left(t,n\right)+GL{U}_{\delta }\left(\beta \left(t,n\right)\right)\right)$$

(47)

(2)

Position-wise feed-forward layer

We applied additional non-linear processing to the outputs of the self-attention layer. This layer also makes use of GRNs:

$$\psi \left(t,n\right)=GR{N}_{\psi }\left(\delta \left(t,n\right)\right)$$

(48)

where the weights of $GR{N}_{\psi }$ are shared across the entire layer. As shown in Figure 5, we also applied a gated residual connection that skips over the entire transformer block, providing a direct path to the sequence-to-sequence layer, yielding a simpler model if additional complexity is not required, as shown below:

$$\tilde{\psi }\left(t,n\right)=LayerNorm\left(\tilde{\varphi }\left(t,n\right)+GL{U}_{\tilde{\psi }} \left(\psi \left(t,n\right)\right)\right)$$

(49)

All the notation is concluded in the Appendix A.

4. Numerical Experiments

4.1. Experiment Settings

4.1.1. Dataset

We collected a mass of trip transaction records from a real-world metro system and constructed a large-scale dataset, which is termed as CQMetro. The overview of the dataset is summarized in

Table 4. Dataset of CQMetro.

Dataset	Notation
City	Chongqing, China
Station	170
Physical Edge	448
Flow volume per day	1.72 M
OD pairs	14,314
Time Step	15 min
Input length	384 (4 days)
Output length	4 (60 min)
Forecasting horizon	2 (30 min)
Training Timespan	2880 (30 days)
Testing Timespan	1440 (15 days)
Validation Timespan	96 (1 day)

.

This dataset was built based on the rail transit system of Chongqing, China. The transaction records were collected from 1 January to 15 February 2019, with daily passenger flow of 1.72 million on average. The total number of OD pairs in the whole network is 14,314 pairs. Each record contains the information of entry/exit station and the corresponding timestamps. In this period, 170 stations operated normally, and they were connected by 224 physical edges (i.e., the sections between stations). For each station, we measured design features every 15 min. The data of the first 30 days were used for training, and the last 15 days were used for training and testing, while OD flows of the following day were used for validation. In particular, 1–3 January and 4–10 February 2019 are New Year’s Day and Chinese New Year holidays.

4.1.2. Details for Implementing GTFNN

We implemented our GTFNN with the deep learning framework PyTorch. The lengths of input and output are listed in Table 4. Hyperparameter optimization was conducted via random search with 60 iterations. The search ranges for all hyperparameters and the selected optimal hyperparameters can be found in

Table 5. Hyperparameter optimization.

Random search range	State size	10, 20, 40, 80, 160, 240, 320
	Dropout rate	0.1, 0.2, 0.3, 0.4, 0.5, 0.7, 0.9
	Minibatch size	64, 128, 256
	Learning rate	0.0001, 0.001, 0.01
	Decay ratio	0.1
	Max. gradient norm	0.01, 1.0, 100.0
	Feature dimensionality	256
Optimal hyperparameters	Random search iterations	60
	State size	320
	Dropout rate	0.3
	Minibatch Size	128
	Learning rate	0.001
	Decay ratio	0.1
	Max Gradient Norm	100
	Num. Heads	4
	Feature dimensionality	256

. We applied Adam [50] to optimize our GTFNN for 200 epochs by minimizing the jointly minimizing function between the predicted results and the corresponding ground-truths. The jointly minimizing function [38], summed across every time step output:

$$\mathcal{L}\left(\mathsf{\Omega },W\right)={\displaystyle \sum }_{y_t\in \mathsf{\Omega }}{\displaystyle \sum }_{\tau =1}^{m_{max}}\frac{QL\left(y_t,\widehat{y}\left(t-\tau ,\tau \right)\right)}{M{m}_{max}}$$

(50)

$$QL\left(y, \widehat{y}\right)=\left|y-\widehat{y}\right|$$

(51)

where $\Omega$ is the domain of training data containing $M$ samples.

During training, dropout is applied before the gating layer and layer normalization.

