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Estimating Urban Road GPS Environment Friendliness with Bus Trajectories: A City-Scale Approach^{ †}

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## Abstract

**:**

## 1. Introduction

- We estimate the GEF of roads at the city scale using the historical GPS trajectories of buses, without the need for extra specialized efforts in GPS data collection. Compared to other methods, this makes our method more scalable, less costly, and more accessible to be transferred to other cities, by only using already existing bus trajectory data. Besides, buses are supposed to run on fixed routes many times a month, which is the prior knowledge for map matching. This helps improve the accuracy and efficiency in the map matching process and reduce the misestimation brought by accidental factors (e.g., the position of satellites, weather).
- We propose a novel three-phase framework for estimating the GEF of urban roads. First, the bus routes’ data and the historical bus GPS data are mapped to the road network based on the map matching algorithm. We calculate the errors of each bus on road segments through which it passes. Secondly, we propose a matrix completion-based method, which makes full use of the correlation between the GPS errors of buses on different road segments and uses the third-party data of urban environment information as regularization to infer the GPS errors of buses on all road segments. Finally, we integrate the errors of buses on all road segments to estimate the GEF.
- We conduct an evaluation and verify our estimated GEF by comparing it with the ground truth collected through field study and the street views on some road segments. The results confirm the effectiveness of our proposed evaluation approach.

## 2. Related Work

#### 2.1. GPS Error and Calibration

#### 2.2. Measuring GPS Positioning Performance

## 3. Basic Concepts

**Definition**

**1.**

**Road network.**The road network is a graph $RN=(Nodes,Edges)$ comprised of a set of roads connected to each other in a graph format. $Edges=\left\{edg{e}_{i}\right\}$ is the set of the edges with each edge associated with a road. $Nodes=\left\{nod{e}_{i}\right\}$ is the set of the nodes with each node associated with an intersection represented by $(i{d}_{i},longitud{e}_{i},latitud{e}_{i})$. Edge set $Edges$ is a subset of the cross product $N\times N$, where N is the number of nodes. Each element $edge(nod{e}_{i},nod{e}_{j})$ in $Edges$ is a street connecting $nod{e}_{i}$ to $nod{e}_{j}$. In this work, the road is depicted as a line without any width. The road network data of Chengdu was downloaded from OpenStreetMap (Please check the official site of OpenStreetMap for more details: http://www.openstreetmap.org/).

**Definition**

**2.**

**Road segment.**A road segment $roa{d}_{i}$ of the road $edg{e}_{j}$ is a continuous part of $edg{e}_{j}$. A road could be divided into several road segments. In this paper, we set the length of a road segment equal to 50 m. The road whose length was less than 50 m was treated as a single road segment.

**Definition**

**3.**

**Bus route.**The bus route $B{R}_{i}$ is a subgraph of the road network graph $RN$. In this paper, there were 184 different bus lines in Chengdu that covered $n=8831$ road segments in road network $RN$. There was always more than one bus running on the same route. For example, the red lines in Figure 1 denote a part of the bus line route.

**Definition**

**4.**

**Bus trajectory.**The trajectory ${G}_{i}=\left\{{g}_{i,t}\right\}(i=1,\cdots ,m)$ of $bu{s}_{i}$ is a sequence of GPS points ${g}_{i,t}$. We used m to denote the number of buses. m equalled 4835 in our work. The GPS point ${g}_{i,t}=(tim{e}_{i,t},latitud{e}_{i,t},longitud{e}_{i,t})$ consists of a time-stamp $tim{e}_{i,t}$, a latitude record $latitud{e}_{i,t}$, and a longitude record $longitud{e}_{i,t}$. For example, the black points in Figure 1 denote the GPS trajectory data of buses.

**Definition**

**5.**

**POI information of the road segment.**The POI information of the road segment is depicted by several different POI categories from the online map. For $roa{d}_{i}$, we constructed a POI feature vector ${c}_{i}=(cn{t}_{1},\cdots ,cn{t}_{num})$, where $num$ denotes the number of different POI categories and $cn{t}_{j}(j=1,\cdots ,num)$ denotes the number of nearby (within 200 m) POI, which belong to category $po{i}_{j}$. Concretely in this paper, there were $num=17$ different POI categories according to the Gaode Online Map (Please check the official site of Gaode Map for more details: http://ditu.amap.com/): catering services, traffic infrastructures, government agency, vehicle sales, corporations, scenic spots, sports services, science education services, shopping services, accommodation services, vehicles services, serviced apartment, finance insurance services, life services, vehicle maintenance, and medical care services.

