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Direct forecasting method for Urban Rail Transit (URT) ridership at the station level is not able to reflect nonlinear relationship between ridership and its predictors. Also, population is inappropriately expressed in this method since it is not uniformly distributed by area. In this paper, a new variable, population per distance band, is considered and a back propagation neural network (BPNN) model which can reflect nonlinear relationship between ridership and its predictors is proposed to forecast ridership. Key predictors are obtained through partial correlation analysis. The performance of the proposed model is compared with three other benchmark models, which are linear model with population per distance band, BPNN model with total population, and linear model with total population, using four measures of effectiveness (MOEs), maximum relative error (MRE), smallest relative error (SRE), average relative error (ARE), and mean square root of relative error (MSRRE). Also, another model for contribution rate of population per distance band to ridership is formulated based on the BPNN model with nonpopulation variables fixed. Case studies with Japanese data show that BPNN model with population per distance band outperforms other three models and the contribution rate of population within special distance band to ridership calculated through the contribution rate model is 70%~92.9% close to actual statistical value. The result confirms the effectiveness of models proposed in this paper.

In transportation planning, ridership modeling and forecasting is the basis for analyzing project viability and sustainability in the long run. Urban Rail Transit (URT) ridership at station level is an important element of URT ridership, which is critical for determining scale of stations and access facilities. Four-step model has been a traditional method for transit modeling [

Forecasting method for URT ridership at station level with multivariate regression models, also known as direct-forecast method, can forecast ridership based on the changes in factors affecting ridership throughout service area of stations [

Considerable research has been conducted on finding factors significantly affecting ridership. Cervero and Kockelman analyzed relationship between travel demand and 3Ds (density, diversity, and design) [

The tendency of using transit declines as the distance from stations/lines increases [

Forecasting models such as the ordinary least squares (OLS) regression model [

The major contributions of this paper are as follows:

Take population or employment per distance band as predictor directly.

Identify key factors affecting ridership through partial correlation analysis.

Formulate BPNN model to reflect exact relationship between URT ridership and its predictors.

Formulate a model of contribution rate of population per distance band to URT ridership using numerical analysis.

Referring to previously mentioned researches, population per distance band, road density, number of shuttle bus lines, land-use mix within station service areas, train frequency in one direction during peak hour, number of lines through station (station on the line), number of park and ride facilities, station type (terminal or not), and distance from station to CBD are selected preliminarily as factors affecting URT ridership at station level. This paper uses data of Tokyo, Japan, to illustrate how values of the above factors are calculated or obtained and the case study is conducted using the same dataset.

The population distance bands include 0-1 km, 1-2 km, 2-3 km, 3-4 km, 4-5 km, 5-6 km, 6 km, and above, according to road network buffer by GIS (Geographic Information System) [

Road density (unit:

Number of shuttle bus lines is used to measure convenience of shuttle bus access, which can be obtained from Google Earth within service area of 200 m radius from station. Both shuttle bus and URT belong to mass transit and they are strongly dependent on each other.

Land-use mix can be expressed by land-use mix ratio. The higher the value, the more diverse the land-use throughout this area and the more likely the fact that inhabitant and employment population are balanced here. The formula is as follows [

Unidirectional peak-hour train frequency is sum of all trains in one direction from any line stopping at a station. Same as number of lines through station, station type (terminal or not) and distance from station to CBD are used to measure station attraction. The station type is a binary variable: “1” indicates terminal and “0” otherwise.

The number of parking and riding facilities is also an important variable. More facilities are likely to attract more drivers, making them choose transit instead. Data for the last 5 variables can be obtained from official websites of railway operation industries in Japan.

Identifying key factors is critical. If all the above variables are used to formulate forecasting model, the performance of model will be unsatisfying because of the correlation between these variables.

Partial correlation analysis [

The partial correlation coefficient needs to be tested for significance further. The null hypothesis is that the partial correlation coefficient to be tested is not significantly different from zero. The corresponding

When degree of freedom is

Data from 129 stations in Tokyo, Japan, are used for case study to get the key predictors. Tokyo is the capital of Japan, encompassing 23 special wards, 26 cities, 5 towns, and 8 villages, which is also called Tokyo Metropolitan Area. The whole area is 2188.67 square kilometers, and the population is 13.23 million. URT is the main commuter traffic mode of Tokyo, whose total line is approximately 1000 km long and amount of total station is approximately 800 excluding the line servicing just suburban area. The selected station for case study is mainly in 23 special wards where the economic and trade activity mainly develops. Figure

URT in Tokyo metropolitan area.