4.1.3. Evaluation Metrics

Following previous works [25,51], we evaluated the performance of methods with Mean Square Error (MSE), Root Mean Square Error (RMSE), Mean Absolute Error (MAE) and Symmetric Mean Absolute Percentage Error (SMAPE), which are defined as:

$$MSE=\frac{1}{n}{\displaystyle \sum }_{i=1}^n{\left({\hat{X}}_i-X_i\right)}^2$$

(52)

$$RMSE=\sqrt{\frac{1}{n}{\displaystyle \sum }_{i=1}^n{\left({\hat{X}}_i-X_i\right)}^2}$$

(53)

$$MAE=\frac{1}{n}{\displaystyle \sum }_{i=1}^n\left|{\hat{X}}_i-X_i\right|$$

(54)

$$SMAPE=\frac{1}{n}{\displaystyle \sum }_{i=1}^n\frac{2\cdot \left|{\hat{X}}_i-X_i\right|}{\left|{\hat{X}}_i\right|+\left|X_i\right|}$$

(55)

where $n$ is the number of testing time steps; for example, if we conduct the forecasting every 60 min, then $n=4$. ${\hat{X}}_i$ and $X_i$ denote the predicted ridership and the ground-truth ridership, respectively. Note that ${\hat{X}}_i$ and $X_i$ have been transformed back to the original scale with an inverted z-score normalization. Our GTFNN is developed to predict the metro ridership of the next two steps. In the following experiments, we would measure the errors of each time interval separately. For the 15 min granularity OD prediction, there may be a true value of 0; thus, SMAPE is used instead of MAPE to describe the relative accuracy of the prediction. Unlike MAPE (Mean Absolute Percentage Error), the SMAPE values range from 0 to 200%. All metrics are close to 0, meaning higher prediction accuracy.

4.2. Comparison with State-of-the-Art Methods

In this section, we compared our GTFNN with four base-line methods including:

Gradient Boosting Decision Trees (GBDT) [52]: GBDT is a weighted ensemble method that consists of a series of weak estimators. We implemented this method with a python package named Sklearn. The number of boosting stages is set to 100, and the maximum depth of each estimator is 4. Gradient descent optimizer is applied to minimize the loss function.

Long Short-Term Memory (LSTM) [53]: This network is a simple Seq2Seq model, and its core module consists of two fully connected LSTM layers. The hidden size of each LSTM layer is set to 256. Its hyper-parameters are the same as ours.

Gated Recurrent Unit (GRU) [54]: With the similar architecture of the previous model, this network replaces the original LSTM layers with GRU layers. The hidden size of GRU is also set to 256. Its hyper-parameters are the same as ours.

Graph-WaveNet [55]: This method develops an adaptive dependency matrix to capture the hidden spatial dependency and utilizes a stacked dilated 1D convolution component to handle very long sequences. We implemented this method with its official code.

4.3. Performance in Different Scenarios

In this part, we discuss the prediction results of different methods on weekends, weekdays and holidays and use four metrics such as MSE, RMSE, MAE and SMAPE for evaluation. Each metric is the average of the predicted results of 14,314 OD pairs, with extreme values excluded. The performances of all methods are summarized in

Table 6. The mean performances of models in forecasting weekdays OD flow.

Metrics	GTFNN	GBDT	LSTM	GRU	Graph-WaveNet
MAE	0.83	1.71	4.57	4.63	2.91
MSE	4.42	7.06	15.49	16.34	10.42
RMSE	2.10	2.66	3.94	4.04	3.23
SMAPE	14.16%	19.41%	35.17%	34.52%	23.25%

,

Table 7. The mean performances of models in forecasting weekends OD flow.

Metrics	GTFNN	GBDT	LSTM	GRU	Graph-WaveNet
MAE	0.78	1.58	4.12	4.26	2.13
MSE	4.02	6.53	13.09	13.19	9.63
RMSE	2.00	2.56	3.62	3.63	3.10
SMAPE	13.21%	18.35%	31.48%	31.54%	20.16%

and

Table 8. The mean performances of models in forecasting holiday OD flow.

Metrics	GTFNN	GBDT	LSTM	GRU	Graph-WaveNet
MAE	4.41	6.60	19.97	20.30	8.89
MSE	35.08	44.44	85.76	85.11	67.32
RMSE	5.92	6.67	9.26	9.23	8.20
SMAPE	47.96%	60.83%	109.86%	85.44%	60.07%

.

