# A Novel Bus Arrival Time Prediction Method Based on Spatio-Temporal Flow Centrality Analysis and Deep Learning

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## Abstract

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## 1. Introduction

## 2. Related Works

## 3. Methodology

#### 3.1. Feature Extraction

#### 3.1.1. Bus Network Basic Features

- Passing Time $PT$$P{T}_{i,j,k}$ is the required time $PT$ in seconds for the jth bus of the route i to arrive at the kth stop from the preceding k− 1th stop. $PT$ is the difference between arrival time $AT$ and departure time $DT$, as shown in Equation (1). We do not compute starting station’s $PT$ as there is no preceding station.$$P{T}_{i,j,k}=A{T}_{i,j,k}-D{T}_{i,j,k-1}$$
- Mean Speed $MS$Mean Speed $MS$ (m/s) is $PT$ divided by Distance d as shown in Equation (2). Here, d is the distance between stop $k-1$ and stop k on bus route i.$$M{S}_{i,j,k}=\frac{P{T}_{i,j,k}}{{d}_{i,k}}$$
- Wait Time $WT$Wait Time $WT$ is the amount of time a bus temporarily stays at the bus stop for loading and unloading passengers or a short break, as shown in Equation (3).$$W{T}_{i,j,k}=D{T}_{i,j,k}-A{T}_{i,j,k}$$
- Inter-Arrival Time IInter-Arrival Time I between buses on a bus route is given in Equation (4). We obtain I by subtracting the time when the j− 1th bus on the bus route i arrived at stop k from the time when the jth bus on i arrived at the same stop.$${I}_{i,j,k}=A{T}_{i,j,k}-A{T}_{i,j-1,k}$$
- Distance dDistance d in meters is the Euclidean distance between stops and $k1$ and $k-1$ on bus route i given their coordinates $(x,y)$ as shown in Equation (5).$${d}_{i,k}=\sqrt{{({d}_{i,{k}_{x}}-{d}_{i,k-{1}_{x}})}^{2}+{({d}_{i,{k}_{y}}-{d}_{i,k-{1}_{y}})}^{2}}$$
- Current Time $CT$Current time $CT$ of $AT$ is the time in which the hour, minute, and second information is converted into seconds.
- Bus Stop Order IndexThe Bus Stop Order Index (BSOI) of k denotes the sequential order of station k for bus route i. Each bus route is composed of bus stops in a different order. For instance, in Figure 2, the BSOI of bus stops A, B, C, and D for route B2 are 1, 2, 3, and 4, respectively.

#### 3.1.2. Bus Flow Centrality Analysis Features

- Individual In Degree $IID$$IID$ refers to a set of times buses took to pass a given stop within a certain amount of time to pass through a specific link. Suppose $A{T}_{i,j,k}$ is given. From a preset time period in the past ($LT$) to $A{T}_{i,j,k}$, we retrieve $PT$ of every bus j on route i passing stop k via $k-1$ (denoted as $P{T}_{i,j,k}$). In other words, $II{D}_{i,j,k}$ can be regarded as a set of buses including j and other buses ahead on the same route i that represent the overall inbound spatial flows upon the point when j passes bus station k. The pseudo-code for obtaining $IID$ is presented in Algorithm 1.An intuitive example is provided in Figure 3. Bus B1 is en route, passing stops in the order of H, D, E, F, G, and C. Figure 3a,b show a snapshot at $A{T}_{i=B1,j=4,k=4}-LT$ and $A{T}_{i=B1,j=4,k=4}$, respectively, where F is the fourth stop for route $B1$. Bus B1-4 on sub-figure a is in between stops D and E. Later on, sub-figure b shows B1-4 arriving at stop F. In this case, the buses involved in producing $II{D}_{i=B1,j=4,k=4}$ are B1-2, B1-3, and B1-4.Spatial flow information is embedded further into a low-dimensional vector. We construct this vector by applying Equations (6)–(8) to a given $IID$. The embedding process is carried out for every stop and every bus. Note that, by using such embedding, we represent not only spatial information but also the different temporal dynamics of every bus on the route.
- Individual Out Degree $IOD$$IOD$ represents the outbound spatial flow when a bus j leaves station k for bus stop $k+1$. Suppose $A{T}_{i,j,k}$ is provided. From a preset time period in the past ($LT$) to $A{T}_{i,j,k}$, we retrieve $PT$ of every bus j on route i that left station k for the next stop $k+1$ (denoted as $P{T}_{i,j,k}$). $IOD$ is expected to be another factor for influencing the time a bus takes to reach the next stop. $IOD$ can be obtained by the running the pseudo-code presented in Algorithm 2.As shown in Figure 3, buses B1-1 and B1-2 that departed stop G are ahead of B1-4. $PT$ of B1-1 and B1-2 can provide a clue for B1-4 on the condition of the link to the next stop.
- Total Out Degree $TOD$$IOD$ is computed for a particular route. On the other hand, $TOD$ is a set of PTs of buses on all routes heading toward the next stop $k+1$ from stations k. $TOD$ represents the aggregate flow pattern on the link to the next stop. Buses not moving to $k+1$ are not considered in constructing $TOD$. For example, in Figure 3, buses on route B3 are not considered for computing $TOD$ for station G via F. It is because the buses on B3 head toward stop I from F instead. Set $TOD$ can be obtained by executing Algorithm 3. $TOD$ is embedded into the latent vector using Equation (6), Equation (7), and Equation (8). Along with $TOD$, we keep IODs separately for all stops and buses to account for microscopic flow patterns that can affect the travel time prediction to the next stations.$$Num\phantom{\rule{0.277778em}{0ex}}of\phantom{\rule{0.277778em}{0ex}}Set=\sum _{i\in Set}^{Set}{C}_{i}$$$$Mean\phantom{\rule{0.277778em}{0ex}}of\phantom{\rule{0.277778em}{0ex}}Set=\frac{1}{Num\phantom{\rule{0.277778em}{0ex}}of\phantom{\rule{0.277778em}{0ex}}Set}\sum _{i\in Set}^{Set}{x}_{i}$$$$Deviation\phantom{\rule{0.277778em}{0ex}}of\phantom{\rule{0.277778em}{0ex}}Set=\sqrt{\frac{1}{Num\phantom{\rule{0.277778em}{0ex}}of\phantom{\rule{0.277778em}{0ex}}Set-1}\sum _{i\in Set}^{Set}{({x}_{i}-\overline{x})}^{2}}$$