**Definition**

**6.**

**Tags of the road segment.**According to the OpenStreetMap, road segments could be categorized by tags: PrimaryLink, LivingStreet, service, residential, SecondaryLink, primary, MotorwayLink, unclassified, motorway, trunk, TrunkLink, tertiary, secondary (Please check the wiki of OpenStreetMap for more details of the tags: http://wiki.openstreetmap.org/wiki/Highway_link/). Each road segment is labelled with only one tag.

**Definition**

**7.**

**Layout information of the road segment.**The layout information of the road segment is depicted by several different floors. For $roa{d}_{i}$, we constructed a layout feature vector ${h}_{i}=(heigh{t}_{1},\cdots ,heigh{t}_{num})$, where $num$ denotes the number of different floors and $heigh{t}_{j}(j=1,\cdots ,num)$ denotes the number of nearby (within 200 m) buildings with j floors. Concretely in this paper, there were $num=60$ different floors within the second-ring road in Chengdu, China.

**Definition**

**8.**

**GPS positioning bias.**The GPS positioning bias refers to the linear distance between the GPS positioning record and the real position of the bus. It ranges from a few meters in open sky environments to over 80m in urban canyons [7]. The positioning bias of a bus on the road could be divided into two orthonormal parts. One is vertical to the road, while the other is parallel with the road. The vertical component is much greater than the parallel component, which can be ignored [7,32]. In this paper, such bias is measured as the vertical distance between the GPS positioning point and the real road where the bus is running.

**Definition**

**9.**

**GPS positioning error.**The real horizontal position of the bus along the roads can be figured out based on map-matching algorithms. However, the width of the actual road cannot be ignored with regard to the GPS positioning bias. It is difficult to tell on which lane the bus is running. As a result, we utilized the standard deviation (std) of the GPS positioning biases to measure the buses’ GPS positioning errors on roads, instead of the mean values of the biases. In this way, the GPS positioning error is defined as the standard deviation (std) of the GPS positioning biases. Such error is affected by satellite ephemeris error, receiver clock error, multipath error, spherical error, receiver measurement noise, and so on. Multipath error is the major component when locating in urban areas. The concepts above are shown in Figure 2.

**Definition**

**10.**

**GPS environment friendliness (GEF).**Multipath error is caused by the delay of the signal arrival due to its reflection off building surfaces in the area. GPS environment friendliness defines the degree to which the multipath phenomenon affects the GPS performance. The GEF depends on the surrounding environment. It is independent of time, weather, the quality of GPS positioning terminal device, and the number of visible GPS satellites. We assumed that different locations within the same road segment shared a similar environment and the same GEF.

## 4. Methodology

#### 4.1. Overview of the Framework

- We utilized the hidden Markov model (HMM)-based map matching algorithm [33,34,35,36] to map the bus trajectories’ data to the roads. The accuracy and efficiency of the map matching process were improved significantly based on the pre-knowledge of bus routes. After the map matching, we constructed a matrix, where the element of the matrix represented the positioning error standard deviation of each bus on each road segment. Note that the route of one bus only covered a small portion of the roads in the city. There were few buses running on any given road. Thus, the matrix to be completed was very sparse.
- We estimated the positioning errors of each bus on each road segment based on the matrix completion algorithm, taking the nearby environment information into consideration. Due to the variance of the quality of the GPS receivers, an incorrect conclusion would be drawn if we estimated the GEF of a road only depending on the buses whose routes covered the road. Ideally, the GEF of a road is supposed to be estimated according to the GPS errors of all buses. Therefore, we needed to complete the matrix that was constructed in the first phase.
- The GEF of each road segment was estimated based on the completion result. The buses whose GPS terminal device had a higher quality would have more weight on the evaluation of the GEF.

#### 4.2. Map Matching-based GPS Error Matrix Construction

- Since the bus is continuously running on the road, the road segment corresponding to the current GPS sampling point should be close to the road segment corresponding to the previous point.

