Fuzzy multivariate NARX model for passenger entrance flow prediction in the Shanghai subway system

Abstract

A fuzzy multivariate based line traffic prediction model and a station traffic proportion inference model are proposed in order to predict line entrance traffic and station traffic in the Shanghai subway system. The nonlinear autoregressive with external input (NARX) is adopted to predict subway line traffic. Correlative features that influence the line traffic time series are identified by time series trend analysis, including meteorological features, time features, and proportion of commuter passenger features. A time series correlation method and fuzzy c-means (FCM) is proposed to simplify the feature set by deleting the features of small coefficients between features and traffic series. We determine the statistics of all stations’ traffic proportion of a subway line in order to construct a proportion matrix, and an eigenvector based station traffic proportion inference model is proposed to predict the future station traffic proportion, which is combined with the subway line traffic prediction results to realize subway station traffic prediction. We evaluate our model on the dataset of smart card records and weather condition dataset of one month in the Shanghai subway system. Experiment results confirm our proposed model’s advantages over baseline approaches.

Keywords

Fuzzy time series NARX eigen recognition subway traffic prediction urban computing

1 Introduction

In recent years, there has been a marked increase in passenger flow on subway systems in China and it has become very important to accurately forecast passenger flow to effectively prevent emergencies that can occur because of large gathered crowds [10]. Subway passenger flow prediction is a research problem that adopts technologies to forecast passenger flow at future intervals, which greatly assists subway system management departments in being able to dynamically adjust train schedules during peak hours [7]. Researchers have proposed many methods to satisfy the need for early warnings of subway traffic congestion. Cai et al., proposed a multiply ARIMA model to predict entrance and exit passenger flow at urban railway stations [1], Sun Y. and S. Clark used time series theory to adopt nonparametric regression models to predict subway transfer station traffic[5, 11]. A passenger source prediction model has been proposed through analysis of passengers’ origins and destinations [3]. Zhou Y. proposed a model to estimate the travel time for subway transfers [13].Wei Y. brought the neural network into passenger flow prediction [12]. As a kind of neural network, nonlinear autoregressive with external input (NARX).[14] is used in wind speed prediction [2] and chaotic flow prediction combined with fuzzy functions [6].

In this paper, we investigate a big data based method of subway passenger entrance flow prediction in the Shanghai subway system. Similar to factors used in bike sharing systems [9], the factors that we use include weather conditions, day of the week, hour of the day, and proportion of the passenger population that consists of commuters. Same as coefficient modeling in [8], we apply fuzzy cluster algorithms to the extracted features and calculate the relationship between factors and passenger traffic. The identified features are dropped if the coefficient is small. Simplified features are set as the external input of NARX to predict the subway line traffic. Because specific line traffic is composed of station traffic, we propose a station traffic contribution model based on eigenvector analysis of the station traffic contribution proportion matrix.

Our main contributions in this paper are:

In predicting passenger flow, we take into account both temporal characteristics, and the influences of weather conditions commuter passenger proportion, day of the week, and hour of the day.

A fuzzy multivariate time series-based NARX model is proposed to forecast subway line traffic. The correlation of multivariate and traffic time series identifies the ultimate correlative features.

A station traffic proportion inference model is proposed to predict the future proportion of station traffic contribution vectors. Station traffic is predicted through combining the results of subway line traffic predictions with the results of station traffic proportion inferences.

Experiments are presented which show supportive evidence for the proposed model of subway traffic prediction.

2 Fuzzy time series and neural net

2.1 Fuzzy multivariate correlation

Both line traffic and station traffic in subway systems are affected by multiple factors. Correlation calculation between relative time series and prediction time series is as shown in [4]. The symbolization method used is the (Statistic Vector based Series Symbolization) SVSS method, which uses the intrinsic statistic characteristics to symbolize the time series. Statistical characteristics include frequency of intervals of equal probability and the relationship between two and three adjacent data points. In the statistics of frequency in intervals of equal probability, first, we must divide the space into intervals of equal probability. Samples are divided into intervals of equal probability I₁, I₂, ... , I_m, where m refers to the interval number. Then, the series is symbolized, and the characteristic vector is expressed as:

