1 Introduction

It is generally noticed that taxicab imposes positive influences on the travel of residents. However, it is difficult to pick up passengers timely for drivers owing to the greater flexibility of route and time [20]. Experienced taxi drivers can pick up passengers continuously, while those lacking experience cannot find the fastest route, leading to problems including low taxi usage, huge income gap among drivers, aggravation of traffic congestion, and waste of energy in the process of locating passengers. Consequently, it is important to construct an effective model to help taxis quickly find passengers by analyzing historical trajectory data. With the popularity of the global positioning system (GPS), taxis in most cities are equipped with GPS sensors sending real-time geographic information and operating conditions of taxis, which ensures the authenticity and reliability of the data. For this reason, many researchers have paid extensive attention to the analysis, mining, and application of taxi GPS trajectory data (e.g., urban planning [2, 5, 24], urban visualization [3, 36], residents’ travel characteristics [8, 40], and traffic flow forecasting [1, 4, 16, 29, 41, 47].

When it comes to searching passengers, experienced taxi drivers quickly and accurately arrive at the destination rather than locate passengers blindly. That is why they have higher incomes than inexperienced ones. However, inexperienced ones cannot look for passengers timely, which causes high cost per unit time and low profitability. To address the problems mentioned above and achieve accurate prediction, the historical taxi GPS trajectory data are employed to model creation and data analysis, which have significant practical applications. On the one hand, accurate and effective passenger location can not only reduce the time and cost of seeking passengers and maximize benefits but also improve taxi service quality according to the needs of passengers. On the other hand, accurate positioning lays a solid foundation for reducing energy consumption and environmental pollution, providing greater accessibility through more direct traffic routes, and improving urban road networks.

Recently, existing studies on passenger location prediction have mainly focused on: (1) Linear models, such as ARIMA and Kalman filtering (KF) [10, 15, 31]; (2) Nonlinear models, such as CNN, SVM, and LSTM [12, 13, 18, 22, 27, 44]. Most of the aforementioned investigations are implemented in the traditional stand-alone environment. However, dealing with the exponential growth of the taxi trajectory data set can easily cause problems, such as time-consuming data transmission and low computational performance. Therefore, we present a parallel GS-SVM algorithm on the Spark distributed computing platform to predict passenger hotspots in this paper. Specifically, our algorithm employs the GS approach to optimize the C (penalty factor) and γ (core parameter) of RBF, and particularly the memory overflow, memory consumption, and other problems that are compensated in the Spark-based parallel distributed environment. Meanwhile, the speed of data transmission and the computing performance are improved in the proposed algorithm.

The main contributions of this paper are summarized as follows:

  • A gridding method of the road network is proposed on the Spark parallel computing platform to locate the passenger position in a complex transportation network efficiently and accurately.

  • A parallel GS-SVM algorithm on Spark is presented to predict passenger hotspots accurately. This proposed algorithm uses the GS approach to optimize the RBF function of SVM and employs the leave-one-out cross-validation method to verify the algorithm’s global optimal parameters, so that the optimized SVM algorithm has a better generalization ability and higher prediction accuracy than the traditional SVM algorithm in passenger hotspot prediction.

  • With massive taxi GPS trajectory data, the parallel GS-SVM algorithm is applied to predict passenger hotspots successfully. Results demonstrate that the accuracy of GS-SVM is significantly superior to that of ARIMA, SVR, LSTM, and CNN.

The remainder of this article is organized as follows. Section 2 reviews existing works on passenger hotspot prediction. The GS-SVM algorithm and its implementation on Spark are illustrated in Section 3. Section 4 reports the extensive experiments and the result analysis, and Section 5 describes the conclusion and the future work.

2 Related work

In this section, we briefly introduce the related works that predict passenger hotspots and then analyze existing problems.

