Towards fusing uncertain location data from heterogeneous sources

Abstract

Properly incorporating location-uncertainties – which is, fully considering their impact when processing queries of interest – is a paramount in any application dealing with spatio-temporal data. Typically, the location-uncertainty is a consequence of the fact that objects cannot be tracked continuously and the inherent imprecision of localization devices. Although there is a large body of works tackling various aspects of efficient management of uncertainty in spatio-temporal data – the settings consider homogeneous localization devices, e.g., either a Global Positioning System (GPS), or different sensors (roadside, indoor, etc.).In this work, we take a first step towards combining the uncertain location data – i.e., fusing the uncertainty of moving objects location – obtained from both GPS devices and roadside sensors. We develop a formal model for capturing the whereabouts in time in this setting and propose the Fused Bead (FB) model, extending the bead model based solely on GPS locations. We also present algorithms for answering traditional spatio-temporal range queries, as well as a special variant pertaining to objects locations with respect to lanes on road segments – augmenting the conventional graph based road network with the width attribute. In addition, pruning techniques are proposed in order to expedite the query processing. We evaluated the benefits of the proposed approach on both real (Beijing taxi) and synthetic (generated from a customized trajectory generator) data. Our experiments demonstrate that the proposed method of fusing the uncertainties may eliminate up to 26 % of the false positives in the Beijing taxi data, and up to 40 % of the false positives in the larger synthetic dataset, when compared to using the traditional bead uncertainty models.

Appendices

Appendix A: Significant times in instantaneous possible location query

In Section 4 we analyze the boundary of the possible locations at a given time instant under the FB model. The detailed significant times calculation will be presented here. Let d _{m i n} and d _{m a x} denote the shortest and longest distance from L ₁ to any point $P(t_{s1},\varepsilon ) \in \overline {P_{1}P_{2}}$.

$$\begin{array}{@{}rcl@{}} t_{i}^{l1}&=& \frac{t_{1} + t_{s}}{2} - \frac{d_{max}}{2v_{max}}\\ t_{i}^{lA}&=& \frac{t_{1} + t_{s}}{2} - \frac{d_{min}}{2v_{max}}\\ t_{i}^{d1}&=& \frac{t_{1} + t_{s}}{2} + \frac{d_{min}}{2v_{max}}\\ t_{i}^{dA}&=& \frac{t_{1} + t_{s}}{2} + \frac{d_{max}}{2v_{max}} \end{array} $$

Appendix B: Enter/exit time calculation for range query

The general case for time t∈[t _i , t _i+1] being a critical point occurs when the intersection of the uncertain region at t with a query rectangle is a single point. In the time interval [t _i , t _s ], the single-point-intersection between disk centered at the first GPS point and query region stands for the entering moment. Similarly, in the time interval [t _s , t _i+1], the single-point-intersection represents exiting moment. Since the query region is represented as polygon in the (X, Y) plane, each edge of the polygon is defined as a segment of 2D line y = a x+b.

The entry boundary of FB is:

$$(x - x_{i})^{2} + (y - y_{i})^{2} = (t - t_{i})^{2}v^{2}_{max}$$

Substituting for y for the equation of the line, we have:

$$(x - x_{i})^{2} + (ax+b - y_{i})^{2} = (t - t_{i})^{2}v^{2}_{max}$$

This yields an equation in x and t:

$$A*x^{2}+x*(B+C*t)+D*t^{2}+E=0$$

Where A, B, C, D, E are constant. Solving for x, as a function of t, we have:

$$x_{1,2}=\frac{-(B+C*t) \pm \sqrt{(B+C*t)^{2}-4*A*(D*t^{2}+E)}}{2*A}$$

To be noted that, we need to check the solution for x against the boundaries of the respect edge of the query region. To find the time for critical point, we set the discriminant to be zero:

$$\sqrt{(B+C*t)^{2}-4*A*(D*t^{2}+E)}=0$$

The real root t _{i n} is the time instant when the uncertain trajectory start to enter the query prism.

In the time interval [t _s , t _i+1], we can use the similar method to find the exiting time t _{o u t} .