The predictions for the weekdays are shown in Table 6. To predict the ridership at the next four consecutive time intervals (60 min), the baseline LSTM obtains a SMAPE score of 35.17% on CQMetro, ranking last among all the methods. Compared to LSTM and GRU, the performance of GBDT and Graph-WaveNet were much better. The prediction ability of the GTFNN model is the best compared with the above models. SMAPE is 14.16%, MAE is only 0.83, and the average prediction error of each period is lower than 1 person. It can be seen that GTFNN fully combines the advantages of the graph neural network model and time series model and has good passenger flow prediction ability.

The predictions for the weekends are shown in Table 7. Similar to working days, the GTFNN model still predicted better than the other models, with a SMAPE of 13.21% and MAE, MSE and RMSE metrics of 0.78, 4.02 and 2.00, respectively. In general, the prediction accuracy of different models for weekend passenger flows is higher than that of weekday passenger flows, with the SMAPE value of different models increasing by about 1–4%.

The predictions for the holidays are shown in Table 8. From the results, the prediction results of different models for holidays are not very satisfactory. The GTFNN model still predicted better than the other models, with SMAPE of 47.96% and MAE, MSE and RMSE metrics of 4.41, 35.08 and 5.92, respectively.

4.4. The Rank of Features

Figure 6 shows the variable importance for the CQMetro dataset. From the model description in Section 2, in the encoder, the predicted features include the observed input feature and known input feature classes. For the encoders, the most known input features are more important than the observed input features, with the $k_t^1$ (Hour of the day) being the most important, at more than 20%. The importance of “passenger flow for the first 2–4 periods” ($k_t^5~k_t^7$) has exceeded 10%. In observed input features, the $o_t^3$ (Max. passenger flow) feature is the most important. In the decoder, the “Passenger flow in the two time steps before the latest update time” ($k_t^5$) is much more important than all the other features, with more than 60%. The importance of the remaining individual features was less than 10%.

5. Discussion

The results of the overall metrics analysis of Section 4.3 showed the excellent performance of the GTFNN model. To further analyze the forecasting performance in different scenarios, this section is designed from two aspects: the prediction results and characteristics of different ODs on weekends and weekdays are analyzed; furthermore, the comparison between forecasting results of ordinary days and holidays is discussed to analyze the applicability of the studied model in forecasting OD passenger flow from different sources.

5.1. Comparison of Different ODs

The following four typical ODs were selected for further discussion and analysis.

(1)

OD 1: This OD consists of a hub-type station and a station in CBD. The selected hub-type station is located close to the city’s high-speed rail passenger hub, mainly serving long-distance passengers entering and leaving the city, and it is an interchange station between the high-speed rail network and the urban rail transit. The station is located in the CBD of the city, which is the most prosperous part of the city, with large passenger flow.

(2)

OD 2: This OD consists of a station in the residential area and a station in CBD. The selected station in the residential area is located in the main residential area of the city and mainly serves the commuting needs of passengers in the residential area. The station in CBD is the same as the station in OD 1.

(3)

OD 3: This OD consists of a station in the residential area and a station in the suburban area. The selected station in the residential area is the same as those in OD 2. The suburban-type station is the starting and ending station of the line, and the station is far away from the city hub, where trains need to make a turnaround, and the daily passenger flow is small.

(4)

OD 4: This OD consists of a hub-type station and station in the suburban area. The selected hub-type station and the station in the suburban area are the same as the stations in OD 1 and OD 3, respectively.

Figure 7, Figure 8, Figure 9 and Figure 10 show the prediction results of four pairs of different ODs on weekdays or weekends. The blue dashed line represents the prediction result, and the red solid line represents the actual flow. The x-axis represents the time step, and the y-axis represents the passenger flow. The time scale represents the first 15 min of the day; thus, the whole day is divided into 1440/15 = 96 periods (since the passenger flow at night is 0, it is not shown in the operation periods).

5.1.1. Forecasting of OD 1 in Weekdays and Weekends

The main purposes of the passenger flow between hubs and CBDs are business activities, shopping, consumption, and attending large events. Hubs and CBDs attract frequent economic activities and commercial activities, which lead to strong population mobility. Passenger flow in these areas is extremely large. As shown in Figure 7, the peak passenger flow exceeds 20 people/15 min on weekdays and 40 people/15 min on weekends.

Table 9. The metrics of forecasting results for OD 1.