Finding passengers has always been one of the hot topics in intelligent transportation systems, and it has also attracted mounting researchers in recent years. Li et al. presented an optimized calculation approach based on a watershed algorithm to divide hotspots in cities and to predict the spatiotemporal variation of passenger hotspots based on the improved ARIMA model [15]. Jamil et al. utilized the automatic ARIMA model to conduct time-series analysis and forecast the hotspot areas for passengers with the spatiotemporal taxi data [10]. Qu et al. developed an effective taxi route recommendation approach, adaptive shortest expected cruising route (ASER), to forecast pick-up probability and capacity of each location using KF [31]. Niu et al. put forward a new neural network, L-CNN (combining with CNN and LSTM), and developed a robust model to predict potential passengers in real time [27]. Yang et al. proposed a novel Wave-LSTM model based on LSTM and Wavelet to predict the passenger flow [44]. Kuang et al. conducted feature embedding by attention-based LSTM and captured the correlations between pick-up and drop-off of taxis using 3D ResNet [13]. Li et al. considered the historical order demand, travel time rate, the demand of neighboring regions, day-of-week, time-of-day, weather, and point of interests, and a combined model based on WT-FCBF-LSTM (Wavelet Transform, Fast Correlation-based Filter, and Long Short-term Memory) was proposed to predict the passenger demand in different regions for different time intervals [18]. Kim et al. aimed to systematically design an explainable deep learning model being capable of assessing the quota system and balancing the demand volumes between two modes. A two-stage interpretable machine learning modeling framework was developed by a linear regression (LR) model, coupled with a neural network, and layered by long short-term memory (LSTM) [12]. Luo et al. introduced a multi-task deep learning (MTDL) model to predict short-term taxi demand at a multi-zone level [22]. To promote efficiency while reducing the emission of hybrid electric buses (HEB), Li et al. proposed a novel predictive energy management strategy with passenger prediction and exhaust emission optimization [17]. Mridha et al. proposed a two-step method predicting the taxi pickup hotspots, during various road closure incidents, by utilizing data of past taxi pickup trends [25]. Huang et al. developed a robust model based on hypertext-induced topic search (HITS), which flawlessly described the relationships of hotspots and drivers’ experience [9]. Ou et al. put forward a novel deep learning framework, STP-TrellisNets, which for the first time augmented the newly-emerged temporal convolutional framework (TrellisNet) for spatial-temporal prediction to accurately predict metro station passenger (MSP) flows [28]. Liu et al. put forward a new density peaks clustering (DPC) approach to finding demand hotspots from a low-frequency and low-quality taxi dataset [19]. Based on an improved DBSCAN algorithm, Mu et al. introduced a taxi pickup recommendation system that analyzed taxis in hotspot areas according to distinct factors [26]. Ke et al. developed a new deep learning (DL) method, fusion convolutional LSTM network (FCL-Net), to deal with these three dependencies within an end-to-end learning framework [11]. Hao et al. put forward an end-to-end DL architecture that can make multi-step predictions for all stations in a large-scale underground system [7]. Markou et al. built a multi-task learning component, and the prediction of getting in a taxi and taxi arrivals were regarded as two related events to extract the characteristics of space and time [23].

Furthermore, Zhang et al. adopted a method to obtain primary positive spatial relationships between POIs from DOPs to represent and model spatial dependencies using multivariate point pattern analysis and presented the MKDE method to predict the service range from the perspective of taxi drivers because temperature sometimes plays an essential role [46]. Saadallah et al. proposed an approach named BRIGHT, and it is a supervised learning framework with the function of predicting demand [32]. The goal of BRIGHT is to offer precise predictions for short-term demand via an innovative time series analysis method, which can address distinct types of concept drift. Zhou et al. put forward an ST-Attn model based on a DL method with a spatiotemporal attention mechanism for MsCPDP, which followed the general encoder-decoder structure for serial data modeling, yet adopted a multiple-output policy without RNN units [49]. Yu et al. developed a deep-learning framework combining an altered DBSCAN and a conditional GAN model, and the spatiotemporal and external dependencies were considered [45]. Li et al. focused on the potential passenger demand prediction of last trains from public traffic data (e.g., FCD data from the taxi and GPS/smart data from the bus) [14]. Xu et al. presented a sequence learning model to predict taxi demands based on the latest demand and correlated information [42]. Liu et al. analyzed the correlation among demand, grid probability of passengers in hotspots, and taxi drivers’ different income levels. After analyzing the taxi GPS data and the income level classification of taxi drivers, the pickup spots were filtrated and matched with all these grids [21]. Qin et al. developed two new hybrids mean integrating seasonal-trend decomposition procedures based on loess (STL) with ESN improved by GOA and Adaboost structure presented to predict monthly passenger flow [30].