$\begin{matrix} I_{1}, I_{2}, \dots, I_{m}, <, =, >, < <, < =, < >, \\ = <, = =, = >, > <, > =, > > \end{matrix}$ (1)

After symbolizing, the relationship between the vectors can be seen in in Equation (2): $ρ = \frac{\sum_{t = 1}^{H} X_{i, t} \times X_{j, t}}{\sqrt{\sum_{t = 1}^{H} X_{i, t}^{2}} \sqrt{\sum_{t = 1}^{H} X_{j, t}^{2}}}$ (2)

Where X_i,t and X_j,t are respectively the t^th value, and H is the length of historic time series. If the relevance is small, the variable must be selected again. Otherwise, the sample series is considered closely relevant to the corresponding series. Then each element in this time series is clustered using the fuzzy c-means (FCM) algorithm, the objective function and cluster number in this paper are calculated usingEquation (3): $\begin{matrix} J = \sum_{i = 1}^{K} \sum_{j}^{H} u_{ij}^{m} {∥ c_{i} - x_{j} ∥}^{2}, u_{ij} \in [0, 1] \\ \sum_{i}^{K} u_{ij} = 1, \forall j = 1, 2, \dots, H \\ K = [|Dmax - Dmin | - 0.1 em / - 0.15 em 1 H - 1∑t = 2H xt - xt - 1] \end{matrix}$ (3)

Respectively, D_max, D_min refer to the maximum and minimum, C₁, C₂, …, C_K are k clusters, c_i is the cluster center point and u_ij denotes whether x_j belongs to cluster C_i. Then subscripts of the fuzzy subsets are used to represent the fuzzy subsets. In this way, each element of a time series is replaced by the fuzzy set subscript.

2.2 Neural net time series

Dynamic neural networks, which include tapped delay lines, are used for nonlinear filtering and prediction. The NARX is a recurrent dynamic network, with feedback connections enclosing several layers of the network. The NARX model is based on the linear ARX model, which is commonly used in time-seriesmodeling. The defining equation for the NARX model is Equation (4).

$\begin{matrix} y (t) = f (y (t - 1), \dots, y (t - d), \\ x (t - 1), \dots, x (t - d)) \end{matrix}$ (4)

Where the next value of the dependent output signal y (t) is regressed on d previous values of the output signal and d previous values of an independent input signal. As shown in Fig. 1, where a two-layer feed forward network is used for the approximation. The the regression performance function used for the NARX network to optimize the network isEquation (5):

$\begin{matrix} {MSE}_{reg} = γ \cdot \frac{1}{n} \sum_{i = 1}^{n} (y_{i} - y_{i}^{⌢})^{2} \\ + (1 - γ) \cdot \frac{1}{n} \sum_{j = 1}^{n} W_{i}^{2} \end{matrix}$ (5)

where γ is the performance ratio, W is the vector of the neurons’ connection weight.

3 Methodology for passenger flow prediction

3.1 Correlative feature identification

Time features: Intuitively, the subway system passenger flow presents different patterns on weekdays and holidays/weekends. Figures 2 and 3 show the passenger entrance traffic of a line and a station. It is obvious that passenger flow on weekdays is similar, consisting of morning rush hours, day hours, evening rush hours, and night hours. While weekends/ holidays have different variation trends. Therefore, we can extract two time features: the hour of the day and the day of the week.

Meteorological features: Intuitively, when there is adverse weather, people tend to use a more convenient form of transportation. As shown in Fig. 4 and Fig. 5, subway traffic is closely related to weather conditions and air quality index. In this paper, two meteorological features are identified: rainfall and air quality index. In Fig. 4, we present the traffic of line 1 in four time spans. It can be seen that rain on 7th Apr at 12am and 15 am lead to traffic decreasing either for line traffic or for the “Lujiazui” station. In Fig. 5, three weekend days are chosen. On 19th Apr, either the line traffic or the station passenger flow is lower than the two other days, which is because the air quality on that day was poor.