Passenger Flow Scenario 1	MAE	MSE	RMSE	SMAPE
weekdays	0.38	0.52	0.72	11.51%
weekends	0.18	0.07	0.27	8.20%

shows the metrics of forecasting results. The passenger flow forecast results on weekends are more accurate compared to those on weekdays, where MAE is reduced by 0.20 and SMAPE is reduced by 3.3%.

5.1.2. Forecasting of OD 2 in Weekdays and Weekends

Jobs or shopping opportunities offered by enterprises located in CBD attract people from near and far, which causes large passenger flow in OD 2. As shown in Figure 8, its traffic peak is close to 15 people/15 min on weekdays, while its traffic peak drops to 10 people/15 min on weekends due to the reduction of commuter traffic. In terms of the forecast metrics (

Table 10. The metrics of forecasting results for residential areas to CBD OD flow.

Passenger Flow Scenario 2	MAE	MSE	RMSE	SMAPE
weekdays	0.26	0.70	0.84	31.23%
weekends	0.28	0.77	0.88	33.47%

), the forecast accuracy of passenger flow on weekdays is slightly higher than that on weekends. In particular, MAE is reduced by 0.02, and SMAPE is reduced by 2.24%.

5.1.3. Forecasting of OD 3 in Weekdays and Weekends

Residential–suburban stations are more diverse in terms of the main purposes of passenger traffic. As usual, suburban stations have smaller passenger flows, but the residential station area selected in this case still has some passenger flows due to its physical proximity to the suburban station. As shown in Figure 9, its peak passenger flow is close to 10 passengers/15 min on weekdays, and drops to 6 passengers/15 min due to the reduced economic activity in the city on weekends. In terms of forecasting metrics (

Table 11. The metrics of forecasting results for residential to suburban OD flow.

Passenger Flow Scenario 3	MAE	MSE	RMSE	SMAPE
weekdays	0.16	0.18	0.41	81.98%
weekends	0.12	0.10	0.32	76.42%

), the forecast accuracy of passenger flow on weekends is slightly higher than that of weekdays. In particular, MAE is reduced by 0.04 and SMAPE is reduced by 2.24%.

5.1.4. Forecasting of OD 4 in Weekdays and Weekends

The main purpose of passenger flow between hubs and suburban stations is to commute to urban hubs or to engage in commercial activities. Since suburban stations are less populated nearby, hubs are extremely attractive to suburban stations located at the edge of the city. The urban functions of suburban areas are largely dependent on the existence of urban hubs; therefore, suburban stations would have lower passenger volumes. As shown in Figure 10, its passenger peak is just over 6 passengers/15 min on weekdays and drops to less than 6 passengers/15 min due to the reduced urban economic activity on weekends. In terms of forecasting metrics (

Table 12. The metrics of forecasting results for hub to suburban OD flow.

Passenger Flow Scenario 4	MAE	MSE	RMSE	SMAPE
weekdays	0.12	0.09	0.29	106.84%
weekends	0.14	0.13	0.36	112.67%

), the accuracy of passenger flow forecasting on weekdays is slightly higher than that on weekends. In particular, MAE is reduced by 0.02, and SMAPE is reduced by 5.83%.

5.1.5. Analyses of the Model Performance

From the above characteristics, we can analyze the following points.

(1)

The predicted and actual values of OD 1 (both the weekdays and weekends) would exceed the other three pairs of ODs. Moreover, because of the great amount of passenger flow, the corresponding prediction ability of OD 1 is much better than the other three pairs of ODs.

(2)

The absolute MAE scores of all ODs are small regardless of the SMAPE difference. It shows that the absolute prediction accuracy of this model is relatively high.

(3)

Because the value of passenger flows in a period is small, small errors could cause big relative errors. The MAE scores of the above four pairs of ODs have little difference, but the larger the passenger flow is, the smaller the SAMPE scores are.

(4)

The SMAPE scores of OD 1 are much lower than the other three pairs of ODs.

In general, it could be seen from the above discussion that for OD prediction, the size of OD passenger flow will affect the evaluation of the relative metrics of the prediction, especially when the passenger flow is small. OD with small passenger flows also shows a poor prediction ability when evaluated with SMAPE. This conclusion has some correlation to the conclusion that “max flow is the most important in observed input features” is obtained from Section 4.4.