#### 4.3. GPS Error Estimation with Additional Environment Information Integration

**Var**was then constructed, where the entry ${v}_{ij}$ denotes the error of $bu{s}_{i}$ on $roa{d}_{j}$. However, there existed no bus that could pass all roads, making this matrix very sparse.

#### 4.3.1. Basic Objective Function of Matrix Completion

**Var**recording the standard deviation of GPS positioning biases, which measured the errors on road segments:

**Var**was to be completed and could be very sparse. The basic objective function of matrix completion was set as [39]:

**Sign**was the same as matrix

**Var**. ${s}_{ij}$ equalled 1 if ${v}_{ij}$ was known. Otherwise, ${s}_{ij}$ equalled 0. ${s}_{ij}={1}_{\left\{(i,j)\right|{v}_{ij}isknown.\}}$. The result of matrix completion was ${\mathbf{LR}}^{T}$. The size of matrix

**L**was $m\times a$, and the size of matrix

**R**was $n\times a$. a was a hyper-parameter of matrix completion. The penalty term $\left|\right|\mathbf{Sign}\xb7{\mathbf{LR}}^{T}-\mathbf{Var}{\left|\right|}_{F}^{2}$ measured the similarity between the completion result and original matrix. ${\left|\right|L\left|\right|}_{F}^{2}+{\left|\right|R\left|\right|}_{F}^{2}$ was the regularization term. $\lambda $ was the hyper-parameter denoting the importance of the penalty term.

#### 4.3.2. Measure the Relative Advantage of GPS Receivers’ Qualities

**Qua**was constructed. To test the equality of variations, we used the F-test [40], initially developed by A.Fisher. The hypothesis was that the means of a given set of normally distributed populations, all having the same standard deviation, were equal. Under the Gaussian assumption, any scaled pair of variations of our sample could form a pivot variable following an F distribution if the null hypothesis was true. Then, we could perform hypothesis tests on any pair of variations at the level of 5%.

**Tran**based on matrix

**Qua**.

**Var**:

**Y**is an arbitrary $m\times n$ matrix:

- The elements in ${E}_{\theta}(\mathbf{Tran}\xb7\mathbf{Var})$: inherit the relative magnitudes of the elements in $\mathbf{Tran}\xb7\mathbf{Var}$, small values for the ideal case, large values for an inappropriate case.
- It guarantees a lower bound of $\left|\right|{E}_{\theta}(\mathbf{Tran}\xb7{\mathbf{LR}}^{T}){\left|\right|}_{F}^{2}$, so that the objective function below has a lower bound. Thus, it is possible to converge when we solve the system iteratively.

#### 4.3.3. Measure the POI Information of Road Segments

**Poi**to describe the similarity of the POI distribution between each of two roads.

#### 4.3.4. Measure the Tag Information of Road Segments

**Tag**$={({t}_{ij})}_{n\times n}$.

#### 4.3.5. Measure the Layout Information around Road Segments

**Dist**$={({d}_{ij})}_{m\times n}$. ${d}_{ij}$ denotes the Euclidean distance between the layout vector of ${h}_{i}$ and ${h}_{j}$. Thus, we could construct matrix

**Layout**$={({l}_{ij})}_{n\times n}$ to describe the similarity of the layout between each of two segments.

#### 4.3.6. Optimization of the Objective Function

Algorithm 1: Matrix completion. |

#### 4.4. Weighted Estimation of GEF

## 5. Experiment

#### 5.1. Dataset Description

#### 5.2. Result of Map-Matching

#### 5.3. Evaluation of the Matrix Completion Result

**Var**were equally divided into k parts $({\mathbf{P}}_{1},{\mathbf{P}}_{2},\cdots ,{\mathbf{P}}_{K})$. For each part ${\mathbf{P}}_{i}$, we covered it and preserved the remaining $k-1$ parts. We applied our completion algorithm to matrix $\mathbf{Var}$ and obtained the completed matrix ${\mathbf{LR}}^{T}$. We calculated the estimate-error [43] according to ${\mathbf{LR}}^{T}$ as follows: ${\xi}_{i}=\frac{{\sum}_{r,t:{v}_{r,t}\in {\mathbf{P}}_{i}}\left|{v}_{r,t}-{\mathbf{LR}}_{r,t}^{T}\right|}{{\sum}_{r,t:{v}_{r,t}\in {\mathbf{P}}_{i}}\left|{v}_{r,t}\right|}$. Enumerate the covered part from ${\mathbf{P}}_{1}$ to ${\mathbf{P}}_{k}$, and calculate the final estimate-error as: $\xi =\frac{{\sum}_{i=1}^{k}{\xi}_{i}}{k}$. Repeat the above operations t times, and calculate the average estimate error as the evaluation result of the completion algorithm. The rank comparing result is shown in Table 2, and we can see that our method outperformed the following baseline methods:

**Naive KNN**: For each empty entry in one row (column), we searched the k nearest rows (columns) whose corresponding entry was not null according to the Euclidean distance. Then, KNN used these non-empty entries to do the estimation.