Commuter proportion features: In Fig. 6, two Wednesdays’ line and station historic traffic are similar, though the air quality on 9th Apr was worse than on 16th Apr. However, the passenger flow on 19th Apr, a rainy day, significantly decreased, compared to the traffic on a weekday 29th Apr, on these two days the corresponding average commuter proportion was respectively 0.24 and 0.21. On weekdays, commuters may still take the subway, regardless of raining or the poor air quality. On weekends/holidays, when the weather is bad, both commuters and other travelers may stay at home or use taxis or personal cars. Therefore, the relative proportion of commuters affects the passenger flow prediction.

3.2 Line traffic prediction methodology

Line traffic prediction is based on the correlative features extracted in Section 3.1. Combined with fuzzy time series theory, the coefficient between line traffic and its different correlative features are calculated as described in Section 2.1, which is set as the input of the neural net time series as described in Section 2.2. The time feature is described in Table 1. According to time feature, the passenger is labeled as “commuter” when the trip is initiated either during the morning rush or evening rush. We compute the commuter proportion as a commute feature. The modeling process is shown as follows.

3.3 Proportion inference model

We can obtain the station traffic contribution proportion matrix M_{s_k,m,t_i,n} for each line in each day using the smart card data, where t_i,j stands for the time slot j of the i^th day in the historical dataset and s_k,m refers to the station m of line k.

We can obtain a set S with M matrix, given that we have smart card expense data for M days. Each matrix is transformed into a vector Γ of size N and placed into the set. $S = {Γ_{1}, Γ_{2}, Γ_{3}, \dots, Γ_{M}}$ (6)

The mean matrix of the set can be calculated using the following formula: $Ψ = \frac{1}{M} \sum_{i = 1}^{M} Γ_{i}$ (7)

Assume that the time slot of the station traffic we want to predict is in the i^th day, then the difference between the predicting matrix and the mean matrix is Φ: $Φ_{i} = Γ_{i} - Ψ$ (8)

Where Ψ is the mean matrix of the set S, and Γ_i is the station traffic contribution proportion vector of the i^th day.

Then we construct a different matrix set A as shown below. The covariance matrix C is obtained in the following manner: $\begin{matrix} A = {Φ_{1}, Φ_{2}, \dots, Φ_{n}} \\ C = \frac{1}{M} \sum_{n = 1}^{M} Φ_{n} Φ_{n}^{T} = {AA}^{T} \end{matrix}$ (9)

Assuming that u_k, λ_k are respectively the eigenvectors and eigenvalues of the covariance matrix C and a set of M orthonormal vectors u can best describe the distribution of the data set. The k^th vector u_k is chosen such that $λ_{k} = \frac{1}{M} \sum_{n = 1}^{M} (u_{k}^{T} Φ_{n})^{2}$ (10)

is a maximum, subject to

$u_{l}^{T} u_{k} = δ_{lk} = {\begin{matrix} \begin{matrix} 1 & \begin{matrix} if & l = k \end{matrix} \end{matrix} \\ \begin{matrix} 0 & otherwise \end{matrix} \end{matrix}$ (11)

Assume that we want to predict the station for the sth time slot in the dth day. First we delete the sth to n column of all the matrixes in set S before vector transformation. Then, predicting the day’s station traffic contribution matrix is projected to eigenvectors: $\begin{matrix} Ω^{T} = [ω_{1}, ω_{2}, \dots, ω_{M}] \\ ω_{k} = u_{k}^{T} (Γ - Ψ) \end{matrix}$ (12)

Lastly, we determine which day’s matrix provides the best description for predicting the day’s station traffic contribution matrix. This is done by minimizing the Euclidean distance. Then the predicting matrix’s s^th column is set to the matching matrix’s sth column which represents the station traffic contribution in the predicting time slot.

4 Experiment

4.1 Datasets and metric

We conducted experiments on the datasets of the Shanghai subway system, which were collected from April 1 to April 30 2015. The dataset information is described in Table 2.

The metrics we adopt to measure the accuracy of the prediction results are the Root Mean Squared Error (RMSE) and Mean Relative Error (MRE).

4.2 Results and comparison

In order to compare the experiment performance of different methods, two baselines are included as shown below

Feature-NARX: the model input contains all features without FCM clustering.

