We investigate the geometry of space-time prisms and their associated potential path areas (for movement that takes place in the plane) in the setting where the prism anchors have a spatial uncertainty area attached to them. Our main result is a characterisation of such space-time prisms and their potential path areas in terms of a generalised distance function between closed subsets of the ambient space. Since the case where this spatial anchor uncertainty takes the form of a closed disk has already been addressed by Kuijpers and Othman (2017), we focus on the more realistic case where the uncertainty areas are ellipses and we apply our general characterisation to this example to obtain explicit descriptions of the uncertain prisms and their potential path areas.

We study large-scale movement on the surface of the earth that is recorded at discrete moments in time and are interested in the uncertainty between recorded locations. Hereto, we define and study space-time prisms for movement on the surface of the earth (modelled as a sphere). We give description of these prisms and their spatial projection onto the sphere (the so-called ‘potential path area’) with respect to spherical coordinates, given by latitude and longitude, and Cartesian coordinates. We also study space-time prisms and their spatial projections on the Mercator map projection. Finally, we show how these concepts can be applied to intercontinental movement. Possible applications include the flight of birds during the migration seasons, the trail of sea or land mammals over longer periods of time and the path of adrift containers on the oceans.

Online Analytical Processing (OLAP) comprises tools and algorithms that allow querying multidimensional databases. It is based on the multidimensional model, where data can be seen as a cube such that each cell contains one or more measures that can be aggregated along dimensions. In a “Big Data” s cenario, traditional data warehousing and OLAP operations are clearly not sufficient to address current data analysis requirements, for example, social network analysis. Furthermore, OLAP operations and models can expand the possibilities of graph analysis beyond the traditional graph-based computation. Nevertheless, there is not much work on the problem of taking OLAP analysis to the graph data model. This paper proposes a formal multidimensional model for graph analysis, that considers the basic graph data, and also background information in the form of dimension hierarchies. The graphs in this model are node- and edge-labelled directed multi-hypergraphs, called graphoids, which can be defined at several different levels of granularity using the dimensions associated with them. Operations analogous to the ones used in typical OLAP over cubes are defined over graphoids. The paper presents a formal definition of the graphoid model for OLAP, proves that the typical OLAP operations on cubes can be expressed over the graphoid model, and shows that the classic data cube model is a particular case of the graphoid data model. Finally, a case study supports the claim that, for many kinds of OLAP-like analysis on graphs, the graphoid model works better than the typical relational OLAP alternative, and for the classic OLAP queries, it remains competitive.

Presently, data concerning moving objects abound. These data mainly consist of time-stamped geographical locations, which are collected by location aware devices, such as Global Positioning System receivers. Space–time prisms are used to model the spatio-temporal space of potential movement in between measured locations (called anchors). They rely on the knowledge of the maximal speed of travel of an object and they capture all space–time paths that respect this speed limit. However, the classic space–time path and prism model is not physically realistic, in the sense that it contains spatio-temporal paths of moving objects can alter their direction and speed instantaneously. Since this is physically impossible, the classical model is not acceptable in applications where mechanics and kinetics are vital. We propose a more realistic version of space–time prisms, in which not only speed but also acceleration is bounded. This additional bound results in a physically realistic model, which we refer to as kinetic prisms. Furthermore, we study how imposing constraints on the speed and heading at anchor points affects the geometry of kinetic prisms. In this paper, we give analytical descriptions of kinetic prisms and algorithms for their construction for movement in one- and two-dimensional space.

Space-time prisms envelop all spatio-temporal locations that moving objects may have visited between two of their known spatio-temporal locations, given a bound on their travel speed. In this context, the known locations are often the result of observations or measurements, and they are called ‘anchor points’. The classic space-time prism, in isotropic two-dimensional space, as well as in transportation networks, assumes that the measurements of these anchor points are exact. Whereas, in many applications, we can assume that time can be measured fairly precisely, this assumption is unrealistic for the spatial components of measured locations (we think of Global Positioning System (GPS) errors, for instance). In this paper, we extend the classical prism from anchor points to circular ‘anchor regions’ that capture the uncertainty or error on their measurement. We define the notion of a space-time prism with uncertain anchor points, called uncertain prism, for short. We study the geometry of uncertain prisms in an arbitrary metric space to make this concept as widely applicable as possible. We also focus on the rims of uncertain space-time prisms, which demarcate the area that a moving object can have visited between two anchor regions (given some local speed limitations).

Online Analytical Processing (OLAP) comprises tools and algorithms that allow querying multidimensional databases. It is based on the multidimensional model, where data can be seen as a cube, where each cell contains one or more measures can be aggregated along dimensions. Despite the extensive cor pus of work in the field, a standard language for OLAP is still needed, since there is no well-defined, accepted semantics, for many of the usual OLAP operations. In this paper, we address this problem, and present a set of operations for manipulating a data cube. We clearly define the semantics of these operations, and prove that they can be composed, yielding a language powerful enough to express complex OLAP queries. We express these operations as a sequence of atomic transformations over a fixed multidimensional matrix, whose cells contain a sequence of measures. Each atomic transformation produces a new measure. When a sequence of transformations defines an OLAP operation, a flag is produced indicating which cells must be considered as input for the next operation. In this way, an elegant algebra is defined. Our main contribution, with respect to other similar efforts in the field is that, for the first time, a formal proof of the correctness of the operations is given, thus providing a clear semantics for them. We believe the present work will serve as a basis to build more solid practical tools for data analysis.

