Nonlinear Model-Based Subway Station-Level Peak-Hour Ridership Estimation Approach in the Context of Peak Deviation

Abstract

Existing techniques for estimation of subway station-level long-term peak-hour ridership (PHR) may produce underestimated PHR values that may result in stations being designed with insufficient capacity during the planning stage; this in turn may increase congestion on the platforms in actual operation. One of the reasons for this potential undesirable outcome is that peak deviation phenomena often arise between stations and lines in subway systems, which could create underestimated PHR values. The default assumption has always been that the peak hour of passenger flow of each station always overlaps with its attributed line. This paper presents a framework of a station-level long-term PHR estimation method calibrated using the peak deviation coefficient (PDC) and a nonlinear model (eXtreme Gradient Boosting). This approach can estimate the PDC values for PHR prediction, and can also quantify the relative importance of PDC associated factors, yielding an explanation of the main causes of peak deviation phenomena. Using a real-world, large-scale passenger flow dataset from Xi’an, China, the approach produces more stable and accurate predictive performance compared with conventional methods (i.e., absolute percentage error controlled within 20% versus 50%, and mean average percentage error reduced by 3.26%–8.35%). Meanwhile, it is found that the relative importance of the unimproved land use ratio ranks in the top four for all types of peak periods; this ratio is a key factor that may be used to mitigate ridership deviations between stations and line peaks. In addition, for subway networks, the influence of land use entropy increases from the morning peak hour to the evening peak hour and weakens across the route from origin to destination.

Keywords

data and data science machine learning (artificial intelligence)public transportation land use rail transit systems stations subway urban rail forecasting

With the rapid increase in highway vehicular travel, more and more urban rail transit systems are being constructed to alleviate road traffic congestion and mitigate air pollution. Given the growing importance of rail transit systems in shaping the future of sustainable transportation, it is critically important for rail and subway planners and designers to estimate future long-term ridership demand accurately. The estimation results of station-level long-term peak-hour ridership (PHR) form the basis of the capacity design of service facilities in subway station planning. PHR calculations will directly affect the levels of a station’s service after it is put into operation, and it is difficult to enhance the capacity of a station after it has been constructed because of the great cost and physical restrictions of stations. An inaccurate PHR estimation result may cause traffic congestion or resource wastage at service facilities once a station has been put into operation. However, while existing research has achieved satisfying results in forecasting passenger flow in regular time, few studies have focused on the accuracy of station ridership during peak hours ( 1 ).

Currently, station-level PHR is generally estimated based on the default assumption that the peak ridership hour of each station overlaps with the peak ridership hour of the attributed line. This assumption is adopted in conventional regional travel forecasting models, such as the four-step model (2, 3). Nevertheless, relying on this assumption may produce underestimated PHR values, because the assumption is not always precisely satisfied by various types of stations. For instance, Liu et al. ( 3 ) found that more than 10 subway lines in Beijing are heavily overcrowded by passengers both in the trains and on the platforms during the morning peak hours, with a load factor that is greater than 100%. One of the reasons for this could be that underestimated ridership values have resulted in flawed subway system design that has limited the total network capacity. Shen ( 4 ) indicated that the predicted station-level PHRs at some stations are underestimated in practical applications, because the peak passenger flow of these stations may be inconsistent with the morning and evening peak hours of the line overall.

Further revealing the underestimation of long-term station-level PHR values, Gu and Ye ( 5 ) found an approximate half-hour deviation between the station peaks and the line peaks for most stations in Osaka, Japan. They found that this inconsistency also existed in Shanghai, China, with a proportion of 30.36% of stations. Yu et al. ( 6 ) indicated that the proportions of stations with peaking deviations in subway lines 1, 3, and 6 in Chongqing, China were respectively 69.57%, 69.44%, and 70.83%. Similar findings can be found in the cities of Beijing ( 7 ), Nanjing ( 8 ), and Shenzhen (9, 10) in China. It is evident that this peak deviation phenomenon often arises between stations and lines in subway systems. The predicted station PHR values may not actually reflect a given station’s maximum ridership.

However, only a few papers have investigated the magnitude of the deviation of peak ridership at stations as a means of calibrating the station-level PHR estimation results. The methods applied in such studies fall into two categories: statistical techniques and linear-based models. For instance, Yu et al. ( 6 ) defined a peak deviation coefficient (PDC) to quantify the magnitude of station-level ridership deviation between the station peak and the line peak and employed a simple statistical technique to analyze the land use characteristics of each type of station based on the clustering results of PDC values. Similar work can be found in Kuanmin et al. ( 11 ). Nevertheless, the findings of these papers are not sufficient to support the station-level PHR forecasting task or the design of micro-level station facilities, as a quantitative PDC relationship model was not provided. With regard to linear-based models, Yu et al. ( 12 ) applied a local regression-based linear method to analyze the determinants of PHR in urban rail transit stations and achieved good fitting results. However, the application of their model is limited to analysis of ridership at existing stations rather than estimation of ridership for stations yet to be constructed, because the requirements for input parameters are relatively high, and it is difficult to develop a detailed approach for the determination of substantial input parameters adapted to local conditions during the planning stage. More recently, Wei et al. ( 13 ) applied three commonly used feature selection-based linear methods (i.e., the Pearson correlation coefficient, recursive feature elimination, and the least absolute shrinkage and selection operator [LASSO] approach) to determine the critical influencing factors for reducing the data collection burden and constructing corresponding quantitative PDC relationship models. They found that LASSO outperformed the other two methods in estimation accuracy and interpretability. Nevertheless, the methods in the aforementioned studies are based on a prior assumption of linear relationship, which may induce erroneous implications for subway design and planning practices.