**Correlation-based KNN**: This was similar to naive KNN. The only difference was that it used the correlation to measure the similarity instead of the Euclidean distance.

**Non-negative matrix factorization (NMF)**[42]: The matrix was factorized into two matrices, with the property that all matrices had no negative elements. Matrix multiplication of the factorized matrices was the completion result.

#### 5.4. Case Study

## 6. Limitation and Future Work

- Although the bus routes could cover most of the primary roads in the city, there were still plenty of bypasses whose GEF could not be estimated. However, our approach could be easily applied to trajectory data of taxis to tackle those bypasses, which is the future work. Besides, using the results of GEF assessment as the training data, environmental attributes could be extracted from urban street view pictures. Those attributes could be employed to estimate the GEF of cities without bus trajectory data.
- There were only a few road segments where we conducted case studies due to the cost. Real-life GPS measurements on more road segments are expected to be collected, which is the future work.
- We intend to apply our approach to location-based services and improve the user experience. Specifically, a model assessing the confidence level of real-time bus location and predicted arriving time could be modified from the GEF evaluation method.

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Bus Line Number | 184 |

Bus Number | 4835 |

Duration | 30 days |

GPS Point Record Number | 62,783,000 |

Sampling Rate of GPS Receiver | 2–4 points/min |

Number of Types of GPS Receivers | >80 |

Length of Road Segment | 50 m |

Road Segment Number | 8831 |

Average of Buses Running on Each Segment | 121 |

Average of Segments Covered by Each Bus Line | 171 |

Number of GPS Points a Bus Recorded on a Segment | >20 |

Methods | Matrix Completion Error | |
---|---|---|

NAKNN | 0.37242 | |

Baseline Approaches | CBKNN | 0.32951 |

NMF | 0.31883 | |

Basic Method (1) | 0.29371 | |

Our Approach | Integrating Layout Information | 0.29348 |

Integrating Layout and Tag and POI | 0.29311 | |

Integrating All Penalty Terms (2) | 0.29220 |

Road | Baseline | Our Approach | std of GPS Biases in Field Tests (m) | ||
---|---|---|---|---|---|

Rank | GEF | Rank | GEF | ||

1 | 87 | Satisfied | 10 | Satisfied | 1.960 |

2 | 5307 | Poor | 1851 | Satisfied | 3.283 |

3 | 5312 | Poor | 2325 | Satisfied | 3.378 |

4 | 5177 | Poor | 326 | Satisfied | 1.492 |

5 | 759 | Satisfied | 6012 | Poor | 9.165 |

6 | 5085 | Poor | 5043 | Poor | 5.035 |

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## Share and Cite

**MDPI and ACS Style**

Ma, L.; Zhang, C.; Wang, Y.; Peng, G.; Chen, C.; Zhao, J.; Wang, J.
Estimating Urban Road GPS Environment Friendliness with Bus Trajectories: A City-Scale Approach. *Sensors* **2020**, *20*, 1580.
https://doi.org/10.3390/s20061580

**AMA Style**

Ma L, Zhang C, Wang Y, Peng G, Chen C, Zhao J, Wang J.
Estimating Urban Road GPS Environment Friendliness with Bus Trajectories: A City-Scale Approach. *Sensors*. 2020; 20(6):1580.
https://doi.org/10.3390/s20061580

**Chicago/Turabian Style**

Ma, Liantao, Chaohe Zhang, Yasha Wang, Guangju Peng, Chao Chen, Junfeng Zhao, and Jiangtao Wang.
2020. "Estimating Urban Road GPS Environment Friendliness with Bus Trajectories: A City-Scale Approach" *Sensors* 20, no. 6: 1580.
https://doi.org/10.3390/s20061580