Which Distance-Decay Function Can Improve the Goodness of Fit of the Metro Station Ridership Regression Model? A Case Study of Beijing

Abstract

Incorporating the distance-decay effects of facility points into the analysis of metro ridership helps generate more precise and actionable strategies for station area renewal. The majority of existing studies, however, calculated the built environment variables based on the same pedestrian catchment areas (PCAs) of metro stations and failed to consider the impact of distance decay from POIs (points of interest) on the accuracy of metro station ridership regression models. The objective of this study is to determine which distance-decay function best improves the fit of the metro ridership regression model and investigate the effect of the built environment on ridership under the optimal distance-decay model. Based on the distribution density of metro stations in Beijing, the research area is divided into three zones with different PCAs. Built environment variables for all metro stations are aggregated according to the PCA scope. Various distance-decay functions are examined to determine how the accuracy of the Multi-scale Geographically Weighted Regression (MGWR) model is affected by built environment variables calculated from POI facilities (Gaussian distance decay, power distance decay, piecewise distance decay). Finally, under optimal distance decay, the MGWR model is used to investigate how the built environment influences metro ridership. The results show the following: (1) The Gaussian distance-decay function improves the goodness of fit of the regression model, resulting in an 11.25% increase in the R² value when compared to the model without a distance-decay function. (2) During morning peak hours, apartment and office density significantly impacts ridership. The proposed research framework is conducive to improving the accuracy of the metro station ridership regression model. Moreover, it facilitates the formulation of targeted strategies for the renewal of the built environment by government managers and planners.

1. Introduction

In recent decades, with the acceleration of urbanization, the ownership of private cars in cities has risen significantly. This has led to serious traffic congestion problems in cities [1,2]. Due to its low pollution and large capacity, rail transport is considered to be one of the most effective forms of transport for relieving traffic congestion [3,4,5]. Therefore, the construction and operation of rail transport have received significant attention from the decision-makers of cities all over the world [5,6]. In China, an increasing number of urban residents choose to travel by rail transit. However, due to the asynchrony between urban development and the planning of metro stations, there has been excessive ridership at some metro stations due to the imbalance between supply and demand. This has forced rail transit operators to impose flow restrictions to enhance safety and comfort [7]. However, this will significantly impact the travelling efficiency of residents. There has been much discussion about how to ensure urban rail transit can meet the commuting needs of city residents [8]. The built environment plays a significant role in urban environments and influences metro ridership [9]. Beijing was the first city in China to operate a metro system, and its network has rapidly expanded over the years, reaching 689 km by the end of 2020—the second longest in the world. However, due to supply–demand imbalances, some stations face severe overcrowding. Therefore, Beijing serves as an excellent case study.

Some scholars have found that the built environment impacts metro ridership [10,11]. Many scholars currently use experience or the Transit-Oriented Development (TOD) theory [12,13] or reference the research results of others [10,14] to establish the metro stations’ pedestrian catchment areas (PCAs) and adopt a unified metro station PCA across the entire city. Existing studies on metro ridership and the built environment have not compared the effects of distance-decay functions. Several global strategies are proposed after analyzing the relationship between the built environment and metro ridership [11,14,15]. Formulating built environment renewal strategies at different scales holds significant guiding importance for subway managers and urban planning scholars.

The purpose of this study is to identify a distance-decay function suitable for analyzing how the built environment affects metro ridership. To achieve this, it focuses on two core research questions: (1) What distance-decay function provides the best fit for regression models on metro ridership? (2) How do built environment variables influence metro ridership under the optimal distance-decay function?

2. Literature Review

As an important component of public transportation, rail transit research has received extensive attention in recent years [3,16,17]. There are five parts to this literature review: (1) PCA determination of metro stations, (2) Distance-decay function in built environment calculation, (3) Explanatory variables for the built environment, (4) Methods for analyzing the built environment’s impact on ridership, and (5) Current gaps and our study.

2.1. PCA Determination of Metro Stations

It is crucial to define the scope of the built environment analysis for metro stations before analyzing its impact on metro ridership [18]. Metro station built environments are typically assessed based on the “maximum” walking distance or the area that is accessible by foot for the majority of users [3,19]. Therefore, the area for environmental analysis around metro stations is usually referred to as pedestrian catchment areas (PCAs). Different PCAs for metro stations will lead to different built environment variable values, so the accuracy of models using different PCAs will vary. Currently, most studies use circular buffer zones centered around metro stations [1,3,8,10,12,20,21,22,23,24] as the PCA. They chose circular buffer zones with radii of 400 m [25], 500 m [20,21], 600 m [12,24,26], 800 m [3,8,10,22], and 1000 m [1,9]. They relied more on pedestrian accessibility [12,22,27], experience [3], and referencing others’ research findings [10] to determine the radius of the circular buffer zone. Studies have already shown that the PCA and service radius of metro stations in different cities vary [28]. Therefore, we cannot simply borrow the PCA of metro stations from other cities. Researchers have also found that metro stations in different city areas have different PCAs [11,29]. By combining the PCAs of metro stations from different zones, Wang et al. could improve the accuracy of the model [11].