Motivated by the current interest in languages for expressing path queries to graph databases, this paper proposes to investigate Walk Logic (WL): the extension of first-order logic on finite graphs with the possibility to explicitly quantify over walks. WL can serve as a unifying framework for path query languages. To support this claim, WL is compared in expressive power with various established query languages for graphs, such as first-order logic extended with reachability; the monadic second-order logic of graphs; hybrid computation tree logic; and regular path queries. WL also serves as a framework to investigate the following natural questions: Is quantifying over walks more powerful than quantifying over paths (walks without repeating nodes) only? Is quantifying over infinite walks more powerful than quantifying over finite walks only? WL model checking is decidable, but determining the precise complexity remains an open problem.

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.

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.

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 Pfoser, D. and Jensen, C.S. 1999. “Capturing the uncertainty of moving-object representations”. In Advances in spatial databases (SSD’99), Hong Kong, China, July 20–23, 1999, Vol. 1651, 111–132. Lecture notes in Computer Science. [Crossref] , [Google Scholar], Egenhofer 2003 Egenhofer, M. 2003. “Approximation of geopatial lifelines”. In SpadaGIS, workshop on spatial data and geographic information systsems, 4SpaDaGIS–Workshop on Spatial Data and Geographic Information Systems, Milano, Italy, March 2003. University of Genova. [Google Scholar], Miller 2005 Miller, H. 2005. A measurement theory for time geography. Geographical Analysis, 37(1): 17–45. [Crossref], [Web of Science ®] , [Google Scholar]) can be used to model the uncertainty about an object's 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.

The paper is talking about moving object databases, a field of spatio-temporal databases and its applications in GIS.In moving object databases, several data models and query languages have been proposed to deal with moving objects whose position is recorded at different moments in time.

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.

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 user's 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.

One of the formalisms to qualitatively describe polylines in the plane are double-cross matrices. In a double-cross matrix the relative position of any two line segments in a polyline is described with respect to a double cross based on their start points. Two polylines are called DC-similar if their double-cross matrices are identical. Although double-cross matrices have been widely applied, a geometric interpretation of the similarity they express is still lacking. In this paper, we provide a first step in the geometric interpretation of this qualitative definition of similarity. In particular, we give an effective characterization of what DC-similarity means for polylines that are drawn on a grid. We also provide algorithms that, given a DC-matrix, check whether it is realizable by a polyline on a grid and that construct, if possible, in quadratic time example polylines that satisfy this matrix. We also describe algorithms to reconstruct polylines, satisfying a given double-cross matrix, in the two-dimensional plane, that is, not necessarily on a grid.

Many association rule‐mining algorithms have been proposed in the last few years. Their main drawback is the huge amount of generated patterns. In spatial association rule mining, besides the large amount of rules, many are well‐known geographic domain associations explicitly represented in geographic database schemas. Existing algorithms have only considered the data, while the schema has not been considered. The result is that also the associations explicitly represented in geographic database schemas are extracted by association rule‐mining algorithms. With the aim to reduce the number of well‐known patterns and association rules, this paper presents a summary of results of a novel approach to extract patterns from geographic databases. A two step‐pruning method is presented to avoid the generation of association rules that are previously known to be uninteresting. Experiments with real geographic databases show a considerable time reduction in both geographic data pre‐processing and spatial association rule mining, with a very significant reduction in the total number of rules.

We address a fundamental question concerning spatio-temporal database systems: “What are exactly spatio-temporal queries?” We define spatio-temporal queries to be computable mappings that are also generic , meaning that the result of a query may only depend to a limited extent on the actual internal representation of the spatio-temporal data. Genericity is defined as invariance under groups of geometric transformations that preserve certain characteristics of spatio-temporal data (e.g., collinearity, distance, velocity, acceleration, …). These groups depend on the notions that are relevant in particular spatio-temporal database applications. These transformations also have the distinctive property that they respect the monotone and unidirectional nature of time. We investigate different genericity classes with respect to the constraint database model for spatio-temporal databases and we identify sound and complete languages for the first-order and the computable queries in these genericity classes. We distinguish between genericity determined by time-invariant transformations, genericity notions concerning physical quantities and genericity determined by time-dependent transformations.

Data aggregation in Geographic Information Systems (GIS) is a desirable feature, although only marginally present in commercial systems, which also fail to provide integration between GIS and OLAP (On Line Analytical Processing). With this in mind, we have developed Piet, a system that makes use of a novel query processing technique: first, a process called sub-polygonization decomposes each thematic layer in a GIS, into open convex polygons; then, another process computes and stores in a database the overlay of those layers for later use by a query processor. We describe the implementation of Piet, and provide experimental evidence that overlay precomputation can outperform GIS systems that employ indexing schemes based on R-trees.