Meanwhile, it can be observed that ignorance of several potential significant influencing factors in the construction of the PDC relationship model is another weakness in the existing research. These factors have been proven to be significantly associated with urban rail transit ridership during peak periods. For instance, using subway station ridership data from Seoul, South Korea, Sung and Oh ( 14 ) found that the number of bus lines is positively and significantly associated with morning peak boarding and alighting ridership. Li et al. ( 15 ) indicated that the impact of several bus lines varies for different types of ridership depending on the peak period and boarding/alighting direction. Li et al. ( 16 ) found that road density is positively and significantly related to the morning peak alighting and evening peak boarding ridership based on rail transit ridership data from Guangzhou, China. Zhao et al. ( 17 ) found that road density is positively related to the afternoon peak ridership at the subway station in Nanjing, China. These deficiencies in the existing research suggest a lack of accuracy in the estimation results and thus also in the reliability of corresponding interpretations.

To fill these research gaps, this study proposes a methodological framework for estimation of nonlinear model-based long-term station-level PHR to enhance the precision of the estimation results and the reliability of the interpretations under peak deviation conditions. This study employs an advanced tree-based model, eXtreme Gradient Boosting (XGBoost), to cope with the complex nonlinear relationship between the PDC and associated factors, and it introduces more comprehensive influencing factors into the PDC modeling. The feature relative importance was calculated by XGBoost to explain the main causes of the peak deviation phenomena. An empirical study was conducted using a real-world, large-scale passenger flow dataset and station attribute data from Xi’an, China to validate the effectiveness and applicability of the proposed approach framework. In addition, a model comparison between XGBoost and a linear-based model (LASSO) was conducted to verify the superiority of the nonlinear model in addressing the research questions of the present study.

The main contributions of the present study concern the following aspects:

The subway long-term station-level PHR estimation approach is calibrated with PDC values using a nonlinear model-based methodology framework. The case study results show that the proposed framework improves the stability and precision of the station-level PHR forecasting results.

The peaking deviation phenomena are interpreted from a reliable and comprehensive perspective, with feature relative importance across different peak periods. This provides critical evidence for planners in prioritizing land use strategies and thereby enhancing the operating efficiency of subway lines.

The quantification process of the PDC is rendered more accurate and reasonable by means of the introduction of more thorough influencing factors. This can serve as an important reference coefficient for subway station capacity design.

Methodology

Approach Framework

This study proposes a methodological framework to extend the existing station PHR forecasting models. The proposed framework introduces the concept of PDC—which measures the degree of deviation of station ridership between the periods of station peaks and line peaks—into the conventional station-level PHR forecasting model to calibrate the PHR estimation results. Thus, the PHR of a station can be calibrated on the basis of the PHR estimation results from conventional approaches by multiplying the estimated value of the PDC. The calculation formulation is given by Equation 1.

R_{S_PHR} = R_{L_PHR} \times C_{SL_PDC}

(1)

where $R_{S_PHR}$ is the calibrated station PHR forecasting value; $R_{L_PHR}$ is the station PHR forecasting value from the conventional approach; $C_{SL_PDC}$ is the estimated value of the PDC, which takes the minimum value of one when the peak of the station completely coincides with the peak of the line; and the larger the PDC value, the more significant the station ridership deviation. The developed approach falls back to the traditional approach when the value of the PDC is equal to one.

The estimation model for the PDC (i.e., $C_{SL_PDC}$ ) is constructed by employing a state-of-the-art tree-based nonlinear XGBoost model, which has been successfully used in recent years with high accuracy and good interpretability in nonlinear prediction tasks in transportation fields (18 –20). The framework of the proposed approach is presented in Figure 1. First, the PDC and four categories of explanatory variables were calculated using the historical data from existing stations, and they were set as dependent and independent variables to build the model. Then, XGBoost was employed to model the complex nonlinear relationship of the PDC. The greedy algorithm and grid search methods were adopted to find the best model structure for making predictions. Next, the PDC estimation and interpretation were implemented based on the best model structure trained by XGBoost. The station-level PHR can be obtained using Equation 1.

Data Preprocessing

Dependent Variables

As the dependent variable, the PDC was calculated based on the collected historical data of existing stations. For a subway station, the PDC denotes the ratio of station ridership during the period of station peaks and line peaks, as in Equation 2.

c_{SL_PD C_{i}} = \frac{r_{S_{i}}}{r_{L_{i}}}

(2)

where $i$ is the number of existing stations, $c_{SL_PD C_{i}}$ is the value of the PDC of existing station I, and $r_{S_{i}}$ and $r_{L_{i}}$ are respectively the ridership levels of existing station i during the station peak hour and the line peak hour.

Independent Variables

On the basis of the insights of the present study and the findings of previous research, the variables that may influence the PDC can be classified into four categories: land use, network structure, accessibility, and station type. Apart from these four categories of factors, some other types of predictors—such as population, employment, precise facility type (i.e., number of joint buses with one section, with two sections, and with three or more sections), and fine-scale land use information (first-level residential land use, second-level residential land use, and third-level residential land use)—were also introduced into the station PHR model for analysis purposes. However, considering that the research purpose is estimation of ridership for stations yet to be constructed, rather than ridership analysis of stations that already exist, in the selection of variables the emphasis was placed on the difficulty and availability of data collection during the planning stage, as this type of data is more important for model generalization and practical applications. On the other hand, these other predictors can be replaced and covered by some of the above-mentioned four categories of factors. For instance, population and employment can be replaced by residential land and office land, respectively ( 12 ). Thus, these other predictors have not been taken into account in the modeling input.