2.2. Distance-Decay Function in Built Environment Calculation

The choice of whether to incorporate a distance-decay function in the built environment calculation is crucial after determining the PCA results of metro stations. Currently, distance-decay functions are commonly used in the planning of public service facilities [30], accessibility calculations [31,32,33], ridership forecasting [34], and the study of built environments and residents’ travel behavior [35]. Studies of the built environment’s impact on ridership rarely consider distance-decay functions. However, existing research has confirmed that different distances from a metro station impact residents’ usage habits [29]. Therefore, we think that the distance of points of interest (POI) from the metro station may impact the station-level ridership. Currently, some scholars use survey data to obtain approximate distance-decay functions [36,37], but this method can lead to significant differences due to variations in the population, which may result in serious deviations from the actual situation. Some scholars also directly use Gaussian functions [38], power functions [29,35], piecewise functions [31,33], and exponential functions [39] to simulate the decay of actual distances. However, they often use a distance-decay function directly without selecting a suitable one that is appropriate for the data and research subjects. To our knowledge, the effects of distance-decay functions on regression model accuracy have not been systematically compared in existing studies.

2.3. Explanatory Variables for the Built Environment

Metro ridership is influenced by a variety of factors. Explanatory variables in the built environment in existing research mainly include land use and density [1,3,8,10,12,20,34,40,41,42,43,44,45], socio-economic characteristics [3,12,20,21,22,23,34,40,41,42,43,46], accessibility [1,3,8,20,21,22,27,40,41,42,44,46], and rail transit services (including rail transit service level and quality) [27,41,46]. However, this selection method does not systematically select explanatory variables, which may overlook some key built environment explanatory variables, and is not conducive to cross-comparison between different studies. Consequently, Cervero et al. introduced a “3D” dimension for the built environment that includes density, diversity, and design [47]. Later, Ewing and Cervero proposed the “5D” dimension of the built environment by incorporating destination accessibility and distance to transit [48]. However, population-related explanatory variables are lacking for the “5D” dimension of the built environment. Based on the “5D” dimensions of the built environment, Ewing et al. added demand management and demographics and proposed the “7D” dimensions [49]. The comparison of 3D, 5D, and 7D is shown in

Table 1. The comparison of 3D, 5D, and 7D.

Explanatory Variables for the Built Environment	Dimension	Used by	Case Study
3D	Density, Diversity, and Design	Cervero et al., 1997 [47]	San Francisco Bay Area, USA
5D	Density, Diversity, Design, Destination accessibility, and Distance to transit	Ewing et al., 2001 [48]	California, USA
		Xi et al., 2024 [50]	Xian, China
		Yang et al., 2024 [51]	Kunming, China
		Huang et al., 2024 [52]	Beijing, China and Tokyo, Japan
7D	Density, Diversity, Design, Destination accessibility, Distance to transit, Demand management, and Demographics	Ewing et al., 2010 [49]
		Wang et al., 2023 [53]	Beijing, China

. Currently, based on the “7D” dimension of the built environment, it is possible to select the variables of the built environment more comprehensively. There have already been some scholars who have tried to select explanatory variables for the built environment based on the “7D” dimension [11].

2.4. Methods for Analyzing the Impact of the Built Environment on Ridership

Scholars have employed a wide variety of models to analyze the impact of the built environment on ridership. The Ordinary Least Squares (OLS) model has been used widely to analyze the impact of the built environment on ridership [3,8,20,21,27,34,40,46], but it cannot identify spatial heterogeneity. The Geographically Weighted Regression (GWR) model [10,22,26,41,54,55,56], the Two-Stage Least Squares (2SLS) model [41], and the Structural Equation (SEM) model [20] have been applied to compare with the OLS model. Scholars currently use the GWR model to study the impact of the built environment on metro ridership. However, whenever multiple explanatory variables are considered, the GWR model fits parameters at an average spatial scale, reflecting the spatial variation in all estimates of parameter values. This can cause some bias in the results [57]. In addition, many scholars currently use machine learning models to study the nonlinear relationship between metro ridership and the built environment [9,11,15,16,25,50,51,58,59,60]. However, nonlinear models may have drawbacks such as instability and overfitting. Furthermore, nonlinear models tend to study the overall impact of the built environment on metro ridership rather than the spatial heterogeneity of that impact. The Multi-scale Geographically Weighted Regression (MGWR) models can investigate not only the overall impact of the built environment on metro ridership but also their spatial heterogeneity. Currently, scholars have used the MGWR model to study how the built environment affects metro ridership [42,43,44].