To sum up, the simple structure of the linear models is one of the advantages. However, the prediction accuracy of these models is relatively low, and poor performance could be perceived when they are used to deal with nonlinear problems. By contrast, the nonlinear model can make up for the shortcoming of the linear model and improve the accuracy of passenger hotspot prediction, but it has some disadvantages, such as complex structure and time-consuming calculation. Meanwhile, the nonlinear model reaches local optimality due to the selection of parameters. It is undoubted that there are still some drawbacks, including high I/O and memory cost, time-consuming data processing, and low computing performance. Moreover, to effectively compensate for the limitations of the traditional single-machine environment in processing large-scale data, such as low effectiveness and poor scalability, several works have migrated the SVM algorithm to the Spark distributed computing platform. For example, Yan et al. put forward a Microblog sentiment classification approach with a parallel Spark-based SVM [43], and Shen et al. adopted the fusion effect of SVM on Spark for speech data mining [35]. Gong et al. developed a parallel one-to-many SVM optimization algorithm on Spark [6]. Wang et al. proposed a parallel Spark-based SVM for intrusion detection [39], and Sai et al. presented a budgeted parallel primal gradient descent Kernel SVM on Spark [33].

To this end, this paper employs a Spark parallel distributed computing platform to remedy the weakness mentioned above effectively and aims to enhance the effectiveness and scalability of prediction by optimizing the SVM algorithm. With historical data, passenger hotspot prediction is mainly used to predict the number of passengers in hotspots, which is a nonlinear problem. It is well known that SVM and deep learning algorithms can effectively cope with nonlinear problems, especially in the field of the visual analysis [34, 38, 48]. However, these methods are rarely applied to predict passenger hotspots. Therefore, this paper utilizes the SVM optimization algorithm to predict passenger hotspots. In particular, parameters of RBF built in the SVM algorithm are not the globally optimal ones. Therefore, we develop the GS approach to optimize parameters of RBF of the SVM algorithm to find the global optimal parameter combinations, which is beneficial for accurately predicting the passenger hotspots.

3 GS-SVM algorithm

In this section, we propose a parallel GS-SVM algorithm to improve the accuracy, effectiveness, and scalability of passenger hotspot prediction on the distributed computing platform with Spark and then implement the parallelization of the GS-SVM algorithm for predicting passenger hotspots under the Spark framework.

3.1 Algorithm overview

As shown in Fig. 1, the passenger hotspot prediction with the parallel GS-SVM algorithm on Spark mainly includes three steps, i.e., data preprocessing, data modeling, and algorithm implementation. In data preprocessing, there are four phases for handling GPS data based on the Spark framework consisting of data extraction, data sorting, grid mapping, and data statistics. Then, in the data modeling, the RBF kernel function parameters of SVM are optimized by the GS approach. Finally, in the algorithm implementation, to improve the efficiency and scalability of passenger hotspot prediction, this work uses RDD under the Spark framework to implement the parallelization of GS-SVM.

Fig. 1
figure 1

The overview of the proposed GS-SVM algorithm

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3.2 Data preprocessing

Taxi GPS information errors occur in some situations, including GPS equipment failure, error operation of taxi drivers, and signal delay. Specifically, the signal is weak when taxis pass through specific places, such as tunnels, leading to the delay in sending messages and the deviation of the taxi’s longitude and latitude. Also, some drivers deliberately set GPS operation status to occupied status to avoid being disturbed during the rest period. To enhance the reliability and accuracy of the algorithm, data preprocessing is needed to eliminate invalid data and incorrect data. The process of data preprocessing is composed of four steps shown in Fig. 2.

Fig. 2
figure 2

Data preprocessing. (a) The flow chart of data preprocessing on Spark, and (b) the flow chart of data on Spark

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  • Step 1: Data extraction.

    Trajectory data in the HDFS file are read, and they are converted into RDD elastic distribution data set on Spark. Then, RDD is divided for filtering out the invalid data, and the required characteristics (taxi ID, operating condition, time, longitude, and latitude) are extracted from GPS trajectories of taxicabs. These fields are sorted by taxi ID (see Algorithm 1).

    figure h

  • Step 2: Data sorting.