Before collecting related station attribution data, the pedestrian catchment areas (PCAs) of rail stations should first be evaluated to determine the range of data collection, including factors such as land use and accessibility. According to previous studies (14, 16, 21), a distance of 800 m is generally regarded as the standard walking distance to delineate the PCAs of subway stations.

Land Use

The proportion of various kinds of land use around stations was deemed an important indicator that influences the distribution characteristics of ridership (3, 22). According to the national urban land use classification standard (GB50137-2011), land use may be classified into nine categories (residence, administration, commerce, primary and middle school, college, recreation, medical, transport hub, and unimproved land) for use in the model. The definition of the proportion of various kinds of land use can be found in Wei et al. ( 13 ).

Land use entropy was chosen as a measure of land use diversity ( 23 ). A large value suggests heterogeneous land use. The formula is given by Equation 3.

E_{i} = - \frac{\sum_{u} p_{i}^{u} \ln (p_{i}^{u})}{\ln (U)}

(3)

where $E_{i}$ is the land use entropy of station i, $p_{i}^{u}$ is the proportion of the u-th category of land use at station i, and U is the total number of land use categories.

Network Structure

The factors of distance from station location to city center $D_{i}$ and the Betweenness centrality $B_{i}$ were selected as network attribute indicators that signify the location attributes of stations and the spatial connectivity to other stations in the subway network (24, 25). The Betweenness centrality of station i is calculated as the number of shortest paths between any two different stations in the network via the station ( 26 ), as shown in Equation 4.

B_{i} = \sum_{o \neq d} \frac{ε_{o, i, d}}{ε_{o, d}}

(4)

where $B_{i}$ is the Betweenness centrality of station i, $ε_{o, i, d}$ is the number of shortest paths from station o to station d via station i, and $ε_{o, d}$ is the number of shortest paths from station o to station d in the subway network.

Accessibility

Feeder buses may further expand a station’s catchment area. Therefore, accessibility variables such as road density $A_{de {n_{road}}_{i}}$ and the number of bus lines $A_{nu {m_{bus}}_{i}}$ (16, 27) around the station were selected as measures of intermodal connection, as they reflect the opportunities that passengers may access to travel to subway stations by means of feeder buses (14 –17).

Station Type

Station type is another important factor that influences temporal distribution of ridership (16, 17). Whether a station transfer and not ${W_{ter}}_{i}$ is chosen, and whether a station terminal and not ${W_{tra}}_{i}$ is chosen, as dummy variables will affect the interpretation of the variation of the PDC.

XGBoost Modeling

XGBoost is an advanced decision-tree-based boosting ensemble method that has been increasingly adopted in transportation research (18 –20). In contrast to the most representative methods in existing PDC estimation literatures (i.e., LASSO), XGBoost is a state-of-the-art nonlinear algorithm that has a more flexible modeling structure with few or no predefined assumptions as to input data; therefore, it can approximate any shape of the nonlinear relationships of the PDC ( 28 ). Compared with single machine learning algorithms, such as the artificial neural network, the Bayesian network, the support vector machine, K nearest neighbors, and multivariate adaptive regression splines, an ensemble model is able to achieve higher prediction performance and computational efficiency, because it combines several machine learning techniques into one surrogate model to reduce variance and deviation (29 –31).

On the other hand, tree-based models have the ability to identify the relative importance of dependent variables to explain the variation of the PDC; this is achieved by the frequency of the variables used in splitting the data across all trees ( 32 ). This offers critical empirical evidence for planners to use as they prioritize land use strategies that are adaptive to the network position of stations and thereby enhance the line operating efficiency of the subway system. Among typical tree-based ensemble techniques such as random forest, gradient boosted decision trees, and XGBoost, XGBoost has demonstrated superior capabilities in prediction accuracy and interpretation power as a result of the application of a gradient boosting framework (33, 34).

Moreover, XGBoost improves model generalization to prevent overfitting and multicollinearity by using a more regularized model formalization ( 34 ). Further, XGBoost requires less effort for data preprocessing and has fewer hyperparameters for tuning ( 32 ). Given the above advantages, XGBoost has been employed to model and predict PDC values based on the aforementioned four categories of explanatory variables.

Tree Generation and PDC Boosting

In XGBoost, the boosting of PDC estimation accuracy is conducted by sequentially adding trees to correct the estimation errors of the previous trees; therefore, the estimation scores of all generated trees sums as the current PDC estimation output. Given a training dataset with n samples (stations), there are dependent variables $c_{SL_PD C_{i}} \in R$ and corresponding independent variables with m features $x_{i} \in R^{m}$ . The process of tree generation and boosting is expressed as Equations 5 and 6, respectively.

{\hat{c}}_{SL_PD C_{i}}^{(k - 1)} = \sum_{k - 1} f_{k - 1} (x_{i})

(5)

{\hat{c}}_{SL_PD C_{i}}^{(k)} = {\hat{c}}_{SL_PD C_{i}}^{(k - 1)} + f_{k} (x_{i}), f_{k} \in F

(6)

where ${\hat{c}}_{SL_PD C_{i}}^{(k - 1)}$ and ${\hat{c}}_{SL_PD C_{i}}^{(k)}$ are the estimation values of the PDC at the k−1 and the k-th iterations (trees), respectively. f is a scoring function used to estimate output in the regression tree algorithm, and $F = {f (x) = w_{q (x)}} (q : R^{m} \to J, w \in R^{J})$ is the space between the regression trees. Here, q represents the structure of each tree that maps a sample to the corresponding leaf index. J is the number of leaves in the tree. Each $f_{k}$ corresponds to independent tree structure q and leaf weight w. Each given sample will be classified into leaves via the decision rules in the tree (given by q). The final estimation is the sum of the score of the corresponding leaves (given by w). We define $I_{j} = {i | q (x_{i}) = j}$ as the instance set of leaf j, and $w_{j}$ as the leaf weight; $f_{k} (x_{i})$ can be expressed as Equation 7.