The collection of moving object data is becoming more and more common, and therefore there is an increasing need for the efficient analysis and knowledge extraction of these data in different application domains. Trajectory data are normally available as sample points, and do not carry semantic information, which is of fundamental importance for the comprehension of these data. Therefore, the analysis of trajectory data becomes expensive from a computational point of view and complex from a user's perspective. Enriching trajectories with semantic geographical information may simplify queries, analysis, and mining of moving object data. In this paper we propose a data preprocessing model to add semantic information to trajectories in order to facilitate trajectory data analysis in different application domains. The model is generic enough to represent the important parts of trajectories that are relevant to the application, not being restricted to one specific application. We present an algorithm to compute the important parts and show that the query complexity for the semantic analysis of trajectories will be significantly reduced with the proposed model.

The research presented in this paper is situated in the framework of constraint databases introduced by Kanellakis, Kuper, and Revesz in their seminal paper of 1990, specifically, the language with real polynomial constraints (FO+poly). For reasons of efficiency, this model is implemented with only linear polynomial constraints, but this limitation to linear polynomial constraints has severe implications on the expressive power of the query language. In particular, when used for modeling spatial data, important queries that involve Euclidean distance are not expressible. The aim of this paper is to identify a class of two‐dimensional constraint databases and a query language within the constraint model that go beyond the linear model and allow the expression of queries concerning distance. We seek inspiration in the Euclidean constructions, i.e., constructions by ruler and compass. We first present a programming language that captures exactly the first‐order ruler‐and‐compass constructions that are expressible in a first‐order language with real polynomial constraints. If this language is extended with a while operator, we obtain a language that is complete for all ruler‐and‐compass constructions in the plane. We then transform this language in a natural way into a query language on finite point databases, but this language turns out to have the same expressive power as FO+poly and is therefore too powerful for our purposes. We then consider a safe fragment of this language and use this to construct a query language that allows the expression of Euclidean distance without having the full power of FO+poly.

We present an algorithm for polyline (and polygon) similarity testing that is based on the double-cross formalism. To determine the degree of similarity between two polylines, the algorithm first computes their generalized polygons, that consist of almost equally long line segments and that approximate the length of the given polylines within an $varepsilon$-error margin. Next, the algorithm determines the double-cross matrices of the generalized polylines and the difference between these matrices is used as a measure of dissimilarity between the given polylines. We prove termination of our algorithm and show that its sequential time complexity is bounded by $Oleft((fracmax(N_1,N_2)varepsilon)^2right)$, where $N_1$ and $N_2$ are the number of vertices of the given polylines. We apply our method to query-by-sketch, indexing of polyline databases, and classification of terrain features and show experimental results for each of these applications.

In frequent geographic pattern mining a large amount of patterns is well known a priori. This paper presents a novel approach for mining frequent geographic patterns without associations that are previously known as non- interesting. Geographic dependences are eliminated during the frequent set generation using prior knowledge. After the dependence elimination maximal generalized frequent sets are computed to remove redundant frequent sets. Experimental results show a significant reduction of both the number of frequent sets and the computational time for mining maximal frequent geographic patterns.

Planar spatial datasets can be modeled by closed semi-algebraic sets in the plane. We establish a characterization of the topological properties of such datasets expressible in the relational calculus with real polynomial constraints. The characterization is in the form of a query language that can only point that can only talk about points in the set and the “cones” around these points.

We study queries to spatial databases, where spatial data are modeled as semi-algebraic sets, using the relational calculus with polynomial inequalities as a basic query language. We work with the extension of the relational calculus with terminating transitive closures. The main result is that this language can express the linearization of semialgebraic databases. We also show that the sublanguage with linear inequalities only can express all computable queries on semilinear databases. As a consequence of these results, we obtain a completeness result for topological queries on semialgebraic databases.

Introduction to Constraint Databases by Peter Revesz, Texts in Computer Science, Springer-Verlag, 2002, ISBN 0-387-98729-0, xiv+393 pages, 112 illustrations, hardcover, list price of $54.95 in the U.S.A. - Volume 3 Issue 6

We consider spatial databases in the plane that can be defined by polynomial constraint formulas. Motivated by applications in geographic information systems, we investigate linear approximations of spatial databases and study in which language they can be expressed effectively. Specifically, we show that they cannot be expressed in the standard first-order query language for polynomial constraint databases but that an extension of this first-order language with transitive closure suffices to express the approximation query in an effective manner. Furthermore, we introduce an extension of transitive-closure logic and show that this logic is complete for the computable queries on linear spatial databases. This result together with our first result implies that this extension of transitive-closure logic can express all computable topological queries on arbitrary spatial databases in the plane.

Abstract We investigate topological properties of subsetsSof the real plane, expressed by first-order logic sentences in the language of the reals augmented with a binary relation symbol forS. Two sets are called topologically elementary equivalent if they have the same such first-order topological properties. The contribution of this paper is a natural and effective characterization of topological elementary equivalence of closed semi-algebraic sets.