2.5. Current Gaps and Our Study

According to the existing research, there are two shortcomings that need to be addressed. Firstly, metro stations in different city areas should have different PCAs in mega-cities like Beijing. Most existing studies, however, use a unified PCA for metro stations, which may reduce the accuracy of the analysis. Secondly, scholars have not taken into account the distance-decay function when calculating built environment variables in the existing research. Additionally, no scholars have compared the accuracy of the metro station-level ridership regression model with different distance-decay functions.

Therefore, we select 13 explanatory variables based on the “7D” dimensions of the built environment as independent variables and use boarding and deboarding ridership during morning peak hours as the dependent variables. Firstly, Beijing is divided into three zones with varying PCAs based on the average distance between metro stations. Secondly, we compare the impact of varying distance-decay functions on the MGWR model accuracy to determine a suitable distance-decay function for this study. Finally, we explore the built environment’s impact on metro ridership and spatial heterogeneity under the recommended distance-decay function.

3. Methods

3.1. Study Scope

Beijing is China’s political, economic, and cultural capital. As the first metro system in China, Beijing’s subway system has grown rapidly over the years. By the end of 2020, the Beijing subway mileage reached 689 km, ranking second in the world. However, some metro stations in Beijing experience excessive ridership due to imbalances between supply and demand. Furthermore, studying Beijing’s built environment can provide some insights into the operation of subways in other Chinese cities and developing countries.

The object of this study is the 292 metro stations that were put into use on 19 lines in Beijing in 2020. This study used ridership data from Beijing’s public transit IC cards. During the five working days of the week from 12 October 2020 to 16 October 2020, we collected the average hourly inbound and outbound ridership for Beijing’s metro lines. An analysis of the boarding and deboarding ridership trends by hour (Figure 1) determined that the morning peak hours in Beijing are from 7:00 to 9:00, while the evening peak hours are from 17:00 to 19:00. This study only examined the relationship between ridership and the built environment during morning peak hours due to the more prominent contradictions during these hours. Figure 2 illustrates the spatial distribution of boarding ridership at metro stations (hereafter referred to as boarding ridership) and deboarding ridership at metro stations (hereafter referred to as deboarding ridership) during the morning peak hours. In this study, metro station ridership serves as an indicator of vitality. Using the natural breakpoint method, we categorized ridership into 5 levels in Figure 2, with Level 1 reflecting lower vitality. The stations with low vitality are highlighted in green.

3.2. Research Framework

Multi-source big data are used in this study to investigate what impact different distance-decay functions have on the model. This study aims to determine a suitable distance-decay function for this study and explore the influence of the built environment on the metro ridership and spatial heterogeneity under the optimal distance-decay function. Figure 3 shows the overall framework for this study. First, based on the distribution pattern of Beijing’s metro stations, Beijing is divided into three zones with different PCAs according to the service ranges of the stations. Secondly, the built environment dataset is constructed by considering three types of distance-decay functions and disregarding the distance-decay function. Thirdly, using boarding and deboarding ridership as the dependent variable and the four constructed datasets as independent variables, we construct four MGWR models. In this study, the most suitable MGWR model is determined by comparing the results of the four MGWR models. Finally, this study examines the impact of the built environment on metro ridership and spatial heterogeneity using the applicable MGWR model.

3.3. The Delineation of the PCA at the Metro Station

Various PCAs influence the results, and metro stations in different cities exhibit distinct PCAs [28]. It is therefore necessary to determine the PCA of metro stations before analyzing the impact of the built environment on ridership. Due to the rapid development of cities, the scale of new districts is often large, and the service range of metro stations is also extensive. It is therefore unreasonable to use a unified PCA for metro stations in the city. According to our analysis, the stations inside the 3rd Ring Road are relatively concentrated. The stations between the 3rd Ring Road and 5th Ring Road are primarily distributed parallel to the ring road, while the stations outside the 5th Ring Road are distributed outwards. Based on their spatial distribution, we categorized Beijing’s metro stations into three zones: those inside the 3rd Ring Road (green shaded area), those between the 3rd and 5th Ring Road (white shaded area), and those outside the 5th Ring Road (gray shaded area) (Figure 4). After the calculation, the average distance between metro stations inside the 3rd Ring Road was found to be 900 m. The average distance between the 3rd and 5th Ring Road was found to be 1000 m, and the average distance outside the 5th Ring Road was found to be 2000 m. We chose the average distances of three areas as the threshold for the decay function. In addition, we used two-thirds of the average distance as the radius to create circular buffer zones for metro stations, known as PCA.

3.4. Construction and Preprocessing of Built Environment Explanatory Variables

3.4.1. Built Environment Explanatory Variables and Data Source

Density, Diversity, Design, Destination accessibility, Distance to transit, Demand management, and Demographics are the “7D” dimensions of the built environment [49]. The “7D” dimension of the built environment offers a comprehensive structure that integrates physical, social, and policy-related dimensions of the built environment relevant to metro ridership. A dataset of the built environment was constructed based on the “7D” dimension of the built environment, which includes 13 explanatory variables (

Table 2. Explanatory variables and data source.