f_{k} (x_{i}) = \sum_{i \in I_{j}} w_{j}

(7)

Regularized Learning Objective

To control for overfitting and to overcome multicollinearity in the PDC estimation process, XGBoost introduces a regularization term to the objective function as an improvement from a simple gradient boosting algorithm. Thus, the model complexity can be controlled by the shrinkage coefficient of the regularization term. The objective function of XGBoost at iteration k is expressed as Equation 8.

\begin{matrix} Ψ^{(k)} = & \sum_{i = 1}^{n} l (c_{SL_PD C_{i}}, {\hat{c}}_{SL_PD C_{i}}^{(k)}) \\ + \sum_{k} Ω (f_{k}) = \sum_{i = 1}^{n} l \\ (c_{SL_PD C_{i}}, {\hat{c}}_{SL_PD C_{i}}^{(k - 1)} + f_{k} (x_{i})) + \sum_{k} Ω (f_{k}) \end{matrix}

(8)

where $\sum_{i = 1}^{n} l (c_{SL_PD C_{i}}, {\hat{c}}_{SL_PD C_{i}}^{(k)})$ is the training loss function at the k-th iteration and describes the fitting performance of the model with training data. $Ω (f_{k})$ is the additional regularization term, defined as Equation 9.

Ω (f_{k}) = γ J + \frac{1}{2} λ \sum_{j = 0}^{J} w_{j}^{2}

(9)

where $w_{j}$ is the weight of the j-th leaf, and $γ$ and $λ$ are the parameters of the regularization term that are used to control the complexity of each leaf and to scale the penalty, respectively.

Identification of the Best Tree Structure

Identifying the best tree structure in XGBoost is important for achieving the best model estimation results for the PDC. This target is conducted by continuously optimizing the objective function until the reduction of the objective function becomes limited. A second-order Taylor expansion is utilized to approximate the objective function of XGBoost, as shown in Equation 10.

\begin{matrix} Ψ^{(k)} & \approx \sum_{i = 1}^{n} [l (c_{SL_PD C_{i}}, {\hat{c}}_{SL_PD C_{i}}^{(k - 1)}) + g_{i} f_{k} (x_{i}) + \frac{1}{2} h_{i} f_{k} {(x_{i})}^{2}] \\ + \sum_{k} Ω (f_{k}) \end{matrix}

(10)

where $l (c_{SL_PD C_{i}}, {\hat{c}}_{SL_PD C_{i}}^{(k - 1)})$ is the loss function for the previous estimation; and $g_{i} = \partial_{{\hat{c}}_{SL_PD C_{i}}^{(k - 1)}} l (c_{SL_PD C_{i}}, {\hat{c}}_{SL_PD C_{i}}^{(k - 1)})$ and $h_{k} (x_{i}) = \partial_{{\hat{c}}_{SL_PD C_{i}}^{(k - 1)}}^{2} l (c_{SL_PD C_{i}}, {\hat{c}}_{SL_PD C_{i}}^{(k - 1)})$ are the first and second derivatives of the loss function, respectively. We simplify the objective function by removing the constant term $\sum_{i = 1}^{n} l (c_{SL_PD C_{i}}, {\hat{c}}_{SL_PD C_{i}}^{(k - 1)})$ , and replace $f_{k} (x_{i})$ and $Ω (f_{k})$ in Equation 10 with Equations 7 and 9, respectively. The objective function is then written as Equation 11.

Ψ^{(k)} = \sum_{j = o}^{J} [(\sum_{i \in I_{j}} g_{i}) w_{j} + \frac{1}{2} (\sum_{i \in I_{j}} h_{i} + λ) w_{j}^{2}] + γ J

(11)

For a fixed tree structure q(x), the optimal weight $w_{j}^{*}$ for leaf j is computed using the derivative of $Ψ^{(k)}$ with respect to $w_{j}$ , as expressed by Equation 12. Then, the optimal weight $w_{j}^{*}$ for leaf j and the corresponding optimal value of the objective function can be solved as Equations 13 and 14.

\frac{\partial Ψ^{(k)}}{\partial w_{j}} = \sum_{j = 1}^{J} [(\sum_{i \in I_{j}} g_{i}) + (\sum_{i \in I_{j}} h_{i} + λ) w_{j}] = 0

(12)

w_{j}^{*} = - \frac{\sum_{i \in I_{j}} g_{i}}{\sum_{i \in I_{j}} h_{i} + λ}

(13)

Ψ^{(k) *} = - \frac{1}{2} \sum_{j = 1}^{T} \frac{{(\sum_{i \in I_{j}} g_{i})}^{2}}{\sum_{i \in I_{j}} h_{i} + λ} + γ J

(14)

Thus, the best base learner $f_{k}^{*} = w_{q (x)}^{*}$ is achieved. Equation 14 can be regarded as a scoring function used to measure the quality of the entire tree structure q. The trees continue to branch off until they grow to the maximum depth. Equation 15 is used to measure the score of the gain after each split to determine when to stop branching. If the gain is smaller than γ, the tree ceases to grow.