    According to the results obtained in the previous step, other taxi IDs with the same operation status can be found. If the operation status of the same taxi ID is 0, it is evident that there is no passenger in the taxi, while 1 represents that the taxi is occupied. When the second operation status of taxi ID is equal to 1, the data characteristics, including ID, operation status, time, longitude, and latitude, are saved and ordered by time (see Algorithm 2).

figure i

  • Step 3: Grid mapping.

    The longitude and latitude ranges are [39.8283918700-39.9909153300] and [116.2611551300-116.4954361600] in this work, respectively. The results first obtained in Step 2 are filtered out in this range. Then, the latitude and longitude of the filtered data are gridded, where the filtered data are divided into 10*10 grids (see Algorithm 3 and Fig. 3).

figure j
Fig. 3
figure 3

Road network grid. (a) 10 × 10 grid, and (b) road network 10 × 10 grid

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  • Step 4: Data statistics.

    The time in the data obtained in Step 3 is one day, and it is divided into data with intervals of fifteen minutes. Next, the number of pick-up hotspots in the same grid at 15-minute intervals is counted (see Algorithm 4).

figure k

3.3 Algorithm design

The core idea of SVM is to construct a suitable hyperplane that is used to classify samples of the same category for addressing the linear inseparability problem. In real-world applications, there are many linearly inseparable problems. To solve them, inseparable data in the low-dimensional space are usually mapped into a high dimensional space, which makes the data linearly separable. This kind of non-linear mapping is mainly realized by kernel function, and the SVM algorithm mainly uses the following four kernel functions:

  1. (1)

    Linear kernel function.

    $$ \begin{array}{@{}rcl@{}} K({X_{i}},{X_{j}}) = {X_{i}}^{T}{X_{j}}, \end{array} $$
    (1)
  2. (2)

    Polynomial kernel function.

    $$ \begin{array}{@{}rcl@{}} K({X_{i}},{X_{j}}) = {(\gamma {X_{i}}^{T}{X_{j}} + r)^{d}}{\text{ }},{\text{ }}\gamma > 0 , \end{array} $$
    (2)
  3. (3)

    Gaussian kernel function (RBF).

    $$ \begin{array}{@{}rcl@{}} K({X_{i}},{X_{j}}) = \exp (- \gamma ||{X_{i}} - {X_{j}}|{|^{2}}),{\text{ }}\gamma > 0 , \end{array} $$
    (3)
  4. (4)

    Sigmoid kernel function.

    $$ \begin{array}{@{}rcl@{}} K({X_{i}},{X_{j}}) = \tanh (\gamma {X_{i}}^{T}{X_{j}} + r), \end{array} $$
    (4)

where γ, r, and d are the kernel parameters. d is used to set the highest degree of the polynomial kernel function, γ is utilized to set the gamma parameter in the kernel function, and r is employed to set the coef0 in the kernel.

When the SVM algorithm is employed for regression prediction, the Gaussian kernel function (RBF) is mainly utilized to project low-dimensional linear data into high-dimensional space. However, SVM is easily affected by the parameters of the built-in radial basis function (RBF), which makes the solution of the equation fall into local optimum. Therefore, this paper uses the grid search (GS) approach to find the optimal parameter combination of the kernel function in SVM, and the optimized SVM algorithm has better generalization ability than the traditional SVM algorithm.

The linear regression function of SVM optimized by the Grid Search approach is given as follows.

$$ \begin{array}{@{}rcl@{}} {f(t)} = {w}{\phi(x)} + {b}, \end{array} $$
(5)

where ϕ(x) is the nonlinear mapping function.

Linear insensitive loss function ε could be defined as follows.

$$ \begin{array}{@{}rcl@{}} L(f(x),y,\varepsilon ) = \left\{ \begin{array}{l} 0 \qquad |y - f(x)| \le \varepsilon \\ |y - f(x)| - \varepsilon \qquad |y - f(x){{| > }}\varepsilon \end{array} \right., \end{array} $$
(6)

where f(x) represents the prediction value returned by the regression function and y denotes the real value.