Table

Correlation analysis of population per distance band and ridership.

Correlation coefficient | 0-1 km | 1-2 km | 2-3 km | 3-4 km | 4-5 km | 5-6 km | Above 6 km | Total |

| ||||||||

Ridership | | | | | | | | |

Correlation analyses of nonpopulation variables and ridership.

Correlation coefficient | Road density | Number of shuttle bus lines | Land-use | Peak-hour unidirectional train frequency | Number of lines through station | Number of park and ride facilities | Station type | Distance To CBD |
---|---|---|---|---|---|---|---|---|

Ridership | | | | | 0.101 | | | 0.04 |

Table

Table

Other than results in Table

Therefore, key predictors affecting ridership in this paper are obtained: population within distance bands (0-1 km, 1-2 km, 2-3 km, 3-4 km, 4-5 km, and 5-6 km), road density, number of shuttle bus lines, land-use mix, peak-hour unidirectional train frequency, and station type (terminal or not).

BPNN model is selected as the forecasting model for URT ridership at station level [

Three-layer BPNN (input-hidden-output) is able to reflect any nonlinearity from input to output and thus is adopted in this paper. The number of key factors affecting ridership is 11, and it is also the number of nodes in input layer. In output layer, there is only 1 node: ridership. The number of nodes in hidden layer is

The mathematical formulation of BPNN is as follows:

For output layer,

For hidden layer,

Steepest gradient descent method is adopted to update the weight and bias at each iteration. The detailed solution steps are described as follows:

Initialize weight and bias of BPNN. At the same time, for BPNN train, it is needed to set prediction accuracy and the maximum number of learning iterations.

Select input (predictors)/output (actual ridership) of any station randomly. Use BPNN with initial weights and bias to forecast and obtain the predicted ridership. Then, MSE can be obtained by comparing the predicted ridership with actual ridership of the station. Calculate

Calculate

Adjust each weight

Use BPNN and weights and bias obtained from Step

Contribution rate model which predicts the contribution of population within specific distance band to ridership at station level is formulated in this section, by fixing the value of other variables. BPNN is able to reflect interrelationship between key predictors and ridership. To obtain contribution rate, population within other distance bands are set to zero. By changing population within specific distance band, we can observe corresponding changes of ridership. Thus, variables other than population (e.g., road density, number of shuttle bus lines, land-use mix, peak-hour unidirectional train frequency, and station type (terminal or not)) need to be known first. Figure

Contribution rate model for ridership of population within 0-1 km band.

The detailed formulas are listed as follows:

The contribution rate model is solved by the following process. Plot dot (

Data of previous 129 stations in Tokyo, Japan, are used as case study and implementing BPNN model to forecast ridership at station level. All variables are normalized due to dimension difference. This paper sets learning rate, prediction accuracy, and maximum number of learning iteration to be 0.8, 0.001, and 30000, respectively. BPNN is trained with data of 117 stations.

The optimal weights and bias are shown in Tables

Weights for nodes in input layer and hidden layer.

Weights | Hidden layer | ||||
---|---|---|---|---|---|

Node 1 | Node 2 | Node 3 | Node 4 | ||

Input | Neural 1 (0-1 km band population) | | | | |

Neural 2 (1-2 km band population) | | | | | |

Neural 3 (2-3 km band population) | | | | | |

Neural 4 (3-4 km band population) | | | | | |

Neural 5 (4-5 km band population) | | | | | |

Neural 6 (5-6 km band population) | | | | | |

Neural 7 (road density) | 0.0877 | 0.0819 | −0.0822 | 0.0778 | |

Neural 8 (number of shuttle bus lines) | 0.0996 | −0.219 | −0.132 | 0.0238 | |

Neural 9 (land use mix) | 2.313 | 1.702 | −1.369 | 1.608 | |

Neural 10 (peak-hour unidirectional train frequency) | 0.0132 | | 0.0069 | −0.007 | |

Neural 11 (station type (terminal or not)) | −0.294 | 1.1325 | 0.869 | 1.168 |

Weights for nodes in output layer and hidden layer.