τ = \frac{1}{2} [\frac{{(\sum_{i \in I_{L}} g_{i})}^{2}}{\sum_{i \in I_{L}} h_{i} + λ} + \frac{{(\sum_{i \in I_{R}} g_{i})}^{2}}{\sum_{i \in I_{R}} h_{i} + λ} - \frac{{(\sum_{i \in I} g_{i})}^{2}}{\sum_{i \in I_{R}} h_{i} + λ}] - γ

(15)

where the first term and second term respectively are the score of the left and right nodes after division. The third term is the score of the present node before division; $I_{L}$ and $I_{R}$ represent the instance sets of the left nodes and right nodes after the split, respectively; and $I = {I_L}\mathop \cup ^\ {I_R}$. $γ$ is the complexity cost of dividing the node into leaf nodes. Typically, a greedy search algorithm that starts from the root node and iteratively adds branches to split the feature is employed in XGBoost to find the best tree structure.

PDC Calculation and Interpretation

PDC Calculation

By employing the constructed best tree structure, the PDC value of a station can be estimated based on its predictor variables (i.e., land use, network structure, accessibility, and station type variables). Assuming a total of K trees (estimators) that have been generated to boost the estimation accuracy, and that $f_{k}^{*}$ denotes the estimation scores of the best tree structure in the k-th tree, the final estimation value of the PDC is calculated by the sum of the estimation scores of K trees, expressed as Equation 16.

{\hat{C}}_{SL_PD C_{i}} = \sum_{k = 1}^{K} f_{k}^{*} (x_{i})

(16)

Feature Relative Importance

XGBoost offers an insightful measure of the importance of contributors to the estimated PDC by calculating the feature relative importance. The importance of a feature is measured by the average gain across all splits in which the feature is used, based on the constructed best model structure. We define $e_{t} = {z | v (z) = t}$ as the set of internal nodes for which feature z was selected as the splitting variable. The relative importance of feature z can be measured by averaging the overall base trees, as shown in Equation 17.

Φ_{z} = \sqrt{\frac{1}{K} \sum_{k = 1}^{K} \sum_{t = 0}^{T - 1} \sum_{z \in e_{t}} τ_{t}^{2}}

(17)

where T is the total number of nodes in the tree, and ${\hat{τ}}_{t}^{2}$ is the squared gains created as a result of using feature z as a splitting variable at internal node t.

Case Study

Study Area and Data Collection

Xi’an, the capital of Shaanxi, was the first city in Northwest China to operate an urban rail transit system. As of April 2021, the Xi’an rail transit system had eight operating routes (lines 1–6, 9 and airport intercity) with 153 stations, and a total operating length of 244 km. The passenger flow intensity reached 1.419 million passenger/km, which ranked fourth in China. The level of the scaled rail transit network and the overall ridership performance confirm Xi’an as a good case study. Figure 2 presents the spatial distribution of the stations along these lines.

The data used in this study was collected from various sources. Data for the subway ridership were obtained from the Xi’an Metro AFC system. The provided ridership dataset covers a period of five weekdays from April 19, 2021 (Monday) to April 23, 2021 (Friday), which is considered a fairly regular week in China without any public holidays or summer/winter holidays and thus can represent the general characteristics of station ridership. Because station ridership has different temporal distribution characteristics during the two peaks (morning and evening) and in the three directions (boarding, alighting, and bidirectional), the PDC was refined into six types with distinct time periods: morning boarding/alighting/bidirectional peak periods (AM_B, AM_A, AM_T) and evening boarding/alighting/bidirectional peak periods (PM_B, PM_A, PM_T), which were then set as dependent variables with the sample size of 153*6 in the modeling input. Accordingly, as shown in Equation 1, the weekday ridership at each station was averaged for six peak periods corresponding to two types of peaks (station peak and line peak).

In relation to the data related to dependent variables, the fine land use type is identified by the point of interest data acquired from the Baidu map API in April 2021, and the construction area for each type of land use can be derived with the assistance of a satellite map that combines manual measurements of the street view. Then, the ratios of nine land use types and land use entropy can be calculated by corresponding formulas. The distance from station location to city center was measured from the Baidu map. The Betweenness centrality was computed by Equation 4 with the assistance of Baidu Maps. Accessibility variables, including road density and the number of bus lines, were collected from Baidu Maps and relied on geographical information system spatial analysis tools. The station type could be judged by visuals from the Xi’an rail transit map.

Data Preprocessing Results

The six types of PDC values of the existing 153 stations were computed as shown in Equation 1. Considering the aggregation characteristics of the PDC values, they were divided into five different ranges, as shown at the top of Table 1. The count results show that despite most of the PDC values equaling one, 33% to 60% of stations had PDC values greater than one; 3% of stations had PDC values that exceeded two during morning boarding peak periods. This demonstrates the existence of peak deviation phenomena and proves that the deviation magnitude varies with the different peak periods. Table 2 shows the descriptive statistics of the dependent variables.

Table 1.

Peak Deviation Coefficient Statistic Results of Existing Stations

Peak period	1 (%)	(1, 1.2] (%)	(1.2, 1.5] (%)	(1.5, 2] (%)	[2, +∞) (%)
AM_B	102 (67)	35 (23)	8 (5)	3 (2)	5 (3)
AM_A	61 (40)	85 (56)	6 (4)	1 (1)	0 (0)
AM_T	87 (57)	53 (35)	8 (5)	4 (3)	1 (1)
PM_B	66 (43)	71 (46)	14 (9)	2 (1)	0 (0)
PM_A	66 (43)	67 (44)	20 (13)	0 (0)	0 (0)
PM_T	86 (56)	60 (39)	6 (4)	0 (0)	1 (1)

Table 2.