To obtain w and b values, slack variables ξi and \({\xi _{i}^{*}}\) are introduced as follows.

$$ \begin{array}{@{}rcl@{}} \left\{ \begin{array}{l} \min {\text{ }}\frac{1}{2}{\left\| w \right\|}^{2} + C\sum\limits_{i = 1}^{l} {({\xi_{i}} + {\xi_{i}}^{*})} \\ {\text{s}}{\text{. t}}{\text{.}}\left\{ \begin{array}{l} {y_{i}} - w{\Phi} ({x_{i}}) - b \le \varepsilon + {\xi_{i}}\\ - {y_{i}} + w{\Phi} ({x_{i}}) - b \le \varepsilon + {\xi_{i}} \qquad {\text{ }}i = 1,2, {\cdots} ,l\\ {\xi_{i}} \ge 0,{\xi_{i}}^{*} \ge 0 \end{array} \right. \end{array} \right., \end{array} $$
(7)

where C is the punishment factor. The bigger the value of C, the greater the sample punishment is when training errors are greater than ε. ε stipulates the error requirement of the regression function. The lower the value of ε, the smaller the error of the regression function.

The Lagrangian function is introduced and transformed into a dual solution as follows.

$$ \begin{array}{@{}rcl@{}} \left\{ \begin{array}{l} \max {\text{ }}\left[ \begin{array}{l} - \frac{1}{2}\sum\limits_{i = 1}^{l} {\sum\limits_{j = 1}^{l} {({\alpha_{i}} - {\alpha_{i}}^{*})({\alpha_{i}} - {\alpha_{i}}^{*})K({x_{i}},{x_{j}})} } - \\ \sum\limits_{i = 1}^{l} {({\alpha_{i}} + {\alpha_{i}}^{*})} {\text{ + }}\sum\limits_{i = 1}^{l} {({\alpha_{i}} - {\alpha_{i}}^{*}){y_{i}}} \end{array} \right]\\ {\text{s}}{\text{. t}}{\text{. }}\left\{ \begin{array}{l} \sum\limits_{i = 1}^{l} {({\alpha_{i}} - {\alpha_{i}}^{*})} = 0\\ 0 \le {\alpha_{i}},{\alpha_{i}}^{*} \le C \end{array} \right. \end{array} \right., \end{array} $$
(8)

where K(xi,xj) = ϕ(xi)ϕ(xj) is the RBF kernel function. The SVM algorithm is very susceptible to the influence of the RBF kernel function parameters C (penalty factor) and γ (kernel parameters), and thus we adopt the GS method to find the optimal parameter combination of the RBF kernel function C and γ. The optimal parameter combination obtained by GS is substituted into (8) to obtain the optimal solution of \({\alpha }{\text { = [}}{\alpha _{1}},{\alpha _{2}}, {\cdots } ,{\alpha _{l}}^{\ }{\text {]}}\) as \({\alpha ^{*}} = [\alpha _{1}^{*},\alpha _{2}^{*}, {\cdots } ,\alpha _{l}^{*}]\), and then there are:

$$ \begin{array}{@{}rcl@{}} {w^{*}} = \sum\limits_{i = 1}^{l} {({\alpha_{i}} - {\alpha_{i}}^{*}){\Phi} ({x_{i}})}, \end{array} $$
(9)
$$ \begin{array}{@{}rcl@{}} {b^{*}} &= \frac{1}{{{N_{nsv}}}}\{ \sum\limits_{0 < {\alpha_{i}} < C} {[{y_{i}} - \sum\limits_{{x_{i}} \in SV} {({\alpha_{i}} - {\alpha_{i}}^{*})K({x_{i}},{x_{j}}) - \varepsilon } ]} \\ &{\text{ + }}\sum\limits_{0 < {\alpha_{i}} < C} {[{y_{i}} - \sum\limits_{{x_{j}} \in SV} {({\alpha_{j}} - {\alpha_{j}}^{*})K({x_{i}},{x_{j}}) + \varepsilon } ]} \}, \end{array} $$
(10)

where Nnsv denotes the number of support vectors. We substitute the optimal solution obtained by (9) and (10) into the linear regression function of the SVM algorithm, and the optimized GS-SVM algorithm is achieved as follows.

