Walied Othman

Modeling uncertainty of moving objects on road networks via space–time prisms

Moving objects produce trajectories, which are typically observed in a finite sample of time‐stamped locations. Between sample points, we are uncertain about the moving objectss location. When we assume extra information about an object, for instance, a (possibly location‐dependent) speed limit, we can use space–time prisms to model the uncertainty of an objects location. Until now, space–time prisms have been studied for unconstrained movement in the 2D plane. In this paper, we study space–time prisms for objects that are constrained to travel on a road network. Movement on a road network can be viewed as essentially one‐dimensional. We describe the geometry of a space–time prism on a road network and give an algorithm to compute and visualize space–time prisms. For experiments and illustration, we have implemented this algorithm in MATHEMATICA. Furthermore, we study the alibi query, which asks whether two moving objects could have possibly met or not. This comes down to deciding if the chains of space–time prisms produced by these moving objects intersect. We give an efficient algorithm to answer the alibi query for moving objects on a road network. This algorithm also determines where and when two moving objects may have met.

Anchor uncertainty and space-time prisms on road networks

Space-time prisms capture all possible locations of a moving person or object between two known locations and times given the maximum travel velocities in the environment. These known locations or ‘anchor points’ can represent observed locations or mandatory locations because of scheduling constraints. The classic space-time prism as well as more recent analytical and computational versions in planar space and networks assume that these anchor points are perfectly known or fixed. In reality, observations of anchor points can have error, or the scheduling constraints may have some degree of pliability. This article generalizes the concept of anchor points to anchor regions: these are bounded, possibly disconnected, subsets of space-time containing all possible locations for the anchor points, with each location labelled with an anchor probability. We develop two algorithms for calculating network-based space-time prisms based on these probabilistic anchor regions. The first algorithm calculates the envelope of all space-time prisms having an anchor point within a particular anchor region. The second algorithm calculates, for any space-time point, the probability that a space-time prism with given anchor regions contains that particular point. Both algorithms are implemented in Mathematica to visualize travel possibilities in case the anchor points of a space-time prism are uncertain. We also discuss the complexity of the procedures, their use in analysing uncertainty or flexibility in network-based prisms and future research directions.

Trajectory databases: Data models, uncertainty and complete query languages

Moving objects produce trajectories. We describe a data model for trajectories and trajectory samples and an efficient way of modeling uncertainty via beads for trajectory samples. We study transformations for which important physical properties of trajectories, such as speed, are invariant. We also determine which transformations preserve beads. We give conceptually easy first-order complete query languages and computationally complete query languages for trajectory databases, which allow to talk directly about speed and beads. The queries expressible in these languages are invariant under speed- and bead-preserving transformations.

An analytic solution to the alibi query in the space-time prisms model for moving object data

Moving objects produce trajectories, which are stored in databases by means of finite samples of time-stamped locations. When speed limitations in these sample points are also known, space–time prisms (also called beads) (Pfoser and Jensen 1999, Egenhofer 2003, Miller 2005) can be used to model the uncertainty about an objects location in between sample points. In this setting, a query of particular interest that has been studied in the literature of geographic information systems (GIS) is the alibi query. This boolean query asks whether two moving objects could have physically met. This adds up to deciding whether the chains of space–time prisms (also called necklaces of beads) of these objects intersect. This problem can be reduced to deciding whether two space–time prisms intersect. The alibi query can be seen as a constraint database query. In the constraint database model, spatial and spatiotemporal data are stored by boolean combinations of polynomial equalities and inequalities over the real numbers. The relational calculus augmented with polynomial constraints is the standard first-order query language for constraint databases and the alibi query can be expressed in it. The evaluation of the alibi query in the constraint database model relies on the elimination of a block of three exªistential quantifiers. Implementations of general purpose elimination algorithms, such as those provided by QEPCAD, Redlog, and Mathematica, are, for practical purposes, too slow in answering the alibi query for two specific space–time prisms. These software packages completely fail to answer the alibi query in the parametric case (i.e., when it is formulated in terms of parameters representing the sample points and speed constraints). The main contribution of this article is an analytical solution to the parametric alibi query, which can be used to answer the alibi query on two specific space–time prisms in constant time (a matter of milliseconds in our implementation). It solves the alibi query for chains of space–time prisms in time proportional to the sum of the lengths of the chains. To back this claim up, we implemented our method in Mathematica alongside the traditional quantifier elimination method. The solutions we propose are based on the geometric argumentation and they illustrate the fact that some practical problems require creative solutions, where at least in theory, existing systems could provide a solution.