Descriptive Statistics of Independent Variables

Categories	Independent variables	Notation	Mean	SD	Min.	Max.
Land use	Residential ratio	$p_{i}^{1}$	0.61	0.29	0.02	0.88
	Administrative ratio	$p_{i}^{2}$	0.11	0.21	0.00	0.38
	Commercial ratio	$p_{i}^{3}$	0.05	0.08	0.00	0.50
	Primary and middle school ratio	$p_{i}^{4}$	0.01	0.01	0.00	0.18
	College ratio	$p_{i}^{5}$	0.02	0.06	0.00	0.33
	Recreational ratio	$p_{i}^{6}$	0.04	0.13	0.00	0.28
	Medical ratio	$p_{i}^{7}$	0.01	0.05	0.00	0.41
	Transport hub ratio	$p_{i}^{8}$	0.05	0.12	0.00	0.78
	Unimproved ratio	$p_{i}^{9}$	0.11	0.23	0.00	0.83
	Land use entropy	$E_{i}$	0.34	0.17	0.01	0.71
Network structure	Distance to city center (km)	$D_{i}$	10.65	6.89	0.24	30.60
Network structure	Betweenness centrality	$B_{i}$	0.10	0.08	0.00	0.48
Accessibility	Road density (km/km²)	$A_{den_roa d_{i}}$	3.43	1.51	0.93	9.07
Accessibility	Number of bus lines	$A_{num_bu s_{i}}$	55	36	1	169
Station types	Terminal dummy	${W_{ter}}_{i}$	0.08	0.26	0	1
Station types	Transfer dummy	${W_{tra}}_{i}$	0.10	0.29	0	1

Note: SD = standard deviation; min. = minimum; max. = maximum.

Hyperparameter Setting

Before inputting the case into XGBoost and modeling the relationship between the PDC values and influential factors, seven important hyperparameters of XGBoost need to be optimized to further improve the modeling performance. The number of trees (k) and the maximum tree depth (max_depth) are used to control tree complexity ( 35 ). A small number of iterations may result in under fitting, while many iterations will increase the risk of overfitting and be difficult to interpret. The maximum tree depth represents the maximum depth a tree can grow. An optimal tree depth can ensure that the base learner is simple but captures the important details of the features in the training set. A trade-off exists between these two hyperparameters. Setting a large tree depth can reduce iterations to ensure convergence. The empirical results of Hastie et al. ( 36 ) have revealed that the number of iterations between four and eight perform well; thus, a maximum tree depth of between three and seven is a reasonable option according to the user guide of Scikitlearn.

The learning rate (learning_rate) adjusts the size of the learning steps to make the boosting process more conservative and robust against overfitting ( 36 ). A learning rate that is too small will result in a local optimum and slow calculation, while one that is too large may miss the optimal value and not converge. In high-dimensional problems, the learning rate should be well below 1 and is typically 0.1 or smaller (36, 37).

The subsample (subsample) and column subsample (colsample_bytree) determine the proportion of individuals and features to be evaluated in each regression tree. These two hyperparameters have the capability of preventing overfitting, mitigating multicollinearity, and reducing the training time of the model. A typical value for both hyperparameters is 0.5 ( 36 ), but a more reasonable value is closer to 1 when the relevant proportion of the high-dimensional data is small ( 34 ).

$γ$ and $λ$ are two regularization hyperparameters used to smooth the final learned weights and prevent overfitting (34, 38). Their increments may make the model more conservative, while a level that is too high will lead to greater training risk for the same number of trees (35, 39). A detailed explanation for each hyperparameter can be obtained from the XGBoost documentation ( 40 ).

On the basis of the above empirical suggestions from previous studies and the sample scale of the present study, the range of the model hyperparameters was set as shown in Table 3. The optimal values of the hyperparameters were obtained by a grid search, which identified the best performance parameter by changing each hyperparameter within an acceptable testing range and employing five-fold cross validation (28, 41). In this work, model hyperparameters were tuned on Python 3.8.6 with computing libraries, which included Numpy 1.19.4, Pandas 1.1.4, Scikitlearn 0.23.2, and XGBoost 1.4.2. The optimal XGBoost hyperparameter combinations identified are presented in Table 3.

Table 3.

Optimal Hyperparameter Combinations and Search Range

Parameters	Ranges; interval	AM_B	AM_A	AM_T	PM_B	PM_A	PM_T
k	[5,15]; 1	7	10	6	7	11	11
max_depth	[4,10]; 1	5	4	4	4	7	4
learning_rate	[0.01,0.3]; 0.01	0.07	0.1	0.08	0.09	0.1	0.09
subsample	[0.5,1]; 0.1	0.6	0.5	0.5	0.9	1	1
colsample_bytree	[0.5,1]; 0.1	0.6	0.6	0.6	0.9	1	0.9
$γ$	[0,0.5]; 0.1	0.2	0.3	0.4	0.1	0.3	0.3
$λ$	[0,0.3]; 0.01	0.01	0.16	0.01	0.04	0.03	0.22

PDC Estimation Results Analysis

PDC Estimation Results and Comparison

To examine the effectiveness of the developed nonlinear model for the PDC estimation, a comparison was conducted with LASSO, the most representative and superior method found in existing the PDC estimation literature ( 42 ). Considering that $R^{2}$ measures model accuracy in relation to goodness of fit, and that root mean square error (RMSE) evaluates model stability in relation to absolute estimation error, the performance of the two models was evaluated by calculating the $R^{2}$ and RMSE values on testing samples (25% of the dataset). Figure 3 shows that the proportions of the variation in the PDC values that are explained by the model increased from 71.3% to 81.4% in LASSO to 95.9% to 98.2% in XGBoost. This indicates that XGBoost achieved a better goodness of fit for the PDC. In relation to the RMSE values, XGBoost exhibited a lower estimation error with mean average percentage error (MAPE) values of 0.016 to 0.031 as compared with the MAPE values of 0.04 to 0.114 in LASSO. This evidence confirms that XGBoost performs better for modeling the complex relationships of the PDC values and is more suitable for accomplishing this research.