$$ \begin{array}{@{}rcl@{}} f(x) = {w^{*}}{\Phi} (x) + {b^{*}}&=& \sum\limits_{i = 1}^{l} {({\alpha_{i}} - {\alpha_{i}}^{*}){\Phi} ({x_{i}}){\Phi} (x){\text{ + }}{b^{*}}} \\ &=& \sum\limits_{i = 1}^{l} {({\alpha_{i}} - {\alpha_{i}}^{*})K({x_{i}},x){\text{ + }}{b^{*}}}. \end{array} $$
(11)

3.4 Algorithm implementation

To improve the efficiency and scalability of predicting passenger hotspots, we implement the parallelization of the GS-SVM algorithm under the Spark framework. The core of Spark is RDD, and the GS-SVM algorithm implemented on Spark with RDD is described in Fig. 4 in detail.

Fig. 4
figure 4

The implementation of GS-SVM on Spark

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As illustrated in Fig. 4, the process of implementing GS-SVM on Spark is mainly composed of three steps.

  • Step 1: Optimizing GS Through a large number of experiments, the parameter range of the RBF kernel function is selected to be within C = [100,300,500,700,900] and γ = [0.001,0.003,0.005,0.007,0.009], and the optimal parameter combination is chosen as C = 900 and γ = 0.001. The leave-one-out cross-validation (LOO-CV) method is used to test the superiority of the parameter combination.

  • Step 2: Verifying optimal parameters We obtain the optimal parameter combination of the GS algorithm by Step 1 C = 900 and γ = 0.001. With the application of the RBF function, the GS-SVM algorithm is experimentally verified via data sets with different sizes, and a parallel GS-SVM algorithm based on Spark is obtained.

  • Step 3: Predicting passenger hotspots A parallel Spark-based GS-SVM algorithm is utilized to capture nonlinear information, which predicts the number of passenger hotspots in the same grid within 15 minutes, and it is compared with other algorithms under different data sets. It is concluded that the prediction accuracy of GS-SVM is better than that of other algorithms in this paper (see Algorithm 5).

figure l

4 Experiments

In this section, compared with several state-of-the-art algorithms, the prediction performance of the proposed GS-SVM algorithm for passenger hotspots is validated using a real-world taxi trajectory data set, and then experimental results are analyzed in detail.

4.1 Experimental setup

This case study is based on a Hadoop distributed computing platform built by a Spark parallel processing framework. Furthermore, all experiments are performed on Ubuntu 18.04 OS using Hadoop 3.1.1 with Java, DL4J, and Spark 2.4.3. Our workstation consists of an Intel Xeon i7-3550 CPU and ECC DDR3 8.0 GB Memory.

In addition, we compare the GS-SVM algorithm with cutting-edge algorithms, including ARIMA, SVR, LSTM, and CNN.

4.2 Data description

The experimental data come from a real-world GPS trajectory data set (about 50 GB) generated by 12,000 taxis of Beijing in November 2012. The data set has more than 900 million GPS trajectory records. The data records of the data set are illustrated in Fig. 5.

Fig. 5
figure 5

GPS trajectory data of taxi

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In particular, for the extensive comparisons, the aforementioned data set is divided into 7 groups (i.e., 1 day: Nov. 1, 5 days: Nov. 1–Nov. 5, 10 days: Nov. 1–Nov. 10, 15 days: Nov. 1–Nov. 15, 20 days: Nov. 1–Nov. 20, 25 days: Nov. 1–Nov. 25, 30 days: Nov. 1–Nov. 30). Furthermore, after data preprocessing, we select passenger hotspot data divided into blocks of 15 minutes of 1 day, 5 days, 10 days, 15 days, 20 days, 25 days, and 30 days in the first grid for prediction. For each group of data sets, this work takes 70% of the data set as the training set and 30% as the test set.