Built Environment Category	Interfering Factor	Descriptive Statistics (Without Considering the Decay Function)			Data Source	Unit
		Max	Med	Min
Density	Density of commercial facilities	639.41	118.25	0.81	https://lbs.amap.com, accessed on 10 October 2021	quantity/km2
	Density of apartment facilities	133.54	22.66	0.65
	Density of public service facilities	102.59	18.38	0.16
	Density of office facilities	990.85	112.64	3.09
	Floor area ratio	3.72	1.23	0	https://map.baidu.com, accessed on 10 October 2021
	Building density	0.38	0.20	0	https://map.baidu.com, accessed on 10 October 2021	m2/km2
Diversity	Mixed utilization of land use	1.25	1.09	0.42	https://lbs.amap.com, accessed on 10 October 2021
Design	Road density	14.36	6.10	0.26	https://map.baidu.com, accessed on 10 October 2021	km/km2
Destination accessibility	Number of entrances and exits	11	4	1	https://www.bjsubway.com, accessed on 13 October 2020	quantity
Distance to transit	Density of bus lines	127.69	33.14	0	https://map.baidu.com, accessed on 10 October 2021	km/km2
	Density of bus stops	14.46	4.55	0.32	https://lbs.amap.com, accessed on 10 October 2021	quantity/km2
Demand management	Density of parking lots	153.00	0.16	36.32	https://lbs.amap.com, accessed on 10 October 2021	quantity/km2
Demographics	Population density	16793.43	12434.21	1072.02	https://hub.worldpop.org, accessed on 10 October 2020	persons/km2

). Data on the road network, building area, and bus lines were sourced from OpenStreetMap (https://map.baidu.com, accessed on 10 October 2021). Data for POI, parking lots, and bus stops were obtained from the AmapAPI (https://lbs.amap.com, accessed on 10 October 2021). The Beijing Subway’s official website (https://www.bjsubway.com, accessed on 13 October 2020) tracks entry and exit points at metro stations. The population data come from Worldpop (https://hub.worldpop.org, accessed on 10 October 2020).

3.4.2. Distance-Decay Function

Based on the actual research context and the characteristics of different decay functions, we selected the Gaussian, power, and piecewise distance-decay functions for analysis, as they represent distinct types of spatial attenuation: gradual, rapid, and threshold-based, respectively. In this study, the densities of commercial facilities, apartment facilities, public service facilities, office facilities, bus stops, and parking lots are considered with distance-decay functions. The calculation formula for the facility point density considering distance decay is

$$D_i={\displaystyle \frac{{\displaystyle \sum _{j=1}^n{F}_j}}{S_i}}$$

(1)

where D_i represents the density of facilities at the i-th metro station, and F_j represents the weighted distance-decay function of the j-th facility point. The distance-decay functions considered here include the Gaussian decay function, power decay function, and piecewise decay function. S_i represents the PCA area of the i-th metro station.

At first, the introduction of the Gaussian decay function [61], power decay function [32,62], and piecewise decay function [31,63] was to address the problem of calculating reachability. The formulas for the Gaussian decay function, power decay function, and piecewise decay function in this study are shown in Equations (2), (3), and (4), respectively.

$$f\left(d_{ij}\right)=\left\{\begin{array}{l}{\displaystyle \frac{e^{-(\frac{1}{2}){(\frac{d_{ij}}{t_0})}^2}-e^{-(\frac{1}{2})}}{1-e^{-(\frac{1}{2})}}},if{d}_{ij}\le d_0\\ 0,if{d}_{ij}>d_0\end{array}\right.$$

(2)

$$f(d_{ij})=\left\{\begin{array}{l}{d}_{ij}^{-\beta },if{d}_{ij}\le d_0\\ 0,if{d}_{ij}>d_0\end{array}\right.$$

(3)

$$f(d_{ij})=\left\{\begin{array}{l}1,if{d}_{ij}\le {\displaystyle \frac{1}{3}}{d}_0\\ 0.66,if{\displaystyle \frac{1}{3}}{d}_0<d_{ij}\le {\displaystyle \frac{2}{3}}{d}_0\\ 0.33,if{\displaystyle \frac{2}{3}}{d}_0<d_{ij}\le d_0\\ 0,if{d}_{ij}>d_0\end{array}\right.$$

(4)

where f(d_ij) represents the weighted value of the facility point. d₀ is the threshold, which is defined as 900 m, 1000 m, and 2000 m in the metro stations inside the 3rd Ring Road, between the 3rd and 5th Ring Road, and outside the 5th Ring Road, respectively. β is the distance attenuation parameter, and the values of β in the existing studies range from 0.9 to 2.29. Since no one has applied power functions to analyze the built environment’s impact on metro ridership in current research, we conducted experiments to determine the value of the distance attenuation parameter. The results showed that in this study, a value of 1 for the distance attenuation parameter resulted in higher model accuracy. Therefore, a distance attenuation parameter of 1 was chosen for this study.