Weights | Hidden layer | ||||
---|---|---|---|---|---|

Node 1 | Node 2 | Node 3 | Node 4 | ||

Output layer | Only one neural (ridership) | | | | |

Bias for layers.

Hidden layer | Output layer | |||
---|---|---|---|---|

Neural 1 | Neural 2 | Neural 3 | Neural 4 | Only one neural |

−7.1365 | −1.9314 | 0.1474 | −1.5717 | 0 |

Comparison of four models.

To verify forecasting accuracy of BPNN model in this paper, its results are compared with that of linear model with population per distance band, linear model with total population, and BPNN model with total population. The other three models are calibrated/trained using the same input/output pattern sets of 117 stations and are implemented for prediction with the same data of other 12 stations. Linear model is calibrated/trained by SPSS. The results are compared using multiple MOEs (unit: %), maximum relative error (MRE), smallest relative error (SRE), average relative error (ARE), and mean square root of relative error (MSRRE). Relative error is the difference between forecasting ridership of one model and actual ridership of each station divided by actual ridership. MRE, SRE, and ARE of a model indicate the maximum, the smallest, and mean values among relative errors of 12 stations, respectively. MSRRE of a model is the mean value of square root of relative errors of 12 stations. Figure

MOEs of four models.

MRE | SRE | ARE | MSRRE | |
---|---|---|---|---|

Linear model with total population | 2704 | 3.3 | 409 | 249 |

BPNN model with total population | 3014 | 0.7 | 369 | 262 |

Linear model with population per distance band | 1484 | 1.2 | 277 | 140 |

BPNN model with population per distance band | 125 | 0.8 | 31 | 13 |

Figure

Mean values of every variable of 129 stations are used as background value. The station type is a binary variable, which makes us not able to take mean value of 129 stations as its background value. Here, it is set as 0. Under the above setting, increase population per distance band from 10000 (unit: persons) to 320000 by 10000 every time. Ridership is predicted by BPNN with weights and bias already obtained in Section

The results are shown in Figure

Curve of contribution rate.

Table

Comparison of results from model and the real contribution rate.

0-1 km | 1-2 km | 2-3 km | 3-4 km | 4-5 km | 5-6 km | ||

| |||||||

Contribution rate | Result from model | 0.161 | 0.069 | 0.0255 | 0.0098 | 0.0067 | 0.0059 |

Actual rate | 0.19 | 0.07 | 0.033 | 0.012 | 0.009 | 0.0066 | |

Relative error (%) | 18.9 | 7.1 | 23.6 | 20.8 | 30 | 16.7 |

On the basis of previous researches, factors affecting URT ridership at station level are summarized and identified. Key factors are then obtained through partial correlation analysis, including population per distance band, road density, number of shuttle bus lines, land-use mix, peak-hour train frequency in one direction, and station type (terminal or not).

BPNN model is formulated to forecast ridership due to the nonlinear relationship between ridership and its predictors. Input (factors affecting ridership)/output (ridership) pattern sets of 117 stations in Tokyo, Japan, are adapted to train the model and data from other 12 stations are used to predict for test. The result obtained from BPNN with population per distance band is compared with that of BPNN with total population, linear model with population per distance band, and linear model with total population. The MOEs, for example, MRE, SRE, MRE, and MSRRE, are used to evaluate the model and results show BPNN model with population per distance band has the best performance. Since the model can reflect the internal relationship between ridership and its affecting factors, when one of factors varies, ridership can be quickly and efficiently predicted.

Based on BPNN model, contribution rate model of population per distance band to ridership is constructed, when setting other nonpopulation variables as background. Results of the case study show the effectiveness of the model. When population within special distance band changes, ridership from this population can be calculated quickly and timely by multiplying corresponding rate and population without performing BPNN model once more. Explanations of the relative errors are also presented.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The work is supported by the National Natural Science Foundation of China under Grant no. 51478198.