PDC Interpretation.

Individual Relative Importance Analysis

Figure 4 presents the relative importance of the top 10 independent variables in the form of percentages, explaining 83% to 97.5% of the variation in the PDC values during different peak periods. Figure 4 indicates that the dominant factors in PDC estimation vary across different peak periods. The factors of boarding direction, unimproved land, land use entropy, and Betweenness centrality make a significant contribution in predicting the PDC for the morning and evening peak hour periods, with joint contributions of 38.9% (morning) and 68.3% (evening). The factors of college land, transport hub land, and number of bus lines make important contributions only for the morning peak hour, with an importance level of more than 10%. The terminal station makes an important contribution only for the evening peak hour, with an importance level of more than 6%.

For the alighting direction, Betweenness centrality, number of bus lines, unimproved land, and road density are the most important explanatory variables for the morning peak hour, with a contribution of more than 9%. For the evening peak hour, primary and middle school land, unimproved land, land use entropy, and medical land are the top factors with the highest relative importance. This suggests that station location and access opportunity account significantly for the deviation in the morning alighting ridership peak, and the non-commuting land is the main driver of the deviation in the evening alighting ridership peak.

As for the explanation of the peaking deviation in bidirectional ridership, unimproved land is the common major factor for both the morning and evening peak hours. Distance to the city center is the critical cause of such deviation during the morning peak hour, whereas residential land and administrative land are the main determinants for the evening peak hour; the importance of these factors is more than 10%. One possible reason for this is that the majority of jobs and schools are located in or close to the city center, so passengers accessing suburban stations need more time to commute. Therefore, they may leave earlier or arrive later than the actual morning peak hour of the line. The peaking deviation during the evening peak hour likely results from the different time distribution of commuting trips undertaken by individuals after work.

In general, it is found that the relative importance of the unimproved land ratio ranks in the top four influencing factors for all types of peak periods, and it accounts for more than 6.2% in the variation of the PDC values. This indicates that the proportion of unimproved land plays an important role in predicting the PDC, regardless of the peak periods. This suggests that the development of unimproved land is the key to avoiding the risks posed by the deviation in station ridership peaks and needs to be prudently planned.

In addition, the relative importance of land use entropy exhibits distinct temporal and directional heterogeneity. Land use entropy has a greater effect during evening peak hours than during morning peak hours (33.6% versus 21.9%), and it has a more significant effect on the boarding direction than on the other two directions (35.8% versus 11%, 9.6%). This indicates that for subway networks, the influence of land use entropy gradually strengthens from the morning peak hour to the evening peak hour and weakens across the path from the origin site to the destination side.

Collective Relative Importance Analysis

In relation to the collective contribution (as shown in Table 4) of each category variable, the land use variables rank the highest with a mean importance of 59.20%, followed by the accessibility variables (with a mean importance of 25.37%), and then the network structure variables (with a mean importance of 13.86%). This indicates that the land use variables are the most important factors that explain peak deviations, and this confirms the necessity of introducing accessibility variables (i.e., the number of bus lines and the road density) into the explanatory variables of the PDC.

Table 4.

Collective Relative Importance of Four Category Variables

Variables category	AM_B	AM_A	AM_T	PM_B	PM_A	PM_T	Average
Land use (%)	78.91	37.12	54.33	29.80	70.85	84.20	59.20
Network structure (%)	10.12	46.45	12.28	6.11	3.94	4.23	13.86
Accessibility (%)	10.93	13.69	29.12	63.30	24.04	11.14	25.37
Station types (%)	0.04	2.75	4.27	0.79	1.17	0.43	1.57

With respect to each peak period, the land use variables are the most important variables in the four peak periods. They have a larger effect on the boarding ridership than on the alighting ridership for the morning peak hour (78.91% versus 37.12%), while this is reversed for the evening peak hour (29.8% versus 70.85%).

Network structural variables have the greatest power in accounting for peak deviations during the morning alighting peak period with a mean importance of 46.45%; this indicates that the fluctuation of the peak deviations may vary with the station’s spatial attributes and may exhibit strong spatial heterogeneity during this period. One possible reason for this is that the destination sites of the morning peak period trips are mainly offices or schools, which are generally unevenly distributed in the network structure and more concentrated in city or sub-city centers.

Accessibility variables have a dominant explanatory power in relation to the peaking deviation during the evening boarding peak period, with an importance of 63.30%. This indicates that evening trips have more flexible times, and passengers prefer to board at stations with convenient access facilities.

Analysis of PHR Estimation Results

Based on the PDC estimation values produced by XGBoost, the station PHR values were calculated via Equation 1. To examine the effectiveness and accuracy of the proposed approach over the conventional approach, a comparison of the three PHR values (i.e., the results of the two approaches and the actual values) was conducted on the peaking deviation stations. Considering the absolute percentage error (APE), measuring the relative estimation error is not easily affected by extreme values and is easy to understand; therefore, APE was adopted to evaluate the model reliability and intuitively interpret the method error at the station level. Figure 5 visualizes the intuitive comparison of the three values, and Figure 6 presents the quantitative comparison using the APE values.

The comparison results reveal that the APE values in the conventional approach appear to be more dispersed and to have an unstable distribution across all peak periods than those produced by the proposed approach. Specifically, the APE values generated by the proposed approach are controlled within 20% across all peak periods; however, the APE values generated by the conventional approach surpassed 20% for some stations during the five types of peak period. Several stations have APE values calculated by the conventional approach that reach as high as 50% during the morning boarding peak period. This confirms that the proposed approach has a more stable and reliable estimation performance than the conventional approach.