3.4.3. Data Processing and Variable Calculation

Based on the built environment explanatory variables constructed in Section 3.4.1, this study uses four data types: point, line, polygon, and other data. Point data include metro stations and POI facilities (office, public service, residential, and commercial) from a map of Beijing from 2021, as well as data on parking lots, bus stops, and metro stations. Line data include road network and bus route data, downloaded from OSM and processed with QGIS, including road classifications, names, lengths, and bus route details. Polygon data consist of building footprint data, including attributes like floor area, property type, and height, crawled from the Baidu Map API. Other data include population data from WorldPop and metro station entrances from the Beijing Metro website. The calculation methods for each variable are shown in

Table 3. Calculation method for explanatory variables.

Built Environment Category	Interfering Factor	Calculation Method
Density	Density of commercial facilities	$D_{i,k}={\displaystyle \frac{N_{i,k}}{S_i}}$ Di,k is the density of POI facilities in the k-th category at Metro Station i, Ni,k denotes the number of the k-th category of facility points within the PCA of Metro Station i, and Si is the area of the PCA for Metro Station i.
	Density of apartment facilities
	Density of public service facilities
	Density of office facilities
	Floor area ratio	$R_i={\displaystyle \frac{A_i}{S_i}}$ Ri is the floor area ratio of the i-th metro station, Ai represents the total above-ground building area within the PCA of Metro Station i, and Si represents the PCA area of Metro Station i.
	Building density	$B_i={\displaystyle \frac{F_i}{S_i}}$ Bi represents the building density of the i-th metro station, Fi is the total building footprint area within the PCA of Metro Station i, and Si represents the PCA area of Metro Station i.
Diversity	Mixed utilization of land use	The calculation formula using the Shannon–Wiener index is as follows: $Di{v}_i={\displaystyle \frac{-{\displaystyle \sum _{i=1}^n\left(K_j\right)\ln }\left(K_j\right)}{\ln \left(P\right)}}$ Kj represents the ratio of the number of POI facilities of a certain type within the PCA of Metro Station i to the total number of POI facilities in that PCA. P denotes the total number of POI facilities within the PCA, and Divi represents the degree of facility diversity within the PCA of Metro Station i.
Design	Road density	$R_i={\displaystyle \frac{L_i}{S_i}}$ Ri is the road density of the i-th metro station, Li represents the total road length within the PCA of Metro Station i, and Si represents the PCA area of Metro Station i.
Destination accessibility	Number of entrances and exits
Distance to transit	Density of bus lines	$B_i={\displaystyle \frac{G_i}{S_i}}$ Bi represents the bus line density of the i-th metro station. Gi represents the total bus route length within the PCA of Metro Station i. Si represents the PCA area of Metro Station i.
	Density of bus stops	$B{S}_i={\displaystyle \frac{Q_i}{S_i}}$ BSi represents the bus stop density of the i-th metro station. Qi represents the total number of bus stops within the PCA of Metro Station i. Si represents the PCA area of Metro Station i.
Demand management	Density of parking lots	$P_i={\displaystyle \frac{T_i}{S_i}}$ Pi represents the parking lot density of the i-th metro station. Ti represents the total number of parking lots within the PCA of Metro Station I. Si represents the PCA area of Metro Station i.
Demographics	Population density	$Po{p}_i={\displaystyle \frac{P{o}_i}{S_i}}$ Popi represents the population density of the i-th metro station. Poi represents the total population within the PCA of Metro Station i. Si represents the PCA area of Metro Station i.

.

3.5. Analysis Methods for the Impact of the Built Environment on Ridership

3.5.1. Multicollinearity Test

The variance influence factor can reflect the severity of multicollinearity [64]. Its calculation formula is as follows:

$$VIF={\displaystyle \frac{1}{1-R^2}}$$

(5)

In the equation, R² represents the coefficient of determination. When the VIF is greater than 10, it is considered evidence of multicollinearity among the explanatory variables, and the variable should be removed.

3.5.2. Multi-Scale Geographically Weighted Regression (MGWR)

The MGWR model was used to analyze the impact of the built environment on metro ridership in this study. The MGWR model not only accurately captures spatial heterogeneity but also determines the optimal bandwidth to quantify scale variations among study factors. The dependent and independent variables were standardized (mean = 0, standard deviation = 1) to ensure comparability. The resulting regression coefficients thus indicate the relative influence of each predictor. Specifically, a one-standard-deviation increase in a standardized independent variable corresponds to an average change in the dependent variable equal to the local regression coefficient. The formula for the MGWR model calculation after standardizing the variables is

$$y_j^{*}={\beta }_{bw0}\left(u_j,v_j\right)+{{\displaystyle \sum }}_{h=1}^n{\beta }_{bwh}\left(u_i,v_i\right)\times x_{jh}^{*}+{\epsilon }_j$$

(6)

$$y_i^{*}={\displaystyle \frac{y_i-\overline{y}}{s_y}}$$

(7)

$$x_{jh}^{*}={\displaystyle \frac{x_{jh}-\overline{x_h}}{s_{xh}}}$$

(8)

