# What is an Appropriate Temporal Sampling Rate to Record Floating Car Data with a GPS?

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## Abstract

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## 1. Introduction

- Measurement error is a property of the measurement system used for recording the movement. For FCD, measurement error refers to the difference between the actual spatial position of the floating car at a specific time and the GPS position estimate at the same time.
- Interpolation error is a property of the discretization of movement. For FCD, interpolation error arises from the difference between the continuous movement of the floating car and the discrete snapshots in the trajectory. Hence, interpolation error is closely connected to the temporal sampling rate at which the data are collected.

- Sampling must reflect the aim of the movement analysis. Which information is needed for the analysis and at which level of detail?
- Sampling must address the characteristics of the measurement system. What is the influence of GPS measurement error when collecting the FCD?
- Sampling must respond to the characteristics of the moving object under observation. What is the influence interpolation error when collecting the FCD?

## 2. Related Work

## 3. FCD and Movement Parameters

#### 3.1. The Experimental Data Set

#### 3.2. Defining the Movement Parameters

Movement Parameter | True | Measured | ||
---|---|---|---|---|

Variable | Definition | Variable | Definition | |

Spatial path | Π | $\mathbf{\Pi}=C$${}^{1}$ | ${\mathbf{\Pi}}^{m}$ | ${\mathbf{\Pi}}^{m}=<{\mathit{P}}_{\mathit{i}},...,{\mathit{P}}_{\mathit{n}}>$ |

Distance | d | $d={\int}_{0}^{1}\left|\mathrm{d}x\right|$${}^{2}$ | ${d}^{m}$ | ${d}^{m}=d({\mathit{P}}_{\mathit{i}},{\mathit{P}}_{\mathit{j}})$${}^{3}$ |

Speed | v | $v=\left|\right|v\left|\right|$${}^{4}$ | ${v}^{m}$ | ${v}^{m}={d}^{m}/\Delta t$${}^{5}$ |

Direction | θ | $\theta =\angle v$${}^{6}$ | ${\theta}^{m}$ | ${\theta}^{m}=\angle \phantom{\rule{4pt}{0ex}}{\mathit{P}}_{\mathit{i}}{\mathit{P}}_{\mathit{j}}$${}^{6}$ |

**Π**; ${}^{3}$ $d(\mathit{A},\mathit{B})$ is the Euclidean distance between

**A**and

**B**; ${}^{4}$ $v=\mathrm{d}\mathrm{x}/\mathrm{d}\mathrm{t}$; ${}^{5}$ $\Delta t={t}_{j}-{t}_{i}=$ constant, for all $i,j$; ${}^{6}$ $\angle b$ is the angle between the vector b and the x-axis.

## 4. Assessing the Influence of Measurement Error

**Figure 1.**The autocorrelation of GPS measurement error in the experimental FCD. The autocorrelation ($\widehat{C}$) decreases as the reference distance (${d}_{0}$) increases. ${d}_{0}$ is the distance between recording two consecutive position estimates according to the car’s CAN bus.

## 5. Assessing the Influence of Interpolation Error

#### 5.1. Rediscretizing the Trajectories

**Figure 2.**Rediscretizing the FCD. The reference movement at $1\phantom{\rule{3.33333pt}{0ex}}\mathrm{Hz}$ is rediscretized to a resolution of $1/3\phantom{\rule{3.33333pt}{0ex}}\mathrm{Hz}$, i.e., $k=3$. In

**a**the moving window is at its initial location and encompasses the movement between $({\mathit{P}}_{\mathbf{1}},{t}_{1})$ and $({\mathit{P}}_{\mathbf{4}},{t}_{4})$. The solid red line represents the reference movement, the dashed red line its rediscretization. In

**b**the moving window has shifted forward , so that the reference movement and its rediscretization are now compared between $({\mathit{P}}_{\mathbf{2}},{t}_{2})$ and $({\mathit{P}}_{\mathbf{5}},{t}_{5})$.

#### 5.2. Metrics for Interpolation Error

**Figure 3.**Path uncertainty and its effect on measured distance ${d}^{m}$. The measured distance ${d}^{m}$ (dashed line) is smaller than the reference distance ${d}_{0}$ (solid line). Interpolation error causes a systematic underestimation of distance.