4.3 Evaluation metrics

To validate the measures of effectiveness (MOEs) of the algorithm, we take MAPE, RMSE, MAE, and ME as four evaluation metrics in this experiment [41], which are defined as follows.

$$ \begin{array}{@{}rcl@{}} {\text{MAPE}} = \frac{1}{n}\sum\limits_{t = 1}^{n} {\frac{{|{X_{t}} - {{\hat X}_{t}}|}}{{{X_{t}}}}} \times 100\%, \end{array} $$
(12)
$$ \begin{array}{@{}rcl@{}} {\text{RMSE}} = \sqrt{\frac{1}{n}\sum\limits_{t = 1}^{n} (\text{X}_{t} - \hat{\text{X}}_{t})^{2},} \end{array} $$
(13)
$$ \begin{array}{@{}rcl@{}} {\text{MAE}} = \frac{1}{n}\sum\limits_{t = 1}^{n} {{{|}}{{\text{X}}_{t}} {{-}}{{{\hat{\text{X}}}}_{t}}{{|}}}, \end{array} $$
(14)
$$ \begin{array}{@{}rcl@{}} {\text{ME}} = \mathop {\max }\limits_{{t=1,\cdots,n}} |{X_{t}} - {\hat X_{t}}|, \end{array} $$
(15)

where Xt denotes the real value of passenger hotspots at the time interval t, \({{\hat X_{t}}}\) represents the prediction value of passenger hotspots, and n is the total number of samples in the provided time. The accuracy of prediction mainly depends on MAPE. The lower the value of MAPE, the higher the accuracy of the algorithm.

4.4 Experimental results

Based on a large number of experiments, C = [100,300,500,700,900] and γ = [0.001,0.003,0.005,0.007,0.009] are defined in this paper. We also calculate the C value on the same data set, change the γ value, and traverse all grids to verify the performance of the parameters, which are illustrated in Table 1 and Fig. 6.

Table 1 Comparisons of parameters on the same data set
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Fig. 6
figure 6

Parameter tuning

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As shown in Table 1 and Fig. 6, when the parameter combination is C = 100, and γ = 0.001, the accuracy of the training set and the test set are 1, and the MAPE value of the GS-SVM algorithm is 0.003, and thus the algorithm performance is the best. However, the cross-validation score is poor. When γ = 0.001 and the values are 300, 500, 700, and 900, the correct rates of the training set and the test set are equal to 1, and the cross-validation scores are also 0.94, 0.95, 0.95, and 0.96, respectively. At this time, the MAPE value of the GS-SVM algorithm is 0.002, and the algorithm performance is the best. Experimental results indicate that when the parameter combination is C = 900 and γ = 0.001, both the correct rate of the training set and the test set are 1. Meanwhile, the cross-validation score reaches the peak, demonstrating that the optimal parameter combination found by the GS approach is the global optimal rather than local optimal.

To further evaluate the superiority of this parameter combination, this paper validates the performance of the GS-SVM algorithm when C = 900 and γ = 0.001 under different data sets, which are illustrated in Table 2.

Table 2 Performance comparisons of GS-SVM (C = 900,γ = 0.001) on different data sets
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It is verified in Table 2 that, when the parameter combination is C = 900 and γ = 0.001, the correct rates of the training set and the test set under different data sets both tend to 1, and the cross-validation score also tends to 1. In particular, the value of MAPE is relatively low. The results mentioned above show that the prediction accuracy of GS-SVM is better with the above parameter combination. Therefore, the GS approach is feasible to find the optimal parameter combination of the SVM algorithm.

To validate the effectiveness and accuracy of the GS-SVM algorithm, we divide the passenger boarding data set on the same grid into 1 day, 5 days, 10 days, 15 days, 20 days, 25 days, and 30 days for prediction. Thus, 70% of the data set is utilized for the training set, and the rest are the test set. The GS-SVM algorithm is compared with existing cutting-edge algorithms with the same data set, illustrated in Table 3, and the experimental results with different data sets are shown in Fig. 7.