$y_j^{*}$ represents the standardized metro ridership at Station j. The geographical location of Station j is denoted by coordinates (u_j,v_j). The term β_bw0 refers to the intercept of the local regression model specific to this station. Each β_bwh denotes the station-specific coefficient for the h-th built environment variable, with its associated optimal bandwidth given by b_wh. The standardized value for the h-th built environment variable at Station j is denoted as $x_{jh}^{*}$, while ${\epsilon }_j$ captures the random error term in the model for Station j. n is the number of factors affecting the built environment. The mean and standard deviation of metro ridership across all stations are represented by $\overline{y}$ and s_y, respectively. For any given built environment variable h, x_jh represents its raw value at Station j, with $\overline{x_h}$ and s_xh indicating the mean and standard deviation of that variable across all stations.

4. Results

4.1. Effect of Different Distance-Decay Functions on Model Accuracy

This study constructed four MGWR models, namely MGWR_1 (without considering the decay function), MGWR_2 (considering the Gaussian decay function), MGWR_3 (considering the power decay function), and MGWR_4 (considering the segmented decay function). The collinearity of the explanatory variables should be determined before modeling. The collinearity results are shown in

Table 4. The multicollinearity test results for the explanatory variables.

Explanatory Variables	MGWR_1	MGWR_2	MGWR_3	MGWR_4
Density of commercial facilities	5.72	6.31	5.74	6.65
Density of apartment facilities	9.05	6.45	5.20	7.51
Density of public service facilities	4.31	5.30	3.92	5.55
Density of office facilities	7.66	8.80	7.24	9.33
Floor area ratio	5.50	7.75	7.32	7.98
Building density	3.97	4.94	4.59	4.96
Mixed utilization of land	1.85	1.56	1.45	1.81
Road density	2.42	2.66	2.56	2.67
Number of entrances and exits	1.25	1.25	1.25	1.25
Density of bus lines	1.87	1.97	1.87	1.97
Density of bus stops	2.04	2.60	1.62	2.47
Density of parking lots	6.03	8.18	3.40	8.22
Population density	4.67	6.34	5.52	6.47

. According to the results, all variables have VIFs less than 10, indicating no evidence of multicollinearity. Therefore, we do not need to exclude any variables when modelling. We used the coefficient of determination (R²) to represent the goodness of fit of the models, and the R² results of each model are shown in

Table 5. The R² for the four MGWR models.

	MGWR_1	MGWR_2	MGWR_3	MGWR_4
Boarding ridership	0.562	0.623 (+11.25%)	0.623 (+11.25%)	0.622 (+11.07%)
Deboarding ridership	0.694	0.716 (+3.77%)	0.682 (−1.16%)	0.713 (+0.33%)

Note: (+11.25%) indicates the R² of the MGWR_2 model increased by 11.25% compared to the R² of the MGWR_1 model.

. As seen from Table 5, for boarding ridership, the accuracy of the model considering the distance-decay function is greater than that of the model without considering it. The R² of both the Gaussian decay function model and the power function decay model is 0.62. The accuracy of the model is improved by 11.25%, while the accuracy of the segmented decay model is improved by 11.07%. For the deboarding ridership, the accuracy of the power decay function model decreased by 1.16% compared to the accuracy of the model without the decay function, while the accuracy of the Gaussian decay function model and the segmented decay function model improved by 3.77% and 3.33%, respectively. Based on the findings, using the Gaussian decay function is most suitable for modelling metro ridership in Beijing. This conclusion offers useful insights for further research into metro ridership modelling and urban planning strategies.

4.2. The Overall Impact of Built Environment Explanatory Variables

Table 6. Coefficient statistics of explanatory variables.

Built Environment Category	Interfering Factor	Boarding Ridership			Deboarding Ridership
		Per (%)	+ (%)	− (%)	Pre (%)	+ (%)	− (%)
Density	Density of commercial facilities	0	0	0	0	0	0
	Density of apartment facilities	90.75% *	100%	0	100% *	100%	0
	Density of public service facilities	100% *	100%	0	0	0	0
	Density of office facilities	0	0	0	100% *	100%	0
	Floor area ratio	0	50.56%	49.44%	100% *	100%	0
	Building density	0	0	0	0	0	0
Density	Mixed utilization of land	69.52% *	100%	0	16.10%	0	100%
Design	Road density	0.00%	0	0	100% *	100%	0
Destination accessibility	Number of entrances and exits	39.04%	100%	0	100% *	100%	0
Distance to transit	Density of bus lines	0	0	0	18.15%	100%	0
	Density of bus stops	71.58% *	96.65%	3.35%	100% *	0	100%
Demand management	Density of parking lots	15.07%	0	0	16.44%	0	100%
Demographics	Population density	14.04%	17.07%	82.93%	100% *	0	100%