In terms of absolute metrics, such as MAE and RMSE, the prediction accuracy of the GTFNN model is high both overall and locally, with MAE scores less than 0.4 for all six pairs of ODs mentioned above. We believe that for the urban rail passenger flow, when the prediction error is less than 1, the impact on the overall line is relatively small, and it is within the acceptable range.

5.2. Comparison between Ordinary Days and Holidays

As the prediction results in Section 4.3 shows, it can be seen that the prediction performance of holidays is worse than that of ordinary days. From the perspective of MAE, the MAE score of holidays (4.41) is 3.58 and 3.63 higher than that of weekdays (0.83) and weekends (0.78), respectively. Likewise, from the perspective of MSE, the score of holidays (35.08) is 30.66 and 31.06 higher than that of weekdays (4.42) and weekends (4.02) and from the perspective of SMAPE, the score of holidays (47.96%) is 33.80% and 34.75% higher than that of weekdays (14.16%) and weekends (13.21%). Although the passenger flow on holidays is larger than that on ordinary days, it is clear that the passenger flow characteristics of holidays are not yet well captured by the model. On the one hand, the model does not consider how to capture the characteristics of holidays (i.e., uncommonly large passenger flow) when designing. Furthermore, there are only a small amount of holidays data in the training data, and the passenger flow characteristics vary between holidays, which poses a great challenge to the model’s prediction during holidays.

6. Conclusions

In this work, we proposed a Graph–Temporal Fused Neural Network (GTFNN) to address the network-level origin–destination (OD) flows online short-term forecasting problem. In order to solve the key issue of online flow forecasting, the proposed GTFNN has made efforts in the four aspects below.

(1)

The GTFNN takes finished OD flow and a series of known and observable features as the input and explores multi-step predictions.

(2)

Unlike previous works that either focus on the spatial relationship or the temporal relationship of OD flows evolution, the proposed method is constructed from capturing both spatial and temporal characteristics.

(3)

In order to learn spatial characteristics, a multi-layer graph neural network model is proposed based on hidden relationships in the rail transit network. Then, we embedded the graph convolution in a Gated Recurrent Unit to learn spatial–temporal features.

(4)

Based on the sequence-to-sequence framework, a Graph–Temporal Fused Deep Learning model was built. In addition, an attention mechanism was attached to the model to fuse local and global temporal dependencies to achieve the prediction of short-term online OD passenger flow.

Experiments based on real-world data collected from Chongqing’s rail transit system showed that the proposed model performed better than other models. For instance, on weekdays for passenger flow forecasting scenarios, the SMAPE score of GTFNN was about 14.16%, with a range from 5% to 20% higher compared to other methods. In addition, the MAE score ranged from 0.1 to 0.8, which is suitable for applications. By comparing some representative ODs, we found that it is more difficult to forecast ODs with small average passenger flow values. OD forecasting for small passenger flows should be one of the next research points.

The proposed model can also analyze weights of different features. The weights of observed input features and known input features were different in the encoder, where the most important feature of observed input features is “hour of the day”, and the most important feature of known input features is “max. passenger flow”.

Obtaining accurate OD passenger flow data on time is vital to support transportation organization in a rail transit system. The accurate OD prediction results allow operators to understand the passenger demand between different ODs of the network at a certain time point in the future, thereby supporting the dynamic and efficient deployment of transportation organization resources. Well-designed train line planning, timetabling and station passenger flow planning can be obtained. From the point of view of the trend of a rail transit passenger flow prediction problem, the traditional station in/out volume prediction can no longer meet the actual application demand. This kind of prediction can only know the station collection and dispersion volume, and the passenger flow distribution on the network is unknown. Accurate prediction of OD will become a popular topic in the future. The method that combines both temporal and spatial relationships into the prediction system studied in this paper will be one of the supports to solve the accurate prediction of OD. Thus, the OD prediction model studied in this paper has strong practical significance.

Nevertheless, there still exist some limitations in the proposed model. By comparing the results of ordinary days and holidays (with uncommonly large passenger flow), we can find that the method studied in this paper still cannot guarantee the accuracy in different passenger flow scenarios. In the future, it is also necessary to optimize the model and algorithm for scenarios where sudden large passenger flows happen to meet the needs of on-time forecasting. In addition, six levels of graph neural networks used in the current model have the same weights. The effect of these graphs on prediction accuracy has not been studied. A good understanding of the importance of different graph relations can help reduce the complexity of the model and improve the training efficiency of the model.