**R**, the point along ${\mathbf{\Pi}}_{\mathbf{0}}$ that is farthest from ${\mathbf{\Pi}}^{\mathit{m}}$. In Figure 2a, for example,

**R**is at $({\mathit{P}}_{\mathbf{3}},{t}_{3})$. The perpendicular (spatial) distance from

**R**to ${\mathbf{\Pi}}^{\mathit{m}}$ is the spatial deviation. Thus,

**R**is bound to be one of the positions along the reference path. Hence, it suffices to calculate the perpendicular distance from the $k-1$ measured positions between start and end point of ${\mathbf{\Pi}}_{\mathbf{0}}$ to ${\mathbf{\Pi}}^{\mathit{m}}$ and to then select the maximum of these.

#### 5.3. Evaluation of Interpolation Error in Real-World FCD

**Figure 4.**Distance difference after a rediscretization of factor k. In (

**a**), the box-plot has whiskers at the $99\%$ quantile; in (

**b**) it has no whiskers.

**Figure 5.**Spatial deviation after a rediscretization of factor k. In (

**a**), the box-plot has whiskers at the $99\%$ quantile; in (

**b**) it has no whiskers.

**Figure 6.**Speed difference after a rediscretization of factor k. In (

**a**), the box-plot has whiskers at the $99\%$ quantile; in (

**b**) it has no whiskers.

**Figure 7.**Speed difference for a rediscretization of factor k mapped to the FCD recorded at $1\phantom{\rule{3.33333pt}{0ex}}\mathrm{Hz}$. The green color indicates a low speed difference ($<1\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}/\mathrm{h}$), the red color a high speed difference ($>20\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}/\mathrm{h}$). A slightly lower sampling rate ($k=5$) already results in a severe loss of information; green and red phases alternate frequently. Especially near to road intersections where the car is decelerating or accelerating speed differs by up to $20\phantom{\rule{3.33333pt}{0ex}}\mathrm{km}/\mathrm{h}$ from the reference.

**Figure 8.**Angular deviation after a rediscretization of factor k. In (

**a**), the box-plot has whiskers at the $99\%$ quantile; in (

**b**) it has no whiskers.

## 6. Temporal Sampling Recommendations for Recording FCD with a GPS

**Table 2.**Temporal sampling recommendations for recording FCD with a GPS. The rationale behind all values is explained in detail in the text.

Movement Parameter | Sampling Rate [Hz] | |
---|---|---|

Spatial Path | Π | 1/3–1/5 |

Distance | d | 1/5–1/10 |

Speed | v | 1–1/2 |

Direction | θ | 1/3–1/5 |

#### 6.1. Path

#### 6.2. (Cumulative) Distance

#### 6.3. Speed

#### 6.4. Direction

**A**and

**B**are two positions in space and ${\mathit{A}}^{\mathbf{\prime}}$ and ${\mathit{B}}^{\mathbf{\prime}}$ are GPS position estimates affected by measurement error.

**Figure 9.**The influence of measurement error on the direction between two uncertain positions. The angle γ is the difference between the vectors $\mathit{A}\mathit{B}$ and ${\mathit{A}}^{\mathbf{\prime}}{\mathit{B}}^{\mathbf{\prime}}$. In (

**a**),

**A**and

**B**are close together; in (

**b**), they are farther apart. The relative position of ${\mathit{A}}^{\mathbf{\prime}}$ to

**A**and ${\mathit{B}}^{\mathbf{\prime}}$ to

**B**does not change. Still γ in (b) is smaller than in (a).

## 7. Dicussion

#### Euclidean Space or Network Space?

- How can our findings support map-matching from two-dimensional space to network space?
- Which movement parameters should rather be calculated from the trajectory in two-dimensional Euclidean space and which in network space?

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Ranacher, P.; Brunauer, R.; Van der Spek, S.; Reich, S. What is an Appropriate Temporal Sampling Rate to Record Floating Car Data with a GPS? *ISPRS Int. J. Geo-Inf.* **2016**, *5*, 1.
https://doi.org/10.3390/ijgi5010001

**AMA Style**

Ranacher P, Brunauer R, Van der Spek S, Reich S. What is an Appropriate Temporal Sampling Rate to Record Floating Car Data with a GPS? *ISPRS International Journal of Geo-Information*. 2016; 5(1):1.
https://doi.org/10.3390/ijgi5010001

**Chicago/Turabian Style**

Ranacher, Peter, Richard Brunauer, Stefan Van der Spek, and Siegfried Reich. 2016. "What is an Appropriate Temporal Sampling Rate to Record Floating Car Data with a GPS?" *ISPRS International Journal of Geo-Information* 5, no. 1: 1.
https://doi.org/10.3390/ijgi5010001