Table 3 MOEs of ARIMA, SVR, LSTM, CNN, and GS-SVM
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Fig. 7
figure 7

Prediction results of GS-SVM on different data sets. (a) 1 day, (b) 5 days, (c) 10 days, (d) 15 days, (e) 20 days , (f) 25 days, and (g) 30 days

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Like many other studies, the prediction accuracy of the algorithm mainly depends on the MAPE value [37] in this work. Combined with Table 3, and Figs. 7 and 8, the MAPE value of the optimized GS-SVM algorithm is far lower than that of SVR, ARIMA, LSTM, and CNN. More specifically, the experimental results with the data set on 1 day, 5 days, 10 days, 15 days, 20 days, 25 days, and 30 days are illustrated in Table 3. With the data set on 30 days, the MAPE value of GS-SVM is 99%, 78.8%, 85.4%, and 96% lower than that of ARIMA, SVR, LSTM, and CNN, respectively. The MAPE value obtained by GS-SVM is 99%, 92.8%, 88.9%, 78.4%, 78.6%, 81.3%, and 78.8% lower when seven groups of data sets are processed, respectively. The aforementioned results show that the SVM algorithm optimized by the GS approach has better accuracy in predicting passenger hotspots.

Fig. 8
figure 8

MOEs. (a) MAPE values of different algorithms on different data sets, and (b) MOEs of 30 days

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To further evaluate the universality of the parallel GS-SVM algorithm, we select 00 grid, 55 grid, 99 grid, and other surrounding grids to produce the mean value of MAPE under the data set on 30 days, and plot the results in Tables 4, 5, and 6, respectively.

Table 4 MOEs values of different grids on the same data set
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Table 5 MOEs values of different grids on the same data set
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Table 6 MOEs values of different grids
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Table 4 explains the value of MAPE for the surrounding grid of 00 grid, 01 grid, 10 grid, and 11 grid with the data set on 30 days. Moreover, Table 5 shows the MAPE values of 44 grid, 45 grid, 46 grid, 54 grid, 56 grid, 64 grid, 65 grid, and 66 grid, which are around the 55 grid with the data set on 30 days. Finally, Table 6 illustrates the MAPE values of 88 grid, 89 grid, and 99 grid, around the 99 grid.

From Tables 4, 5, and 6, it is evident that the average values of MAPE generated from the parallel GS-SVM algorithm are 0.7%, 0.28%, and 0.27%, respectively. These experimental results show that it is reliable to predict passenger hotspots between different grids in different areas of passenger hotspots with the parallel GS-SVM algorithm on Spark.

5 Conclusions

In this paper, a parallel GS-SVM algorithm based on Spark was proposed to predict passenger hotspots with real-world GPS trajectories of taxicabs, which was beneficial for locating passengers timely and accurately. Specifically, the urban traffic road network was gridded on Spark, and passenger hotspots in the same grid were counted at an interval of 15 minutes. Then, the GS approach found the optimal combination of C (penalty factor) and γ (nuclear parameter) in the RBF function, and it was validated by cross-validation methodology. When the optimal combination of the RBF function is C = 900 and γ = 0.001, both the correct rates of the training set and the test set are 1, and the cross-validation score reaches the peak, which indicates that finding the optimal parameter combination in the GS approach is the global optimal rather than the local optimal.

Finally, this combination was substituted into the RBF function. Compared with ARIMA, SVR, LSTM, and CNN, the MAPE value generated from the proposed GS-SVM algorithm was decreased by 99.0%, 92.8%, 88.9%, 78.4%, 78.6%, 81.3%, and 78.8% in seven groups of different data sets. The experimental results demonstrated that our algorithm had a high prediction accuracy and effectively process large-scale traffic data on the Spark platform. In addition, to validate the universality of GS-SVM, we predicted the surrounding grids by fixing a grid. The surrounding MAPE values of 00 grid, 55 grid, and 99 grid are 0.7%, 0.28%, and 0.27%, respectively, which verified the reliability of GS-SVM.

This work considers the number of passenger hotspots in each grid within 15 minutes but does not consider the impacts of weather factors, traffic conditions, and passenger mobility. For example, the travel rate of passengers on sunny days may be higher than on cloudy and rainy days. When the traffic conditions are good, the travel rate of residents is high, and passenger mobility may affect the accuracy of the number of passengers in the grid. On the other hand, this paper only employs the GPS trajectory data of taxis in Beijing to validate the universality of the algorithm, so it is unknown that the optimal combination of the parameters is still applicable under different data sets. In future work, we will incorporate the impacts of weather, traffic conditions, and passenger mobility into predicting passenger hotspots and utilize taxi GPS trajectory data in different cities to evaluate the algorithm.