FCM-NARX: the model input contains all features with FCM clustering.

CL-NARX: our coefficient limited NARX model that takes into consideration the coefficient between feature and traffic series.

The NARX model generated in our experiment is by MATLAB software as shown in Fig. 7. Owing to space limitation, we only present the line 2 traffic prediction comparison graph as shown in Fig. 8. The station traffic contribution of each subway line is described in Table 3. The overall comparison of the traffic prediction error is in Table 4.

As shown in Table 3, the fitting degree of all station traffic contribution is 0.9768, which means that the stations’ traffic contribution of a specific line is relatively stable. As shown in Table 4, the prediction error of the station traffic increases by 0.07% when we use the FCM clustering method for the extracted features. While, when we apply the coefficient limitation to features, the error of line traffic and station traffic decreases by about 22.16% and 0.79%,respectively.

5 Conclusion

We proposed a subway system traffic prediction methods respectively for line and station passenger entrance flow. Before modeling, affecting features are extracted: rain, air quality index, whether it is a weekday or holiday, and time interval of day, and a correlation calculation method is proposed. For line traffic prediction, affecting features are selected by coefficient limitation after being clustered by FCM, and the NARX model is adopted with the features as external input. Passenger flow traffic of each subway station is predicted by combining the stations’ traffic proportion prediction and the line traffic prediction results. Experiments on the Shanghai subway dataset shows that our proposed model shows more prediction accuracy. Further work needs to be done to test the model on more datasets and compare it with more baselines.

Footnotes

Acknowledgments

This research is supported by China Postdoctoral Science Foundation under Grant No. 2015M71369, NSFC No. 51278221 and NSFC No. 51378076.

References

Cai

, Yao

, Wang

, et al., Prediction of urban railway station’s entrance and exit passenger flow based on multiply ARIMA model, Journal of Beijing Jiaotong University38(2) (2014), 135–140.

Cadenas

, Rivera

, Campos-Amezcua

, et al., Wind speed prediction using a univariate ARIMA model and a multivariate NARX model, Energies9(2) (2016), 109–.

Wang

, Li

, Liu

, et al., Vulnerability analysis and passenger source prediction in urban rail transit networks, PloS One8(11) (2013), e80178.

Huang

, Yang

and Cui

, Fuzzy time series-based cross-layer optimization scheme for cooperative communication in wireless network, Journal of Intelligent & Fuzzy Systems29(6) (2015), 2801–2807.

Clark

, Traffic prediction using multivariate nonparametric regression, Journal of Transportation Engineering129(2) (2003), 161–168.

Goudarzi

, Jafari

, Moradi

M.H.

, et al., NARX prediction of some rare chaotic flows: Recurrent fuzzy functions approach, Physics Letters A380(5–6) (2016), 696–706.

Tsai

T.H.

, A self-learning advanced booking model for railway arrival forecasting, Transportation Research Part C39(2) (2014), 80–93.

Feng

, Zhang

, Gan

, et al., Random coefficient modeling research on short-term forecast of passenger flow into an urban rail transit station, Transport31(1) (2016), 94–99.

, Zheng

, Zhang

, et al., Traffic Prediction In A Bike-Sharing System, Sig spatial International Conference on Advances in Geographic Information Systems, ACM, (2015).

10.

Sun

, Leng

and Guan

, A novel wavelet-SVM short-time passenger flow prediction in Beijing subway system, Neurocomputing166(C) (2015), 109–121.

11.

Sun

, Zhang

and Yin

, Passenger flow prediction of subway transfer stations based on nonparametric regression model, Discrete Dynamics in Nature & Society2014(3) (2014), 1–8.

12.

Wei

and Chen

M.C.

, Forecasting the short-term metro passenger flow with empirical mode decomposition and neural networks, Transportation Research Part C Emerging Technologies21(1) (2012), 148–162.

13.

Zhou

, Lin

, Yi

, et al., Time prediction model of subway transfer, Springerplus5(1) (2016), 1–11.

14.

Z.Y.

and Best

, Structure optimisation of input layer for feed-forward NARX neural network, International Journal of Modelling Identification and Control25(3) (2016), 217.