Note: “Per” indicates the percentage of metro stations significantly affected; “+” indicates the percentage of metro stations with a positive impact among the stations with significant impact. “−” indicates the percentage of metro stations with a negative impact among the stations with a significant impact. “*” indicates that the variable is a significant influence variable.

is a summary statistical table of the coefficients of the MGWR_2 model, including the proportion of metro stations with significant influence at a 99% confidence level, the proportion of metro stations with a positive coefficient of significant influence, and the proportion of metro stations with a negative coefficient of significant influence. The following can be seen from the table: (1) For boarding ridership, more than half of the metro stations are significantly influenced by factors such as residential facility density, public service facility density, floor area ratio, land-use mix, and bus stop density, indicating strong correlations between these variables and metro usage levels. (2) Deboarding ridership is significantly associated with several factors, including apartment facility densities, office facility densities, road network density, the number of station entrances/exits, bus stop density, and population density. (3) Among the significant influencing factors of boarding ridership, the density of bus stops has both positive and negative correlations with metro ridership, but other variables have positive or negative correlations with ridership. Among the significant influencing factors of deboarding ridership, each variable has a positive or negative correlation with ridership. Comparing boarding and deboarding ridership, we find that not only are the explanatory variables of significant influence inconsistent, but the positive and negative effects of their variables on ridership are also inconsistent. This proves that it is necessary to analyze boarding ridership and deboarding ridership separately in the morning peak hour.

To better explore the overall impact of explanatory variables with significant influence on ridership, we calculated the average values of the positive and negative coefficients of those explanatory variables, as shown in Figure 5 and Figure 6. The following can be seen from the figure: (1) For boarding ridership, apartment facility density exhibits the strongest positive correlation with ridership, with mixed utilization of land as the second most influential factor (Figure 5a). It is suggested that increasing these two explanatory variables may lead to an increase in ridership. The only negative correlation is with the density of bus stops (Figure 5b). (2) For deboarding ridership, the density of office facilities and the floor area ratio emerge as the two most influential variables exhibiting positive correlations, as shown in Figure 6a. Moreover, the average coefficients of these two variables are substantially higher than those of the other explanatory variables, indicating their relatively stronger impact on deboarding ridership. In the negative correlation, the average coefficient of population density is the largest (Figure 6b). (3) A comparative analysis of the coefficient estimates for boarding and deboarding ridership during the morning peak hours reveals that apartment and office facility densities are the most influential factors, suggesting that metro ridership management can be effectively informed by adjustments in these two aspects. These findings enable more targeted interventions in metro station areas, allowing for better adjustment of ridership patterns and ultimately improving the overall efficiency of the metro system.

4.3. Spatial Heterogeneity of the Impact of the Built Environment on Metro Ridership

Based on the results of the MGWR model, the coefficients of MGWR are visualized. We selected three variables with significant correlation in boarding ridership and deboarding ridership, respectively, for analysis, while other significant explanatory variables were shown in Figure A1 of Appendix A.

For boarding ridership, the correlation of the density of apartment facilities with ridership is generally positive (Figure 7a). The spatial distribution of the coefficients of the mixed utilization of land is similar to that of apartment facility density (Figure 7b). This also proves that for the overlapping parts of the stations selected according to these two variables, we can consider both variables to improve the vitality of the surrounding stations. The density of bus stops has both positive and negative correlations with ridership and has a strong agglomeration effect. Stations of positive correlation by bus stop density are distributed in the north and west, while stations of negative correlation by bus stop density are distributed in the east (Figure 7c).

For deboarding ridership, the correlation of the density of office facilities with ridership is positive on the whole, and all stations are significant (Figure 7d). The correlation of the floor area ratio on ridership is positive on the whole, and the stations that are greatly affected by the positive floor area ratio are located north of the 5th Ring Road (Figure 7e). The overall correlation of population density with ridership is negative (Figure 7f).

By comparing the results of boarding and deboarding ridership, we find that most of the metro stations that are greatly affected by the explanatory variables are located on the outskirts of the city, while the ridership of metro stations in the center of the city is mostly less affected by the explanatory variables. These findings enable government officials and urban planners to develop more targeted built environment renewal strategies for individual metro stations.

5. Discussion

5.1. Evaluating Distance-Decay Functions in Metro Ridership Modeling

This study compares four models—without attenuation, Gaussian, piecewise, and power decay functions—using R² as the evaluation metric for both boarding and deboarding ridership during morning peak hours. The results indicate that the Gaussian attenuation function yields the highest R², confirming it as the most effective distance-decay function for analyzing the impact of Beijing’s built environment on metro ridership. This aligns with our first research objective and expectations, as the Gaussian function reflects realistic spatial interaction patterns without over-penalizing distant influences.