Algorithm 1: Individual In Degree $IID$. |

Algorithm 2: Individual Out Degree $IOD$. |

Algorithm 3: Total Out Degree $TOD$. |

#### 3.1.3. Contextual Features

#### 3.2. Models

## 4. Evaluation

- Buses pass through stops in order;
- A bus x cannot overtake the other bus y ahead if x and y are on the same route;
- Algorithm 4 implements linear interpolation between two points. However, interpolation cannot be carried out for the time intervals, with consecutive nulls appearing at the beginning or end of the sequence.

Algorithm 4: Interpolation function. |

#### 4.1. Measuring Prediction Performance

#### 4.2. Measuring the Predictive Performance of Random Forest Models

#### 4.3. Measuring the Predictive Performance of Multi-Linear Regression Models

#### 4.4. Measuring the Predictive Performance of LSTM Models

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

BFC | Bus Flow Centrality; |

IID | Individual In Degree; |

IOD | Individual Out Degree; |

TOD | Total Out Degree; |

PT | Passing Time; |

AT | Arrival Time. |

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**Figure 1.**The overall procedure for bus flow centrality analysis (BFC) and training for bus arrival time prediction.

**Figure 2.**Buses on the basic road structures and the network per bus routes. Symbols A, B, C, and D represent bus stops.

**Figure 3.**Snapshot of a sample bus network and snapshot focusing on the bus stop F. Symbols A through I represent bus stops.

**Figure 4.**Using LSTM for modeling the correlation between the tensors of comprehensive features and the bus arrival time prediction at every time window.

Model | Data Type | Data Range | MAPE (%) | |||
---|---|---|---|---|---|---|

GPS | Flow Embedding | Contextual | Temporal Range | Spatial Range | ||

ALSTM [11] | ✓ | - | - | 1 month | 1 route | 4 |

Weighted LSTM [16] | ✓ | - | ✓ | 8 month | 1 route | 4.89 |

LSTM [32] | ✓ | - | - | 12 month | 1 route | 3.6 |

ConvLSTM [12] | ✓ | - | - | 6 month | 1 route | 4.19 |

Ensemble ML [41] | - | - | ✓ | 1 month | 1 route | 19.64 |

LSTM-RNN [38] | - | - | - | 1 month | 47 routes | 11.75 |

DA-RNN [40] | - | - | ✓ | unknown | 4 routes | 18 |

BFC-LSTM | - | ✓ | - | 1 month | 100+ routes | 1.19 |

- | ✓ | ✓ | 1 month | 100+ routes | 2.90 |

Data | Status | Time (by Bus Stop ID) | Temperature | Precipitation | Day | Holiday |
---|---|---|---|---|---|---|

Mean | - | - | −1.72 | 0.00 | - | - |

Median | - | - | −1.7 | 0.0 | - | - |

Max | 1 (Arrival) | 1-January-2018 00:00:00 | 9.5 | 1.0 | 1 | 1 |

Min | 0 (Depart) | 1-December-2017 00:00:00 | −16.9 | 0.0 | 0 | 0 |

Unit | - | DD-MM-YYYY hh:mm:ss | °C | mm | - | - |

Time Interval | Each Bus | 1-s | 1-min | 1-min | 1-day | 1-day |

Type | Binary | Int | Float | Float | Binary | Binary |

Model | Passing Time Feature | Basic Features | BFC Features | Contextual Features | MAPE (%) |
---|---|---|---|---|---|

Linear Regression | ✓ | - | - | - | 54.6 |

Random Forest | ✓ | - | - | - | 20.8 |

Multi-Linear Regression | ✓ | ✓ | ✓ | - | 18.8 |

- | - | 22.4 | |||

LSTM | ✓ | ✓ | ✓ | - | 1.19 |

✓ | ✓ | 2.91 | |||

- | - | 31.7 | |||

- | ✓ | 34.9 | |||

GRU | ✓ | ✓ | ✓ | - | 2.03 |

- | - | 34.1 | |||

ALSTM | ✓ | ✓ | ✓ | - | 4.71 |

- | - | 38.7 |

**Table 4.**Performance of BFC-based models against previous non-BFC-based methods in terms of the number of perceptrons in the neural networks.