On the other hand, the MAPE during the six peak periods was reduced from 5.34% to 14.73% in the conventional approach to 2.08% to 6.38% in the proposed approach. MAPE decreases of 3.26% to 8.35% were achieved by the proposed approach as compared with the results of the traditional approach. These results demonstrate that the proposed XGBoost-based approach has wider applicability and a higher estimation precision for station-level PHR forecasting than the traditional method.

Key Findings and Discussion

The result of comparison analysis shows that the nonlinear model can significantly enhance the fitting and predictive accuracy of the PDC in all time periods considered. This confirms the existence of nonlinearity in the PDC. This implies that the determinants would not have a significant influence on the PDC unless a certain level is reached, and the effect may become saturated when the amount exceeds a certain range.

Some interesting and meaningful findings were generated from the feature relative importance of the estimated PDC. First, the different influencing mechanisms of the PDC in distinct time periods suggest that station capacity determinations should be considered based on the refined PDC analysis, rather than simply in relation to the determinants of a single type of PDC.

Second, in relation to the land use variables, the relative importance of the unimproved land ratio ranks in the top four influencing factors for all types of peak periods. Planners should pay more attention to the allocation of improved land, which is crucial for mitigating the deviation of the two peaks to improve the line operating efficiency in the subway. Land use entropy, the other significant land use variable in explaining the variation of the PDC, has a more significant effect on the boarding direction than on the alighting direction and a greater effect during evening peak hours than during morning peak hours. This suggests that land use diversity plays an important role in pushing go-to-work trips from residential land during the morning peak period and go-to-home trips from administrative land during the evening peak period, and it has more pushing power during the evening peak hours. Primary and middle school land and college land, both of which are attributed to the category of educational land, exhibit differentiated influencing mechanisms of temporal and directional PDC. This evidence is consistent with the findings of Yu et al. ( 6 ). Planning for station-area land use and determining station capacity should separately consider the effect of primary and middle school land and college land.

Third, both number of bus lines and road density provide the main explanation for the variation of the PDC in four peak periods. This suggests that the coordination between bus and subway should be given enough attention in the estimation of station PHR and the determination of subway station capacity. On the other hand, although road density is one of the major factors affecting peak deviation, reasonable ratios of different road hierarchies around stations should be carefully kept in mind, as road supply also stimulates car ownership and usage ( 43 ).

Last, as for the collective contribution of each category variable, network structure variables exhibit the greatest power in accounting for peak deviations in the morning alighting direction. This implies the number of job opportunities is unevenly distributed in the network structure and more concentrated in city or sub-city centers. Urban planners should place some industrial zones in suburban areas to provide more employment opportunities. This policy can balance the ridership distribution in the subway network from a transportation planning perspective, while reducing the risk of station peaking deviations caused by long-distance commuting passengers. Accessibility variables were found to have a dominant explanatory power in relation to the peaking deviations in the evening boarding direction. This finding indicates that subway trip activities or trip chains are more complex after work, and some users first travel to conveniently accessed stations with entertainment venues and shopping centers before going home. Potential implications of this finding include estimating the station-level PHR by considering accessibility-related features and emphasizing the importance of intermodal connections that affect the temporal distribution of ridership in subway stations.

Conclusion and Future Research

To improve the reliability and precision of the station-level long-term PHR estimation results, this study introduces the PDC to validate and calibrate the PHR estimation results obtained from the conventional approach. XGBoost was employed to enhance the estimation precision and interpretation of the PDC. The forecast results show that the proposed approach produced more stable and accurate PHR estimation results across all peak periods as compared with the conventional method (i.e., APE controlled within 20% versus 50%, MAPE reduced by 3.26%–8.35%). This demonstrates that the station PHR forecasting results obtained using the proposed approach are more reliable and precise in accordance with the station design. In addition, a linear-based method, LASSO, was implemented to conduct a PDC estimation comparison with XGBoost. The comparison results indicated that the nonlinear model can enhance the fitting and predictive accuracy. This confirms the existence of nonlinearity in the PDC.

In relation to the main causes of the peaking deviation, the main findings are as follows. (i) The dominant factors in PDC estimation vary across different peak periods and ridership directions, whereas unimproved land is the most important cause of station-level ridership peaking deviation across all peak periods (i.e., more than 6.2%). This suggests that subway planners should pay attention to the planning of improved land. (ii) For subway networks, the influence of land use entropy gradually increases from the morning peak hour to the evening peak hour and weakens across the path from origin to destination. (iii) The joint contribution of land use variables ranks the highest, followed by accessibility variables and network structure variables. This confirms the dominant role of land use variables in determining the PDC and calculating the necessity of accessibility variables—such as the number of bus lines and road density—as the influencing factors of the calculation of the PDC.

Future research will focus on determining the times of peak ridership at each station to provide quantitative evidence that enhances the operational management in subway systems and prevents subway station congestion.

Footnotes

Acknowledgements

The authors would like to thank Xi’an Rail Transit Group Co., Ltd. for the station ridership data.

Author Contributions

The authors confirm contribution to the paper as follows: study conception and design: J. Wei, Y. Cheng; data collection: J. Wei, K. Chen, M. Wang, C. Ma; analysis and interpretation of results: J. Wei, Y. Cheng; draft manuscript preparation: J. Wei, Y. Cheng, K. Chen, X. Hu. All authors reviewed the results and approved the final version of the manuscript.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Jie Wei

Yanqiu Cheng

Xianbiao Hu

Data Accessibility Statement

Station ridership data used during the study were provided by Xi’an Metro Co., Ltd. Direct requests for these materials may be made to the provider.

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