From a practical perspective, Beijing’s uneven subway development—often lagging behind urban expansion—makes power functions unsuitable due to their rapid decay. In contrast, Gaussian decay preserves influence over wider distances, consistent with findings from Gutiérrez et al. [34]. Our use of a variable PCA (pedestrian catchment area) by city zone (900 m, 1000 m, 2000 m) instead of a uniform threshold improves spatial realism and addresses the limitations of past studies [11,26]. Additionally, we contribute a novel method of threshold determination: rather than relying on fixed distances or complex Voronoi-based delineations [10,14,15], we use average inter-station distances for different zones. This approach supports practical TOD implementation by aligning PCA with functional urban form.

5.2. The Influence of Built Environment on Metro Ridership Under Distance-Decay

This study examines the impact of 13 built environment variables under the Gaussian decay framework. The variables are organized under the 7D framework (Density, Diversity, Design, Destination accessibility, Distance to transit, Demand management, and Demographics). Among them, the density of apartment facilities shows the strongest positive correlation with boarding ridership, while office facility density dominates in explaining deboarding ridership [11,65,66]—reflecting Beijing’s spatial mismatch between housing and employment [11,67]. Interestingly, population density exhibits a negative association with deboarding ridership, diverging from prior studies [11,53]. This may stem from data limitations, as our population data represents the residential, not working, population. Since office districts have fewer residential units, they register lower resident density despite high deboarding ridership.

We also found that the density of apartment facilities, the density of office facilities, the mixed utilization of land, and road density consistently show significance, reinforcing their roles in shaping ridership patterns. These factors indicate that higher land use density and better connectivity drive more frequent metro usage, as people are more likely to rely on public transit when residential, commercial, and transportation options are concentrated and easily accessible. Conversely, bus stop density exhibits a weak and often negative correlation with ridership. This implies that passengers favor non-bus modes (e.g., bikes, e-scooters, or cars) for first-/last-mile connectivity. This is further supported by the small and unstable coefficients associated with bus stop density in the MGWR outputs. The built environment variables show spatial heterogeneity. The MGWR results reveal that suburban stations are more sensitive to built environment characteristics than central ones, suggesting targeted strategies. Moreover, interactions between variables—e.g., between land use mix and road density—may compound effects on ridership, a promising direction for future research.

Finally, while the 7D framework provides a structured lens for analysis, it has limitations. For instance, it underrepresents temporal dynamics and socio-cultural factors such as travel behavior or modal preferences. In our Beijing case, additional “D”s—such as digital connectivity or demographic segmentation—may be necessary to capture emerging urban trends.

5.3. Policy Implications

The causes of ridership are complicated, and the impact of the built environment on metro ridership is also complicated. However, the analysis results of this study may support planning decisions in the following ways.

Firstly, from the overall planning level, the office and residence facility densities have a great impact on ridership. Due to the serious separation of work and residence in Beijing, the distribution of boarding and deboarding ridership is uneven. Existing research has confirmed that rail transit can accelerate the separation of work and housing [68], so policymakers need to consider the overall function of the city. Therefore, we propose to reduce the concentration of offices and residences in a certain part of the city at the level of Beijing’s master plan, particularly to increase employment opportunities in the city periphery and to increase residential areas in the city center as much as possible. In addition, we propose to plan a polycentric structure in Beijing.

Secondly, there is the regional planning level. For boarding ridership, increasing the distribution of bus stops and land use mix around metro stations in the northwest could enhance the vitality of low-vitality stations during urban renewal. For deboarding ridership, increasing road density and residential facilities around metro stations in the north could boost the vitality of low-activity stations. For metro stations in the east, increasing road density and apartment facilities while reducing bus stop distribution could improve the vitality of low-vitality stations.

Finally, from the station level. The station-level coefficients of MGWR enable the development of location-specific built environment optimization strategies. A key finding emerges from stations exhibiting significant coefficients (whereby built environment characteristics demonstrate pronounced effects on ridership): these stations are disproportionately concentrated in suburban zones. This spatial pattern underscores the strategic importance of directing renewal interventions toward urban peripheries.

6. Conclusions

This study employs the MGWR model, incorporating different distance-decay functions to evaluate the influence of the built environment on metro ridership. The results suggest that the Gaussian distance-decay function yields the best fit, providing a valuable reference for improving metro ridership modeling accuracy in Beijing. The MGWR model, which considers the Gaussian distance-decay function, is utilized to explore the influence of the built environment on passenger flow and spatial heterogeneity. Notably, the density of apartment and office facilities significantly influences ridership, with a strong spatial heterogeneity. This approach facilitates the development of targeted urban renewal strategies at global, regional, and local scales. Moreover, the framework is transferable to other cities, offering a replicable method for analyzing transit–land use interactions and informing evidence-based planning.

There are some limitations of this study. The actual catchment areas of metro stations are difficult to define due to diverse access modes like walking, cycling, and ride-hailing. Future research could use mobile data or travel surveys to improve accuracy. The lack of socio-economic data, such as on income and car ownership, also limits the analysis; obtaining such data would enhance model depth. In addition, future work should explore how built environment factors interact and consider time-based variations to better capture ridership patterns.