Number of Perceptrons | Previous Model MAPE (%) | BFC Model MAPE (%) | ||||
---|---|---|---|---|---|---|

LSTM | GRU | ALSTM | LSTM | GRU | ALSTM | |

64 | 35.0 | 37.8 | 42.8 | 1.64 | 2.76 | 5.87 |

128 | 31.7 | 34.1 | 39.1 | 1.19 | 2.03 | 4.92 |

256 | 32.1 | 34.3 | 38.7 | 1.22 | 2.11 | 4.71 |

512 | 31.9 | 34.4 | 39.0 | 1.21 | 2.08 | 4.74 |

MAPE (%) | T | ||||||
---|---|---|---|---|---|---|---|

25 | 50 | 75 | 100 | 150 | 200 | ||

D | 10 | 22.08 | 21.97 | 21.95 | 21.93 | 21.92 | 21.91 |

15 | 21.29 | 21.13 | 21.13 | 21.03 | 21.00 | 20.98 | |

20 | 21.25 | 21.02 | 21.02 | 20.90 | 20.87 | 20.85 | |

25 | 21.23 | 21.01 | 21.01 | 20.89 | 20.86 | 20.84 | |

30 | 21.26 | 21.03 | 21.03 | 20.91 | 20.86 | 20.84 | |

35 | 21.24 | 21.02 | 21.02 | 20.90 | 20.86 | 20.84 |

BFC Features LT (min.) | With Contextual Features MAPE (%) | Without Contextual Features MAPE (%) |
---|---|---|

- | 22.41 | 22.36 |

5 | 21.26 | 21.22 |

10 | 20.36 | 20.35 |

15 | 20.03 | 20.00 |

20 | 19.66 | 19.66 |

30 | 19.24 | 19.23 |

45 | 18.92 | 18.92 |

60 | 18.83 | 18.83 |

120 | 18.84 | 18.84 |

180 | 18.84 | 18.84 |

LT (min.) | With Contextual Features | Without Contextual Features | ||||
---|---|---|---|---|---|---|

MAE | RMSE | MAPE (%) | MAE | RMSE | MAPE (%) | |

5 | 3.84 | 8.29 | 3.71 | 1.62 | 4.23 | 1.65 |

10 | 3.09 | 6.50 | 3.05 | 1.15 | 2.91 | 1.19 |

15 | 2.94 | 5.75 | 3.01 | 1.16 | 2.92 | 1.21 |

20 | 2.94 | 5.80 | 2.98 | 1.22 | 2.99 | 1.26 |

30 | 2.88 | 5.68 | 2.90 | 1.32 | 3.10 | 1.37 |

45 | 3.00 | 5.80 | 3.10 | 1.30 | 3.12 | 1.37 |

60 | 3.07 | 5.98 | 3.09 | 1.37 | 3.21 | 1.42 |

120 | 3.08 | 6.23 | 3.16 | 1.43 | 3.49 | 1.47 |

180 | 3.36 | 6.51 | 3.42 | 1.39 | 3.25 | 1.47 |

Number of Time Windows | MAE | RMSE | MAPE (%) |
---|---|---|---|

5 | 1.16 | 2.92 | 1.19 |

10 | 1.33 | 3.27 | 1.37 |

15 | 1.38 | 3.38 | 1.44 |

20 | 1.36 | 3.44 | 1.40 |

25 | 1.42 | 3.38 | 1.47 |

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**MDPI and ACS Style**

Lee, C.; Yoon, Y.
A Novel Bus Arrival Time Prediction Method Based on Spatio-Temporal Flow Centrality Analysis and Deep Learning. *Electronics* **2022**, *11*, 1875.
https://doi.org/10.3390/electronics11121875

**AMA Style**

Lee C, Yoon Y.
A Novel Bus Arrival Time Prediction Method Based on Spatio-Temporal Flow Centrality Analysis and Deep Learning. *Electronics*. 2022; 11(12):1875.
https://doi.org/10.3390/electronics11121875

**Chicago/Turabian Style**

Lee, Chanjae, and Young Yoon.
2022. "A Novel Bus Arrival Time Prediction Method Based on Spatio-Temporal Flow Centrality Analysis and Deep Learning" *Electronics* 11, no. 12: 1875.
https://doi.org/10.3390/electronics11121875