Kinetic space-time prisms

The space-time path and prism demarcate the estimated and potential locations (respectively) of a moving object with respect to time. The path is typically formed through linear interpolation between sampled locations of a moving object, while the prism is the envelope of all possible paths between two locations given the maximum speed of travel. The classic path and prism, however, are not physically realistic since they imply the ability of the object to make instantaneous changes in direction and speed without acceleration and deceleration. This is not acceptable in applications where kinetics is vital for scientific understanding such as animal ecology, vehicles moving through media such as ships through water and planes through air, human-powered movement such as bicycling and walking and environmental applications of transportation such as energy consumption and emissions modeling. In this paper we demonstrate how imposing an upper bound on acceleration, as well as information such as the initial speed and heading, affects the geometry of the space-time prism. We discuss how to calculate kinetic paths and prisms in one-dimensional and two dimensional space, and provide examples comparing the kinetic prisms and classical prisms.

Analytical methods for error propagation in planar space–time prisms

The space–time prism demarcates all locations in space–time that a mobile object or person can occupy during an episode of potential or unobserved movement. The prism is central to time geography as a measure of potential mobility and to mobile object databases as a measure of location possibilities given sampling error. This paper develops an analytical approach to assessing error propagation in space–time prisms and prism–prism intersections. We analyze the geometry of the prisms to derive a core set of geometric problems involving the intersection of circles and ellipses. Analytical error propagation techniques such as the Taylor linearization method based on the first-order partial derivatives are not available since explicit functions describing the intersections and their derivatives are unwieldy. However, since we have implicit functions describing prism geometry, we modify this approach using an implicit function theorem that provides the required first-order partials without the explicit expressions. We describe the general method as well as details for the two spatial dimensions case and provide example calculations.

From A to B, randomly: a point-to-point random trajectory generator for animal movement

For applications in animal movement, we propose a random trajectory generator (RTG) algorithm that combines the concepts of random walks, space-time prisms, and the Brownian bridge movement model and is capable of efficiently generating random trajectories between a given origin and a destination point, with the least directional bias possible. Since we provide both a planar and a spherical version of the algorithm, it is suitable for simulating trajectories ranging from the local scale up to the (inter-)continental scale, as exemplified by the movement of migrating birds. The algorithm accounts for physical limitations, including maximum speed and maximum movement time, and provides the user with either single or multiple trajectories as a result. Single trajectories generated by the RTG algorithm can be used as a null model to test hypotheses about movement stimuli, while the multiple trajectories can be used to create a probability density surface akin to Brownian bridges.

Analyzing Trajectories Using Uncertainty and Background Information

A key issue in clustering data, regardless the algorithm used, is the definition of a distance function. In the case of trajectory data, different distance functions have been proposed, with different degrees of complexity. All these measures assume that trajectories are error-free, which is essentially not true. Uncertainty is present in trajectory data, which is usually obtained through a series of GPS of GSM observations. Trajectories are then reconstructed, typically using linear interpolation. A well-known model to deal with uncertainty in a trajectory sample, uses the notion of space-time prisms (also called beads ), to estimate the positions where the object could have been, given a maximum speed. Thus, we can replace a (reconstructed) trajectory by a necklace (intuitively, a a chain of prisms ), connecting consecutive trajectory sample points. When it comes to clustering, the notion of uncertainty requires appropriate distance functions. The main contribution of this paper is the definition of a distance function that accounts for uncertainty, together with the proof that this function is also a metric, and therefore it can be used in clustering. We also present an algorithm that computes the distance between the chains of prisms corresponding to two trajectory samples. Finally, we discuss some preliminary results, obtained clustering a set of trajectories of cars in the center of the city of Milan, using the distance function introduced in this paper.

Map matching and uncertainty: an algorithm and real-world experiments

A common problem in moving object databases (MOD) is the reconstruction of a trajectory from a trajectory sample (i.e., a finite sequence of time-space points). A typical solution to this problem is linear interpolation. A more realistic model is based on the notion of uncertainty modelled by space-time prisms, which capture the positions where the object could have been, when it moved from a to b. Often, object positions measured by location-aware devices are not on a road network. Thus, matching the users position to a location on the digital map is required. This problem is called map matching. In this paper we study the relation between map matching and uncertainty, and propose an algorithm that combines weighted k-shortest paths with space-time prisms. We apply this algorithm to two real-world case studies and we show that accounting for uncertainty leads to obtaining more positive matchings.

Privacy through Uncertainty in Location-Based Services

Location-Based Services (LBS) are becoming more prevalent. While there are many benefits, there are also real privacy risks. People are unwilling to give up the benefits - but can we reduce privacy risks without giving up on LBS entirely? This paper explores the possibility of introducing uncertainty into location information when using an LBS, so as to reduce privacy risk while maintaining good quality of service. This paper also explores the current uses of uncertainty information in a selection of mobile applications.